Alessandro Avellone

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.

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

Almost Duplication-Free Tableau Calculi for Propositional Lax Logics

In this paper we provide tableau calculi for the intuitionistic modal logics PLL and PLL1, where the calculus for PLL1 is duplication-free while among the rules for PLL there is just one rule that allows duplication of formulas. These logics have been investigated by Fairtlough and Mendler in relation to the problem of Formal Hardware Verification. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Namely, we enlarge the language containing the usual sings T and F with the new sign Fc. PLL and PLL1 logics are characterized by a Kripke-semantics which is a “weak” version of the semantics for ordinary intuitionistic modal logics. In this paper we establish the soundness and completeness theorems for these calculi.

Synthesis of Programs in Abstract Data Types

In this paper we propose a method for program synthesis from constructive proofs based on a particular proof strategy, we call dischargeable set construction. This proof-strategy allows to build proofs in which active patterns (sequences of application of rules with proper computational content) can be distinguished from correctness patterns (concerning correctness properties of the algorithm implicitly contained in the proof). The synthesis method associates with every active pattern of the proof a program schema (in an imperative language) translating only the computational content of the proof. One of the main features of our method is that it can be applied to a variety of theories formalizing ADTs and classes of ADTs. Here we will discuss the method and the computational content of some principles of particular interest in the context of some classes of ADTs.

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 Formal Framework for Synthesis and Verification of Logic Programs

In this paper we present a formal framework, based on the notion of extraction calculus, which has been applied to define procedures for extracting information from constructive proofs. Here we apply such a mechanism to give a proof-theoretic account of SLD-derivations. We show how proofs of suitable constructive systems can be used in the context of deductive synthesis of logic programs, and we state a link between constructive and deductive program synthesis.

On maximal intermediate predicate constructive logics

We extend to the predicate frame a previous characterization of the maximal intermediate propositional constructive logics. This provides a technique to get maximal intermediate predicate constructive logics starting from suitable sets of classically valid predicate formulae we call maximal nonstandard predicate constructive logics. As an example of this technique, we exhibit two maximal intermediate predicate constructive logics, yet leaving open the problem of stating whether the two logics are distinct. Further properties of these logics will be also investigated.

Improvements to the Tableau Prover PITP

In this paper we discuss the new version of PITP, a procedure to decide propositional intuitionistic logic, which turns out at the moment to be the best propositional prover on ILTP. The changes in the strategy and implementation make the new version of PITP faster and capable of deciding more formulas than the previous one. We give a short account both of the old optimizations and the changes in the strategy with respect to the previous version. We use ILTP library and random generated formulas to compare the implementation described in this paper to the other provers (including our old version of PITP).

How to avoid the formal verification of a theorem prover

The purpose of this papers to show a technique to automatically certify answers coming from a nontrustable theorem prover. As an extreme consequence, the development of non-sound theorem provers has been considered and investigated, in order to evaluate their relative efficiency on particular classes of difficult theorems. The presentation will consider as a case study, a tableau-based theorem prover for first-order intuitionistic logic without equality [1].