The ksmt Calculus Is a $$\delta $$-complete Decision Procedure for Non-linear Constraints

Abstract

is a CDCL-style calculus for solving non-linear constraints over the real numbers involving polynomials and transcendental functions. In this paper we investigate properties of the calculus and show that it is a $$\delta $$ δ -complete decision procedure for bounded problems. We also propose an extension with local linearisations, which allow for more efficient treatment of non-linear constraints.

Combining Constraint Solving and Bayesian Techniques for System Optimization

“…, x n ) may occur in the abbreviated form In this paper we consider formulas over ⟨R, 0, 1, F , P ⟩, where P are the usual order predicates <, ≤, =, etc. and F contains addition, multiplication with rational constants and can also contain non-linear functions supported by SMT solvers including polynomials, transcendental functions such as combinations of sine, cosine, exponentials, solutions of differential equations and more generally computable functions [de Moura and Bjørner, 2008;Brauße et al, 2019;Brauße et al, 2021;Cimatti et al, 2013;Gao et al, 2012]. We extend functions F by functions definable by formulas: F D , i.e., we assume f ∈ F D is represented by a formula F (x 1 , .…”

Handling Polynomial and Transcendental Functions in SMT via Unconstrained Optimisation and Topological Degree Test

“…Other approaches, e.g. dReal [19] and ksmt [5], rely on the notion of δsatisfiability [18], which guarantees that there exists a perturbation (up to some δ > 0 specified by the user) of the original formula that is satisfiable. 1 iSAT3 relies on a similar notion and, when not able to prove satisfiability nor to detect conflicts, returns a candidate solution.…”

“…x def = {x 1 , • • • , xm } ∈ Lφ , it is trivial to check whether x is a model for φ by substituting the variables with their values into the formula. 5 If x is not a model we can try to look in the surroundings of x. An idea is to reduce φ to a linear under-approximation by forcing all the multiplications to be linear, similarly to what is done in [7] equation (3), in the context of the incremental linearization approach (we will refer to this techique as check-crosses).…”

Bayesian Optimisation with Formal Guarantees

“…We only assume that the SMT solver supports quantifier-free fragment including formulas F and θ. Let us note that SMT solvers can handle a range of non-linear functions including polynomials, transcendental functions such as combinations of sine, cosine, exponentials and solutions of differential equations [12], [13], [14], [15].…”