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# Predicate transformer semantics

Predicate transformer semantics is an extension of Floyd-Hoare Logic invented by Dijkstra and extended and refined by other researchers. It was first introduced in Dijkstra's paper "Guarded commands, nondeterminacy and formal derivation of programs". It is a method for defining the semantics of an imperative programming language by assigning to each command in the language a corresponding predicate transformer. A predicate transformer is a total function mapping between two predicates on the state space of a program.

The canonical predicate transformer of sequential imperative programming is the so-called "weakest pre-condition" wp(S,R). Here S denotes a list of commands and R denotes a predicate on the space, called the "post-condition". The result of applying this function gives the "weakest pre-condition" for S terminating with R true. An example is the following definition of the asignment statment:

$wp(x := E, R) = R_E^x$

This gives a predicate that is a copy of R with the value of x replaced by E.

An example of a valid calculation of wp for assignments with integer valued variables x and y is:

wp(x: = y - 5,x > 10) = (y - 5 > 10) = (y > 15)

This means that in order for the "post-condition" x > 10 to be true after the assignment, the "pre-condition" y > 15 must be true before the assignment. This is also the "weakest pre-condition", in that it is the "weakest" restriction on the value of y which makes x > 10 true after the assignment.

Dijkstra also defined alternative (if) and repetitive (do) constructs as well as a composition operator (;) using wp. The alternative and repetitive constructs used guarded commands to influence execution. Because of the rules he imposed on the definition of wp, both constructs allow for non-deterministic execution if the guards in the commands are non disjoint.

Unlike many other semantic formalisms, predicate transformer semantics was not designed as an investigation into foundations of computation. Rather, it is intended to provide programmers with a methodology to develop their programs as "correct by construction" in a "calculational style". This style was advocated by Dijkstra and others, and also developed further in a higher order logic setting by R.J Back in the Refinement Calculus.

Although the most common and most widely discussed because of their relevance to sequential programming, "weakest pre-conditions" are not the only predicate transformers. For example, Leslie Lamport has suggested win and sin as predicate transformers for concurrent programming.

## References

• Edsger W. Dijkstra. "Guarded commands, nondeterminacy and formal derivation of programs". Communications of the ACM, 18(8):453-457, August 1975. [1]
• Leslie Lamport. "win and sin: Predicate Transformers for Concurrency". ACM Transactions on Programming Languages and Systems, 12(3), July 1990. [2]
• Ralph-Johan J. Back et al. "Refinement Calculus: A Systematic Introduction, 1st edition". 1998. ISBN 0-387-98417-8
• Edsger W. Dijkstra: "A Discipline Of Programming" (A systematic introduction with examples.) ISBN 0-613-92411-8
03-10-2013 05:06:04