Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T06:12:49.196Z Has data issue: false hasContentIssue false
  • English
  • Français

A graphical representation of relational formulae withcomplementation∗

Published online by Cambridge University Press:  27 February 2012

Domenico Cantone ,
Andrea Formisano ,
Marianna Nicolosi Asmundo  and
Eugenio Giovanni Omodeo
Domenico Cantone
Affiliation:
Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy. cantone@dmi.unict.it; nicolosi@dmi.unict.it
Andrea Formisano
Affiliation:
Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy; formis@dmi.unipg.it
Marianna Nicolosi Asmundo
Affiliation:
Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy. cantone@dmi.unict.it; nicolosi@dmi.unict.it
Eugenio Giovanni Omodeo
Affiliation:
Dipartimento di Matematica e Informatica, Università di Trieste, Via Valerio 12/1, 34127 Trieste, Italy; eomodeo@units.it
Get access

Abstract

We study translations of dyadic first-order sentences into equalities between relationalexpressions. The proposed translation techniques (which work also in the conversedirection) exploit a graphical representation of formulae in a hybrid of the twoformalisms. A major enhancement relative to previous work is that we can cope with therelational complement construct and with the negation connective. Complementation ishandled by adopting a Smullyan-like uniform notation to classify and decompose relationalexpressions; negation is treated by means of a generalized graph-representation offormulae in ℒ+, and through a series of graph-transformation rules whichreflect the meaning of connectives and quantifiers.

Keywords

Algebra of binary relationsquantifier eliminationgraph transformation
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 46 , Issue 2: 12th Italian Conference on Theoretical Computer Science(ICTCS) , April 2012 , pp. 261 - 289
Copyright
© EDP Sciences 2012

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

J.G.F. Belinfante, Gödel’s algorithm for class formation, in Proc. of CADE’00, edited by D. McAllester. Lect. Notes Comput. Sci. 1831 (2000) 132–147.
C. Brown and G. Hutton, Categories, allegories and circuit design, in Proc. of 9th IEEE Symp. on Logic in Computer Science. IEEE Computer Society Press (1994) 372–381.
C. Brown and A. Jeffrey, Allegories of circuits, in Proc. of Logic for Computer Science (1994) 56–68.
D. Cantone, A. Cavarra and E.G. Omodeo, On existentially quantified conjunctions of atomic formulae of ℒ+, in Proc. of International Workshop on First-Order Theorem Proving (FTP97), edited by M.P. Bonacina and U. Furbach (1997).
Cantone, D., Formisano, A., Omodeo, E.G. and Zarba, C.G., Compiling dyadic first-order specifications into map algebra. Theoret. Comput. Sci. 293 (2003) 447–475. Google Scholar
Curtis, S. and Lowe, G., Proofs with graphs. Sci. Comput. Program. 26 (1996) 197–216. Google Scholar
de Freitas, R., Veloso, P.A.S., Veloso, S.R.M. and Viana, P., On graph reasoning. Inf. Comput. 207 (2009) 228–246. Google Scholar
N. Dershowitz and J.-P. Jouannaud, Rewrite systems, in Handbook of Theoretical Computer Science B : Formal Models and Semantics (B) (1990) 243–320.
Duffin, R.J., Topology of series-parallel graphs. J. Math. Anal. Appl. 10 (1965) 303–318. Google Scholar
D. Dougherty and C. Gutiérrez, Normal forms and reduction for theories of binary relations, in Proc. of Rewriting Techniques and Applications, edited by L. Bachmair. Lect. Notes Comput. Sci. 1833 (2000).
Dougherty, D. and Gutiérrez, C., Normal forms for binary relations. Theoret. Comput. Sci. 360 (2006) 228–246. Google Scholar
M.C. Fitting, First-order Logic and Automated Theorem Proving, Graduate Texts in Computer Science, 2nd edition. Springer-Verlag, New York (1996).
Formisano, A. and Nicolosi Asmundo, M., An efficient relational deductive system for propositional non-classical logics. JANCL 16 (2006) 367–408. Google Scholar
A. Formisano and E.G. Omodeo, An equational re-engineering of set theories, in Selected Papers from Automated Deduction in Classical and Non-Classical Logics, edited by R. Caferra and G. Salzer. Lect. Notes Comput. Sci. 1761 (2000) 175–190.
A. Formisano and M. Simeoni, An Agg application supporting visual reasoning, in Proc. of GT-VMT’01 (ICALP 2001), edited by L. Baresi and M. Pezzè. Electron. Notes Theoret. Comput. Sci. 50 (2001).
Formisano, A., Omodeo, E.G. and Temperini, M., Goals and benchmarks for automated map reasoning. J. Symb. Comput. 29 (2000) 259–297. Google Scholar
A. Formisano, E.G. Omodeo and M. Simeoni, A graphical approach to relational reasoning, in Proc. of RelMiS 2001 (ETAPS 2001), edited by W. Kahl, D.L. Parnas and G. Schmidt. Electron. Notes Theoret. Comput. Sci. 44 (2001).
A. Formisano, E.G. Omodeo and M. Temperini, Instructing equational set-reasoning with Otter, in Proc. of IJCAR’01, edited by R. Goré, A. Leitsch and T. Nipkow (2001).
A. Formisano, E.G. Omodeo and M. Temperini, Layered map reasoning : An experimental approach put to trial on sets, in Declarative Programming, edited by A. Dovier, M.-C. Meo and A. Omicini. Electron. Notes Theoret. Comput. Sci. 48 (2001) 1–28.
W. Kahl, Algebraic graph derivations for graphical calculi, in Proc. of Graph Theoretic Concepts in Computer Science, WG ’96, edited by F. d’Amore, P.G. Franciosa and A. Marchetti-Spaccamela. Lect. Notes Comput. Sci. 1197 (1997) 224–238.
Kahl, W., Relational matching for graphical calculi of relations. Inform. Sci. 119 (1999) 253–273. Google Scholar
M.K. Kwatinetz, Problems of expressibility in finite languages. Ph.D. thesis, University of California, Berkeley (1981).
R.M. Smullyan, First-order Logic. Dover Publications, New York (1995).
A. Tarski and S. Givant, A formalization of Set Theory without variables, Amer. Math. Soc. Colloq. Publ. 41 (1987).
J. von Neumann, Eine Axiomatisierung der Mengenlehre. J. Reine Angew. Math. 154 (1925) 219–240. English translation, edited by J. van Heijenoort. From Frege to Gödel : a source book in mathematical logic, 1879–1931. Harvard University Press (1977).