Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-16T23:22:42.258Z Has data issue: false hasContentIssue false

The algebra of multirelations

Published online by Cambridge University Press:  18 March 2013

C. E. MARTIN
Affiliation:
Department of Computing and Communication Technologies, Oxford Brookes University, Wheatley, Oxfordshire, OX33 1HX, United Kingdom Email: [email protected], [email protected]
S. A. CURTIS
Affiliation:
Department of Computing and Communication Technologies, Oxford Brookes University, Wheatley, Oxfordshire, OX33 1HX, United Kingdom Email: [email protected], [email protected]

Abstract

Multirelational semantics are well suited to reasoning about programs involving two kinds of non-determinism. This paper lays the categorical foundations for an algebraic calculus of multirelations.

Type
Paper
Copyright
Copyright © Cambridge University Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. (1991) Domain Theory in Logical Form. Annals of Pure and Applied Logic 51 177.CrossRefGoogle Scholar
Back, R. J. R. and von Wright, J. (1992) Combining angels, demons and miracles in program specifications. Theoretical Computer Science 100 (2)365383.CrossRefGoogle Scholar
Back, R. J. R. and von Wright, J. (1998) Refinement Calculus: A Systematic Introduction, Springer-Verlag.CrossRefGoogle Scholar
Backhouse, R. C., de Bruin, P., Malcolm, G.Voermans, T. S. and van der Woude, J. C. S. P. (1991) Relational catamorphisms. In: Möller, B. (ed.) Constructing Programs from Specifications, North-Holland287318.Google Scholar
Barr, M. and Wells, C. (1990) Category theory for computing science, Prentice-Hall.Google Scholar
Bird, R. S. and de Moor, O. (1997) The Algebra of Programming, Prentice Hall.Google Scholar
Davey, B. A. and Priestley, H. A. (2002) Introduction to Lattices and Order, second edition, Cambridge University Press.CrossRefGoogle Scholar
Dijkstra, E. (1975) Guarded commands, nondeterminacy and formal derivation of programs. Communications of the ACM 18 (8)453457.CrossRefGoogle Scholar
de Moor, O., Gardiner, P. and Martin, C. (1991) Factorizing predicate transformers in a topos (unpublished note).Google Scholar
de Morgan, A. (1856) On the symbols of logic, the theory of the syllogism, and in particular of the copula, and the application of the theory of probability to some questions in the theory of evidence. Transactions of the Cambridge Philosophical Society 9 79127.Google Scholar
Düntsch, O., Orłowska, E. and Rewitsky, I. (2010) Structures with Multirelations, their Discrete Dualities and Applications Fundamenta Informaticae 100 7798.CrossRefGoogle Scholar
Floyd, R. W. (1967) Assigning meanings to programs. Proceedings Symposia in Applied Mathematics, AMS 19 1931.CrossRefGoogle Scholar
Fokkinga, M. (1994) Monadic maps and folds for arbitrary datatypes. Memoranda Informatica, Department of Computer Science, University of Twente (94-28).Google Scholar
Freyd, P. and Ščedrov, A. (1993) Categories, Allegories, Springer-Verlag.Google Scholar
Gardiner, P. H. B., Martin, C. E. and de Moor, O. (1994) An algebraic construction of predicate transformers. Science of Computer Programming 22 (1–2)2144.CrossRefGoogle Scholar
Hoare, C. A. R. (1969) An axiomatic basis for computer programming. Communications of the ACM 12 (10)576583.CrossRefGoogle Scholar
Johnstone, P. T. (2002) Sketches of an Elephant: A Topos Theory Compendium, Oxford University Press.Google Scholar
Maddux, R. D. (1996) Relation-Algebraic Semantics. Theoretical Computer Science 160 (1)185.CrossRefGoogle Scholar
Martin, C. E. (1995) Towards a calculus of predicate transformers. In: Wiedermann, J. and Hájek, P. (eds.) Mathematical Foundations of Computer Science 1995: 20th International Symposium, MFCS ‘95. Springer-Verlag Lecture Notes in Computer Science 969 489498.CrossRefGoogle Scholar
Martin, C. E. and Curtis, S. A. (2006) Nondeterministic folds. In: Uustalu, T. (ed.) Proceedings of the 8th Mathematics of Program Construction conference. Springer-Verlag Lecture Notes in Computer Science 4014 274298.CrossRefGoogle Scholar
Martin, C. E. and Curtis, S. A. (2009) Supplement to the paper ‘Monadic maps and folds for multirelations in an allegory’. Available at http://cms.brookes.ac.uk/staff/SharonCurtis/publications/mf-supp.pdf.CrossRefGoogle Scholar
Martin, C. E. and Curtis, S. A. (2010) Monadic maps and folds for multirelations in an allegory. In: Butterfield, A. (ed.) Unifying Theories of Programming: Second International Symposium, UTP 2008. Springer-Verlag Lecture Notes in Computer Science 5713 102121.CrossRefGoogle Scholar
Martin, C. E., Curtis, S. A. and Rewitzky, I. (2007) Modelling angelic and demonic nondeterminism with multirelations. Science of Computer Programming 65 (2)140158.CrossRefGoogle Scholar
Morgan, C. C. (1994) Programming from Specifications, second edition, Prentice-Hall.Google Scholar
Morgan, C. C., Gardiner, P., Vickers, T. and Robinson, K. (1994) On the Refinement Calculus, first edition, Springer-Verlag.CrossRefGoogle Scholar
Morris, J. (1987) A theoretical basis for stepwise refinement and the programming calculus. Science of Computer Programming 9 2873067.CrossRefGoogle Scholar
Morris, J. M. and Tyrrell, M. (2008a) Dually Nondeterministic functions. ACM Transactions on Programming Languges and Systems 30 (6).Google Scholar
Morris, J. M. and Tyrrell, M. (2008b) Modelling higher-order dual nondeterminacy. Acta Informatica 45 441465.CrossRefGoogle Scholar
Naumann, D. A. (1998) A Categorical Model for Higher Order Imperative Programming. Mathematical Structures in Computer Science 8 351399.CrossRefGoogle Scholar
Naumann, D. A. (2001) Ideal models for pointwise relational and state-free imperative programming. In: Proceedings of the 3rd ACM SIGPLAN international conference on Principles and practice of declarative programming 4–15.CrossRefGoogle Scholar
Plotkin, G. D. (1979) Dijkstra's predicate transformers and Smyth's powerdomains. In: Bjorner, D. (ed.) Abstract Software Specifications: Proceedings 1979 Copenhagen Winter School. Springer-Verlag Lecture Notes in Computer Science 86 527553.CrossRefGoogle Scholar
Rewitzky, I. (2003) Binary multirelations. In: Schmidt, G., de Swart, H., Orlowska, E. and Roubens, M. (eds.) Theory and Application of Relational Structures as Knowledge Instruments 2929 259274.Google Scholar
Smyth, M. B. (1983) Power Domains and Predicate Transformers: A Topological View. In: Diaz, J. (ed.) Proceedings of the 10th Colloquium on Automata, Languages and Programming. Springer-Verlag Lecture Notes in Computer Science 154 662675.CrossRefGoogle Scholar
Stone, M. H. (1936) The theory of representations for Boolean algebras. Transactions of the American Mathematical Society 40 37111.Google Scholar
Tarski, A. (1941) On the calculus of relations. Journal of Symbolic Logic 6 (3)7389.CrossRefGoogle Scholar