TLA^{+} in Practice and Theory
Part 3: The (Temporal) Logic of Actions
This is part 3 in a fourpart series. Part 1, Part 2, Part 4. A video of a 40minute talk that covers parts of this series.
If you find TLA^{+} and formal methods in general interesting, I invite you to visit and participate in the new /r/tlaplus on Reddit.
In part 2 we learned how we specify data and operations on data. In this post we will finally learn how we specify and reason about processes, or dynamic systems — in particular, algorithms — in TLA^{+} and how we check their correctness. While there is only very little syntax left to learn, our study of the theory of representing programs as mathematical objects is now just beginning. Most of our discussion will not concern TLA^{+} specifically, but TLA, Lamport’s Temporal Logic of Actions.
This post is the longest in the series by far. Covering in depth a mathematical theory — even a relatively simple one — for reasoning about computation is not light reading. But I will repeat my warning from part 1: very little of the theory and its background presented here is necessary in order to use TLA^{+} to specify and verify realworld software.
A Bit of Historical Background
The need to mathematically reason about nontrivial programs was recognized very early on in the history of computation. Possibly the earliest attempt at a method for mathematical reasoning about programs — the first formal method — was devised by Alan Turing in his paper Checking a Large Routine^{1} and presented in a talk in 1949, who, interestingly, used a similar notation, that of primed variables denoting a variable’s value following a program step, to that of Lamport’s TLA. Unfortunately, it seems that talk, like much of Turing’s work, was ignored at the time and probably did not influence later ideas^{2}. A very similar, yet somewhat expanded approach to that of Turing’s, was invented by Robert Floyd in 1967 and then presented in a more formal way by Tony Hoare in 1969 as what is now known as FloydHoare logic (or just Hoare logic).
Turing’s paper begins thus:
How can one check a routine in the sense of making sure that it is right?
In order that the man who checks may not have too difficult a task, the programmer should make a number of definite assertions which can be checked individually, and from which the correctness of the whole programme easily follows.
Turing’s, Floyd’s and Hoare’s method uses what’s known as “assertional reasoning”; its main idea — listing at each point in the program which facts, assertions, about the program’s state^{3} are true at that point.
Lamport, who in 1977 began working on expanding Floyd and Hoare’s techniques to concurrent programs, was an early believer in the idea of program verification. In 1979 he wrote a letter to the editor of Communications of the ACM regarding their recently introduced rules for publishing algorithms:
For years, we did not know any better way to check programs than by testing them to see if they worked… But the work of Floyd and others has given us another way. They taught us that a program is a mathematical object, so we can apply the reasoning methods of mathematics to deduce its properties… After Euclid, a theorem could no longer be accepted solely on the basis of evidence provided by drawing pictures. After Floyd, a program should no longer be accepted solely on the basis of how it works on a few test cases. A program with no demonstration of why it is correct is the same as a conjecture — a statement which we think may be a theorem. A conjecture must be exceptionally interesting to warrant publication. An unverified program should also have to be exceptional to be published… The ACM should require that programmers convince us of the correctness of the programs that they publish, just as mathematicians must convince one another of the correctness of their theorems. Mathematicians don’t do this by giving “a sufficient variety of test cases to exercise all the main features,” and neither should computer scientists.
One point of contention between Lamport and others who were working on reasoning about programs at the time from a more linguistic point of view, like Tony Hoare and Robin Milner, was about the representation of the program’s control state, namely the program counter and (if applicable) the call stack. Those who work on programming language theory are adamantly reluctant — to this day — to make the program’s control’s state explicit, while Lamport claims that it makes reasoning about concurrency significantly easier.
Amir Pnueli’s introduction of temporal logic into computer science was somewhat of a watershed moment for software verification, resulting in not one but two Turing Awards, one to Pnueli and one to Clarke, Emerson and Sifakis for the invention of temporal logic model checkers^{4}. Pnueli’s logic offered “a unified approach to program verification… which applies to both sequential and parallel programs.”^{5} Note that “parallel programs” also refers to nonterminating interactive programs, where it’s useful to consider the user as a concurrent process.
Lamport was impressed with temporal logic but became disillusioned with its practical application when he saw his colleagues “spending days trying to specify a simple FIFO queue — arguing over whether the properties they listed were sufficient. I realized that, despite its aesthetic appeal, writing a specification as a conjunction of temporal properties just didn’t work in practice.”
The Temporal Logic of Actions (TLA), which forms the core of TLA^{+}, was invented in the late ’80s and is the culmination of Lamport’s ideas about how to reason about algorithms and software or hardware systems. TLA differs from earlier uses of temporal logics in two important ways. The first is that Lamport, a staunch believer in the power of ordinary math, made TLA restrict the need for temporal reasoning to a bare minimum:
TLA differs from other temporal logics because it is based on the principle that temporal logic is a necessary evil that should be avoided as much as possible. Temporal formulas tend to be harder to understand than formulas of ordinary firstorder logic, and temporal logic reasoning is more complicated than ordinary mathematical … reasoning.^{6}
Another difference is that temporal logic is usually used to reason about programs written in some programming language, while TLA is a universal mathematical notation in which one writes both the algorithm and its claimed properties as formulas in the same logic. This makes TLA a program logic, and a particularly powerful one at that. Lamport writes:
Classical program verification, begun by Floyd and Hoare, employs two languages. The program is written in a programming language, and properties of the program are written in the language of formulas of some logic. Properties are derived from the program text by special proof rules, and from other properties by reasoning within the logic… A program logic expresses both programs and properties with a single language. Program $\Pi$ satisfies property $P$ if and only if $\Pi \implies P$ is a valid formula of the logic.
Algorithms and Programs
Because algorithms are described in TLA rather differently from what you may be used to in programming, this would be a good point to ask, what is an algorithm?
In that 1979 letter to the editor of CACM, Lamport explains how he sees the difference between an algorithm and a program:
The word “algorithm” usually means a general method for computing something, and “program” means code that can be executed on a computer.
Bob Harper, a programming language theorist, disagrees and writes that:
Algorithms are programs. The supposed distinction only arose because of ultracrappy programming languages.
Let’s examine Harper’s assertion. Suppose we read on Wikipedia a description of Tony Hoare’s famous Quicksort algorithm — which we will formally specify later in this post — and implement it in, say, BASIC and Pascal. Do the two programs encode the same algorithm or not? If we say they do, then Harper’s statement is taken to mean “algorithms are programs modulo some equivalence,” but this requires us to define what it means for programs to be equivalent, or “essentially the same”, and it is then that equivalence — which is bound to be far from trivial — that captures the essence of what an algorithm is rather than a specific program. Indeed, a paper I will discuss briefly in part 4, defines an algorithm precisely in this way, an equivalence relation on programs, and shows that it is the resulting “quotient” that captures the more important essence of what we mean when we say algorithm. On the other hand, if we do not consider the two programs to be the same algorithm — perhaps because the two languages have some differences in implementation, concerning, say, how arrays are represented in memory, and we consider those differences significant — then the very same Lisp program could actually encode several different algorithms depending on what version of compiler or interpreter we use to run it, as different versions may have implementation details that are just as different as those between Pascal and BASIC^{7}.
As an example, consider Euclid’s algorithm for computing the greatest common divisor of two natural numbers. I picked this example because the algorithm is one of the oldest known that is still in use (according to Wikipedia, it was published in 300BC), it is very simple, and because it is a favorite example of Leslie Lamport’s in his TLA^{+} tutorials and talks. This is the algorithm: if the two numbers are equal, then they’re equal to their GCD; otherwise, we instead consider the pair comprising the smaller of the two numbers and the difference between the two numbers, and repeat.
Now, here’s a simple question that echoes a favorite topic of discussion in some online programming forums: is Euclid’s algorithm imperative or purefunctional? We cannot answer this question because the algorithm doesn’t talk about specific memory operations, doesn’t say anything about the scope and mutation rules for variables, and doesn’t dictate or forbid the use function abstractions when programming it. Nevertheless, it is no doubt an actual algorithm. We can even reason about it and prove its correctness — so we understand exactly what it means and why it works — yet it leaves unspecified this particular detail, which turns out to be irrelevant to understanding how and why the algorithm works. The lack of this detail is not a flaw, let alone one due to Euclid’s choice of an ultracrappy language to describe his algorithm (although, personally, Greek would not be my first choice).
An algorithm provides some details and leaves others unspecified; we say that an algorithm works if it does what it claims to do no matter how we fill in the gaps in the description^{8}. A program, being a description of an algorithm in a (formal) programming language, is a description at a specific level of detail fully dictated by the language in which it is expressed. It can provide neither more nor less detail than that required by the language. A Python program cannot define precisely how its data is to be laidout in memory, and when writing a mergesort algorithm, it cannot leave out the detail of whether it is to be done sequentially or in parallel (some languages may give more leeway, but even then they have some fixed range of specificity). When Euclid’s algorithm is implemented in some programming language, we usually have no trouble determining whether the algorithm is imperative or purefunctional, yet, regardless of the answer to that, we also have no trouble determining whether or not the program actually implements Euclid’s algorithm. Qualities that seem so essential to discussions and endless debates about programming, are irrelevant detail for the algorithms our programs are ultimately written to carry out.
To drive this point home, let’s consider Quicksort again. This is the algorithm according to Wikipedia :
 Pick an element, called a pivot, from the array.
 Partitioning: reorder the array so that all elements with values less than the pivot come before the pivot, while all elements with values greater than the pivot come after it (equal values can go either way). After this partitioning, the pivot is in its final position. This is called the partition operation.
 Recursively apply the above steps to the subarray of elements with smaller values and separately to the subarray of elements with greater values.
I claim that none of the steps of this algorithm is directly expressible as a program in virtually any wellknown programming language. The first step requires picking an element in the array; but which element is it and how is it picked? There are many different choices. The second step requires reordering the elements of the array, but there are many different reorderings that satisfy the partitioning requirement. The third step calls for recursively applying the steps to two subarrays, but doesn’t mention in which order, or whether in any sequential order at all; maybe we can work on the two subarrays in parallel.
Without those details, we simply cannot write a program that implements Quicksort. However, those details are not missing from the algorithm, which is complete as given; they simply do not matter, or, in other words, their particulars are not assumed. The algorithm works no matter which element we pick as a pivot; it works no matter which particular reordering we choose as long as its a proper partition; it works no matter in what order we execute the recursive steps or even if we run them in parallel. In fact, any description that does fill in those details — and therefore any Quicksort program — cannot possibly fully describe Quicksort, but merely one of its many possible implementations.
This is not a limitation of one specific programming language or another, ultracrappy or not. It is a limitation of all programming languages, as those are formalisms designed to program deterministic computers, while algorithms are often best specified — as in the case of Quicksort — as nondeterministic. Various kinds of implicit nondeterminism in programming languages (e.g. concurrency) are not rich enough to express the nondeterminism we use to describe algorithms, and even programming languages that aim to express nondeterminism explicitly (like Prolog) fail at this, because they all must still describe a deterministic executable if they are to serve as programming languages, and it is impossible to simulate arbitrary nondeterminism on deterministic machines without changing at least one of algorithm’s core properties, namely its computational complexity.
If we could describe the general Quicksort algorithm, we could verify that it is correct (i.e. that it sorts its input). Once we’ve done that, we no longer need to prove that a particular implementation of the algorithms is correct — we just need to prove that it is indeed an implementation of the algorithm. TLA^{+} allows precisely this form of verification, namely, verifying that some property holds for a highlevel specification, and then verifying that a lowlevel specification implements the highlevel one.
Now, you may think that as a programmer, rather than a researcher who publishes algorithms in scientific journals, you do not care about the distinction between an algorithm and a program because you’re only interested in programs, anyway. But you would be wrong. You are probably required to design large software systems, not just code isolated subroutines, and a system resembles an algorithm much more than a program. When you design the system, you don’t care about the details of which text search product will be used, which message queue will be picked, which database will store your data, and which MVC framework will be used on the client. You may very well be interested in some of their properties — for example, whether or not the message queue can drop messages, the consistency guarantees of the database, the performance of the search, etc. — but there are many more details you don’t care about. Nevertheless, you still need to know that the system will meet its requirements no matter which component is chosen (as long as it satisfies the requirements you do care about). You therefore need to test the correctness of a design that cannot be detailed enough to actually run, and it must not be: if it is too detailed, its behavior may accidentally depend on some database quirk that all of a sudden goes away in the next release. Even if your components are all picked in advance and you’re sure that they won’t change, as I said in part 1 verifying a specification that is very detailed, spanning everything from the codelevel to highlevel architecture is currently infeasible in practice. The ability to specify and verify a highlevel design without supplying lowlevel details is necessary for a formal method to scale.
TLA gives a quite satisfying answer to the question of what an algorithm is (and how algorithms of different levels of detail relate to one another): an algorithm is quite literally the collection of all its implementations. Another way to look at it is that an algorithm is the common element among all of its implementation; yet another is that an algorithm is an abstraction — a less detailed description — of its implementations and a refinement — a more detailed description — of algorithms it implements itself, namely its own abstractions. This yields a partialorder relation on all algorithms, called the abstraction/refinement relation.
Some programming languages that include specification mechanisms (like Eiffel, SPARK and Clojure) also allow you to specify what an algorithm does in some languagespecific level of detail, and those specifications also denote the set of all possible implementations, but they still make a very clear distinction between a specification and a program, the latter being the only thing that’s executable. The same is true in sophisticated research programming languages (like Agda and Idris) that make use of dependent types as a specification language. Even though the type definitions (like specification contracts) allow full use of the language’s syntax, there is a clearcut, binary distinction between a specification — a type — which can be at some relatively flexible level of detail, and a program or a subroutine — a type inhabitant — whose level of detail is predetermined by the language’s particular semantics. Types and their inhabitants, even in dependentlytyped languages, are two distinct semantic categories. You cannot execute a type, nor can a subroutine directly serve as a specification for another subroutine. TLA makes no such distinction; one algorithm can implement another, and “running” an algorithm means examining all of its possible behaviors, whether there are very few of them (as in a program) or very many (as in a nondetailed algorithm).
An algorithm is, then, something more general than a program. Every program is an algorithm, but not every algorithm is a program. Nevertheless, there is no theoretical distinction between a program and an algorithm; they differ in measure not in quality, and what constitutes sufficient detail to qualify as a program depends on the choice of a programming language. In fact, just as an algorithm can implement another algorithm by adding more detail, and a program can implement an algorithm, so too can a program implement another program, although that usually requires both programs to be written in different languages. For example, two different Lisp compilers produce two different machine code programs, both implementing the same Lisp program. I will therefore be using the two terms, algorithm and program, somewhat interchangeably unless it is clear from context that only one applies.
One of the nice things TLA^{+} teaches you is to think of an algorithm as an abstract concept. There is no computer (unless you want to simulate it), and no compilation — just a mathematical description, not unlike a mathematical model of a physical system. This lets you carefully think about what the algorithm is all about, or what it really does, but you can choose at what level of detail you want to describe it, what level of detail captures the essence of the algorithm. If you wish, you can say, “this algorithm sorts” adding no additional information; if you wish you can describe what’s required to happen all the way down to logic gates in the processor and their power consumption. This provides us with a nice separation of concerns: you can think about how an algorithm or a system works separately from how to cleanly organize the code that implements it.
Despite this long introduction, I still haven’t said anything about what kind of thing an algorithm is. Even if we forget about the difficulty of expressing an algorithm as a program, and suppose that there were some other precise way of describing an algorithm, that still doesn’t tell us anything about what an algorithm is. Even though virtually every story could be translated from its original language to English and then printed in black ink on white paper, it would be very reductive to say that a story is black English letters printed on white paper. It is also unsatisfying to say that a story is the common element to printed stories, spoken stories and acted stories. A story has some essential elements, and so does an algorithm, so let’s start listing them. An algorithm is clearly not a function, as both bubblesort and mergesort compute the same functions, yet we consider them to be different algorithms. An algorithm is some series of steps – which we can call a process or a behavior – that take place through time. It doesn’t have to be actual physical time, but the steps depend on one another and are subject to causality, so we can call whatever dimension is related to causality “time”. But an algorithm does not necessarily describe a single process. For example, bubblesorting two different lists would yield two different behaviors, or sequences of steps. So, if nothing else, one thing we can say for sure is that an algorithm is one or more behaviors. It turns out that this simple characterization can take us quite far.
