Satisfiability of Non-linear (Ir)rational Arithmetic

Abstract

Abstract. We present a novel way for reasoning about (possibly ir)rational quantifier-free non-linear arithmetic by a reduction to SAT/SMT. The approach is incomplete and dedicated to satisfiable instances only but is able to produce models for satisfiable problems quickly. These characteristics suffice for applications such as termination analysis of rewrite systems. Our prototype implementation, called MiniSmt, is made freely available. Extensive experiments show that it outperforms current SMT solvers especially on rational and irrational domains.

SAT Modulo Linear Arithmetic for Solving Polynomial Constraints

“…Each of these tables has four columns, which for every solver show the number and total time in seconds of "sat", "unsat" and "unknown" answers (first to third columns), and the number of timeouts (fourth column). We have set the timeout to 1200 seconds for those problems coming from the SMT-LIB and to 60 seconds for the problems in the family m2int, which is the timeout used in [45]. Tables 1, 2 and 3 clearly show the superiority of our solver, although CVC3 performs slightly better with unsatisfiable problems on the leipzig and m2int families, as could be expected since it applies methods that focus on proving unsatisfiability.…”

“…Namely, we have considered the tools Z3 [35], CVC3 [6], and minismt and minismtbv [45]. While the former two solvers implement techniques based on proving unsatisfiability, the latter two are based on proving satisfiability, like our approach.…”

“…Besides, a family of problems m2int considered in [45] that arises in a similar application to the latter has also been included.…”

“…An interesting feature of this approach is that, in contrast to SAT translations [19,20], by having linear arithmetic built into the language, negative values can be handled without additional codification effort. Other recent techniques that incorporate arithmetic natively are similar in this sense (such as [45], which is based on bit-vector arithmetic). Another remarkable characteristic of our method is that, although it targets satisfiability, if enough variables are bounded it also allows one to reason about unsatisfiability, as pointed out in the example above.…”

Uncurrying for Termination and Complexity

“…To see this, we encoded the required condition as a satisfaction problem in non-linear arithmetic over the integers. MiniSmt [46] can prove this problem unsatisfiable by simplifying it into a trivially unsatisfiable constraint. Details can be inferred from the website mentioned in Footnote 4 on page 31.…”