A High-Level View of TLA+

# A High-Level View of TLA+

Leslie Lamport

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## Introduction       [show]

TLA+ is a language for modeling software above the code level and hardware above the circuit level.  It has an IDE (Integrated Development Environment) for writing models and running tools to check them.  The tool most commonly used by engineers is the TLC model checker, but there is also a proof checker.  TLA+ is based on mathematics and does not resemble any programming language.  Most engineers will find PlusCal, described below, to be the easiest way to start using TLA+.

TLA+ models are usually called specifications.  They are called models in this introduction.

## PlusCal [show]

PlusCal is a language for writing algorithms—especially concurrent and distributed ones.  It is meant to replace pseudocode with precise, testable code.  PlusCal looks like a simple toy programming language, but with constructs for describing concurrency and nondeterminacy.  It is infinitely more expressive than any programming language because any mathematical formula can be used as a PlusCal expression.  A PlusCal algorithm is translated into a TLA+ model that can be checked with the TLA+ tools.  Because it looks like a programming language, most engineers find PlusCal easier to learn than TLA+.  But because it looks like a programming language, PlusCal cannot structure complex models as well as TLA+ can.

## Models       [show]

Computers and computer networks are physical objects whose behaviors are described by continuous physical laws.  They differ from most other kinds of physical objects in that their behaviors are naturally modeled as sets of discrete events.  Programming, software engineering, and most of computer science is concerned with models in which a behavior of a system is described as a set of discrete events. No model is a completely accurate description of a real system.  A model is a description of some aspect of the system, written for some purpose.

TLA+ is state-based, meaning that it models an execution of a system as a sequence of states, where an event is represented by a pair of consecutive states.  We call a sequence of states a behavior; and we call a pair of consecutive states a step rather than an event.  A system is modeled as the set of behaviors describing all of its possible executions.

## Modeling Above the Code Level       [show]

TLA+ is used to model systems above the code level.  To see what this, means consider Euclid's algorithm for computing `GCD(M,N)` , the greatest common divisor of two positive integers  `M`  and  `N` . Here is the algorithm:
• Let  `x`  equal  `M`  and  `y`  equal  `N` .
• Repeated subtract the smaller of  `x`  and  `y`  from the larger.
• Stop when  `x`  and  `y`  have the same value. That value is the GCD of  `M`  and  `N` .
This description is above the code level.  Code to compute `GCD(M,N)` would have to specify additional details such as the type of  `M`  and  `N`  (is it  `int`?  `long`?  `BigInteger`?) and what to do if  `M`  or  `N`  is not positive (throw an exception? return an error value?).

A programmer who didn't know Euclid's algorithm might decide to compute `GCD(M,N)` by a naive algorithm that sets  `x`  to the minimum of  `M`  and  `N`  and keeps decreasing  `x`  until it divides both  `M`  and  `N` .  The best coder in the world will not produce a good GCD program by coding the naive algorithm.  Moreover, thinking in terms of code makes it harder for a programmer to find a better algorithm.  Finding a good algorithm requires thinking above the code level.

No one writes a piece of code without first having a high-level model of what the code should do and how it should do it.  A programmer never starts by deciding to declare some variables, adding a  while  statement, then adding an  if  statement, and so on—only discovering when finished that she's written a sorting program.  But programmers rarely start with a precise model of the code. Having only a vague, incomplete model leads to basic design errors that the best coding won't correct.

TLA+ is a language for writing precise high-level models of what code does and how it does it.  Most programmers believe that precise models are good only for tiny well-defined problems like computing the GCD, but are useless for implementing complex systems.  They're wrong.  The more complex a system is, the more important it is to make it as simple as possible.  In complex systems, simplicity isn't achieved by coding tricks.  It's achieved by rigorous thinking above the code level.

In one industrial project, starting with a TLA+ model reduced the size of a real-time operating system's code by a factor of ten.  Such a reduction in code size isn't obtained by better coding; it comes from thinking rigorously above the code level.

Writing a model above the code level doesn't prevent coding errors.  Many methods and tools have been developed for finding coding errors, and they should be used.  But they are not good for finding errors in the high-level model from which the code is derived.  And it's impossible to test that a high-level model is implemented correctly if the model is just a vague idea in the programmer's mind, with no precise description.

Testing the code is not an effective way to find fundamental design errors—especially in concurrent and distributed systems.  Moreover, a design error found after the code has been written is usually fixed with an ad hoc patch that is unlikely to eliminate all instances of the problem and is likely to introduce new errors.  Design errors should be caught by writing a precise high-level model, before the code is written.

