Attilio Colagrossi

Deduction and abduction using a sequent calculus

A sequent calculus for automated reasoning is a particular sequent calculus that constitutes a single uniform method to perform different types of logical inferences in first order theories.

A normalization algorithm for truncated p-adic arithmetic

This paper presents a new algorithmic approach to cope with the problems related to the generation and the manipulation of the pseudo-Hensel-codes in the p-adic arithmetic. After reviewing some classical properties and the results of the Hensel code arithmetic, a new algorithm to manipulate pseudo-Hensel-codes is presented, discussed and compared with two existing methods. The lower cost of the proposed new algorithm will result from the comparison.

p-adic arithmetic: a tool for error-free computations

In this paper we propose the use of the p-adic arithmetic as a basic computational tool for a symbolic computation system in the framework of the TASSO project. This arithmetic has been chosen for two main reasons.

A sequent calculus for automated reasoning in symbolic computation systems

Abstract In this paper the problem of reasoning on properties of mathematical objects is considered in the context of symbolic computation. Automated reasoning mechanisms are proposed as a new basic computing tool in a symbolic computation system. These mechanisms are aimed to support the semantical correctness of a computation by allowing for verification of properties of mathematical objects introduced in the system and for generation and abduction of new properties of mathematical objects resulting from computations. The main objective of this paper is to define an extended sequent calculus to deal with generative and abductive logic problems, as well as with verificative problems, within a single methodological and computational environment. The implementation aspects of the proposed automated reasoning apparatus are also discussed. Examples of execution are presented and possible further applications are hinted.

A Desk-Top Sequent Calculus Machine

This paper presents an implementation of a sequent calculus to be used as a flexible tool for automated deduction. The proposed implementation represents a desk-top machine used in interactive mode to solve verification as well as generation and abduction problems. The object oriented design is described and some implementation remarks are given. Examples of working sessions are presented.

Computing real zeros of polynomials with parametric coefficients

The problem of localizing real zeros of polynomials with parametric coefficients is considered.An informative solution to this problem is proposed and an algorithm based on a generalization of Sturms method for univariate polynomials over the reals is presented. Thus, given a polynomial P(x,y), where x is the variable and y a parameter, and a real interval Ix for the variable x, the algorithm furnishes a list (eventually empty) of real intervals Iiy for the parameter y, such that there exist i real simple zeros of P(x,y) in the two dimension interval determined by Ix and Iiy.

An Approach to Class Reasoning in Symbolic Computation

A methodology, following the object-oriented paradigm, for modeling mathematical structure by axiomatization is presented. A suitable programming language is defined in terms of its class construct and inheritance. The class construct allows for expressing also the properties of the structure represented by the class. By reasoning about the properties defined in classes (class reasoning) logical relations among the related mathematical structures can be evaluated. We show how class reasoning can be applied in order to rearrange and/or modify a hierarchy of (classes representing) mathematical structures.

Linear Algebraic Approach for Computing Polynomial Resultant

This paper presents a linear algebraic method for computing the re sultant of two polynomials. This method is based on the computation of a determinant of order equal to the minimum of the degrees of the two giv en polynomials. This method turns out to be preferable to other known linear algebraic methods both from a computational point of view and for a total generality respect to the class of the given polynomials. Some relationships of this method with the polynomial pseudo-remainder operation are also discussed.