Time in Computation
One of my goals in this series is to compare the mathematical theory of TLA^{+} with others mathematical theories of computation and programming. A much talkedabout mathematical theory of programming, especially among programminglanguage researchers and enthusiasts, is the theory of functional programming. At its core, functional programming of the pure variety chooses to ignore the notion of time in computation, rather representing a program as a composition of functions. This is a major contrast with TLA, which represents the passage of (logical) time very explicitly. It is a direct result of this representation of time that programs with side effects, pure computation, batch programs, interactive programs, sequential programs, concurrent programs, parallel programs, distributed programs — any kind of program imaginable — are represented in exactly the same way, and reasoning about all kinds of programs is also done in the same way. In fact, it is impossible to formally tell the difference between a program that does pure calculations and one that has lots of sideeffects in TLA^{9}. In fact, it is also impossible to formally tell the difference between sequential, concurrent and parallel programs in TLA. The difference between them lies entirely in how we choose interpret TLA formulas.
I believe that the first thorough, formal treatment of time in the context of programming was the situation calculus, introduced by John McCarthy (Lisp’s inventor) in his 1963 paper, Situations, Actions, and Causal Laws^{10}, as part of his research of AI. He writes:
Intuitively, a situation is the complete state of affairs at some instant of time. The laws of motion of a system determine from a situation all future situations. Thus a situation corresponds to the notion in physics of a point in phase space. In physics, laws are expressed in the form of differential equations which give the complete motion of the point in phase space.
Around the same time, Arthur Prior was working on temporal logic, a simpler formalism than McCarthy’s which was later introduced into computer science by Amir Pnueli, in his paper The Temporal Logic of Programs. Pnueli describes his formalism as one “in which the time dependence of events is the basic concept”. TLA is simpler still.
While I pointed out in part 1 that preference for different formalisms is often a matter of aesthetic taste, the fact that all kinds of computation are expressed in TLA in one simple manner — because of the treatment of time — that questions of sideeffects or concurrency not only do not pose any serious difficulty but are entirely a nonissue, and that reasoning about different kinds of programs can be done in exactly the same straightforward way, at least suggests that TLA is a formalism that describes computation in a very “natural” way, meaning there is little friction between the formalism and what it seeks to model. This harmony, as we’ll see, comes at no cost in terms of our ability to talk about programs in very abstract or very concrete ways.
Introduction to TLA
TLA describes algorithms not as programs in some programming language but as mathematical objects, detached from any specific syntactic representation, just like the number four represents the same mathematical object whether it is written as the arabic numeral 4 or as the roman numeral IV^{11}. TLA views algorithms and computations as discrete dynamical systems, and forms a general logical framework for reasoning about such discrete systems that is similar in some ways to how ordinary differential equations are used to model continuous dynamical systems, but with constructs that are of particular interest to computer scientists.
In The Temporal Logic of Actions, Lamport writes:
A[n]… algorithm is usually specified with a program. Correctness of the algorithm means that the program satisfies a desired property. We propose a simpler approach in which both the algorithm and the property are specified by formulas in a single logic. Correctness of the algorithm means that the formula specifying the algorithm implies the formula specifying the property, where implies is ordinary logical implication.
We are motivated not by an abstract ideal of elegance, but by the practical problem of reasoning about real algorithms. Rigorous reasoning is the only way to avoid subtle errors… and we want to make reasoning as simple as possible by making the underlying formalism simple.
TLA itself does not specify how data and primitive operations on it are defined, but only concerns itself with describing the dynamic behavior of a program, assuming we have some way of defining data and data operations. TLA therefore requires a “data language” or a “data logic”. In part 2 we saw the data logic provided by TLA^{+}.
Let’s now look at how Euclid’s algorithm is expressed in TLA^{+}. Note that we could use Euclid’s algorithm to define a GCD operator as we learned in part 2, like so:
But this does not describe an algorithm in TLA^{+}; it is just another mathematical definition of the greatest common divisor, and not a very good one, as it is unclear that the greatest common divisor is the object being defined. Rather, this is Euclid’s algorithm in TLA^{+}:
assuming that we treat either $x$ or $y$ as the “output” once the algorithm terminates (there are some missing details here, like how $Init$ and $Next$ are combined to form a single formula, as well an important detail of how termination is modeled, but this is “morally” correct, and, in fact, all you need to try the algorithm in the TLC model checker).
Your first objection may be that unlike my original English description of the algorithm, in this one you can tell that the algorithm is imperative because it contains the expression $x’ = x  y$ , which looks like an imperative assignment. It is not. All it means is that the variable, i.e. the name, $x$ will refer to the value $x  y$ in the next program step. If you look at a pure functional implementation of the algorithm, you see that the same happens there:
gcd(x:Int, y:int) =
if x = y then x else
if x > y then gcd(xy, y) else
if x < y then gcd(x, yx)
The variables x
and y
stand for different values at different times in the program’s execution, yet no imperative assignment is involved.
Another objection may be that, like a program, the above specification is only one particular description of the algorithm, where others could be chosen. For example, instead of two separate variables, $x$ and $y$, I could have used a single variable assigned a pair of values. However, that seemingly different description would actually be equivalent to the one above; it would still be the same algorithm in a very precise sense. TLA defines an equivalence relation under which two formulas that describe the same algorithm informally also describe the same algorithm formally. The specification above is Euclid’s algorithm, or, rather, it describes the mathematical object that is the algorithm; it is equivalent to all other descriptions of the algorithm, but not to different algorithms for computing the GCD (in part 4 we’ll learn how we can define exactly what it means for different algorithms to be the same or different).
A property of an algorithm can be something like “returns a positive, even integer”, or “runs in $O(n\log n)$”, or “finds the greatest common divisor”. Properties are things that, in the programming languages I mentioned above, you’d normally describe in a contract (Eiffel, SPARK, Clojure) or as a type (Agda, Idris). But in TLA, just as one algorithm can specify another by being a more abstract, less detailed, description — or, in other words, an algorithm can be a “type” — so too can a type describe an algorithm; as we’ll see, there is no distinction in TLA between an algorithm and an algorithm property.
TLA describes sequential, concurrent, parallel and even quantum algorithms — and their properties — all using a simple formalism on top of the logic that specifies data and operations/relations on it (like the one we’ve seen in part 2), of which only equality, =
, is strictly required by TLA. It has just four constructs — $’$, $\Box$, , $\EE$ — only one of which, prime, is absolutely necessary in practice if all you want to do is specify programs and check their correctness in the TLC modelchecker. This minimalism is a testament to the power and elegance of the logic.
The Standard Model, Safety and Liveness
We will describe TLA in two iterations: first a simple but naive version of TLA which Lamport calls “raw” TLA, or rTLA, and then, after describing the problems with rTLA, we’ll get to TLA, which fixes those problems.
TLA (and rTLA), is a logic, and a logic has both syntax and semantics. I will start with the semantics or, rather, with the conceptual framework that will form the semantics of the logic. This is a natural place to start because this was also the chronological order of development of the ideas behind TLA.
Computation As a Discrete Dynamical System
There are many ways to describe computation. One way is to describe a computation as a mathematical function. But this description is too imprecise. For example, a bubblesort algorithm and a mergesort algorithm both compute the same function (from a list of, say, integers, to a sorted list of the same integers), but if our formalism equated computation with functions, we would not be able to tell the two apart, and we know that the two have different properties — like their computational complexity — that we may be interested in, and that a universal formalism must be able to reason about. In addition, modeling important classes of algorithms, like interactive and concurrent systems, as functions is inconvenient, requiring a different treatment from sequential computations. For example, while even the most complex of compilers can be described as a single function from input to output, a trivial clock program that continuously displays the time cannot.
Instead, we will describe a computation as a discrete dynamical process. A dynamical process is some quantity that is a function of time. If time is continuous, we say that the dynamical process is continuous; if time is discrete (and so most likely denotes some logical time rather than physical time), the process is discrete. A continuous process can be expressed mathematically as a function of a positive realvalued time parameter. A discrete process is a function of a positive integer parameter or, put simply, a sequence, usually called a behavior or a trace.
But a sequence of what exactly? We have a few choices. We can have a sequence of what the program does at each step; that would make it a sequence of events (usually, the word action is used instead of event, but I will use event so as not to confuse it with the different concept of action in TLA). An event can be thought of as some output that the program emits or an input it consumes; it can be thought of as an effect. Another alternative is to describe a computation as a sequence of program states. Yet another is as a sequence of states and events (each event occuring at the transition between two consecutive states).
Process calculi (like the πcalculus or CSP) use a trace of events to describe a computation. This choice was made because process calculi use the languagebased, or algebraic approach to specification, and programming languages don’t like to explicitly mention the program’s state, which includes control information like the program counter (the index of which instruction is to be executed next). But this choice has a major drawback on reasoning. Consider the following two flowcharts describing two programs (branching indicates nondeterministic choice):
Both programs generate the same two possible traces: $\set{a \to b, a \to c}$, so they are similar in some sense, yet they are clearly different in another. This means that an algorithm cannot be uniquely defined by its set of possible traces, and we will therefore need some other — more complicated — notion of equivalence^{12}. After all, a definition of any kind of mathematical object must include a notion of equivalence that defines when two objects of that kind are the same.
This leaves us with representing a computation as a sequence of states or a sequence of states and events. We note that a sequence of states and events $s_1 \xrightarrow{e_1} s_2 \xrightarrow{e_2} s_3…$ can be easily represented as this sequence of states alone: $\seq{s_1, } \to \seq{s_2, e_1} \to \seq{s_3, e_2}\ldots$ So we’ll pick the simpler option, use a sequence of states. This is also nicely analogous to continuous systems, which are also functions from time to state.
The Standard Model
We define a computation — an execution of an algorithm — as a behavior. A behavior is an infinite sequence of states, and a state is an assignment, or mapping, of variables to values. What values a variable can take is defined separately by what we’ll call the data logic; in part 2 we’ve seen that in the data logic of TLA^{+}, a value is any set of ZFC set theory. While behaviors are always infinite^{13}, we call behaviors that have a terminal state that is reached in some prefix of the infinite sequence and then never changes terminating behaviors. An abstract system is a collection of behaviors. This is what Lamport calls the standard model.
In this post, will use the term “set” loosely, and interchangeably with “collection”, but note that a collection of behaviors may or may not be a formal set in the set theory sense, depending on the data logic. For example, in TLA^{+}, since a variable can take any set as a value, a collection of behaviors may be “too large to be a set” in ZFC, meaning it may not be a set that can be constructed using the ZFC axioms, but rather a proper class.
An algorithm is a kind of an abstract system because we can think of it as something that generates different behaviors, different executions. Note that not every abstract system, i.e., a set of behaviors, can be reasonably called an algorithm because we require of algorithms that they have some finite description, and by a simple cardinality argument, there are many more sets of behaviors than there are possible strings of finite length. In addition, there may be other limitations (such as computability) that depend on what operations our data logic allows. But this is only a starting point. We will gradually refine the notion of an algorithm so that it nicely coincides with our intuitions as well as other definitions.
How does an algorithm generate more than one behavior? Because an algorithm allows for nondeterminism, which simply means that the specification allows for more than one state at some step of the program. What is the source of the nondeterminism? Input from the environment is one such source. Notice that we didn’t define the algorithm in any parameterized way, like “a function of an integer argument”. If an algorithm is a deterministic subroutine with a single integer parameter called, say, x
, then in our standard model the algorithm will be the set of all of the routine’s behaviors, one for each possible integer value of x
. Another source of nondeterminism may be the behavior of the operating system as it schedules processes. At any step, the OS may execute an instruction of one thread of the application or of another. However, the most important source of nondeterminism is our own choice of level of detail. We may choose to specify an algorithm down to the level of machine instructions or we may leave some details unspecified. The most important insight is that all of these kinds of nondeterminism are actually of all of the same kind — the last. When we specify anything, we describe only certain aspects of its behavior; nondeterminism is all the rest.
A property of a behavior is any collection of behaviors. For example, “the x
variable is a natural number between 3 and 5” is the set of behaviors whose x
variable is always between 3 and 5.
Now, what is a property of an algorithm or a system? If a property of behaviors is a set of behaviors, then a property of systems should be a set of systems, namely a set of sets of behaviors. However — and this is a crucial point in the design and theory of TLA — if we are willing to restrict the kind of algorithm properties we allow, we can simplify things greatly. If we only allow discussing properties that are true of an algorithm if and only if they are true of all its behaviors — those that say what an algorithm must or must not do in every execution — then we can define a property of an algorithm to also be just a set of behaviors. The algorithm satisfies a property iff it — i.e. the set of its behaviors — is a subset of the property. There is no ambiguity.
How can we enforce this restriction? By making our logic firstorder. As we saw in part 2, in a firstorder logic, the model of any formula is a set of objects in the logic’s structure. What does this buy us? This means that properties of behaviors, algorithms, and properties of algorithms are all the same kind of objects — a collection of behaviors. This is an extremely powerful idea. One of the selling points of research programming languages with dependent types is that the algorithm properties they specify — their types — can make full use of the language syntax. Semantically, however, types and programs are completely distinct: you can’t run a type and you can’t use a program as a property (at least, not in the same sense as in TLA). This separation makes sense for a programming language, as it ensures that programs are specified at a level that’s fit for efficient execution, but unifying programs and programs properties makes a formalism for reasoning simpler.
Safety and Liveness
Let us now define two kinds of properties: safety and liveness. We don’t need to say whether we’re talking about a property of behaviors or a property of algorithms, as the two are the same in our framework. A safety property, intuitively speaking, specifies what a behavior must never do. For example, “x
is always an even number”, is a safety property that is violated if x
is ever not an even number. A liveness property says what a behavior must eventually do. For example, “x
will eventually be greater than 100”. Where sequential algorithms are concerned, partial correctness, meaning the assertion that if the program terminates then the result it produces will be the expected one, is a safety property. The only interesting liveness property of a sequential algorithm is termination, meaning the assertion that the program will eventually terminate. Interactive and concurrent programs, however, may have many interesting liveness properties, such as, “every request will eventually yield a response”, or “every runnable process will eventually run”. Worstcase computational complexity, of time or space, is a safety property, as it states that the algorithm must never consume more than some specific amount of time or memory. It may seem obvious that the safety property of worstcase time complexity implies the liveness property of termination, but actually, whether it does or not depends on the specific mathematical framework used (to see how it may not, consider that there may be different relationships between a state in a behavior and what constitutes a program step in the context of time complexity, or different ways to map the states in the behavior to time; but we’ll get to all that later).
The names “safety” and “liveness” in the context of software verification were coined by Lamport in his 1977 paper Proving the Correctness of Multiprocess Programs, and their interesting topological properties, which we will now explore, were discovered by Bowen Alpern and Fred Schneider in their 1987 paper, Recognizing Safety and Liveness. This part is among the least important part of the theory of TLA^{+} for practical purposes, but I personally find the view of computation from the perspective we’ll now explore to be extremely interesting.
Notice the following characteristics of safety and liveness properties: because a safety property says that nothing “bad” must ever happen, a behavior violates it iff some finite prefix of it (remember, behaviors are always infinite) violates the property, as the “bad” thing must happen at some finite point in time in order for the property to be violated. In contrast, a liveness property — something “good” must eventually happen — has the quality that any finite prefix of a behavior could be extended to comply with the property by adding appropriate states that eventually fulfill the property.
If $\sigma$ is some behavior, we will let $\sigma^n$ denote the prefix of length $n$ of the behavior. If we have an infinite sequence of behaviors, $\sigma_1, \sigma_2, …$ we’ll say that the behavior $\sigma$ is their limit, namely $\sigma = \lim_{i \to \infty}\sigma_i$ or just $\sigma = \lim \sigma_i$, iff, intuitively, the prefixes of the behaviors converge to $\sigma$. More precisely, $\sigma = \lim \sigma_i$ iff for every $n \geq 0$ there exists an $m \geq 0$ such that $i > m \implies \sigma_i^n = \sigma^n$. If $S$ is a set of behaviors, we say that the behavior $\sigma$ is a limit point of $S$, iff there are elements $\sigma_1, \sigma_2, …$ in $S$ such that $\sigma = \lim \sigma_i$.
This definition of a limit allows us to turn our universe of behaviors into a topological space^{14}, by defining a set $S$ of behaviors to be closed iff it contains all of its limit points. We then define the closure of any set $S$ of behaviors, denoted $\overline{S}$, as the set of all limit points of $S$. $\overline{S}$ is the smallest closed superset of $S$, and $S$ is closed iff $S = \bar{S}$. We will say that a set $S$ of behaviors is dense iff any behavior whatsoever in the space is a limit point of S.
