Simplification of Quantifier-free Formulae over Ordered Fields

Abstract

Given a quantifier-free first-order formula over the theory of ordered fields, our aim is to find an equivalent first-order formula that is simpler. The notion of a formula being simpler will be specified. An overview is given over various methods combining elements of field theory, order theory, and logic. These methods do not require a Boolean normal form computation. They have been developed and implemented in reduce for simplifying intermediate and final results of automatic quantifier elimination by elimination sets. Their applicability is certainly not restricted to the area of quantifier elimination.

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

“…All examples discussed in this article have been computed with an implementation of VS in Redlog that is limited to formulas where for the elimination of a quantifier the total degree of the corresponding quantified variable does not exceed 2. Nevertheless we will encounter significantly higher degrees, because VS implementations are supplemented with powerful simplification strategies; see, e.g., [20] and [36,Section 5]. The nonlinear bound of 2 is based on work by Weispfenning in the early 1990s, which also theoretically discussed arbitrary degree bounds [73].…”

“…a The original reaction system. b A visualization of the reaction; 20 here P 1 = P by rescaling time with a factor δ M and adapting all other constants. In that course α n ϑ n−1 has been replaced with α.…”

“…The first line of our formula ϕ 9 collects the above-mentioned positivity conditions on the variables and parameters. The second line 20 This image is taken from [3]. Franois Boulier kindly offered their source code and permitted its reproduction here.…”

Investigating Generic Methods to Solve Hopf Bifurcation Problems in Algebraic Biology

“…We do not go into technical details on this here but indicate a few major points, where positivity can be exploited. To start with, recall that the practical applicability of substitution methods depends crucially on efficient and powerful simplification of intermediate and final results [16]. These simplification methods can be considerably improved by positivity assumptions on all variables:…”

“…-There is an established simplification strategy which checks for terms that are (strict) trivial squaresums [16]. These are by definition sums of monomials with positive coefficients and even degree in all variables (and a positive absolute summand).…”