Optimization Modulo Theories (OMT) is an extension of SMT which allows for finding models that optimize given objectives. OPTIMATHSAT is an OMT solver which allows for solving a list of optimization problems on SMT formulas with linear objective functions-on the Boolean, the rational and the integer domains, and on their combination thereof-including (partial weighted) MAXSMT. Multiple and heterogeneous objective functions can be combined together and handled either independently, or lexicographically, or in linear or min-max/max-min combinations. OPTIMATHSAT provides an incremental interface, it supports both an extended version of the SMT-LIBV2 language and a subset of the FLATZINC language, and can be interfaced via an API. In this paper we describe OPTIMATHSAT 1 and its usage in full detail.

NOTE This is an extended version of a paper published at TACAS 2015 [24].Abstract. In the last decade we have witnessed an impressive progress in the expressiveness and efficiency of Satisfiability Modulo Theories (SMT) solving techniques. This has brought previously-intractable problems at the reach of stateof-the-art SMT solvers, in particular in the domain of SW and HW verification. Many SMT-encodable problems of interest, however, require also the capability of finding models that are optimal wrt. some cost functions. In previous work, namely Optimization Modulo Theory with Linear Rational Cost Functions -OMT(LRA ∪ T ), we have leveraged SMT solving to handle the minimization of cost functions on linear arithmetic over the rationals, by means of a combination of SMT and LP minimization techniques. In this paper we push the envelope of our OMT approach along three directions: first, we extend it to work with linear arithmetic on the mixed integer/rational domain, by means of a combination of SMT, LP and ILP minimization techniques; second, we develop a multi-objective version of OMT, so that to handle many cost functions simultaneously or lexicographically; third, we develop an incremental version of OMT, so that to exploit the incrementality of some OMTencodable problems. An empirical evaluation performed on OMT-encoded verification problems demonstrates the usefulness and efficiency of these extensions.

Abstract. Optimization Modulo Theories (OMT) is an extension of SMT which allows for finding models that optimize given objectives. (Partial weighted) MAXSMT -or equivalently OMT with Pseudo-Boolean objective functions, OMT+PB-is a very-relevant strict subcase of OMT. We classify existing approaches for MAXSMT or OMT+PB in two groups: MAXSAT-based approaches exploit the efficiency of state-of-the-art MAXSAT solvers, but they are specific-purpose and not always applicable; OMT-based approaches are general-purpose, but they suffer from intrinsic inefficiencies on MAXSMT/OMT+PB problems. We identify a major source of such inefficiencies, and we address it by enhancing OMT by means of bidirectional sorting networks. We implemented this idea on top of the OPTIMATHSAT OMT solver. We run an extensive empirical evaluation on a variety of problems, comparing MAXSAT-based and OMT-based techniques, with and without sorting networks, implemented on top of OPTI-MATHSAT and νZ. The results support the effectiveness of this idea, and provide interesting insights about the different approaches.

No abstract

Optimization Modulo Theories (OMT) is an important extension of SMT which allows for finding models that optimize given objective functions, typically consisting in linear-arithmetic or pseudo-Boolean terms. However, many SMT and OMT applications, in particular from SW and HW verification, require handling bit-precise representations of numbers, which in SMT are handled by means of the theory of Bit-Vectors (BV) for the integers and that of Floating-Point Numbers (FP) for the reals respectively. Whereas an approach for OMT with (unsigned) BV has been proposed by Nadel & Ryvchin, unfortunately we are not aware of any existing approach for OMT with FP. In this paper we fill this gap. We present a novel OMT approach, based on the novel concept of attractor and dynamic attractor, which extends the work of Nadel & Ryvchin to signed BV and, most importantly, to FP. We have implemented some OMT(BV) and OMT(FP) procedures on top of OPTIMATHSAT and tested the latter ones on modified problems from the SMT-LIB repository. The empirical results support the validity and feasibility of the novel approach. * We would like to thank the anonymous reviewers for their insightful comments and suggestions, and we thank Alberto Griggio for support with MATHSAT code. the bitwise representation of the objective, ordered from the most-significant bit (MSB) to the least-significant bit (LSB).In this paper we address -for the first time to the best of our knowledge-OMT for the theory of signed Bit-Vectors and, most importantly, for the theory of Floating-Point Arithmetic (OMT(FP)), by exploiting some properties of the two's complement encoding for signed BV and of the IEEE 754-2008 encoding for FP respectively.We start from introducing the notion of attractor, which represent (the bitwise encoding of) the target value for the objective which the optimization process aims at. This allows us for easily leverage the procedure of [31] to work with both signed and unsigned Bit-Vectors, by minimizing lexicographically the bitwise distance between the objective and the attractor, that is, by minimizing lexicographically the bitwise-xor between the objective and the attractor.Unfortunately there is no such notion of (fixed) attractor for FP numbers, because the target value moves as long as the bits of the objective are updated from the MSB to the LSB, and the optimization process may have to change dynamically its aim, even at the opposite direction. (For instance, as soon as the minimization process realizes there is no solution with a negative value for the objective and thus sets its MSB to 0, the target value is switched from −∞ to 0+, and the search switches direction, from the maximization of the exponent and the significand to their minimization.)To cope with this fact, we introduce the notions of dynamic attractor and attractor trajectory, representing the dynamics of the moving target value, which are progressively updated as soon as the bits of the objective are updated from the MSB to the LSB. Based on these ideas, we present novel OMT...