Back to safety and liveness: Let $P$ be some safety property (i.e. a set of behaviors). If a behavior $\sigma$ does not satisfy $P$, i.e. $\sigma \notin P$, then the property is violated by some finite prefix of the behavior. Let’s say that the first state that violates the behavior is at index $n$. By the definition of convergence, if some subset $\sigma_i \subseteq P$ were to converge to $\sigma$ then all elements of the sequence of $\sigma_i$ would need to coincide with $\sigma^n$ starting at some index, but that would mean that all of the elements after that index would no longer be in $P$ and we get a contradiction. This means that any $\sigma \notin P$ cannot be a limit point of $P$, which means that $P$ contains all its limit points; a safety property is therefore a closed set. In fact, this works in the other direction, too: every closed set can be distinguished by a finite prefix, and so every closed set is a safety property. Now, let $L$ be some liveness property, and let $\sigma$ be any behavior whatsoever. As we said, any finite prefix of any behavior could be extended by adding more states to it so that it is in $L$. This means that for any finite prefix $\sigma_i$ of $\sigma$, we can create a behavior $\hat{\sigma_i}$ such that $\hat{\sigma_i} \in L$ but a prefix of length $i$ of $\hat{\sigma_i}$ is equal to $\sigma_i$ i.e. $\hat{\sigma_{i_i}} = \sigma_i$. This means that $\sigma = \lim \hat{\sigma_i}$. But as $\sigma$ is completely arbitrary, this means that every behavior in the entire space is a limit point of $L$; a liveness property is a dense set. This works in the other direction, too: if $L$ is some dense set, and $S$ is any set, every finite prefix of a behavior in $S$ could be extended by adding states so that it lies in $L$; a dense set is a liveness property.
It is a theorem that every set in a topological space is equal to the intersection of some closed set and some dense set. This means that any system, and therefore any algorithm, is an intersection of a safety property, stating what “bad” things must never happen, and a liveness property, stating what “good” thing must eventually happen.
Temporal Formulas
Let us now turn our attention to the syntax of the logic. TLA (and its simplified version, rTLA, which we’re starting from) is a temporal logic, which is a kind of modal logic. Whereas in the ordinary, nonmodal logic we saw in part 2, a model is an assignment of values to variables that makes the formula true, in modal logic there are multiple modalities, or worlds, in which variables can take different values; $x = 1$ in one world and $x = 2$ in another. A variable that can have different values in different modalities is called a flexible variable. A variable that must take the same value in all modalities is called a rigid variable. In temporal logic, the different modalities represent different points in time. $x = 1$ at one point in time and $x = 2$ in another.
There are two basic kinds of temporal logic. The first, called linear temporal logic, or LTL, views time as linear, and the time modalities are points on a line. The second, computation tree logic, or CTL, views time as branching towards many possible futures, and the time modalities are nodes of a tree. Interestingly, LTL and CTL are incomparable, meaning there are things that can be expressed in one and not the other^{15}. There is, however, a consensus that LTL is easier to use, and probably more useful^{16}, and TLA is a lineartime logic (although it only borrows one construct from LTL), as that fits with our view of computations as behaviors, which are sequences rather than trees.
I will call the flexible variables of our logic temporal variables, and like the rigid variables — declared with the keyword or introduced with the quantifiers $\A$ and $\E$ — temporal variables are declared with the keyword (or its synonym ), or introduced with a temporal quantifier. A temporal variable may have a different value at each state of a computation, i.e. it may change during a single behavior – an execution of an algorithm. It is crucial to remember, however, that while s may not change during a behavior, they may be different for different behaviors.
Whereas a model for a formula in an ordinary logic is an assignment of values to variables that satisfies the formula, a model for a temporal formula is an assignment of values to variables in all modalities, i.e., all states of a behavior. The model, or meaning, or formal semantics of a temporal formula is, then, the set of behaviors that satisfy it. We will explain how a temporal formula is constructed by building it up from different kinds of simple expressions.
Our expressions have four syntactic levels. An expression that does not refer to any temporal variable, either directly or through some definition it refers to, is called a constant expression (level0). An expression that refers to temporal variables, either directly or through a definition (and doesn’t also contain the operators we haven’t yet discussed) is called a state function (level1), because it is some function of the state (remember, the state is the value of all temporal variables). For example, if $x$ and $y$ are temporal variables (declared with ), then the expression $x$ is a state function, as is $x * 2$, as is $x + y$. So is the expression $x < y$, but because that state function denotes a boolean value^{17} we will call it a state predicate (which is just a name for a booleanvalued state function).
Some TLA^{+} constructs, like declarations, can only be constant expressions; we cannot a state predicate or a formula of any of the higher levels we will now see.
Now things get more interesting. In TLA, we can write an expression that is not a function of a single state, but of two states. A state is an assignment of values to temporal variables, and when we want to talk about two different states, we need to consider two different assignments to any variable. We will refer to the variables in one state, say $x$, by denoting it as usual, and to its value in the second state with $x’$ (read, “$x$ prime”). We can now write expressions or predicates about two states, e.g. $x’ > x$, which says that the value of $x$ in the primed state is greater than the value of $x$ in the unprimed state. Why would we want to do that? As every TLA expression relates to a modality — or a point in time. By writing expressions relating two states, we can describe the relationship between the system’s states at two different points in time, and thus describe its evolution. The “current” state is always defined, and as our temporal modalities are arranged as an infinite sequence, a “next” state is also always defined. Unprimed variables will refer to the current state, and primed variables would refer to the next state. If the expression $e$ is a state function (or a state predicate), then $e’$ is equal to the expression $e$ with all the temporal variables in it — and in definitions it references — primed. So the value of the state function $e’$ is the value of the state function $e$ in the next state. An expression that contains primed variables or primed subexpressions is called a transition function or an action expression, or, if it is a predicate, a transition predicate (level2). Transition predicates are better known as actions, the very same actions that give TLA its name and form its very core.
Only constant and statefunction (i.e. level0 and level1) expressions can be primed. Priming an expression that already contains primed subexpressions or other temporal operators we’ll get to soon, is a syntax error, and is illformed. Of course, as a constant has the same value in all states, priming it has no effect.
So actions, or transition predicates, are just predicates of two states, one denoted with unprimed variables and the other with primed variables. If the predicate is true, then the primed state is a possible nextstate of the current, unprimed state.
For example, the expression $x’ = 1$ is a transition predicate — an action — which states that the value of $x$ in the next state will be 1. The expression $x’ = x + 1$ is also an action, this time one that says that the value of $x$ will be incremented by 1, as is $x’ = x + y$, or $x’ \in Nat \land x’ < y’$, which says that the next value of $x$ will be some natural number which is less than the next value of $y$.
If $A$ is an action, $\ENABLED A$ is a state predicate stating that there exists some next state that satisfies A. For example, if:
then $A$ could be read as “if $x$ is even in the current state, then in the next state, $x$ could be either $\frac{x}{2}$ or $\frac{x}{2}$”. $A$ specifies a next state only if $x$ is currently even, so, in this case, $\ENABLED A \equiv x \%2 =0$.
For the sake of completeness, I will mention the action composition operator, $\cdot$ , even though it is unsupported by any of the TLA^{+} tools, and its use is discouraged. If $A$ and $B$ are actions, then $A \cdot B$ is the action that is true for the transition $s \rightharpoonup t$ iff there exists a state $u$ such that $A$ is true for $s \rightharpoonup u$ and $B$ is true for $u \rightharpoonup t$; basically, it’s an $A$ step followed by $B$ step rolled into a single transition.
So far we’ve been talking about expressions, but now have the basic components to start talking about formulas. If $F$ is a formula and $\sigma$ is a behavior, we will write $\sigma \vDash F$ if $F$ is true for the behavior, or $\sigma$ satisfies F. The meaning, or semantics of $F$ (which we write as $\sem F$) is all behaviors $\sigma$ such that $\sigma \vDash F$. I will denote — unlike in the previous section — the (i+1)’th state of $\sigma$ as $\sigma_i$, so $\sigma = \sigma_0 \to \sigma_1 \to \sigma_2 \to \ldots$ Also, $\sigma^{+n} = \sigma_n \to \sigma_{n+1} \to \ldots$, namely the suffix of $\sigma$, with its first $n$ states removed.
We will define wellformed formulas recursively. If $F$ and $G$ are formulas, then so is $F \land G$, and a behavior $\sigma$ satisfies $F \land G$ iff it satisfies both $F$ and $G$ (formally $\sigma \vDash (F \land G) \equiv (\sigma \vDash F) \land (\sigma \vDash G)$). $\neg F$ is a formula, and is satisfied by a behavior $\sigma$ iff $\sigma$ does not satisfy $F$. Similarly, we can define the meaning of all other connectives, but let’s take a look at $\implies$: $\sigma \vDash (F \implies G) \equiv (\sigma \vDash F) \implies (\sigma \vDash G)$. This means that if $F$ and $G$ are formulas, $F \implies G$ iff every behavior that satisfies $F$ also satisfies $G$, or, in other words, the set of behaviors defined by $F$ are a subset of those of $G$’s. This relation forms the core of what it means for an algorithm to implement another or satisfy a property, and we’ll take a closer look at it later.
Now, if $P$ is a state predicate then it is also a formula, and $\sigma \vDash P$ iff the predicate is satisfied by the first state of the behavior. For example, if $x$ is a temporal variable, the formula $x = 3$ denotes all behaviors in which $x$ is 3 in the first state.
If $A$ is an action — namely a transition predicate, or a predicate of two states — then it can also serve as a formula, and $\sigma \vDash A$ iff the action is satisfied by the first two states of the behavior. For example, if $x$ is a temporal variable, the formula $x = 3 \land x’ = x+1$ denotes all behaviors in which $x$ is 3 in the first state and 4 in the second. The formula $x \in Nat \land x’ = x+1$ denotes all behaviors in which $x$ is some natural number in the first state and is incremented by 1 in the second.
Now lets introduce the temporal operator $\Box$, borrowed from LTL. If $F$ is a formula then $\Box F$ is also a formula, and $\sigma \vDash \Box F$ iff $\sigma^{+n} \vDash F$ for all $n$ (i.e. $\sigma \vDash \Box F \equiv \A n \in Nat : \sigma^{+n} \vDash F$).
Therefore, if $P$ is a state predicate, then $\Box P$ means that $P$ is true for every state, because $\sigma \vDash \Box P$ iff $\sigma^{+n} \vDash P$ for all $n$ and the first state of $\sigma^{+n}$ is $\sigma_n$. If $A$ is an action, then $\Box A$ is a formula that means that $A$ is true for every pair of consecutive states, $\seq{\sigma_i, \sigma_{i+1}}$, because $\sigma \vDash \Box A \equiv \A n \in Nat : \sigma^{+n} \vDash A$, and the first two states of $\sigma^{+n}$ are $\seq{\sigma_n, \sigma_{n+1}}$.
How the $\Box$ operator works on general formulas, and how to quickly understand temporal formulas, will become clear when we analyze this example, taken from Specifying Systems:
So a behavior satisfies $\Box((x = 1) \implies \Box(y>0))$ iff if $x = 1$ at some state $n$, then $y > 0$ at state $m + n$ for all $m \in Nat$. The box operator therefore means “always” or “henceforth”, and the above formula means that if $x$ is ever one, then $y > 0$ from then on.
If $F$ is some formula, we define the dual temporal operator $\Diamond$ (diamond) as: $\Diamond F \equiv \neg \Box \neg F$. $\Diamond F$ therefore means “not always not F”, or, more simply, eventually F.
The temporal operators can be combined in interesting ways, so, for example, $\Diamond \Box F$, or “eventually always F”, means that in all behaviors that satisfy this formula, $F$ will eventually be true forever. As termination is defined as a state that stutters forever, such a formula can specify termination, e.g., for a formula with some tuple of variables $v$, the proposition $\E t : F \implies \Diamond\Box (v=t)$ states that the behaviors of $F$ terminate^{18}. $\Box \Diamond F$, or “always eventually F” means that at any point in time $F$ will be true sometime in the future, or, in other words, $F$ will be true infinitely often. Or we can define an operator (built into TLA^{+}) that says that “$F$ leads to $G$”, meaning that if $F$ is ever true, then $G$ must become true some time after that, like so:
Because $\Box$ means “always” and $\Diamond$ means “eventually” you may — quite naturally — be tempted to think that $\Box$ defines only safety properties, while $\Diamond$ defines only liveness properties, but you would be wrong on both counts when the formula contains actions (and that’s a problem which will require addressing!), as we’ll see in the next section.
The following pairs of tautologies hold for the temporal operators:
The operators $\Box$ and $\Diamond$ are duals, and from any temporal tautology we can obtain another by substituting
and reversing the direction of all implications.
TLA has another, rarely used, temporal operator, $\whileop$, that cannot be directly defined in terms of the operators we’ve introduced and which we’ll cover in part 4.
Finally, just as in addition to declaring free ordinary, i.e. rigid, or constant variables with a declaration, we can also introduce bounded ordinary variables with the quantifiers $\A$ and $\E$, so to in addition to declaring free temporal variables with , we can introduce bounded, or quantified, temporal variables with temporal quantifiers.
The existential temporal quantifier, $\EE$, is the temporal analog of regular existential quantification. $\EE x : F$ , where $x$ is a temporal variable, basically says, “there exists some temporal assignment of $x$ such that $F$”; instead of a single value for $x$, it asserts the existence of a value for $x$ in each state of the behavior.
For those of you interested in the finer details of formal logic, the existential temporal quantifier behaves like an existential quantifier because it satisfies the same introduction and elimination rules:
There is also the dual universal temporal quantifier, $\AA x : F\defeq \neg\EE x: \neg F$. The existential temporal quantifier is used to hide internal state; we will talk a lot about that in part 4. The universal temporal quantifier is hardly ever used.
As with the ordinary quantifiers, tuples of variables can also be used with a temporal quantifier, so I will usually write $\EE v : …$ to denote the general form of the existential temporal quantifier with some tuple of variables. Unlike ordinary quantifiers, however, the TLA^{+} syntax does not support bounded temporal quantification (e.g. $\EE x \in Int : F$ is illegal).
It is also possible to define a temporal operator in the same vein, but Lamport writes that he did not add it to TLA^{+} because it is not necessary for writing specifications.
Note that the four expression levels form a hierarchy. A constant predicate (level0) is a degenerate form of a state predicate (level1), which is a degenerate form of an action (level2), one that states that if the predicate holds in the current state then any next state is possible, which is a degenerate form of a temporal formula (level3). TLA^{+} enforces levels at the syntax level and applying operations to expressions with a level for which they are not defined is a syntax error; this is the second kind of property TLA^{+} enforces at the syntax level similarly to typed languages (the first was the enforcement of operator arity we saw in part 2).
Also note that the operators $’$ and $\Box$ (and so $\Diamond$, too) are different from the “ordinary” logical operators we saw in part 2. If the value of the variable $x$ is 3 at some point in time, then $x’$ is not the same as $3’$ (which, 3 being constant, is equal to 3). Similarly, if the value of $A$ is at some point in time, then $\Box A$ is not the same as . This is a feature (called referential opacity) of modal logic, and, in fact of a larger class of logics that modal logics belong to, called intensional logic. Intensional logics have great expressive power – i.e., they can say more things about their universe of discourse than ordinary, nonintensional (or extensional) logics can – in our case, being able to talk about systems at various points, of our choosing, in their behavior over time – but this power comes at the cost of certain inferences not being available (as we’ll see in the chapter, Invariants and Proofs).
Another Take on Algorithms
Actions and $\Box$ are combined to specify algorithms in the following way. Take a look at the formula (assuming $\VARIABLE x$),
This formula defines the following behavior: in the first state $x$ is 0 (recall how state predicates are interpreted), and then it is incremented by 1 at every step. We’ve defined what is clearly an algorithm by specifying an initial state, $x = 0$, and a transition, or a next state relation, $x’ = x + 1$. Using a formula of the form $Init \land \Box Next$ , where $Init$ is the initial condition and $Next$ is an action, we’ve defined a state machine! Now we are ready to refine the definition of an algorithm expressed in TLA: an algorithm is a state machine of the form above. This definition is still wrong, but we’ll fix it later.
We will look at state machines in much more detail and see what a powerful mechanism for defining algorithms they are, but it’s crucial to point out that the expression $x’ = x + 1$ is absolutely not an assignment in the imperative sense, meaning, it is not x := x + 1
. Rather, it is a relation on states called the next state relation, that we can denote with $\rightharpoonup$, such that $s \rightharpoonup t$ iff the state $t$ can follow the state $s$ in a behavior. This relation is written in TLA as an action, which is a predicate on two states given as two sets of variables: unprimed and primed. We could have just as well written $x’  x = 1$ (this is not quite true in TLA^{+}, as we’ll see in a moment, but it is “morally” true).
To drive this point home, let me give a few more examples. We can write an action, $x’ \in \set{1, 1}$ that means that in the next state, the value of $x$ will be either 1 or 1, but we can write an equivalent action like so: $x’ \in Int \land (x^2)’ = 1$. This is because, as per our definition of the prime operator, $(x^2)’ = (x’)^2$. We can write the action $x’ = x + 1 \land y’ = y$, but we can write it equivalently as $\seq{x, y}’ = \seq{x + 1, y}$. It may be hard for people used to programming languages to grasp, but this works not because there is some kind of clever binding or destructuring or unification going on here, but merely because $\seq{x, y}’$ is the same as $\seq{x’, y’}$ and $\seq{x’, y’} = \seq{x + 1, y}$ is just a predicate that is true iff $x’ = x + 1 \land y’ = y$. It’s all just simple math.^{19}
The reason $x’  x = 1$ cannot quite be used instead of $x’ = x + 1$ in the paragraph before last lies in the untyped data logic of TLA^{+} (and so it is outside the scope of our present discussion, but I think it is worth an extra paragraph here, as it may be a source of confusion in real TLA^{+} specifications). It is the same reason that in ordinary arithmetic the equation $y = ax$ is not equivalent to $y/x = a$, even if we assume $x$, $y$ and $a$ are all real numbers – we cannot simply divide both sides by $x$, because $y/x$ is undefined when $x$ is 0. Similarly, in our case $x’ = x + 1$ is not equivalent to $x’  x = 1$. In the first equation, equality can always be used because it is defined for all sets, and we can assume $x$ is an integer because it was 0 in the initial state, and it remains an integer by induction. However, we cannot subtract $x$ from both sides because $x’$ may not be an integer (remember, an action is a predicate on all possible pairs of states), and subtraction may not be defined for it. Another way of thinking about it is that $x’ = \str{hi}$ may be a solution to the equation (in the sense that it cannot be ruled out), because $\str{hi}\; x$ is undefined, which means it is some unknown value, which means it could be 1. Therefore, to write a proper action using the second form, we must write $x’ \in Int \land x’  x = 1$, which is equivalent to $x’ = x + 1$ (assuming the initial state predicate requires $x = 0$, or even more generally, $x \in Int$; otherwise it is equivalent to $x’ \in Int \land x’ = x + 1$. In short, when writing an action, it must be defined for all potential next states (whether or not they are possible next states, i.e. next states for which the action is true), and to make sure it means what we want it to mean, we must ensure all operations used in an action are defined for all potential next states. There is, indeed, an asymmetry here, as we can make do with the operations being defined for all current states that are actually possible in the specification.
That actions are transition predicates has an important consequence. What does an imperative assignment statement like x := x + 1
