Quantifier Elimination for Real Algebra -- the Quadratic Case and Beyond

We address the problem of building intelligent systems to reason about linear arithmetic constraints. We develop, along the lines of Logic Programming, a unifying framework based on the concept of Parametric Queries and a quasi-dual generalization of the classical Linear Programming optimization problem. Variable (quantifier) elimination is the key underlying operation which provides an oracle to answer all queries and plays a role similar to Resolution in Logic Programming. We discuss three methods for variable elimination, compare their feasibility, and establish their applicability. We then address practical issues of solvability and canonical representation, as well as dynamical updates and feedback. In particular, we show how the quasi-dual formulation can be used to achieve the discriminating characteristics of the classical Fourier algorithm regarding solvability, detection of implicit equalities and, in case of unsolvability, the detection of minimal unsolvable subsets. We illustrate the relevance of our approach with examples from the domain of spatial reasoning and demonstrate its viability with empirical results from two practical applications: computation of canonical forms and convex hull construction.

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Model Completeness, Uniform Interpolants and Superposition Calculus

Journal of Automated Reasoning ◽

10.1007/s10817-021-09596-x ◽

2021 ◽

Author(s):

Diego Calvanese ◽

Silvio Ghilardi ◽

Alessandro Gianola ◽

Marco Montali ◽

Andrey Rivkin

Keyword(s):

Automated Reasoning ◽

Quantifier Elimination ◽

Effective Solution ◽

Model Completeness ◽

Present Contribution ◽

First Order ◽

Reduction Strategies ◽

Infinite State Systems ◽

Infinite State ◽

Superposition Calculus

AbstractUniform interpolants have been largely studied in non-classical propositional logics since the nineties; a successive research line within the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. This further research line is motivated by the fact that uniform interpolants offer an effective solution to tackle quantifier elimination and symbol elimination problems, which are central in model checking infinite state systems. This was first pointed out in ESOP 2008 by Gulwani and Musuvathi, and then by the authors of the present contribution in the context of recent applications to the verification of data-aware processes. In this paper, we show how covers are strictly related to model completions, a well-known topic in model theory. We also investigate the computation of covers within the Superposition Calculus, by adopting a constrained version of the calculus and by defining appropriate settings and reduction strategies. In addition, we show that computing covers is computationally tractable for the fragment of the language used when tackling the verification of data-aware processes. This observation is confirmed by analyzing the preliminary results obtained using the mcmt tool to verify relevant examples of data-aware processes. These examples can be found in the last version of the tool distribution.

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3 Quantifier elimination algorithm to boolean combination of ∃∀-formulas in the theory of a free group

Groups and Model Theory ◽

10.1515/9783110719710-003 ◽

2021 ◽

pp. 87-126

Author(s):

Olga Kharlampovich ◽

Alexei Myasnikov

Keyword(s):

Free Group ◽

Quantifier Elimination ◽

Boolean Combination ◽

Elimination Algorithm

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