Wim H. Hesselink
Abstract
Abstract
Dijkstra's language of guarded commands is extended with recursion and transformed into algebra. The semantics is expressed in terms of weakest preconditions and weakest liberal preconditions. Extreme fixed points are used to deal with recursion. Unbounded nondeterminacy is allowed. The algebraic setting enables us to develop efficient transformation rules for recursive procedures. The main result is an algebraic version of the rule of computational induction. In this version, certain parts of the programs are restricted to finite nondeterminacy. It is shown that without this restriction the rule would not be valid. Some applications of the rule are presented. In particular, we prove the correctness of an iterative stack implementation of a class of simple recursive procedures.
- [Apt81] Ten Years of Hoare's Logic. A Survey — Part 1ACM Transactions on Programming Languages and Systems19813431483Google Scholar
Digital Library
- [Bac87] Back, R. J. R.: A calculus of refinements for program derivations.Dept. Comp. Sci. Abo Finland, Ser. A,54, (1987).Google Scholar
- [BBK87] On the consistency of Koomen's fair abstraction ruleTheoretical Computer Science19875112917610.1016/0304-3975(87)90052-1Google ScholarDigital Library
- [Bak80] Bakker, J. W. de:Mathematical Theory of Program Correctness, Prentice-Hall, 1980.Google Scholar
- [BaW81] Bauer, F. L. and Wossner H.:Algorithmische Sprache und Programmentwicklung, Springer-Verlag, 1981.Google Scholar
- [Dij76] Dijkstra, E. W.:A Discipline of Programming, Prentice-Hall, 1976.Google Scholar
- [Dij82] Dijkstra, E. W.:Selected Writings on Computing: a Personal Perspective, Springer-Verlag, 1982.Google Scholar
- [DiS90] Dijkstra, E. W. and Scholten, C. S.:Predicate Calculus and Program Semantics, Springer-Verlag (1990).Google Scholar
- [Gri81] Gries, D.:The Science of Programming., Springer-Verlag, 1981.Google Scholar
- [Hes88a] Interpretations of Recursion Under Unbounded NondeterminacyTheoretical Computer Sceience198859211234Google ScholarDigital Library
- [Hes89a] Predicate Transformer Semantics of General Recursion (WHH 8)Acta Informatica198926309332Google ScholarDigital Library
- [Hes88b] Hesselink, W. H.: An Algebraic Calculus of Commands (WHH 13). Tech. Rep. CS 8808, Groningen University, 1988.Google Scholar
- [Hes89b] Hesselink, W. H.: Modalities of Nondeterminacy. Draft, September 1989.Google Scholar
- [Hoa71] Hoare, C. A. R.: Procedures and Parameters: an Axiomatic Approach. In:Symposium on Semantics of Algorithmic Languages, E. Engeler (ed.), Lecture Notes in Math. 188, pp. 102–116, Springer-Verlag, 1971.Google Scholar
- [Hoa78] Some Properties of Predicate TransformersJ. ACM197825461480Google ScholarDigital Library
- [Kel55] Kelley, J. L.:General Topology. Van Nostrand, 1955.Google Scholar
- [Lak76] Lakatos, I.:Proofs and Refutations: the Logic of Mathematical Discovery J. Worrall and E. Zahar, (eds), Cambridge University Press, 1976.Google Scholar
- [Man74] Manna, Z.:Mathematical Theory of Computation. McGraw-Hill, 1974.Google Scholar
- [Mor87] Data Refinement by MiraclesInformation Processing Letters1987/8826243246Google Scholar
- [MoG88] Morgan, C. C. and Gardiner, P. H. B.: Data refinement by calculation. Draft, July 1988.Google Scholar
- [Nel87] A generalization of Dijkstra's CalculusACM Transactions on Programming Languages and Systems198911517561Google ScholarDigital Library
- [TaZ71] Takeuti, G. and Zaring, W. M.:Introduction to Axiomatic Set Theory. Graduate texts in mathematics 1, Springer-Verlag, 1971.Google Scholar
Index Terms
- Command algebras, recursion and program transformation
Recommendations
Partial algebras and complexity of satisfiability and universal theory for distributive lattices, boolean algebras and Heyting algebras
Characterizations are given for the classes of partial subalgebras of distributive lattices, boolean algebras and Heyting algebras. Thereby, complexity results are obtained for the satisfiability of quantifier-free first-order sentences in these ...
Lattice-valued finite state machines and lattice-valued transformation semigroups
This paper provides a generalization of known results about fuzzy finite state machines, fuzzy transformation semigroups and their relationship by broading the truth values domain from the interval [0,1] to a complete lattice endowed with a t-norm and a ...
CIA structures and the semantics of recursion
FOSSACS'10: Proceedings of the 13th international conference on Foundations of Software Science and Computational StructuresFinal coalgebras for a functor serve as semantic domains for state based systems of various types. For example, formal languages, streams, non-well-founded sets and behaviors of CCS processes form final coalgebras. We present a uniform account of the ...
Comments