say about the next value of the variable y
? Such an assignment statement means that only the value of x
changes and nothing else, and so y
remains unchanged. But what does the action $x’ = x + 1$ say about the next value of $y$? Well, the transition predicate holds true of the state transition $[x: 1, y: 5] \rightharpoonup [x: 2, y: 5]$ (and not of $[x: 1, y: 5] \rightharpoonup [x: 3, y: 5]$), but it also holds true for $[x: 1, y: 5] \rightharpoonup [x: 2, y: 500]$ or $[x: 1, y: 5] \rightharpoonup [x: 2, y: \str{hi}]$. In other words, because it says nothing about the next value of $y$, any value is allowed. To say that $y$ doesn’t change, we can write $x’ = x + 1 \land y’ = y$. We can also specify that several variables don’t change by writing, $\seq{x, y, z}’ = \seq{x, y, z}$, but TLA^{+}’s keyword lets us write instead $\UNCHANGED y$ or $\UNCHANGED \seq{x, y, z}$.
What if we want to specify a program that takes the initial value as an input and counts up from there? We could define our algorithm using a parameterized definition like so:
But it is better and more useful to model inputs from the environment as unknowns, or nondeterministic quantities in our algorithm, and instead write:
This is still a single algorithm, but now it has many possible behaviors, one for each initial value, or input.
Now let’s see how we can say things about our algorithms. We’ll define the following algorithm that models an hour clock, and add another definition, that states that the hour is always a whole number between 1 and 12:
We can now make the following claim:
That $HourClock$ implies $WithinBounds$ means that $HourClock$ algorithm satisfies the $WithinBounds$ property, but you can also think of it in terms of behaviors. $HourClock$ is the collection of all behaviors where $h$ counts the hour, and $WithinBounds$ is the collection of all behaviors where $h$ takes any value between 1 and 12 at each step. The behaviors of $HourClock$ are all behaviors of the property, or $\sem{HourClock} \subseteq \sem{WithinBounds}$.
But remember that there is no real distinction between a property of an algorithm and an algorithm; both are just collections of behaviors. To help you see that more clearly, we’ll define the same property differently:
$CrazyClock$ and $WithinBounds$ are equivalent, i.e., $WithinBounds \equiv CrazyClock$ — they specify the exact same set of behaviors — but the way $CrazyClock$ is defined makes it easier for us to think of it in terms of a (nondeterministic) algorithm: it starts by nondeterministically picking a starting value between 1 and 12, and then, at every step, picks a new value between 1 and 12.
It is therefore true that $HourClock \implies CrazyClock$, but both of them are algorithms! We say that $HourClock$ implements, or is an instance of $CrazyClock$ and that $CrazyClock$ is a specification of $HourClock$, or that $HourClock$ refines $CrazyClock$ and that $CrazyClock$ is an abstraction of $HourClock$. But it all means the same: all the behaviors of $HourClock$ are also behaviors of $CrazyClock$.
Logical implication in TLA corresponds with the notion of implementation. In part 4 we’ll explore this general notion in far greater detail, and see that implication can denote very sophisticated forms of implementation by manipulating the subformula on the righthand side of the implication connective.
Two Serious Problems
The logic we’ve defined, rTLA, suffers from two problems. Suppose we want to write an algorithm for a clock that shows both the hour and the minute (remember that a conjunction list, like that appearing after the $\Box$ operator, is read as if it is enclosed in parentheses):
We would very much like it to be the case that $\vdash Clock \implies HourClock$ because a clock that shows both the minute and the hour is, intuitively at least, an instance of a clock that shows the hour. Unfortunately, this isn’t true in rTLA, because while the behaviors of $HourClock$ change the value of $h$ at each step, in $Clock$’s behaviors, $h$ changes only once every 60 steps.
There is also a problem of a slightly more philosophical nature (although it, too, has pragmatic implications). The way we’ve identified the notion of an algorithm with formulas has a very unsatisfying consequence. To see it, let’s revisit our analogy between how continuous dynamical systems are specified with ODEs and how discrete dynamical systems are specified in TLA.
The following continuous dynamical system, $x(0) = 0 \land \dot{x} = 10$ specifies a system that begins at 0, and grows linearly with a slope of 10. It is analogous to this rTLA formula: $x = 0 \land \Box(x’ = x + 10)$. We could even make this clearer by the operator $d(v) \defeq v’  v$ and rewriting the above formula as $x = 0 \land \Box(d(x) = 10)$^{20}.
However, ODEs can be given boundary conditions rather than initial conditions. This ODE specifies the same system: $\dot{x} = 10 \land x(10) = 100$. Could the same be done in rTLA? Absolutely. We’ll make time explicit ($\VARIABLES x, t$):
Or, if you prefer, we can use the equivalence $\Box(A \land B) \equiv \Box A \land \Box B$ to rewrite the formula as:
So, even though we don’t tell our “algorithm” where to start — the initial condition is $x \in Nat$ — it somehow knows that it must start at 0 because that’s the only way it will get to 100 at step 10.
While this may be fine for describing a behavior, it is not how we normally think of an algorithm (although this sort of nondeterminism — called “angelic” nondeterminism^{21}, as it helps the algorithm do the “right thing” — is frequently used in theoretical computer science to study complexity classes; in fact, NP problems are precisely those that can be solved in polynomial time with the help of angelic nondeterminism). When we think of algorithms we assume that it’s not a process that’s allowed to use information from the future.
We could get rid of nondeterminism altogether, but nondeterminism is what gives specifications their power, as it allows us to to say exactly what we know or what we care about, and leave out things we don’t know or details we don’t care about. Also, there are kinds of nondeterminism that fit perfectly with our notion of algorithms, such as nondeterministic input, or nondeterministic scheduling by the OS.
Even the kind of nondeterminism we’ve seen only poses a problem in those formulas we’d like to consider algorithms rather than just general properties of behaviors. We would certainly like to use such nondeterminism in properties that we’d show our algorithms satisfy, i.e., we’d like to show that:
but it would be nice to be able to tell whether a formula is a “proper” algorithm or not.
We could separate algorithms and properties into two separate semantic and possibly syntactic categories. Indeed, this is precisely the road taken by specification languages such as Coq, which use dependent types for specification: arbitrary nondeterminism is allowed only in types. This is pretty much required from programming languages that need to be able to compile programs — but not specifications of their properties — into efficient machine code, so it makes sense for the syntax rules to impose this constraint on descriptions of algorithms (namely, the body of the program).
But from a specification language designed for reasoning, not programming this would require a complicated formalism whereas we want simplicity, and it would also make the important ability of showing that one algorithm implements another much more complicated if not nearly impossible.
This problem is related to why it is not true that $\Box$ specifies only safety properties and $\Diamond$ specifies only liveness properties. In fact, $\Box$ can imply liveness properties and $\Diamond$ can imply safety properties. For example:
and if
then
because for $x$ to reach 10 by the time the time $t$ runs out, it must always increment.
Luckily, one simple solution solves all those problems.
Stuttering Invariance and Machine Closure
We’ll now see how TLA fixes the two problems we encountered with rTLA. Much of the analysis in this section is taken from the paper by Martín Abadi and Leslie Lamport, The Existence of Refinement Mappings, 1991, although the paper doesn’t talk about any formal logic, only about abstract behaviors.
As before, we’ll begin with the semantics.
Invariance Under Stuttering
For a behavior $\sigma$ we’ll define $\natural\sigma$ — the stutterfree form of $\sigma$ — obtained by replacing any finite subsequence of consecutively repeating states with just a single instance of the state. For example $\natural \seq{a,b,b,b,c,d,d,…,d,…} = \seq{a,b,c,d,d,…,d,…}$. Note that the trailing repetition of an infinite sequence of d’s is not replaced. We now define an equivalence relation on behaviors. Behaviors $\sigma$ and $\tau$ are equivalent under stuttering (we write $\sigma \simeq \tau$) iff $\natural \sigma = \natural \tau$. Furthermore, for a set $S$ of behaviors $\Gamma(S)$ is the set of all behaviors that are stutteringequivalent to those of $S$. We say that $S$ is closed under stuttering, if $\Gamma(S) = S$, i.e., if for every behavior in $S$ all behaviors that are stutteringequivalent to it are also in $S$ ($\sigma \in S \land \sigma \simeq \tau \implies \tau \in S$).
Now on to syntax. We say that a formula $F$ is invariant under stuttering iff its model — i.e. the collection of all behaviors that satisfy it — is closed under stuttering. A stutteringinvariant formula cannot distinguish between two stutteringequivalent behaviors (meaning, it cannot be for one and for the other).
Because state functions/predicates only talk about one state at a time and are therefore trivially stuttering invariant, only actions can potentially break stuttering invariance. The formula $x = 1 \land \Box(x’ = x + 1)$ is not invariant under stuttering because it distinguishes between $\seq{1, 2, 3, 4, …}$ and $\seq{1, 1, 1, 2, 3, … }$, as it admits the first but not the second. However, the formula $x = 1 \land \Box(x’ = x + 1 \lor x’ = x)$ doesn’t, and is indeed stuttering invariant.
The following formula is also invariant under stuttering:
because all the added $\Diamond$ clause requires is that an increment of $x$ occurs at least once at some point, namely, that $x$ isn’t always 1. It still cannot distinguish between behaviors that are stuttering equivalent.
If $A$ is an action of the kind we’ve seen so far, and $e$ is some state function, we’ll define the actions:
In practice, $e$ is virtually always just a variable or a tuple of variables. It’s common in TLA^{+} specifications to define a tuple of all temporal variables, $vars \defeq \seq{x ,y, z}$, and write $[A]_{vars}$.
Now, instead of distinct categories for algorithms and properties that complicate matters considerably, all TLA does is enforce a syntax that ensures that all TLA formulas are invariant under stuttering, and all that takes are the following syntax rules:
 When $\Box$ is immediately followed by an action, the action must be of the form $[A]_e$
 When $\Diamond$ is immediately followed by an action, the action must be of the form $\seq{A}_e$
Additionally, an action is not a wellformed formula on its own; it must be immediately preceded by a temporal operator ($\Box$ or $\Diamond$).
So the formula $\Box(x’ = x + 1)$ is illformed in TLA (but is wellformed in rTLA) — it is a syntax error. Note that $\Box(x > 0)$ is still fine, as $x > 0$ is not an action but a state predicate (there are no instances of $x’$).
We similarly change the definition of $\EE$ to mean $\EE x : F$ is true for a behavior $\sigma$ iff it is true for a behavior $\tau$ obtained from $\sigma$ by changing the values of $x$ and adding or removing stuttering steps.
These rules ensure the following: every TLA formula is invariant under stuttering, and therefore cannot distinguish between two behaviors that are stutteringequivalent.
This solves our first problem. Let’s revisit our clocks and fix their definition so that they’re valid TLA by placing the actions inside $[\ldots]_e$:
Indeed, now $Clock \implies HourClock$ (we’ll later see how we can show similar facts using syntactic manipulation).
The precise significance of the conjuncts ($\Box\Diamond(…)$) will be explained later (and we’ll see a shortcut for writing them), but because $\Box[A]_e$ can mean that the action $A$ is never taken (i.e. the initial state stutters forever), those conjuncts ensure the liveness property that the clocks keep ticking an infinite number of times. They never get stuck forever.
Stuttering invariance buys us something else, too. If we think of each state in a behavior as a snapshot in a moment of time, and of the transition between the states as the ticking of some universal clock, when we specify the behavior of variable $x$ there is always room for some other variable to change any (finite) number of times between any changes to $x$ – as we’ve seen in the clock example. But this means that we can now view a TLA specification not as specifying a single system defined by a state of some given temporal variables, but as specifying all of them, and by all of them, I mean all discrete dynamic systems whatsoever. Every state in a behavior is an assignment of values to all variables, not just the variables we mention in our formula. Thanks to stuttering invariance, a TLA formula doesn’t impose any speed or pace on the algorithm or property relative to all others. A TLA formula mentions how just a small finite subset of an infinite set of temporal variables change, and essentially says “I don’t know anything about any other variables”; it really means that it allows any assignment of any value to any of the variables not mentioned, including changes to unmentioned variables that may occur between changes to those mentioned. All systems exist in the same logical universe, which allows us to compare and compose them in interesting ways. Every formula describes a tiny bit of the behavior of the entire universe. Everything else is part of the nondeterminism of the formula — anything that the formula doesn’t determine.
In fact, every TLA formula specifies the entire world, which includes both the system and its environment. Proving that a formula entails some property guarantees that if the world behaves according to the specification, then the behaviors it exhibits will have the property. A particular kind of TLA formulas that explicitly separates the environment from the system and explicitly specifies their interaction is called an “open system specification,” and will be covered in part 4, but all formulas specify both the system and the environment, whether or not they are explicitly separated in the specification. The necessity for invariance under stuttering when specifying systems – i.e. things that exist in the physical world – as opposed to purely abstract mathematical systems is nicely explained in this short note by Lamport.
Machine Closure and Fairness
Now, we’ve seen that a formula of the form $\Box[A]_e$ only specifies a safety property and cannot imply a liveness property (as it can stutter forever), but the converse — that $\Diamond F$ cannot imply a safety property — isn’t true, which brings us to the second problem.
When we rewrite our rTLA formula analogous to the ODE we saw above to get a wellformed TLA formula we get:
This seems to avoid the problem, because now, instead of a behavior that knows the future and picks the right starting value for $x$ , we get many behaviors, some starting with $x=0$ and hitting 100 when $t=10$ and then continuing on incrementing $x$ forever (or stuttering), while others begin with other values, and necessarily stutter forever at some point before the tenth step. Say we start at $x = 2$ and get $x = 102$ when $t = 10$. At that point, the predicate inside the $[]_{\seq{t, x}}$ will be false, but the brackets add an implicit $\lor \seq{t, x}’ = \seq{t, x}$, which means that the behavior will stay at $t = 10, x = 102$ forever.
It doesn’t matter even if we write our formula like so:
However, we can reclaim the power to see the future if we change the $\Box$ in the last conjunct, a safety property, into a $\Diamond$:
We will call the initial condition and the transition relation (i.e. the $\Box[Action]$ part) — in this case, $t = 0 \land x \in Nat\land \Box[t’ = t +1 \land x’ = x + 10]_{\seq{t, x}}$ — the machine property of the specification, and any conjuncts starting with $\Diamond$, the liveness property. It is easy to see that the machine property is a safety property. The problem is this: while $\Diamond(t = 10 \land x = 100) $ in our example is indeed a liveness property, namely every finite prefix of any behavior can be extended so that it satisfies $\Diamond(t = 10 \land x = 100) $, not every behavior that satisfies the machine property can be extended while still complying with the machine property. This particular liveness property interferes with the safety condition of the machine property and rules out even finite prefixes — in fact it rules out all behaviors that don’t begin with $x$ behaving like $\seq{0, 10, 20, 30}$. It implies the safety property that $x$ must be 0 in the initial state. When a liveness property, concerned with what will eventually happen, interferes with a safety property, concerned with what must hold at every state and rules out behaviors that comply with the safety property, that means that information about the future is allowed to affect the present, and that is not how algorithms should be defined.
What we’d want is to make sure that the liveness property does not interfere with the machine property, which implies that every finite prefix of any behavior that satisfies the machine property can be extended so that it satisfies the liveness property. If we denote the set of behaviors that comprise the machine property as $M$ and the liveness property as $L$, we would like it so that $L$ would not specify any safety property (i.e. a closed set) not already implied by $M$. What we want is that $\overline{M \cap L} = M$. If that condition holds, we say that the pair, $\seq{M, L}$ is machine closed, or that $L$ is machine closed with respect to $M$ (or simply that $L$ is machine closed, if it is clear which machine property we’re talking about). A liveness property that is machine closed with respect to its machine property, is also called a fairness property.
A liveness property specifies what must eventually happen. A fairness property must do so in a way that does not dictate precisely what states are allowed at any specific time. The way this is achieved is by having the liveness property specify only that the algorithm moves forwards — makes progress — without specifying any targets it must progress towards. This can be intuitively thought about as specifying the job of the operating system and/or the CPU. Fairness can be thought of as the assumptions we make about how the OS schedules our program. Indeed, when we write a program, we operate under the assumption that it will be scheduled by an OS, but we also assume that the OS will treat our program fairly, and guarantee that it makes progress if it can. In TLA, if we care about liveness and want to prove what our program will eventually do, we must make this assumption explicit, and if our program is multithreaded, we must state our assumptions about how multiple threads are scheduled (we will see how to model threads/processes in the next section). Both the algorithm and the scheduler can still make nondeterministic choices, but machine closure dictates that those choices can in no way be restricted by what has not yet happened, i.e., we allow nondeterminism, but not angelic nondeterminism. Fairness just guarantees progress.
Note that machineclosure is no longer a property of the model of an algorithm — a set of behaviors — but of how that algorithm is written as an intersection of safety and liveness properties. Indeed, two formulas can be equivalent, yet one would be machineclosed and the other won’t. If we replace $x \in Nat$ with $x = 0$ in the initial condition of the specification above, we’ll get one that’s equivalent to it (i.e., admits the very same set of behaviors), yet is now machineclosed, as the liveness property no longer places additional restrictions on the safety property. This is to be expected, because the problem of angelic nondeterminism demonstrates that a set of behaviors is not enough to fully capture the concept of an algorithm.
But how do we know that our liveness property is a fairness property and that our specification is machineclosed? The answer is that we don’t in general, but TLA^{+} has two builtin fairness operators, and if our liveness property is a conjunction of those operators, then we’re guaranteed that it’s machineclosed. The operators, for weak fairness and for strong fairness take an action $A$ as an argument, and ensure that it will eventually occur if it is enabled “often enough”. They only differ in what “often enough” means.
Weak fairness will ensure $A$ eventually occurs if it remains enabled from some point onward; in other words, the weak fairness condition is violated if at some point $\ENABLED \seq{A}_e$ remains true without $\seq{A}_e$ ever occurring. Here is the definition of :
Strong fairness requires $\seq{A}_e$ to occur infinitely often if $\ENABLED \seq{A}_e$ is true infinitely often, although it need not remain enabled constantly:
Strong fairness implies weak fairness, i.e., $\SF_e(A) \implies \WF_e(A)$.
When we specify a sequential algorithm and are interested in liveness, $Fairness$ can be just $\WF_{vars}(Next)$, where $vars$ is a tuple of all of our temporal variables. In the next section, when we learn how threads/processes are simulated, we’ll see why it may be necessary to have more than one / conjunct. Intuitively, because the $Next$ action will result in a step of one or several processes, we would need a more finegrained specification of how the scheduler schedules the different threads, namely, we would need to state that each thread/process is scheduled fairly, meaning that every thread makes progress.
