Enhancing Linear Algebraic Computation of Logic Programs Using Sparse Representation

Algebraic characterization of logic programs has received increasing attention in recent years. Researchers attempt to exploit connections between linear algebraic computation and symbolic computation to perform logical inference in large-scale knowledge bases. In this paper, we analyze the complexity of the linear algebraic methods for logic programs and propose further improvement by using sparse matrices to embed logic programs in vector spaces. We show its great power of computation in reaching the fixed point of the immediate consequence operator. In particular, performance for computing the least models of definite programs is dramatically improved using the sparse matrix representation. We also apply the method to the computation of stable models of normal programs, in which the guesses are associated with initial matrices, and verify its effect when there are small numbers of negation. These results show good enhancement in terms of performance for computing consequences of programs and depict the potential power of tensorized logic programs.

New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

ASP (): Answer Set Programming with Algebraic Constraints

Weighted Logic is a powerful tool for the specification of calculations over semirings that depend on qualitative information. Using a novel combination of Weighted Logic and Here-and-There (HT) Logic, in which this dependence is based on intuitionistic grounds, we introduce Answer Set Programming with Algebraic Constraints (ASP($\mathcal A \mathcal C$)), where rules may contain constraints that compare semiring values to weighted formula evaluations. Such constraints provide streamlined access to a manifold of constructs available in ASP, like aggregates, choice constraints, and arithmetic operators. They extend some of them and provide a generic framework for defining programs with algebraic computation, which can be fruitfully used e.g. for provenance semantics of datalog programs. While undecidable in general, expressive fragments of ASP($\mathcal A \mathcal C$) can be exploited for effective problem solving in a rich framework.