Àngel J. Gil

Logics preserving degrees of truth from varieties of residuated lattices

A wrong argument in the proof of one of the main results in the paper is corrected. The result itself remains true. The right proof incorporates the basic ideas in the originally alleged proof, but in a more restricted construction.

On the infinite-valued Łukasiewicz logic that preserves degrees of truth

AbstractŁukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.

On Gentzen Systems Associated with the Finite Linear MV-algebras

In this paper we obtain a characterization of the algebraizability of an m-dimensional Gentzen system in line with the characterization obtained for m-dimensional deductive systems and the characterization of 2-dimensional Gentzen systems. We also prove that if S(m) is the finite linear MV-algebraof m elements, then the m-dimensional Gentzen system obtained by using the sequent calculi associated with S(m) is equivalent to the m-valued Łukasiewicz logic Ł m and to the equational consequence relation associated with S(m). Taking the two-element Boolean algebra we obtain the expected result concerning the relationship between the sequent calculus LK, the Classical Prepositional Calculus and the variety of Boolean algebras.

Protoalgebraic Gentzen Systems and the Cut Rule

In this paper we show that, in Gentzen systems, there is a close relation between two of the main characters in algebraic logic and proof theory respectively: protoalgebraicity and the cut rule. We give certain conditions under which a Gentzen system is protoalgebraic if and only if it possesses the cut rule. To obtain this equivalence, we limit our discussion to what we call regular sequent calculi, which are those comprising some of the structural rules and some logical rules, in a sense we make precise. We note that this restricted set of rules includes all the usual rules in the literature. We also stress the difference between the case of two-sided sequents and the case of many-sided sequents, in which more conditions are needed.

A Strong Completeness Theorem for the Gentzen systems associated with finite algebras

ABSTRACT In this paper we study consequence relations on the set of many sided sequents over a propositional language. We deal with the consequence relations axiomatized by the sequent calculi defined in [2] and associated with arbitrary finite algebras. These consequence relations are examples of what we call Gentzen systems. We define a semantics for these systems and prove a Strong Completeness Theorem, which is an extension of the Completeness Theorem for provable sequents stated in [2]. For the special case of the finite linear MV-algebras, the Strong Completeness Theorem was proved in [10], as a consequence of McNaughtons Theorem. The main tool to prove this result for arbitrary algebras is the deduction-detachment theorem for Gentzen systems.

Efficient Algorithms for Description Problems over Finite Totally Ordered Domains

Given a finite set of vectors over a finite totally ordered domain, we study the problem of computing a constraint in conjunctive normal form such that the set of solutions for the produced constraint is identical to the original set. We develop an efficient polynomial-time algorithm for the general case, followed by specific polynomial-time algorithms producing Horn, dual Horn, and bijunctive formulas for sets of vectors closed under the operations of conjunction, disjunction, and median, respectively. Our results generalize the work of Dechter and Pearl on relational data, as well as the papers by Hebrard and Zanuttini. They complement the results of Hahnle et al. on multivalued logics and Jeavons et al. on the algebraic approach to constraints.

A sentence compression module for machine-assisted subtitling

We present in this paper a sentence compression module used in a machine-assisted subtitling application developed in the European e-content project e-title. Our approach to compression and the architecture of the system are motivated by the commercial and multilingual nature of the project, that is, the need to output reasonable compressions and the ability to add new strategies, genres and languages easily. The compression module currently works for the Catalan and English languages and uses the Constraint Grammar engine for linguistic preprocessing and for the linguistically motivated compression rules, thus providing a homogenous format throughout the compression process. The compression rules were implemented based on a corpus of automatically aligned pairs of films for both languages. We performed for both languages an automatic quantitative evaluation of the compression using the aligned corpus and a qualitative manual evaluation of grammaticality and informativeness.

Efficient Algorithms for Constraint Description Problems over Finite Totally Ordered Domains

Given a finite set of vectors over a finite totally ordered domain, we study the problem of computing a constraint in conjunctive normal form such that the set of solutions for the produced constraint is identical to the original set. We develop an efficient polynomial-time algorithm for the general case, followed by specific polynomial-time algorithms producing Horn, dual Horn, and bijunctive constraints for sets of vectors closed under the operations of conjunction, disjunction, and median, respectively. We also consider the affine constraints, analyzing them by means of computer algebra. Our results generalize the work of Dechter and Pearl on relational data, as well as the papers by Hebrard and Zanuttini. They also complete the results of Hahnle et al. on multivalued logics and Jeavons et al. on the algebraic approach to constraints. We view our work as a step toward a complete complexity classification of constraint satisfaction problems over finite domains.

On the efficiency and sensitivity of a pyramidal classification algorithm

In this paper we propose a Pyramidal Classification Algorithm, which together with an appropriate aggregation index produces an indexed pseudo-hierarchy (in the strict sense) without inversions nor crossings. The computer implementation of the algorithm makes it possible to carry out some simulation tests by Monte Carlo methods in order to study the efficiency and sensitivity of the pyramidal methods of the Maximum, Minimum and UPGMA. The results shown in this paper may help to choose between the three classification methods proposed, in order to obtain the classification that best fits the original structure of the population, provided we have an a priori information concerning this structure.

On Gentzen Relations Associated with Finite-valued Logics Preserving Degrees of Truth

When considering m-sequents, it is always possible to obtain an m-sequent calculus VL for every m-valued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiple-valued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2-Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a reduct that is a distributive lattice or a pseudocomplemented distributive lattice. We also show that the sentential logic naturally associated with the provable sequents of this algebraizable Gentzen relation is the logic that preserves degrees of truth with respect to the original algebra (in contrast with the more common logic that merely preserves truth). Finally, for some particular logics we obtain 2-sequent calculi that axiomatize the algebraizable Gentzen relations obtained so far.