Grazyna Mirkowska

Complete axiomatization of algorithmic properties of program schemes with bounded nondeterministic interpretations

Propositional algorithmic logic PAL is a propositional counterpart of algorithmic logics. It investigates properties of program connectives: begin...end, while...do, if..then..else, or .... PAL supplies tools for reasoning about programs constructed from program variables by means of program connectives and about their algorithmic properties. Sound rules of inference and tautologies of PAL are as important in analysis of programs (e.g. verification) as tautologies of classical propositional calculus. On the other hand propositional algorithmic theories are of highest interest, since they can capture properties of data structures and also algorithmic properties of behaviours of concurrent systems. We proved that the semantical and the syntactical consequence operations coincide (completeness property) and that the model existence theorem holds.

PAL - Propositional Algorithmic Logic

The aim of propositional algorithmic logic is to investigate the properties of program connectives. Complete axiomatic systems for deterministic as well as for nondeterministic interpretations of program variables are presented. They constitute basic sets of tools useful in the practice of proving the properties of program schemes. Propositional theories of data structures, e.g. the arithmetic of natural numbers and stacks, are constructed. This shows that in many aspects PAL is close to first-order algorithmic logic. Tautologies of PAL become tautologies of algorithmic logic after replacing program variables by programs and propositional variables by formulas. Another corollary to the completeness theorem asserts that it is possible to eliminate nondeterministic program variables and replace them by schemes with deterministic atoms.

On the propositional algorithmic logic

The aim of propositional algorithmic logic PAL is to investigate properties of program connectives and to develop tools useful in the practice of proving properties of program schemes. The tautologies of PAL become tautologies of algorithmic logic after replacing program variables by programs and propositional variables by formulas.

The Deducibility Problem in Propositional Dynamic Logic

The problem of whether an arbitrary formula of Propositional Dynamic Logic (PDL) is deducible from a fixed axiom scheme of PDL is Π 1 1 -complete. This contrasts with the decidability of the problem when the axiom scheme is replaced by any single PDL formula.

The Algebraic Specifications do not have the Tennenbaum Property

It is commonly believed that a programmable model satisfying the axioms of a given algebraic specification guarantees good properties and is a correct implementation of the specification. This convinction might be related to the Tennenbaums property[Ten] of the arithmetic: every computable model of the Peano arithmetic of natural numbers is isomorphic to the standard model. Here, on the example of stacks, we show a model satisfying all axioms of the algebraic specification of stacks which can not be accepted as a good model in spite of the fact that it is defined by a program. For it enables to ”pop” a stack infinitely many times.

On Applications of Algorithmic Logic

In this survey we contained a short presentation of algorithmic logic AL and two sections devoted to selected applications of algorithmic logic. First, we argue that the goal of formal definition of a programming language can be fully achieved with the tools offered by AL. Next, we show possible applications of AL in production of modular software. The interaction between theoretical tools (AL) and software ones (LOGLAN programming language) seems to provide a reasonable way toward creation of reliable software from prefabricated modules and their specifications. It seems that in this way one can speed up the task of production of software.

The Representation Theorem for Algorithmic Algebras

Algorithmic algebras form semantical base for semantics of algorithmic logics as Boolean algebras do for propositional calculus. The paper contains the proof of the following statement : every separable algorithmic algebra is representable by an algebra of computations.