Guarded commands

The Guarded Command Language (GCL) is a programming language defined by Edsger Dijkstra for predicate transformer semantics in EWD472. It combines programming concepts in a compact way. It makes it easier to develop a program and its proof hand-in-hand, with the proof ideas leading the way; moreover, parts of a program can actually be ''calculated''.

An important property of GCL is nondeterminism. For example, in the if-statement, several alternatives may be true, and the choice is made at runtime, when the if-statement is executed. This frees the programmer from having to make unnecessary choices and is an aid in the formal development of programs.

GCL includes the multiple assignment statement. For example, execution of the statement is done by first evaluating the righthand side values and then storing them in the lefthand variables. Thus, this statement swaps the values of and .

The following books discuss the development of programs using GCL:

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Guarded command

A guarded command consists of a boolean condition or guard, and a statement "guarded" by it. The statement is only executed if the guard is true, so when reasoning about the statement, the condition can be assumed true. This makes it easier to prove the program meets a specification.

Syntax

A guarded command is a statement of the form G → S, where
* G is a proposition, called the guard
* S is a statement

Semantics

* If G evaluates to true, S is eligible to be executed. In most GCL constructs, multiple guarded commands may have guards that are true, and exactly one of them is chosen ''arbitrarily'' to be executed.
* If G is false, S will not be executed.

skip and abort

skip and abort are important statements in the guarded command language. abort is the undefined instruction: do anything. It does not even need to terminate. It is used to describe the program when formulating a proof, in which case the proof usually fails. skip is the empty instruction: do nothing. It is often used when the syntax requires a statement but the state should not change.

Syntax

skip

abort

Semantics

* skip: do nothing
* abort: do anything

Assignment

Assigns values to variables.

Syntax

v := E
or
v, v, ..., v := E, E, ..., E

where
* v are program variables
* E are expressions of the same data type as their corresponding variables

Catenation

Statements are separated by one semicolon (;)

Selection: if

The selection (often called the "conditional statement" or "if statement") is a list of guarded commands, of which one is chosen to execute. If more than one guard is true, one statement whose guard is true is arbitrarily chosen to be executed. If no guard is true, the result is undefined, that is, equivalent to abort. Because at least one of the guards must be true, the empty statement skip is often needed. The statement if fi has no guarded commands, so there is never a true guard. Hence, if fi is equivalent to abort.

Syntax

if G0 → S0
□ G1 → S1
...
□ Gn → Sn
fi

Semantics

Upon execution of a selection, the guards are evaluated. If none of the guards is ''true'', then the selection aborts, otherwise one of the clauses with a ''true'' guard is chosen arbitrarily and its statement is executed.

Implementation

GCL does not specify an implementation. Since guards cannot have side effects and the choice of clause is arbitrary, an implementation may evaluate the guards in any sequence and choose the first ''true'' clause, for example.

Examples

Simple

In pseudocode:
if a < b then set c to True
else set c to False

In guarded command language:
if a > b → c := true
□ a < b → c := false
fi

Use of skip

In pseudocode:
if error is True then set x to 0

In guarded command language:
if error → x := 0
□ error → skip
fi

If the second guard is omitted and error is False, the result is abort.

More guards true

if a ≥ b → max := a
□ b ≥ a → max := b
fi

If a = b, either a or b is chosen as the new value for the maximum, with equal results. However, the implementation may find that one is easier or faster than the other. Since there is no difference to the programmer, any implementation will do.

Repetition: ''do''

Execution of this repetition, or loop, is shown below.

Syntax

do G0 → S0
□ G1 → S1
...
□ Gn → Sn
od

Semantics

Execution of the repetition consists of executing 0 or more ''iterations'', where an iteration consists of arbitrarily choosing a guarded command whose guard is true and executing the command . Thus, if all guards are initially false, the repetition terminates immediately, without executing an iteration. Execution of the repetition do od, which has no guarded commands, executes 0 iterations, so do od is equivalent to skip.

Examples

Original

Euclidean algorithm

a, b := A, B;
do a < b → b := b - a
□ b < a → a := a - b
od

This repetition ends when a = b, in which case a and b hold the greatest common divisor of A and B.

Dijkstra sees in this algorithm a way of synchronizing two infinite cycles a := a - b and b := b - a in such a way that a≥0 and b≥0 remains true.

Extended Euclidean algorithm

a, b, x, y, u, v := A, B, 1, 0, 0, 1;
do b ≠ 0 →
q, r := a div b, a mod b;
a, b, x, y, u, v := b, r, u, v, x - q*u, y - q*v
od

This repetition ends when b = 0, in which case the variables hold the solution to Bézout's identity: xA + yB = gcd(A,B) .

Non-deterministic sort

do abubble sort