A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems

Abstract

We present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded Reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two transformations turn any system of linear mixed constraints into a bounded system, i.e., a system for which termination can be achieved easily. Existing approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from proofs, only explore a finite search space after application of our two transformations. Instead of generating a priori bounds for the variables, e.g., as suggested by Papadimitriou, unbounded variables are eliminated through the two transformations. The transformations orient themselves on the structure of an input system instead of computing a priori (over-)approximations out of the available constants. Experiments provide further evidence to the efficiency of the transformations in practice. We also present a polynomial method for converting certificates of (un)satisfiability from the transformed to the original system.

Spass-Satt

“…Branch and bound alone is an incomplete decision procedure and only guarantees termination on bounded problems, i.e., problems where all variables have an upper and a lower bound. For this reason, we developed two transformations that reduce any unbounded problem into an equisatisfiable problem that is bounded [7]. The transformed problem can then be solved with our branch-and-bound implementation because it is complete for bounded problems.…”

“…In this section, we quickly explain how our theory solver and SAT solver interact. To this end, we list inFigure 3the main interface functions of our SAT solver and theory solver and show through3 These instances belong to the dillig[16], CAV-2009[16], slacks[23], 20180326-Bromberger[7], and prime-cone benchmark families[23], which together contain more than 1483 instances of absolutely unbounded problems 4. These instances belong to the 20180326-Bromberger[7], arctic-matrix[14], cut lemmas[20], slacks[23], and tropical-matrix[14] benchmark families.…”

Extending a Verified Simplex Algorithm

“…Here, the integration of techniques like branch and bound or Gomory cuts would be interesting for the support of LIA, or, more generally, mixed integer problems [DdM06, Chapter 4 of Technical Report]. In order to formally prove termination, a recent transformation [Bro18] might be useful which turns arbitrary mixed integer problems into bounded ones.…”

Lookahead-Based SMT Solving

“…The experiments in this paper done on QF LIA show that the lookahead heuristic can be used for guiding the search in non-convex SMT theories to accomplish in some cases termination of a straightforward, incomplete implementation. An orthogonal approach involves altering the cutting plane algorithm, as proposed in, e.g., [1,13].…”

“…This implementation could be made more efficient by using more sophisticated methods for choosing the cuts. Fore more details, see, for example [13,1,6]. While we plan to study the relationship of more sophisticated cuts in the future in conjunction with the lookahead approach, we now prefer to keep the solver implementation simple to better understand the effect of lookahead in isolation.…”