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Register a new userAxioms for behavioural congruence of single-pass instruction sequences
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Added by
Kees Middelburg
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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In program algebra, an algebraic theory of single-pass instruction sequences, three congruences on instruction sequences are paid attention to: instruction sequence congruence, structural congruence, and behavioural congruence. Sound and complete axiom systems for the first two congruences were already given in early papers on program algebra. The current paper is the first one that is concerned with an axiom system for the third congruence. The presented axiom system is especially notable for its axioms that have to do with forward jump instructions.
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Earlier work on program and thread algebra detailed the functional, observable behavior of programs under execution. In this article we add the modeling of unobservable, mechanistic processing, in particular processing due to jump instructions. We model mechanistic processing preceding some further behavior as a delay of that behavior; we borrow a unary delay operator from discrete time process algebra. We define a mechanistic improvement ordering on threads and observe that some threads do not have an optimal implementation.
A program is a finite piece of data that produces a (possibly infinite) sequence of primitive instructions. From scratch we develop a linear notation for sequential, imperative programs, using a familiar class of primitive instructions and so-called repeat instructions, a particular type of control instructions. The resulting mathematical structure is a semigroup. We relate this set of programs to program algebra (PGA) and show that a particular subsemigroup is a carrier for PGA by providing axioms for single-pass congruence, structural congruence, and thread extraction. This subsemigroup characterizes periodic single-pass instruction sequences and provides a direct basis for PGAs toolset.
Single-pass instruction sequences under execution are considered to produce behaviours to be controlled by some execution environment. Threads as considered in thread algebra model such behaviours: upon each action performed by a thread, a reply from its execution environment determines how the thread proceeds. Threads in turn can be looked upon as producing processes as considered in process algebra. We show that, by apposite choice of basic instructions, all processes that can only be in a finite number of states can be produced by single-pass instruction sequences.
We present a formal system for proving the partial correctness of a single-pass instruction sequence as considered in program algebra by decomposition into proofs of the partial correctness of segments of the single-pass instruction sequence concerned. The system is similar to Hoare logics, but takes into account that, by the presence of jump instructions, segments of single-pass instruction sequences may have multiple entry points and multiple exit points. It is intended to support a sound general understanding of the issues with Hoare-like logics for low-level programming languages.
In this paper, we study the phenomenon that instruction sequences are split into fragments which somehow produce a joint behaviour. In order to bring this phenomenon better into the picture, we formalize a simple mechanism by which several instruction sequence fragments can produce a joint behaviour. We also show that, even in the case of this simple mechanism, it is a non-trivial matter to explain by means of a translation into a single instruction sequence what takes place on execution of a collection of instruction sequence fragments.
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