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Some results on intermediate constructive logics.

Some techniques for the study of intermediate constructive logics are illustrated. In particular a general characterization is given of maximal constructive logics from which a new proof of the maximality of MV (Med- vedevs logic of finite problems ) can be obtained. Some semantical notions are also introduced, allowing a new characterization of MV, from which a new proof of a conjecture of Friedmans and a new family of principles valid in MV can be extracted.

An improved refutation system for intuitionistic predicate logic

In this paper a refutation calculus for intuitionistic predicate logic is presented where the necessity of duplicating formulas to which rules are applied is analyzed. In line with the semantics of intuitionistic logic in terms of Kripke models a new signFCbeside the SignsT andF is added which reduces the size of the proofs and the involved nondeterminism. The resulting calculus is proved to be correct and complete. An extension of it for Kuroda logic is given.

Optimization techniques for propositional intuitionistic logic and their implementation

This paper presents some techniques which bound the proof search space in propositional intuitionistic logic. These techniques are justified by Kripke semantics and are the backbone of a tableau based theorem prover (PITP) implemented in C++. PITP and some known theorem provers are compared using the formulas of ILTP benchmark library. It turns out that PITP is, at the moment, the propositional prover that solves most formulas of the library.

Refutation Systems for Propositional Modal Logics

Now we are working on extensions to first order modal logics and to modal logics with intuitionistic basis. A comparison with the work of Wallen [Wal] is planned.

Generalized Tableau Systems for Intemediate Propositional Logics

Given an intermediate propositional logic L (obtained by adding to intuitionistic logic INT a single axiom-scheme), a pseudo tableau system for L can be given starting from any intuitionistic tableau system and adding a rule which allows to insert in any line of a proof table suitable T-signed instances of the axiom-scheme. In this paper we study some sufficient conditions from which, given a well formed formula H, the search for these instances can be restricted to a suitable finite set of formulae related to H. We illustrate our techniques by means of some known logics, namely, the logic D of Dummett, the logics PR k (k≥1) of Nagata, the logics FIN m (m≥1), the logics G n (n≥1) of Gabbay and de Jongh, and the logic KP of Kreisel and Putnam

Constructive Theories with Abstract Data Types for Program Synthesis

The research explained in this paper originates from program synthesis in the frame of intuitionistic logic [6] and has been furtherly developed as a study involving, on the one hand, constructive proofs as programs [12], on the other hand the possibility of providing axiomatizations of mathematical structures (abstract data types) compatible with constructive logical principles [3].

Program Specification and Synthesis in Constructive Formal Systems

Constructive mathematics has been proposed by many authors as a theoretical basis for program synthesis, and various implementations of this idea have been developed. However, the main problem in implementation is how to build a real environment for software development. In this paper, we present the main features of a logical system we are studying which provides specification tools and a deductive system for deriving programs from their specifications. Our aim is to use this system as a starting point for a real programming environment.

ESBC: an application for computing stabilization bounds

We describe the application ESBC to perform the timing analysis of a combinatorial circuit. The circuit is described by formulas of Classical Logic and the delays of propagation of the signals in a gate are represented by a kind of valuation form semantics. ESBC computes the exact stabilization times at which the output signals stabilize.

A Space Efficient Implementation of a Tableau Calculus for a Logic with a Constructive Negation

A tableau calculus for a logic with constructive negation and an implementation of the related decision procedure is presented. This logic is an extension of Nelson logic and it has been used in the framework of program verification and timing analysis of combinatorial circuits. The decision procedure is tailored to shrink the search space of proofs and it is proved correct by using a semantical technique. It has been implemented in C++ language.

A Constructive Logic Approach to Database Theory

In this paper we propose an approach to database theory based on a constructive logic. The semantics here assumed is a particular one; it is based on the notion of info(K,F) (the information type of F), where K is the set of constants of a first order language L, F is a formula of L and info(K,F) is the set of all the possible pieces of information (within L) on the “truth” of F.