Truth table invariant cylindrical algebraic decomposition. (September 2016)

• Record Type: Journal Article
• Authors: Bradford, Russell; Davenport, James H.; England, Matthew; McCallum, Scott; Wilson, David
• Is Part Of: Journal of symbolic computation. Volume 76(2016)
• Page Start: 1
• Page End: 35
• Publication Date: 2016-09
• Subjects: 68W30 -- 03C10; Cylindrical algebraic decomposition -- Equational constraint; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.11.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD). In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallum's theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We also explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint. The algorithm is implemented fully inMaple and we present both promising results from experimentation and a complexity analysis showing the benefits of our contributions.