We can now refine yet again our definition of an algorithm, and say that an algorithm in TLA — or, at least a realizable algorithm — is anything that can be specified by a formula of the form
where $v$ is a tuple of temporal variables, and $w$ is a tuple of a subset of those variables. Of course, each of the components can be absent (and often the temporal quantifier is absent). This is called the normal form of a TLA formula. Such a formula is guaranteed to be machineclosed, provided that $Fairness$ is a conjunction of a finite number of $\WF/\SF(A)$ forms, such that $A \implies Next$^{22}.
One can ask why we would want to allow nonmachine closed specification at all. For example, in formulas that specify algorithms — namely those that contain actions rather than just state predicates — we could syntactically allow only the use of / and no other liveness property. Lamport and others answer this question in a short note, and say that it is often beneficial and elegant to specify an algorithm at a highlevel that doesn’t need to be realizable in code, where, in order to show how the algorithm works, we may want to pretend that it can see into the future. Only lowerlevel specifications, those that could be directly translated into program code need to be machine closed, and, of course, we may want to show how a low level, machine closed specification implements a high level, non machine closed one (general relations between algorithms will be covered in part 4).
But if we want some way to tell syntactically whether or not we’re specifying realizable algorithms, at least we’ve now isolated our problems to liveness properties and offered a simple sufficient condition for realizability — use only / as liveness conditions (and actions that imply $Next$, but that comes naturally in most cases, as we’ll see in the section about concurrency).
There is another quirk involving liveness with existential quantification, that I find interesting. Consider the following specification:
Recall that formulas of the form $Init \land \Box[Next]_v$ are safety properties, and because they’re also stuttering invariant, they do not imply any liveness properties. However, with the existential quantifier we get a specification with the single free temporal variable $m$, which admits all behaviors that increment $m$ and eventually stop; in other words, it admits only terminating behaviors that increment $m$. Formally, $Spec \implies \E k : \Diamond\Box(m = k)$. But termination is clearly a liveness property, and so this specification is not equivalent to any machine machineclosed specification involving $m$ and no other variables. The specification $m = 0 \land \Box[m’ = m + 1]_m$ (with only a machine property and no liveness) is not equivalent to our specification as it admits a behavior that increments indefinitely, while the specification $m = 0 \land \Box[m’ = m + 1]_m \land \E k : \Diamond\Box(m = k)$, which is equivalent to $Spec$, is not machineclosed as the liveness property rules out behaviors allowed by the machine property (with $m$ incrementing indefinitely).
As it turns out, the cause of this phenomenon is the combination of the quantifier, which hides the $n$ variable, and what happens at the very first step of the algorithm. That single transition from $m=0 \to m=1$ hides an infinite nondeterministic choice of $n$. If, instead, the nondeterministic choice were finite, say between 0 and 1000, as in:
Then $Spec$ would be equivalent to the following formula that only mentions $m$:
To see why, recall that equivalence on behaviors is stutteringinvariant, so a behavior that stutters and then resumes incrementing, e.g. $\seq{1, 2, 2, 2, 3, \ldots}$, is equivalent to one that doesn’t stutter, e.g. $\seq{1, 2, 3, \ldots}$, unless the stuttering goes on indefinitely. So the formula allows $m$ to be incremented as long as it’s < 1000, but doesn’t require it to; it may stop incrementing at any value less than 1000.
A specification that only allows hidden (i.e., of a bound temporal variable) deterministic choice from a finite set at every step is said to have finite invisible nondeterminism, or FIN. Like machine closure, FIN is a necessary condition for an important result regarding the existence of something called a refinement mapping, which we’ll learn about in part 4 .
The Unreasonable Power of Liveness
I will end this chapter with an example of how problematic, or powerful (depending on your point of view), liveness properties can be. Consider the definition:
where $M$ is an arbitrary formula, and $state$ is a tuple of variables. Note that we’re using an ordinary quantifier, not a temporal one, so $t$ is a constant. You may think that $Halts$ is suspicious because it tells you if an arbitrary algorithm halts and must therefore be undecidable, but there is nothing wrong with defining what it means to halt, and that is all the definition does; it does not claim that there exists an algorithm for deciding halting. Termination is an important property that we may wish to prove for some specific algorithms, and in order to prove it, we must be able to state it. That, in isolation, is an example of a perfectly good use of $\Diamond$.
You might fear that we can use $Halts$ to construct an algorithm to decide halting, as if it were a subroutine we could call, but that is not so simple. $Halts$ is a (parameterized) temporal formula, a level3 expression, and so according to our syntax rules we can’t use it in an action, as in $answer’ = Halts(M, vars)$.
However, consider the specification of an “algorithm” that decides things (meaning, answers “yes” or “no”):
Our decider simply answers for each element in the given set $S$ either “yes” or “no”. It does so simply by using the supplied operator $Is$ which provides the decider with the answer. Seems innocuous enough, except that our decider doesn’t use $Is$ inside an action like so: $answer’ = \IF Is(x) \;\THEN \str{yes}\ELSE \str{no}$, but, instead, uses it in a liveness condition that is very much not machine closed — it fully determines the action’s nondeterministic choice. The final two conjuncts say this: $answer$ will eventually be “yes” or “no”, and it will be “yes” iff $Is$ is true. In the action, we just pick “yes” or “no” nondeterministically, but in the liveness property we “pull” the answer in the right direction. This allows $Is$ to be a temporal formula, bypassing the restriction of using temporal operators in actions.
Using the decider we could define:
Now $H$ “decides” — i.e., forms an “algorithm” for deciding halting. Note that we have not used any noncomputable operations in the data logic. The only source of noncomputability here is the use of $\Diamond$ in the decider in a way that isn’t machineclosed.
Issues of fairness, however, truly matter only for concurrency experts. When writing real TLA^{+} specifications, you’ll find that you rarely use the $\Diamond$ operator (in 1000 lines of specifications, it is common to see $\Diamond$ or $\leadsto$ appear just once or twice), and when you do, it will be to specify very simple and very natural liveness properties (such as: “when we receive a request, we will eventually send a response”). If you only care about safety properties, you can ignore fairness altogether. Indeed, when we use the TLC model checker to check a formula $F$ in normal form, if we only ask it to check for safety properties (i.e. that $\vDash F \implies \Box P$), it will ignore any fairness conjuncts in $F$ as they can have no effect on safety.
It is entirely possible and reasonable to write complete specifications of large realworld systems without once worrying about liveness^{23}. In fact, the entire discussion of temporal formulas and liveness appears only in section 2, “More Advanced Topics”, of Lamport’s book, Specifying Systems. The book even has a short section entitled “The Unimportance of Liveness”, which I’ll reproduce here:
[I]n practice the liveness property [of a specification]… is not as important as the safety part. The ultimate purpose of writing a specification is to avoid errors. Experience shows that most of the benefit from writing and using a specification comes from the safety part. On the other hand, the liveness property is usually easy enough to write. It typically constitutes less than five percent of a specification. So, you might as well write the liveness part. However, when looking for errors, most of your effort should be devoted to examining the safety part.
Liveness can indeed be complicated. Lamport et al. end the note about the importance of allowing nonmachineclosed specifications that I mentioned above, with these words:
[A]rbitrary liveness properties are “problematic”. However, the problem lies in the nature of liveness, not in its definition.
One cannot avoid complexity by definition. — Stephen Jay Gould
We have now completed defining TLA, and explored its trickier nuances (all but one, actually, which we’ll save for part 4). For those of you who may be interested in a technical discussion of this logic’s expressivity, see Alexander Rabinovich, Expressive completeness of temporal logic of action (sic)^{24}, 1998, and Stephan Merz, A More Complete TLA, 1999.
State Machines
Let us now emerge from the depths of TLA, and turn to more mundane things. In a section called “Temporal Logic Considered Confusing” of Specifying Systems, Lamport writes:
Temporal logic is quite expressive, and one can combine its operators in all sorts of ways to express a wide variety of properties. This suggests the following approach to writing a specification: express each property that the system must satisfy with a temporal formula, and then conjoin all these formulas… This approach is philosophically appealing. It has just one problem: it’s practical for only the very simplest of specifications — and even for them, it seldom works well. The unbridled use of temporal logic produces formulas that are hard to understand. Conjoining several of these formulas produces a specification that is impossible to understand.
Algorithms As State Machines
In practice, we write our TLA^{+} specifications as state machines, and check or prove whether they imply some simple properties, usually written as very simple temporal formulas, or whether they implement other, higherlevel specifications also written as state machines (we will cover the implementation relation in part 4).
In Computation and State Machines (2008), Lamport writes:
State machines provide a framework for much of computer science. They can be described and manipulated with ordinary, everyday mathematics—that is, with sets, functions, and simple logic. State machines therefore provide a uniform way to describe computation with simple mathematics.
A state machine — and, note, we are not talking about finite state machines, but about state machines with possibly an infinite number of states that can describe any algorithm — is made of an initial set of states, a transition relation relating a current state to one or more next states, and possibly a fairness property that is important for demonstrating liveness, especially when specifying concurrent algorithms.
State machines are written in TLA^{+} in the normal form we’ve seen:
Where $Init$ is the property, or condition, specifying the initial states, the $Next$ action specifies the nextstate relation, $Fairness$ is the fairness property and the optional existential quantifier is used to hide internal state, as we’ll see in part 4.
Very Simple State Machines
Nearly all of the effort of writing specifications — and most lines — goes into writing the nextstate relation expressed by the $Next$ action of the normal form. Obviously, it’s broken up into a hierarchy of pieces divided over many definitions — just like in programming — but the basic building blocks are simple.
Just as the values our variables can take are not the concern of TLA, but rather of the data logic, so too are the operations performed on those values, which form the primitive operations of our algorithms, namely, what transformations on the data can be performed at each state. In TLA^{+}, any of the builtin or userdefined operators and functions we learned about in part 2 can serve as primitive operations.
Suppose $x$ is a temporal variable. The action that says “if x is even, divide it by two”, can be written as $x \%2 = 0 \land x’ = x/2$. This is a common pattern. Note that it is not the same as the action $x \%2 = 0 \implies x’ = x/2$. The reason is that an action is a predicate on two states. In the first action, the formula is true — i.e. the primed state is a nextstate of the unprimed state — iff $x$ is even and $x’ = x/2$. In the second action, if $x$ is even and $x’ = x/2$ then the predicate is indeed true, but it is also true if $x$ is not even and $x’$ is any value whatsoever, because an implication is true if its left operand is false; this is rarely what you want.
It is very common for an action to be a disjunction (“or”) of conjuncts (“and”), like so:
In this particular case, the same action can be written equivalently as:
but this is only because $\A x : x \%2 = 0 \lor x \% 2 \neq 0$ so the left operand of one and only one of the $\implies$ connectives is guaranteed to be true at any time.
Of course, that action can also be equivalently written as:
or as
We can now write the Collatz conjecture in TLA^{+}:
A property akin to (simple) type correctness in typed programming languages is modeled in TLA^{+} as a simple invariant (a state predicate that is asserted to hold in every state). Most TLA^{+} specifications contain a definition that looks like:
Type preservation is then stated as a simple safety property: $Spec \implies \Box TypeOK$.
Nondeterminism
Nondeterminism plays a central role in TLA. Indeed, it is a crucial tool for analyzing programs in any formalism^{25}. It can express unimportant detail ($x$ is one of those values, doesn’t matter which), IO (the user can enter any of these values), or concurrency (the scheduler may schedule this or that operation). But they all mean the same thing: we don’t know or don’t care about some details.
At the semantic level, nondeterminism is simply the existence of more than one behavior for a specification. In TLA^{+}, nondeterminism is so crucial that every specification is nondeterministic with respect to the values of all variables that it doesn’t mention; nondeterminism is the default, and that we must explicitly opt out of by writing something we do know. Remember, all specifications exist in a single universe, and their model is all behaviors of all possible variables that satisfy a formula that mentions just a small finite set of them.
In the syntax, we can express nondeterminism in the initial condition, actions or any property, by writing propositions that are true for multiple values of a variable, using disjunction, set membership or existential quantification. For example, the following actions are all equivalent, and all forms are commonly found in real specifications:
Of course, “expressing nondeterminism” is a very bad choice of words, as that is really the exact opposite of what the expressions above do: they restrict nondeterminism. An empty formula is the same as : it allows all behaviors. In TLA, like in most logics, anything you write just reduces nondeterminism. $x’ = 2$ reduces it a lot — it makes the choice of $x$ completely deterministic, but it leaves all other infinity of variables deterministic; $x’ \in 1..3$ reduces it just a bit less. The action (or ) says that at the next step, either $x$ and $y$ will not change, or that $x$ will be incremented and $y$ will not changed. The action , which is equivalent to , says that at the next step, either $x$ will not change or that $x$ will be incremented; in either case, $y$ (and $z$ or any other variable) could be any value whatsoever. Not mentioning a variable at all is full nondeterminism. Once you mention it, it’s just a matter of how much that nondeterminism is to be restricted.
Level of Detail
In part 1 I quoted Lamport saying:
… The obsession with language is a strong obstacle to any attempt at unifying different parts of computer science. When one thinks only in terms of language, linguistic differences obscure fundamental similarities.
To see how programming languages obscure important similarities and highlight less important detail — necessary only for technical reasons — let’s consider the following C/Java examples, adapted from Computation and State Machines (the paper from which the above quote was taken), for computing the factorial:
fact1(int n) { int f = 1;
for (int i = 2; i <= n; i++)
f = f*i;
return f; }
fact2(int n) { int f = 1;
for (int i = n; i > 1; i)
f = f*i;
return f; }
fact3(int n) { return (n <= 1) ? 1 : n * fact3(n  1); }
Most programmers will say that fact1
and fact2
are similar, because they’re both iterative or imperative, whereas fact3
is recursive or “functional”. However, those are just similarities or differences in the style of expressing computations; they don’t mean that the computations thus expressed are different. It is actually fact2
that differs most from the other two, and it yields the same result only because multiplication is commutative. We could see that by replacing multiplication with a noncommutative operation like subtraction; the first and third program would give the same result, while the second would give a different one.
In TLA^{+}, we would write the algorithms in state machines (I’ll ignore fairness, which isn’t necessary for safety properties). The algorithm carried out by fact2
would be:
Whereas the algorithm in fact1
and fact3
is:
Of course, one may object that the use of recursion in fact3
is not an irrelevant detail at all as it may have important consequences (say, on memory consumption), and so deserves a much more detailed specification, like this one:
After all, who decides which details are important and which are relevant? The answer is that you do, depending on what you want to model, and what you want to model depends on what properties of your system or algorithm you want to reason about. In any event, TLA^{+} gives the very notion of detail precise meaning, and, as we’ll see in part 4, $Fact3$ is an implementation of $Fact1$ — it is a more detailed (refined) description of $Fact1$ — but not of $Fact2$. Whether or not the extra detail is important, depends on what we want to do with it.
For example, modeling the stack may be important if we want to say something about the space complexity of the algorithm. First, we need to add a measure of space complexity:
We can now say something about the worstcase space complexity:
Note how we restrict $Fact3$ to a single computation on a single input by conjoining it with $N = n$, which restricts $Fact3$’s initial condition $N \in Nat$.
Even though that’s sufficient, let’s do something a little more general (but probably less useful — my intent is to show the power of the formalism, not to teach best practices), that captures the exact definition of complexity:
Then, to express the fact that the space complexity of $Fact3$ is exponential (the size of the input is the logarithm of the integer argument), we can write:
However, as we’ll see in part 4, $Fact1$ (and so $Fact3$ as well) and $Fact2$, are both implementations of the following specification of $FactAlg$,
because they are extensionally equal (note that $fact$ is not an algorithm but a function).
By the way, it is much nicer to write the factorial specification as follows:
but I’ve chosen the version to make some modifications in part 4 easier.
State machines, then, give us a very general, mathematical framework of reasoning about algorithms. Unlike state machines you may be familiar with — Turing machines or finitestate automata — the state machines that are specified in TLA^{+} are completely abstract in the sense that a state can be any mathematical structure (definable in the data logic), and a primitive step, defined as an action, can be as limited or as powerful as we desire (and as the data logic allows).
Many other models of computation can also be easily described as state machines in TLA^{+}. For example, lambda calculus (or any other abstract rewriting system) is normally thought of as a state machine — where each step is a betareduction — only when combined with an evaluation strategy such as callbyname or callbyvalue. However, even without a specific evaluation strategy, lambda calculus is easily described as a state machine in TLA^{+}^{26} that is nondeterministic with respect to an evaluation strategy. In fact, we could do that to prove the confluence of lambda expressions.
Concurrency
In part 2 we saw how to model data at different levels of details, and in the previous section we saw how we can model a stack if we find it to be of importance. But how do we model processes or threads?
Suppose we have the following two specifications:
What does the specification $SpecX \land SpecY$ mean? Well, let’s write it down, and do some basic manipulations on the formula:
So initially both $x$ and $y$ are 0. Then, every action must satisfy both $[NextX]_x$ and $[NextY]_y$, so at every step $x$ is either incremented or stays the same and $y$ is either incremented or stays the same. This means that every step increments either $x$, $y$, both, or neither (if neither, then it’s a stuttering step with respect to $\seq{x, y}$). $SpecX \land SpecY$ specifies what is essentially two processes, $SpecX$ and $SpecY$ running concurrently.
By the way, note that because $SpecX$ and $SpecY$ are really the same process operating on different variables, we could have written a parameterized specification like so:
In fact, we can take that a step further, and generalize this composition to any number of processes:
But for simplicity, we’ll continue with the nonparameterized version, and take a closer look at the action part of the specification, now repeating the reasoning we did before using syntactic manipulation:
We can therefore define:
Where $Spec$ is equivalent to $SpecX \land SpecY$.