## Modeling Concurrent Systems       [show]

A concurrent system is one that we think of as composed of multiple concurrently operating components called processes A distributed system is a concurrent system in which we think of the processes as being spatially separated, usually communicating with one other by sending messages.

In a state-based model, a state represents the entire physical state of a system.  Some people find it hard to believe that one can or should model a distributed system in terms of a single global state.  Over 40 years of experience has taught me that this is the most generally useful way to model distributed algorithms and systems.

## State Machines       [show]

Like many state-based methods, TLA+ describes a set of behaviors with two things:
1. An initial condition that specifies the possible starting states.
2. A next-state relation that specifies the possible steps (pairs of successive states).
They specify the set of behaviors whose first state satisfies the initial condition and whose every step satisfies the next-state relation.

This kind of model is often called a state machine.  (A finite-state machine is a state machine with a finite set of possible states.  Finite-state machines are not nearly as useful as general state machines.)  A Turing machine is an example of a state machine.  In a deterministic Turing machine, the next-state relation allows at most one next state for any state, and it allows no next state for a terminating state.

The simplest and most practical method of precisely describing the semantics of a programming language, called operational semantics, essentially consists of showing how to compile a program in the language to a state machine.  Given an operational semantics, any program in the language can be viewed as a state machine.  I suspect that most programmers intuitively think of a program in that way.

The next-state action specifies what steps may happen; it doesn't specify what steps, if any, must happen.  That requires an additional condition, called a fairness property.  A state machine that models a sequential program usually includes the fairness property that some step must be taken (the behavior must not stop) if the next-state relation allows a step to be taken.  Models of concurrent and distributed programs often have more complicated fairness properties.

A state-machine model without a fairness condition can be used to catch errors of commision, in which the system does something wrong.  It can't be used to catch errors of omission, in which the system fails to do something.  In practice, errors of commission are more numerous and harder to find than errors of omission.  Often, engineers don't bother adding fairness conditions.  Therefore, you should first learn to write the initial condition and next-state relation in your TLA+ models.  Later, you can learn to write fairness conditions.

## Checking Properties [show]

One reason for modeling a system is to check if it does what we want it to.  We do that by checking if the model satisfies properties that we believe assert that the system does what it should.  TLA+ can assert, and its tools can check, only that some property of an individual behavior is true of every possible behavior of the model.  Thus, TLA+ cannot assert that 99% of all possible behaviors terminate in a correct state.  However, it can assert (and its tools can check) that every possible behavior terminates in a correct state if its initial state belongs to a particular set containing 99% of all possible initial states.

The most useful type of property to check is an invariance property, which asserts that something is true of every state of every possible behavior.  Often, an engineer will check only invariance properties of a model.

For a model containing a fairness condition, you should also check simple properties asserting that something eventually happens—for example, that every execution eventually halts.  Those properties, called liveness properties, are easily expressed in TLA+.

The rich variety of properties that we want to check for concurrent systems can't all be expressed as invariance and simple liveness properties.  They can be expressed as state machines (possibly with fairness conditions).  A state machine can be viewed as the property that is satisfied by the possible behaviors of the state machine.  We can check whether another state machine satisfies this property.  If it does, we say that the other state machine implements the state machine.

In TLA+ there is no formal distinction between a state machine and a property.  Both are described by mathematical formulas.  A state machine is a formula having a particular shape, different from the shape of an invariance or liveness property.  Both satisfying a property and implementing a state machine mean that one formula implies another.

Today, most engineers check only invariance properties and simple liveness properties.  However, even if you never do it, knowing how it is done explains what it means for a program to implement a model, which can help you avoid making errors in your code.

## The TLA+ Language [show]

TLA+ is based on mathematics and does not resemble a programming language.  Most engineers are familiar with programming languages, but not with precise mathematical notation.  We naturally find what we're familiar with to be simpler than anything else.  It's hard for me to believe that English is not inherently simpler than German.  Upon first seeing a TLA+ model, some engineers find TLA+ intimidating.  Read the Industrial Use of TLA+ page to see that TLA+ is not very hard to learn.

Using TLA+ teaches you that math is inherently more expressive than programming languages because it can describe a value without having to describe how the value is computed.  For example, it can describe the greatest common divisor (GCD) of two numbers as the largest positive integer that divides both numbers.  This makes it possible to write a model for a specific purpose, abstracting away irrelevant details such as how to calculate the GCD.  (Putting code in a procedure doesn't abstract it away;  it just makes you go elsewhere to read the code, requiring that you understand the semantics of procedure call as well as the code.)

Starting with PlusCal provides a gentle entry to TLA+. Even if you know TLA+, it's easier to write some models in PlusCal rather than directly in TLA+. But to get the full benefit of thinking mathematically above the code level, you should learn TLA+.