Carel S. Scholten

Predicate calculus and program semantics

This text gives a self-contained foundation of predicate transformer semantics by making extensive use of the predicate calculus. The semantics of the repetitive construct is defined in terms of weakest and strongest solutions in terms of the weakest precondition and the weakest liberal precondition, the notion of determinacy is defined it is shown how to cope unbounded nondeterminacy without using transfinite induction.

A Class of Simple Communication Patterns

We consider a finite, undirected graph each node of which contains a process. Processes contained in nodes directly connected by an edge of the graph are called each other’s neighbours.

On-the-fly darbage collection: an exercise in cooeration

A technique is presented which allows nearly all of the garbage detection and collection activity to be performed by an additional processor, operating concurrently with the processor carrying out the computation proper. Exclusion and synchronization contraints between the processors have been kept weak.

The calculus of boolean structures

In this chapter we develop the calculus of boolean structures in a rather algebraic fashion. We do so for a variety of reasons. Firstly, we have to introduce the reader to the repertoire of general formulae that will be used throughout the remainder of this booklet. Secondly, by proving all formulae that have not been postulated, we give the reader the opportunity of gently familiarizing himself with our style of conducting such calculational proofs. Thirdly, we wish to present this material in a way that does justice to how we are going to use it. Since value-preserving transformations are at the heart of our calculus, so are the notions of equality and function application; hence our desire to develop this material with the equality relation in the central role. (It is here that our treatment radically departs from almost all introductions to formal logic: it is not uncommon to see the equality —in the form of “if and only if”— being introduced much later as a shorthand, almost as an afterthought.)

The strongest postcondition

From the (rather operational) introduction of Chap. 7, we recall (wlp.S)⋆.X: holds in precisely those initial states for which there exists a computation under control of S that belongs to the class “finally x”.

On substitution and replacement

With respect to substitution and replacement, two extreme attitudes seem to prevail. Either the manipulations are deemed so simple that the author performs them without any explanation or statement of the rules, or the author tries to give a precise statement of the most general rules and ends up with 200 pages of small print, in which all simplicity has been lost. Neither is entirely satisfactory for the purposes of this little monograph.

Semantics of straight-line programs

This is a monograph about a theory of programming language semantics. Programming language definitions traditionally consist of two parts, called its “syntax” and its “semantics”, respectively.

On our proof format

This chapter describes how we present our calculational proofs and, to a certain extent, why we have chosen to adopt this format.

Semantics of repetitions

The reason for the inclusion of the previous chapter is that we shall use extreme solutions of equations in predicates to define the semantics of our next compound statement, known as the “repetition”. We shall first study it in its simple form, in which it is composed of a guard B and a statement S . It is denoted by surrounding the guarded statement B → S by the special bracket pair do…od . For the sake of brevity we shall call the resulting compound statement in this chapter “D0” , i.e.,

Some properties of predicate transformers

In this chapter we define and explore a number of properties that predicate transformers may or may not enjoy. It is a preparation for the later chapters in which we analyse in terms of these properties the predicate transformers that will be used to define programming language semantics. The purpose of that later analysis is to justify the procedures followed in proving properties of programs.