Can we also combine $\WF_x(NextX) \land \WF_y(NextY)$ into $\WF_{\seq{x, y}}(Next)$? Absolutely not! The latter specifies that if $Next$ is continually enabled, it will eventually occur — and as $Next$ is always enabled, it means that it will occur infinitely often — which will only guarantee that either $x$ or $y$ will be incremented infinitely often, but maybe only $x$. If our processes are scheduled by a fair scheduler, then both $x$ and $y$ will be incremented infinitely often, which is what $\WF_x(NextX) \land \WF_y(NextY)$ specifies.
If we’d like to forbid a simultaneous change to both $x$ and $y$ to get an interleaving specification we could remove the first disjunct ($NextX \land NextY$) from $Next$. This would result in $Spec$ being equivalent to:
The last conjunct forbids the simultaneous change to $x$ and $y$.
More generally, if is the state function representing process $A$’s internal state (e.g., a tuple of $A$’s internal state variables), is the state function representing process $B$’s internal state, and is the state function representing their shared state (modeling some communication channel). Their conjoined specification would be:
(where specifying that the shared state can only be modified by one of these processes)
Then usually we’d be interested in an interleaving specification, where an $A$ step cannot simultaneously be a $B$ step. In such a specification we have and and therefore we have:
Similarly, we get
The conjunction of the two specifications, then, denotes a parallel composition of processes, but it is equivalent to what appears like a specification of a single process with a nondeterministic action, specified as a disjunction of subactions.
Lamport writes:
The idea that the execution of a sequential algorithm can be described as a sequence of states seems to be due to Turing. It is so widely accepted that most engineers are happy to describe sequential algorithms this way… I think the idea that the execution of a concurrent algorithm (or a concurrent system) can also be described by a sequence of states was implicit in the first paper on concurrent algorithms: Dijkstra’s seminal 1965 paper on mutual exclusion. That idea is not widely known, and many engineers need to be shown how to describe concurrent systems in that way. When they learn how to write such descriptions in TLA^{+}, they find them quite useful.
In other words, there is no difference between multiple processes and a single process with a certain kind of nondeterminism. This has important consequences when reasoning about algorithms, because it means we can use the same proof techniques for both sequential and concurrent algorithms. Lamport’s paper, Processes are in the Eye of the Beholder explores this idea further.
When writing specifications of concurrent systems, instead of writing conjunctions of specifications of each process, we normally write a single $Next$ action, which is a disjunction of subactions representing different processes. As we’ve seen, the two forms are logically equivalent, but if we care about liveness we must remember to specify weak (or strong) fairness of each subaction. Here’s an example:
Notice how $A \implies Next$ and $B \implies Next$, and so our fairness condition $\WF_{vars}(A) \land \WF_{vars}(B)$ satisfies the condition given in the section Machine Closure and Fairness that guarantees that our specification is machineclosed.
It is often the case that we have multiple instances of processes (or machines) each running the same program. We can model that like so:
Note how the existential quantifier is used to nondeterministically select one of the processes. This works because actions are transition predicates. Let’s see that with a concrete example:
Now, the transition $\seq{7,5,8,19} \rightharpoonup \seq{7,6,8,19}$ is a possible transition, because there exists a $p \in Processes$ — in this case $p=2$ — such that $x’ = [x \;\EXCEPT ![self] = @ + 1]$. Does this specification require interleaving or does it allow multiple processes to run at once? After all, an expression with an existential quantifier may be true if more than one element satisfies the predicate. This example does not allow multiple processes to increment their counter simultaneously because the expression $x’ = [x \;\EXCEPT ![self] = @ + 1]$ states that just one index in the sequence is changed, and cannot be true for two different values of $self$. Again, notice that for every $p \in Processes$, $Run(p) \implies Next$ and so our fairness condition satisfies the condition in the Machine Closure and Fairness, and so our specification is machineclosed.
In general, for an interleaving specification (which is what we want most of the time) under normal conditions, the composition of two specifications $A \land B$ becomes — i.e., the conjunction of the temporal formula becomes a disjunction in the action — and similarly, a composition of an arbitrary number of specifications, $\A i \in Component: P(i)$, becomes — i.e., the universal quantification in temporal formula becomes an existential quantification in the action.
Concurrency is yet another example where programming languages emphasize differences that are important details when the goal is efficient compilation but unimportant for reasoning, and obscure similarities that help with reasoning about algorithms. Because a composition of specifications is equivalent to a specification with a single action, it is in general easier to reason about (but not to program!) a complex system when it is expressed with a single action. Lemport makes this argument in his paper, Composition: A Way to Make Proofs Harder.
Let’s now consider a more complicated example of concurrency. We have a producer adding elements to a FIFO queue, and a consumer removing them:
What happens when we conjoin the two specifications? Clearly, the initial condition poses no problem because $ProducerInit \land ConsumerInit \equiv buf = \seq{}$. But $ProducerNext$ says that at every step $buf$ must be appended to or not change, and $ConsumerNext$ says that at every step $buf$ must have an element removed or not change. Formally:
The result is that $ProducerSpec \land ConsumerSpec$ allows only behaviors where $buf$ never changes. The two specifications must be made more flexible to allow each other to change their shared state. We’ll make the following adaptations:
Notice how each action protects its own “private” variable ($put/get$), yet allows modifications to the shared variable $buf$, provided that they comply with its behavior, namely adding or removing an element at a time. Now
From the perspective of the consumer (and similarly, from that of the producer), when we’ve added the disjunct in $ConsumerNext1$ we’ve really just used nondeterminism to model input. We didn’t place any restrictions on the input, while the actual producer only puts integers in the queue; a producer that produces strings would have worked as well. You can think of IO, concurrency and nondeterminism as different ways of looking at the same thing. An interactive algorithm is a concurrent algorithm; whether the user’s behavior is modeled as a separate process or as “inline” nondeterminism within a single action is just a matter of notation.
Lamport writes:
Mathematical manipulation of specifications can yield new insight. A producer/consumer system can be written as the conjunction of two formulas, representing the producer and consumer processes. Simple mathematics allows us to rewrite the same specification as the conjunction of n formulas, each representing a single buffer element. We can view the system not only as the composition of a producer and a consumer process, but also as the composition of n bufferelement processes. Processes are not fundamental components of a system, but abstractions that we impose on it. This insight could not have come from writing specifications in a language whose basic component is the process.
Plotkin’s Parallel Or
Using logical conjunction to denote the parallel composition of concurrent components — or of a system component with input from the environment — is just a special case of composition. We will explore composition more generally in part 4, but as an exercise, let’s consider the following problem: how do we specify a program that takes two programs as arguments, and terminates iff one of them terminates. This problem is known as Plotkin’s parallel or, after Gordon Plotkin, who studied it in the context of reduction strategies for programming languages. The general idea is to simulate each program for a single step in an interleaved fashion.
We can do this without requiring each program’s “source code” to be represented as data, even though programs in TLA are expressed as opaque formulas, and we have no mechanism (like Lisp’s macros), to quote them and access their internal structure:
where $Spec0$ and $Spec1$ are the formulas specifying the two programs (for simplicity, we’re assuming they don’t have liveness conditions^{27}), and $state0$ and $state1$ are the tuples of temporal variables used in each, respectively (assuming that they don’t share variables; in part 4 we’ll see how we can rename variables so that we can enforce this). The internal variable $s$, introduced with the temporal existential quantifier, serves to alternate between one program to the other. As soon as one program terminates — i.e., no longer changes its state — the switching mechanism cannot switch to the other program, and so gets stuck, and the combined program terminates as well.
The ability to do this without quoting or encoding code as data in any way, stems directly from our choice of representing programs as dynamical systems rather than as a mapping of inputs to outputs.
Also, note how $Alternator$, which would be a higherorder combinator in other formalisms, as it “operates” on other programs, is composed with them in a very firstorder way (plus,the composition is, obviously, completely commutative: we could also equivalently write $Spec 0\land Alternator \land Spec1$ or $Alternator \land Spec0 \land Spec1$). In part 4 we’ll see why this form of composition through conjunction is so general.
More Complex Examples
While I won’t describe how to write a large, complex specification (in part 1 I listed some resources containing tutorials and examples) — and so you’ll have to take my word, and Amazon’s technical report I mentioned in part 1, that TLA^{+} specifications scale very gracefully to large, complex systems — I do want to show a couple of more complex example than the ones we’ve seen, algorithms that actually do something interesting. Being able to easily read and understand these specifications does take a bit of practice, but my intention here is just to give a sense of how a real program can be described as a logical formula
I’ve chosen two examples given by Lamport, both of sorting algorithms. The specifications will make use of some of the definitions we’ve defined in part 2, such as $Toset$, $Permutations$, $Ordered$ and $Image$. We’ll assume that both specifications are preceded by the following declarations and definitions, which establish that $S$ is some set with a partial order defined on it:
The first example is an exchange sort (taken from this talk by Lamport). Informally, the algorithm is this: as long as the array is not sorted, pick a pair of consecutive elements that are out of order and swap them. Here it is in TLA^{+}:
Note how highlevel our description is. We don’t specify how we find the pair to swap or how we tell that the array is not yet sorted. Of course, we could add those details, but this level of description captures very clearly the gist of how exchangesort works.
The next example is Quicksort. For a stepbystep explanation of how the specification is written, see this explanation by Lamport. This specification is also at a very high level, attempting to describe only the essence of Quicksort and little else. It describes Quicksort like so: maintain a set of unsorted array regions; remove a region from the set and, if it’s not empty, partition it around some pivot point so that it now contains two subregions, the left with elements smaller or equal to the elements in the right; add those two subregions to the set.
For the partitioning operation, the spec relies on the following mathematical definition of the set of all possible legal partitions, i.e. the set of sequences obtained from $B$ by permuting $B[low]..B[hi]$ such that the subarray left of $lo \leq pivot < hi$ contains only elements that are smaller than in the subarray on the right:
Here is the spec:
Note how the specification makes no mention of recursion in the familiar programming sense — that of a subroutine calling itself, storing data on the stack, and yet captures the essence, the partitioning around a pivot, without specifying how exactly the problem is divided in order to conquer.
This specification makes use of two different forms of “ignoring detail”. One is the use of nondeterminism in the choice of the region to sort and the particular partition; the other is the specification of the partitioning operation as a mathematical definition using the data logic rather than as a computation.
It is easy to see how nondeterminism is lack of detail in a precise way. For example, because $x’ = 1 \implies x’ \in \set{0, 1}$, clearly $(x = 0 \land \Box[x’ = 1]_x) \implies (x = 0 \land \Box[x’ \in \set{0, 1}]_x)$. But in what precise sense is the use of a mathematical definition rather than a computation “ignoring detail”? We will answer that question in part 4.
However, you may rightly object that the specification does not exactly match my description of Quicksort in the section Algorithms and Programs as it performs the partitions sequentially, while doing it in parallel is essentially allowed by the algorithm. To specify the algorithm precisely, instead of picking just one pair from $U$ at each step we nondeterministically pick any number of pairs at once. Doing so requires just a little bit more work, but I’ve chosen to keep the presentation sequential so that it would be a little easier to read for those new to TLA^{+}.
Some algorithms — especially those that are specified at the code level — may be easier for programmers to specify in PlusCal, a pseudocodelike language that can be written inside a comment block in a TLA^{+} file, and automatically translated to a readable TLA^{+} specification. While I will show a tiny example of PlusCal later, I will not cover it, as it is little more than syntax sugar around TLA^{+}. The resources I linked to in part 1 teach PlusCal (except for Specifying Systems, which had been written prior to the development of PlusCal).
HigherOrder Computation
Some formalisms, particularly functional programming languages, make common use of socalled higherorder programs. In those languages, a higherorder subroutine (or “function”), is one that takes as an argument another function or one that returns a function. “Higherorder” programs are firstorder in TLA, similarly to how “higherorder” continuous dynamical systems are expressed as firstorder systems of ordinary differential equation. In the section on Plotkin’s parallel or we’ve seen how we can express a computation that interacts and controls other computations, and in part 4 we’ll see a general way of expressing subroutines and functional compositions, which, in TLA, is also expressed as a firstorder composition, just like in the parallel or example. However, precisely modeling subroutines (AKA, “functions”) and their interaction is not required in order to express “higherorder” computations in TLA^{+} in a simple manner. Let’s examine a higherorder subroutine very familiar to functional programmers, and see how it is expressed in TLA^{+}.
The following is a somewhat unidiomatic implementation of the familiar map
function in Standard ML (I’m not using patternmatching to make the code more readable to those not familiar with the construct). The function takes as parameters a function, f
, and a list, ls
, and returns a list of element, each of which is the result of applying f
to the corresponding element in the input argument ls
.
fun map(f: 'a > 'b, ls: 'a list): 'b list
if l = nil
then nil
else (f (hd ls)) :: (map f (tl ls))
This subroutine is recursive: at each recursive step the variable ls
refers to the tail of ls
in the previous recursive step, and the output elements are accumulated on the stack, later to be assembled into a list data structure back to front.
How would we specify this program in TLA^{+}? A specification of a program represents all possible executions, or behaviors, of that program. As this program is deterministic, we just a get a different behavior for different values of the subroutine’s parameters, f
, and ls
. Different values of ls
correspond to a different list being traversed, but what different behaviors do different values of f
yield?
What f
can represent depends on the programming language. In ML, thanks to typesafety, the compiler would guarantee that if f
returns a value, it would be of type 'b
, and that’s pretty much it. f
may not return a value at all — for example, it may not terminate or it may throw an exception — and it is not guaranteed to return the same value for the same argument (for example, it can randomize its responses).
Let’s describe the program in TLA^{+}. We will not bother with modeling the different representations on the output list first on the stack and then as a list data structure, and we will ignore exceptions (they’re easy to model, but would distract us from the core of this treatment):
The last line encodes our only assumption that, if f
returns a value, it will be one in the set $B$. Note that we are not assuming that f
terminates, as we’ve not added a liveness condition, and so our program may get stuck at any step.
Suppose, however, that the program were written in (again, unidiomatic) Haskell, like so:
map :: (a > b) > [a] > b
map f ls =
if null ls
then []
else (f (head ls)) : (map f (tail ls))
Functions in Hakell do not mean the same thing as they do in ML, because the language provides further guarantees (asuming they’re not circumvented with an escape hatch — a possibility we will choose to ignore in order to show a somewhat different specification). Because Haskell enforces purity, it guarantees that, while f
may throw an exception or not terminate, it behaves a little more like an actual function in the sense that if it returns a value, it will always return the same value for a given argument. This guarantee reduces the nondeterminism in our specification. To describe this expected behavior, we will introduce a function, $fn$, that will allow us to limit the algorithm’s behavior. $fn$ is determined in the first step and then never changes, and is used to represent f
’s functionlike behavior, and ensures that if the algorithm terminates, it will yield the equal output elements for equal input elements.
It is important to note that $fn$ is a function and so does not represent the subroutine f
; it is just used to specify some aspect of its behavior. This specification still captures the possibility that f
may not terminate, as there are no liveness conditions.
If the program were written in a total functional language that guarantees f
’s termination, we would specify it by adding a fairness condition on the algorithm. In part 4 we will explore the abstraction/refinement relation, which forms the very core of TLA’s power, and using which we could justify using the function $fn$ as an abstraction of the subroutine f
, but there is no need to think of $fn$ in this way (although the abstraction/refinement relation serves as the deep explanation to why TLA can specify anything at all).
What We Can and Cannot Specify
Once we have a system specification, we make statements about its behavior that are normally of the form:
i.e., $\vdash Spec \implies P$, Where $P$ can be a property that our specification is supposed to satisfy, or another system specification it implements (we will talk at length about relations on algorithms in TLA in part 4).
What kind of properties do we want to verify? Very often those will be simple safety properties, of the kind $\Box I$, where $I$ is a state predicate, for example, partial correctness, like $\Box(done \implies Ordered(A))$ or $\E c \in Nat : \Box(x + y = c)$. Such properties are called invariants. We can also make some more interesting statements. For example, $\Box[x’ > x]_x$, which states that the value of the variable $x$ can only increase.
We can also write liveness properties like termination, $\Diamond done$, or, “for every request there will eventually be a response” — $\A n \in Nat : \Box(request.id = n \implies \Diamond (response.id = n))$. This can be written as $\A n \in Nat : request.id = n \leadsto response.id = n$. However, we’ll probably model $request$ and $response$ as sequences to which requests and responses are being added so:
We can also express more complex properties like worstcase complexity, as we’ve seen above. In practice, we’d just add a counter variable directly to the specification, but as we’re dealing with theory, we can write:
which increments $time$ whenever any of the specification’s variables change, and then something like:
to state that the worstcase time complexity is quadratic (compare with the even more general complexity operator we defined above).
However, because TLA is a firstorder logic, any property we write in a formula must be true for all behaviors of the specification. Remember $\vdash F$, iff all behaviors satisfy $F$. This works for all safety and liveness properties, but we can’t, for example, express a property about the average complexity of an algorithm, as that would be a property on the set of behaviors of the algorithm, and not of each and every one; in otherwords, averagecase complexity is a secondorder property.
Similarly, we cannot express as a TLA formula a (wrong) statement like: for every property, if two different algorithms satisfy the property, then they must share at least one behavior. It is questionable whether such statements are of any interest at all to engineers (probably not), and in any case, we can express secondorder propositions not as formulas but as an construct (which we’ve seen in part 2, and will see in the context of TLA in the next section).
