Optimization Modulo the Theory of Floating-Point Numbers

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...

Optimization Modulo the Theories of Signed Bit-Vectors and Floating-Point Numbers

“…This is an extension of a paper published at CADE 2019 conference [44]. We would like to thank the anonymous reviewers for their insightful comments and suggestions, and we thank Alberto Griggio for support with MathSAT code.…”

“…In this paper (as in [44]) we address-for the first time to the best of our knowledge-OMT for objectives in the theory of signed Bit-Vectors and, most importantly, in the theory of Floating-Point Arithmetic, by exploiting some properties of the two's complement encoding for signed BV and of the IEEE 754-2008 encoding for FP respectively. (We consider the former as a straightforward extension of [32], and the latter as our main contribution.…”

From MiniZinc to Optimization Modulo Theories, and Back (Extended Version)

“…In the short term, we plan to address the inefficient handling of Pseudo-Boolean constraints over the rationals revealed by the experimental evaluation in Section §5.2. In order to deal with those FLATZINC constraints that require non-linear arithmetic, we envisage an opportunity to either extend OPTIMATHSAT with proper handling of the non-linear arithmetic theory [21] or to experiment with an encoding based on the floating-point theory [60]. This objective goes hand in hand with the extension of OMT2MZN to deal with other SMT theories.…”

“…This has brought previouslyintractable problems to the reach of state-of-the-art SMT solvers. Optimization Modulo Theories (OMT), [42,53,36,34,56,16,39,60], is an extension to SMT that allows for finding a model of a first-order formula ϕ that is optimal with respect to some objective function expressed in some background theory, by means of a combination of SMT and optimization procedures. State-of-the art OMT tools allow optimization in a variety of theories, including linear arithmetic over the rationals (OMT(LRA)) [53] and the integers (OMT(LIA)) [16,56], bit-vectors (OMT(BV)) [39] and floating-point numbers (OMT (FP)) [60].…”

Optimization Modulo Non-linear Arithmetic via Incremental Linearization

“…These problems are grouped under the umbrella term of Optimization Modulo Theories -OMT [32,34,8]. OMT techniques have been conceived for a variety of theories, including LRA [34,8], LIA [8,36], BV [31], FP [39]. In general, they work by performing sequences of incremental SMT calls, possibly combined with theory-specific optimization techniques for the conjunctive fragment of the given theory, which progressively tighten the range of values of the objective function.…”