Efficient Solution of a Class of Quantified Constraints with Quantifier Prefix Exists-Forall

Abstract

In various applications the search for certificates for certain properties (e.g., stability of dynamical systems, program termination) can be formulated as a quantified constraint solving problem with quantifier prefix exists-forall. In this paper, we present an algorithm for solving a certain class of such problems based on interval techniques in combination with conservative linear programming approximation. In comparison with previous work, the method is more general-allowing general Boolean structure in the input constraint, and more efficientusing splitting heuristics that learn from the success of previous linear programming approximations.

A boundary approach for set inversion

“…3) When a box has two faces of two different colors, it means that it is crossed by the boundary of the solution set. This information which can be useful, for projection problems [15] and quantified problems [12], was not provided by existing approaches. 4) The procedure can easily be implemented in an HC4revised procedure where the function to be inverted is represented by a DAG (directed Acyclic Graphs).…”

Instantaneous Localization

“…Now, as shown in [9], the use of these symmetries is more interesting when we deal with projection problems where quantifier elimination is needed. This type of projection problem is indeed much more difficult to solve with classical interval approaches [5].…”

Delta-Decision Procedures for Exists-Forall Problems over the Reals

“…The procedures are known to have very high computational complexity (double exponential [7]), and can not handle problems with transcendental functions. Quantified constraints over real numbers have been studied in constraint programming [8,9,10,11,12,13]. In particular, the work in [10,11,12] develops quasi-decision procedures for solving quantified constraints over the reals with a numericallyrelaxed notion of completeness [14] that is closely related to the notion of deltacompleteness here.…”

“…Quantified constraints over real numbers have been studied in constraint programming [8,9,10,11,12,13]. In particular, the work in [10,11,12] develops quasi-decision procedures for solving quantified constraints over the reals with a numericallyrelaxed notion of completeness [14] that is closely related to the notion of deltacompleteness here. In comparison, the focus of our work (apart from improving scalability) can be seen as an extension of the same line of work that further parameterizes the procedures with explicit bounds on the numerical errors, which requires the design of various new techniques such as double-sided error control (Section 3.2).…”