Abstract
Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multi-algebras and partial algebras, analogous to the classical presentation of algebras over a signature Σ as cartesian functors from the algebraic theory of Σ, Th(Σ), to Set. The functors we introduce are based on variations of the notion of theory, having a structure weaker than cartesian, and their target is Rel, the category of sets and relations. We argue that this functorial presentation provides an original abstract syntax for partial and multi-algebras.
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Research partly supported by the EC TMR Network GETGRATS (General Theory of Graph Transformation Systems) through the Technical University of Berlin and the University of Pisa.
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References
R. Bruni, F. Gadducci, and U. Montanari. Normal forms for partitions and relations. In J. L. Fiadeiro, editor, Recent Trends in Algebraic Development Techniques, LNCS. Springer Verlag, 1999. This volume.
A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gsmonoidal categories. Applied Categorical Structures, 1998. To appear.
A. Corradini and F. Gadducci. Rewriting on cyclic structures. Technical Report TR-98-05, Dipartimento di Informatica, Pisa, 1998. Extended abstract in Fixed Points in Computer Science, satellite workshop of MFCS’98.
F. Gadducci. On the Algebraic Approach to Concurrent Term Rewriting. PhD thesis, University of Pisa-Department of Computer Science, 1996.
F. Gadducci and R. Heckel. An inductive view of graph transformation. In F. Parisi-Presicce, editor, Recent Trends in Algebraic Development Techniques, volume 1376 of LNCS, pages 219–233. Springer Verlag, 1998.
U. Hensel and D. Spooner. A view on implementing processes: Categories of circuits. In M. Haveraaen, O. Owe, and O. Dahl, editors, Recent Trends in Data Types Specification, volume 1130 of LNCS, pages 237–255. Springer Verlag, 1995.
H.-J. Hoenke. On partial algebras. In Universal Algebra, volume 29 of Colloquia Mathematica Societatis János Bolyai, pages 373–412. 1977.
H.-J. Hoenke. On partial recursive definitions and programs. In M. Karpinski, editor, Fundamentals of Computation Theory, volume 56 of LNCS, pages 260–274. Springer Verlag, 1977.
B. Jacobs. Semantics of weakening and contraction. Annals of Pure and Applied Logic, 69:73–106, 1994.
G.M. Kelly and R.H. Street. Review of the elements of 2-categories. In G.M. Kelly, editor, Sydney Category Seminar, volume 420 of Lecture Notes in Mathematics, pages 75–103. Springer Verlag, 1974.
A. Kock and G.E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283–313. North Holland, 1977.
Y. Lafont. Equational reasoning with 2-dimensional diagrams. In H. Comon and J-P. Jouannaud, editors, Term Rewriting, volume 909 of LNCS, pages 170–195. Springer Verlag, 1995.
F.W. Lawvere. Functorial semantics of algebraic theories. Proc. National Academy of Science, 50:869–872, 1963.
S. Mac Lane. Categories for the working mathematician. Springer Verlag, 1971.
E. Robinson and G. Rosolini. Categories of partial maps. Information and Computation, 79:95–130, 1988.
E.G. Wagner, S.L. Bloom, and J.W. Thatcher. Why algebraic theories? In M. Nivat and J.C. Reynolds, editors, Algebraic methods in semantics, pages 607–634. Cambridge University Press, 1985.
M. Walicki and M. Bialasik. Categories of relational structures. In F. Parisi-Presicce, editor, Recent Trends in Algebraic Development Techniques, volume 1376 of LNCS, pages 418–433. Springer Verlag, 1998.
M. Walicki and S. Meldal. Algebraic approaches to nondeterminism: An overview. ACM Computing Surveys, 29:30–81, 1997.
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Corradini, A., Gadducci, F. (1999). Functorial Semantics for Multi-algebras. In: Fiadeiro, J.L. (eds) Recent Trends in Algebraic Development Techniques. Lecture Notes in Computer Science, vol 1589. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48483-3_6
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DOI: https://doi.org/10.1007/3-540-48483-3_6
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