It is possible to specify a probabilistic machine in TLA^{+} like so:
where $A$, $B$ and $C$ are actions. However, it is unfortunately impossible to specify a probabilistic property, such as “$x$ is greater than 0 with probability 0.75”. This is, I think, the one glaring omission of the formalism. It seems like it may be possible to add this ability to TLA while keeping it firstorder with the addition of a single probability operator, as described in a paper by Zoran Ognjanović, Discrete Lineartime Probabilistic Logics: Completeness, Decidability and Complexity (2006).
Invariants and Proofs
TLA has a relatively complete proof theory for formulas written in the normal form. If $F$ and $G$ are such formulas and $F \implies G$ is valid, i.e., $\vDash F \implies G$, then it is provable by the TLA proof rules, i.e., $\vdash F \implies G$, provided that the necessary assertions about the actions in $F$ and $G$ — written in the data logic — can be proven^{28}.
Because TLA is a temporal logic, it suffers from some of the shortcomings of temporal logic. In ordinary logic, $(F \vdash G) \vdash (F \implies G)$. However, this is not true in temporal logic (or in any modal logic for that matter). If $F$ is a temporal formula, then $\vdash F$ iff $F$ is true for all behaviors (via a completeness theorem for temporal logic). $F \vdash G$ means that if $F$ is true for all behaviors, then it is provable that so is $G$. This means that the proof rule $F \vdash \Box F$ is valid because if $F$ is true of all behaviors, then it is also true for all suffixes of all behaviors (because a suffix of a behavior is a behavior). But $F \nRightarrow \Box F$, or otherwise, $x = 1 \implies \Box(x = 1)$, which is not true. So in temporal logic, $(F \vdash G) \nvdash (F \implies G)$. This happens because implication, like all the logical connectives, “lives” in a modality, whereas $\vdash$ talks about all modalities.
Because of this, Lamport writes:
[This] principle lies at the heart of all ordinary mathematical reasoning. That it does not hold means temporal logic is evil. Unfortunately, it is a necessary evil because it’s the best way to reason about liveness properties. TLA is the least evil temporal logic I know because it relies as much as possible on nontemporal reasoning.
As with the data logic, we can write secondorder theorems to serve as proof rules for TLA using the construct. In part 2 we learned that the keyword (which is synonymous with or just ) introduces an arbitrary constant (level0) name. Similarly, we can introduce names referring to state, action and temporal expressions with (or ), (or ), or (or ) respectively. Remember that , , and form a hierarchy, where an object of one kind is also an instance of all following kinds. We can then state secondorder theorems like the following:
While TLA has various proof rules, some of which apply to liveness like the one above, we will not concern ourselves with proving liveness at all but only with safety properties. In fact, we will not concern ourselves with any of the temporal proof rules but one.
As a verification tool, TLA belongs in the family of methods making use of assertional reasoning, which means that we assert that certain propositions are true at various states of the program. Another forms of reasoning include behavioral reasoning, which makes statements about the program’s entire behavior^{29}.
It is easy to reason behaviorally in an assertional framework by adding a variable which holds a sequence of all the program’s states so far; at each step, the new state is added to the sequence. Such a variable is called a history variable.
Lamport writes:
What a program does next depends on its current state, not on its history. Therefore, a proof that is based on a history variable doesn’t capture the real reason why a program works. I’ve always found that proofs that don’t use history variables teach you more about the algorithm.
The best known assertional reasoning method for sequential algorithms is the FloydHoare method, that attaches an assertion to each of the programs control points (lines). Those assertions are commonly written as Hoare triplets, ${P}C{Q}$, where C is a program statement, $P$ is an assertion of the precondition, namely a proposition about the state of the program before the execution of $C$, and $Q$ is a postcondition, a proposition about the state of the program after the execution of $P$. The FloydHoare method is used to prove what’s known as partial correctness, namely a property of the program that holds true if and when the program terminates.
If $\vdash Spec \implies \Box P$ then $P$ is said to be an invariant of the program — something that holds true at every step. As we’ve seen, partial correctness is an invariant of the form $Terminated \implies P$.
In general, a Hoare triplet can be written in TLA as $P \land C \implies Q’$, where $P$ and $Q$ are state predicates expressing pre and postconditions respectively, and $C$ is an action. In TLA, a program’s control points are made explicit, using a variable that we can call $pc$ (for program counter) that corresponds with a label or a line of the program. If our program contains a statement $c: C; d: … $ where $c$ and $d$ are labels or line numbers, we can write the Hoare triplet as $(pc = c) \land (pc’ = d) \land P \land C \implies Q’$. Looking at the program as a whole, the collection of all Hoare triplets can be expressed as the invariant:
Where $P(c)$ is presumably written as:
where the assertions do not mention the variable $pc$ (as Hoare triplets cannot).
The FloydHoare method is relatively complete for simple programming languages, i.e., if the individual assertions can be proven, then so can partial correctness, but it is not relatively complete for languages with procedures, because partial correctness may require proof that makes use of the state of the callstack, which the FloydHoare method cannot directly mention.
The FloydHoare method has been generalized for concurrent programs by the OwickiGries method, which attaches predicates for each control point of each process. Similarly to the FloydHoare method, OwickiGries assertions can be written in TLA as:
where $c$ is a control point and $p$ is a process (recall how we modeled multiple processes in the section Concurrency above).
In a note about his 1977 paper, Proving the Correctness of Multiprocess Programs, Lamport writes:
In the mid70s, several people were thinking about the problem of verifying concurrent programs. The seminal paper was Ed Ashcroft’s Proving Assertions About Parallel Programs, published in the Journal of Computer and System Sciences in 1975. That paper introduced the fundamental idea of invariance. I discovered how to use the idea of invariance to generalize Floyd’s method to multiprocess programs. As is so often the case, in retrospect the idea seems completely obvious. However, it took me a while to come to it. I remember that, at one point, I thought that a proof would require induction on the number of processes.
When we developed our methods, Owicki and I and most everyone else thought that the OwickiGries method was a great improvement over Ashcroft’s method because it used the program text to decompose the proof. I’ve since come to realize that this was a mistake. It’s better to write a global invariant. Writing the invariant as an annotation allows you to hide some of the explicit dependence of the invariant on the control state. However, even if you’re writing your algorithm as a program, more often than not, writing the invariant as an annotation rather than a single global invariant makes things more complicated. But even worse, an annotation gets you thinking in terms of separate assertions rather than in terms of a single global invariant. And it’s the global invariant that’s important. Ashcroft got it right. Owicki and Gries and I just messed things up. It took me quite a while to figure this out.
Because TLA allows us to reason globally about the program’s state, there is no need to use just invariants of the forms above — those are appropriate for reasoning about programs written in a programming language — rather, we can generalize assertional reasoning completely. That reasoning would work equally well for concurrent and sequential programs, as they are the same kind of object in our model (processes are in the eye of the beholder etc.). A general proof method that works well for many kinds of algorithms, is the inductive invariant method, which works as follows.
Suppose we have a specification $Spec \defeq Init \land \Box[Next]_v \land Fairness$, and want to prove some invariant (safety property) $P$ — i.e. $Spec \implies \Box P$ — like partial correctness. We find a state predicate, $Inv$, such that $Inv \implies P$, but also $Init \implies Inv$ and $Inv \land Next \implies Inv’$. In other words, $Inv$ holds in the initial states, and it is preserved by state transitions. $Inv$ is then called an inductive invariant. We make use of the following proof rule:
If $Init\implies Inv$, using the proof rule above we can conclude that $Init \land \Box[Next]_v \implies \Box Inv$. In other words, if $Inv$ is an inductive property, then it is an invariant. And since it’s an invariant and $Inv \implies P$ then so is $P$, and we conclude $Spec \implies \Box P$.
Formal TLA^{+} proofs based on inductive invariants all have the following structure:
Where PTL stands for propositional temporal logic, and means use of temporal deduction rules. Specifically, the mechanical proof system TLAPS uses a solver for propositional temporal logic to checks proofs by PTL.
Let’s look at a very simple example. Suppose our algorithm is:
The safety property we wish to prove would also be a simple one:
While $P$ is an invariant (which we can conclude by observation in this trivial case), unfortunately it is not an inductive invariant because $P \land Next \nRightarrow P’$, for example if $x = 3$. Instead, the inductive invariant we’ll use to prove $P$ will be:
Now, the formal proof (which can be mechanically checked by TLAPS):
Finally, let’s consider a short exercise given by Lamport in his note Teaching Concurrency:
Consider N processes numbered from 0 through N − 1 in which each process i executes
x[i] := 1 y[i] := x[(i  1) mod N]
and stops, where each
x[i]
initially equals 0. (The reads and writes of eachx[i]
are assumed to be atomic.) This algorithm satisfies the following property: after every process has stopped,y[i]
equals 1 for at least one process i. It is easy to see that the algorithm satisfies this property; the last process i to writey[i]
must set it to 1. But that process doesn’t sety[i]
to 1 because it was the last process to writey
. What a process does depends only on the current state, not on what processes wrote before it. The algorithm satisfies this property because it maintains an inductive invariant. Do you know what that invariant is? If not, then you do not completely understand why the algorithm satisfies this property.
Because this is clearly a codelevel algorithm, it is more convenient to write it in PlusCal, a pseudocodelanguage that is translated to TLA^{+}. The language should be selfexplanatory, except for the labels, that specify atomicity: all statements between two consecutive labels are executed atomically:
Note that I’ve chosen to make reading $x$ before writing $y$ in $p2$ atomic.
When writing this PlusCal algorithm in a special comment box, the TLA^{+} toolbox translates it into the following TLA^{+} specification:
The property we wish to verify is $AtLeastOne1$:
The challenge is finding an inductive invariant, but the TLC model checker can assist with that. We begin with a guess for $Inv$. Assuming that our algorithm is written in the normal form, $Spec \defeq Init \land [Next]_v \land Fairness$, first we make sure that $Inv$ is truly an invariant by letting the model checker verify that $Spec \implies \Box Inv$. Then, $Inv$ is an inductive invariant iff $Init \implies Inv$ (also easy to check) and $Inv \land [Next]_v \implies \Box Inv$. As $Inv \land [Next]_v$ is itself written in normal form, only one where $Inv$ is the initial condition, this, too, can be checked by the model checker (TLC, the model checker included in the TLA^{+} tools can only check temporal formulas containing actions if they’re written in normal form).
In our case, the inductive invariant, $Inv$ is:
The secret to how the algorithm ensures that $AtLeastOne1$ holds is in the last conjunct of $Inv$.
Because TLA allows us to explicitly mention the program’s control state in the inductive invariant (e.g., the control point — the program counter — or the state of the call stack), it is more convenient than FloydHoare or OwickiGries. The inductive invariant method it is relatively complete with respect to proving invariants (i.e., if the assertions about the action can be proven, so can the invariant). For a more thorough discussion of the completeness of the inductive invariant method and comparison with other approaches see Lamport’s Computation and State Machines.
As to the utility and necessity of explicitly mentioning control state, Lamport writes:
There used to be an incredible reluctance by theoretical computer scientists to mention the control state of a program. When I first described the work … to Albert Meyer, he immediately got hung up on the control predicates. We spent an hour arguing about them — I saying that they were necessary (as was first proved by Susan [Owicki] in her thesis), and he saying that I must be doing something wrong. I had the feeling that I was arguing logical necessity against religious belief, and there’s no way logic can overcome religion.
In practice, however, when specifying complex algorithms — let alone large and complex realworld software systems — formal proofs become very costly and are rarely what you need. Finding an inductive invariant of a complex algorithm is not easy. Using the model checker is likely to be at least an order of magnitude cheaper and give you sufficient confidence (TLC checks not only safety but also liveness properties). It almost always yields a better return on investment. Currently, formal proofs are used mostly by academics either to prove the correctness of a novel general algorithm or to demonstrate the viability of the approach in principle. In industry, formal proofs are used either for very small and very critical modules, or (especially in hardware) to tie together results about the correctness of a composition of modules, each verified with a model checker or some other automatic method. Even if you choose to write a formal proof, you should check your theorem in a modelchecker first. As Lamport says, it’s much easier to prove something if it’s true.
Algorithms and Programs Revisited
Algorithms and programs are both specifications of discrete systems that differ in their measure of nondeterminism. The difficulty of verifying a specification — no matter whether using an automated tool like a modelchecker or an SMT solver, or a semiautomated tool like an interactive proof assistant — directly depends on how much detail the specification contains, or, more precisely, on how many different states an algorithm can have. This is a result of computational complexity theory.
An interesting question is whether, instead of verifying a very detailed specification, we can generate it from a less detailed one. Alas, the two problems are equivalent. Like almost all problems in program analysis — like verification or superoptimization — program generation is uncomputable in general.
To see why, suppose we had an algorithm that, given a specification $S$, produces a program $P$ such that $P$ implements $S$ — we will write that $P \implies S$ — if such a program can exist (i.e., if the specification does not contain some contradiction). But a program is also a specification, so we can pass it as input, too. Now, given any program $Q$, we give our algorithm as input the specification $Q \land Halts(Q)$. Being able to generating a program or declare that the specification is contradictory, therefore, requires the ability to decide halting.
And like other problems that are uncomputable in the general case, we cannot satisfactorily avoid the difficulty in practice by adding restrictions (like requiring that the algorithm produces output only if the specification is consistent or by somehow ensuring it in advance) for reasons explained in this post. Problems that are uncomputable in general, are often intractable even in very restricted circumstances. In practice, there are various research tools that can generate code from specification, but their practical capabilities are nothing to write home about, and rarely rise above being a party trick.
Conclusion
The temporal logic of actions, forms the core of TLA^{+}. Like other elements of TLA^{+}, its core logic was designed with great attention to practicality, informed by years of experience. Lamport writes:
TLA differs from other temporal logics because it is based on the principle that temporal logic is a necessary evil that should be avoided as much as possible. Temporal formulas tend to be harder to understand than formulas of ordinary firstorder logic, and temporal logic reasoning is more complicated than ordinary mathematical (nonmodal) reasoning.
A typical TLA specification contains one $\Box$, … and a few and/or formulas—even if the specification is hundreds of lines long. Most of the specification consists of definitions of predicates and actions. Temporal reasoning is reduced to a bare minimum; it is used almost exclusively for proving liveness properties. Formalizing liveness proofs is temporal logic’s forte.
I hope it is clear by now that most questions about semantics that are pertinent to discussions of formal reasoning about programming languages are inconsequential in TLA^{+}. The semantics of TLA^{+} is so straightforward that it bears little resemblance to programming languages: temporal formulas mean sets (more correctly, classes) of infinite state sequences; a state predicate means a set (class) of states; an action means a relation on states. Composition works using the usual propositional logic semantics (intersection, union, inclusion) of the semantic objects, etc.. Two TLA formulas $F$ and $G$ describe the same sets of behaviors iff $\vdash F \equiv G$. Discussions of denotational and operational semantics, full abstraction etc., that fuel long discussions in programming language theory are all irrelevant in TLA^{+}.
Programming languages are complex because they have to be; a mathematical formalism for reasoning about programs doesn’t have to be. Just as ODEs can be used to reason about continuous systems generally, without getting into the specifics of the language used to describe their construction, so can TLA be used to generally reason about discrete processes. Doing so gives insight into the similarity between different systems, rather than having technical details obscure them. Being able to specify algorithms or systems rather than programs allows to reason at any level of detail, and to scale up the size of systems we can reason about.
Lamport writes:
In a mathematical approach, there is no distinction between programs, specifications, and properties. They are all just mathematical formulas. … Perhaps the greatest advantage of specifying with mathematics is that it allows us to describe systems the way we want to, without being constrained by ad hoc language constructs. … Mathematical manipulation of specifications can yield new insight.
We’ve seen how TLA’s explicit treatment of time makes the difficulty of reasoning about sideeffects disappear. Side effects and “pure calculation” are just different interpretation of a computation’s behavior. Can this view of time be used in programming as well? The answer is yes, and there are programming languages that also model (logical) time explicitly. It turns out that those languages make reasoning about sideeffects as easy as reasoning about computation, because their theory — like TLA — naturally embraces both as essential features of computation. Indeed, those are the languages that, to date, yield programs that are the easiest to formally analyze among all programming languages. Unsurprisingly, that programming style — called synchronous programming and developed in the 1980s through the work on languages like Esterel and Statecharts — is used in industry where formal verification is crucial, namely safetycritical realtime systems. It should be noted that the style was developed both for its amenability to formal reasoning as well as for informal reasoning, meaning ease of design and communication among developers.
Until recently, that style has been largely confined to that specialized niche, but has now started slowly making its way towards generalpurpose programming, in languages such as Céu and Eve. The latter shares even more similarity with TLA^{+}, as it is not only a synchronous language, but also a logicprogramming language. This is unsurprising as Eve is based on a language semantics called Dedalus, which has been inspired, in part, by TLA^{+}. Notice how easy it is to express global correctness properties even in a language like Eve, designed to be beginnerfriendly. Synchronous languages naturally accommodate interesting, ergonomic higherlevel abstractions, like behavioral programming. I hope more attention would be paid to this programming style, and that it would find its way out of the highassurance software niche, and into mainstream programming, as it may turn out to have something important to contribute.
Next week we’ll look at how TLA^{+} supports encapsulation and composition, at general notions of equivalence on algorithms, at the abstraction/implementation relation, and at how TLA^{+} compares with similar notions in other formalisms. We will also see some advanced uses of TLA^{+} for specifying realtime and hybrid, or cyberphysical systems — systems that interact with the physical world and its continuous quantities, with sensors and actuators.

