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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

Abstract

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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

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