Many-Sorted Logic

Lecture Notes in Computer Science

The theory of abstract algebraic logic aims at drawing a strong bridge between logic and universal algebra, namely by generalizing the well known connection between classical propositional logic and Boolean algebras. Despite of its successfulness, the current scope of application of the theory is rather limited. Namely, logics with a many-sorted language simply fall out from its scope. Herein, we propose a way to extend the existing theory in order to deal also with many-sorted logics, by capitalizing on the theory of many-sorted equational logic. Besides showing that a number of relevant concepts and results extend to this generalized setting, we also analyze in detail the examples of first-order logic and the paraconsistent logic C1 of da Costa.

We assume the opinion by which "tranlation into classical logic" is a reliable methodology of Universal Logic in the task of comparing different logics. What we add in this paper, following Manzano [9], is some evidence for adopting the slightly different paradigm of "tranlation into many-sorted classical logic." Our own methodology, splitted into three levels of translation, is discussed in some detail.

2010

Fujiwara defined, through the concept of family of basic mappingformulas between single-sorted signatures, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras. Subsequently he also defined an equivalence relation, the relation of conjugation, on the families of basic mapping-formulas. In this article we extend the theory of Fujiwara to the, not necessarily similar, many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures under which are subsumed the standard signature morphisms, the derivors of Goguen-Thatcher-Wagner, and the basic mapping-formulas of Fujiwara.

cs.wpi.edu

The Schoenfinkel-Bernays-Ramsey class is a fragment of first-order logic with the Finite Model Property: a sentence in this class is satisfiable if and only if it is satisfied in a finite model. Since an upper bound on the size of such a model is computable from the sentence, the satisfiability problem for this family is decidable. Sentences in this form arise naturally in a variety of application areas, and several popular reasoning tools explicitly target this class. Others have observed that the class of sentences for which such a finite model theorem holds is richer in a many-sorted framework than in the one-sorted case. This paper makes a systematic study of this phenomenon in the general setting of order-sorted logic supporting overloading and empty sorts. We establish a syntactic condition generalizing the Schoenfinkel-Bernays-Ramsey form that ensures the Finite Model Property. We give a linear-time algorithm for deciding this condition and a polynomial-time algorithm for computing the bound on model sizes. As a consequence, model-finding is a complete decision procedure for sentences in this class. Our algorithms have been incorporated into Margrave, a tool for analysis of access-control and firewall policies, and are available in a standalone application suitable for analyzing input to the Alloy model finder.

2015

Point-free foundation of geometry and multi-valued logic

2011

It has been a common belief that the standard results of universal algebra as developed since the work of Birkhoff and others in the thirties carry over without much change to the framework of many-sorted algebras. Perhaps the only exception widely noticed by the community is the care needed in the treatment of many-sorted equational logic. However, while the standard results remain valid in essence in the many-sorted frameworks, some nuances and technicalities require considerably more care in formulation and ∗This work was funded in part by by the Polish Ministry of Science and Higher Education, grant N206 493138, and by the European IST FET programme under the IST-2005-016004 SENSORIA project. proof of the results. We give some examples of this, indicating how equational calculus, Birkhoff’s variety theorem and interpolation results should be adjusted for many-sorted algebras.

1993

Many-valued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomatizability on the one hand, and algebraical/model theoretic investigations on the other. The proof theory of many-valued systems has not been investigated to any comparable extent.