Reproduced in F. L. Morris and C. B. Jones in An Early Program Proof by Alan Turing, 1984. The original, with multiple significant printing errors, is in the Turing Archive. ↩

This was in part Turing’s own fault, who lost his audience when he wrote numbers on the blackboard in reverse, as he’d been used to when programming the Manchester computer. ↩

If you are a functional programmer, please don’t confuse “program state”, with what’s know as “state”, or “mutable state” in functional programming. I hope to make the distinction clear later. ↩

See part 2 for a brief explanation of what a model checker is. ↩

Amir Pnueli, The Temporal Logic of Programs, 1977 ↩

The claim that the language semantics is preserved across different compiler versions is not sufficient to resolve the issue, as it just means that equivalence under the details of the published semantics is the crucial one, but that level of detail in the semantics may be arbitrary, and we have no a priori reason to prefer that equivalence over others. ↩

One may object and say that some minimal level of detail is required or an algorithm for finding the greatest element in a list could be “find the greatest element in the list.” However, insisting on any minimal level of detail would get us into even greater trouble, because we do accept “find the greatest element in the list” as a description of sufficient detail if it appears as a step in some larger algorithm. ↩

By which I mean that there is no mechanical syntactic rule that can tell us — or even hint — whether a TLA specifications specifies a pure computation or one with sideeffects. ↩

On this paper Alan Kay remarked, “you don’t need monads”. ↩

This is not true for programs written in any programming language, where it is generally hard to represent exactly the same concept in different languages. A BASIC subroutine is something quite different from a Haskell function. In TLA, we can choose to represent both as the same mathematical object or choose to model their precise differences. ↩

We will briefly discuss equivalence in the process calculi in part 4. ↩

Just as differential equations normally specify functions over an infinite duration of time. ↩

The same could also be done by using a metric space where the distance between two behaviors is $1/n$ when the earliest state in which they differ is the nth one ↩

The temporal logic CTL*, however, is a superset of both, and can express anything either logic can. ↩

See Moshe Vardi, Branching vs. Linear Time: Final Showdown, 2002 ↩

Assuming, of course, that the relation $\lt$ is indeed defined on the values of $x$ and $y$. ↩

Although this is rarely if ever done in practice. Usually we declare a boolean variable called $done$ and specify termination as $\Diamond done$. ↩

Note that the TLA^{+} tooling, like the model checker TLC, may not be equally accommodating to the equivalent ways of writing the actions in the examples above, and may strongly prefer the simpler forms over the more complicated ones. ↩

Again, this is not quite correct in TLA^{+}. To correct it, we’d need to say $x’ \in Real \land d(x) = 10$. ↩

The term was coined by Tony Hoare. ↩

To see why this is a necessary condition, consider our running example with $\land \Box(t = 10 \implies x = 100) \land \WF_t(t’ = t + 1)$. ↩

Or even temporal formulas at all, as the model checker supports checking specifications given as two separate $Init$ and $Next$ formulas, as well as verify safety invariants given only as $P$ instead of $\Box P$, so there’s even no need to know about $\Box$ and $[]_e$. ↩

An ωlanguage is definable in TLA if and only if it is stuttering closed and definable in monadic second order logic. ↩

For example, in formalisms based on type theory, types express nondeterminism. $e : Int$, is the static assertion that the expression $e$ can be any integer, and we don’t know or don’t care which. ↩

The syntax may be a little unpleasant, as we’d need to encode lambda expressions as tuples or records. ↩

This is because each program could specify — as we’ve seen in the previous section — that it is to be scheduled by a fair scheduler, whereas our intent is the exact opposite. Nevertheless, a specification that handles that case, too, is just a little more complicated. Consider it an exercise, and I’ll post the solution in part 4. ↩

An extended version of TLA that is equally expressive but with a proof system that is complete for all TLA formulas can be found in Stephan Merz, A More Complete TLA, 1999. An encoding of the TLA* logic introduced in that paper in Isabelle/HOL can be found here. ↩

There’s also equational reasoning which uses mathematical (or denotational) semantics to assign a certain abstract mathematical object to each expression in the program, and then uses substitution and equality as with algebraic equations. Speaking of equational reasoning in TLA is irrelevant as TLA is not a programming language but a logic. All logical statements can be understood as algebraic equations — and inequalities — over a boolean algebra. Unlike a programming language, the semantics of a logic are so simple that it makes sense to point out equational reasoning. It becomes subsumed in all forms of logical reasoning. ↩