Pushing the Envelope of Optimization Modulo Theories with Linear-Arithmetic Cost Functions

Abstract

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.

On the Synthesis of Guaranteed-Quality Plans for Robot Fleets in Logistics Scenarios via Optimization Modulo Theories

“…Though there exist successful symbolic reasoning methods towards solving the underlying scheduling problem [9,12], a disadvantage of these methods is that they provide no guarantees about the quality of the solution. A promising solution to this problem is offered by the recently emerging field of Optimization Modulo Theories (OMT), where Satisfiability Modulo Theories (SMT) solving is extended with functionalities towards optimization [15,18,20].…”

On Optimization Modulo Theories, MaxSMT and Sorting Networks

“…They consist of 18996 automatically-generated formulas which encode the problem of computing the lexicographically-optimum realization of a constrained goal model [23], according to a prioritized list of (up to) three objectives obj 1 , obj 2 , obj 3 . A solution optimizes lexicographically obj 1 , ..., obj k if it optimizes obj 1 and, if more than one such obj 1 -optimum solutions exists, it also optimizes obj 2 ,..., and so on; both OMT-based and MAXSAT-based techniques handle lexicographic optimization, by optimizing obj 1 , obj 2 , ... in order, fixing the value of each obj i to its optimum as soon as it is found [13,14,32,31]. In this experiment, we set the timeout at 100 seconds.…”

“…Eventually, OMT(LRA ∪ T ) has been extended so that to handle costs on the integers, incremental OMT, multi-objective, and lexicographic OMT and Pareto-optimality [19,18,13,32,14,31]. To the best of our knowledge only four OMT solvers are currently implemented: BCLT [18], νZ (aka Z3OPT) [13,14], OPTIMATHSAT [32,31], and SYMBA [19]. Remarkably, BCLT, νZ and OPTIMATHSAT currently implement also specialized procedures for MaxSMT, leveraging to SMT level state-of-the-art MaxSAT procedures; in addition, νZ features a Pseudo-Boolean T -solver which can generate sorting circuits on demand for Pseudo-Boolean inequalities featuring sums with small coefficients when a Pseudo-Boolean inequality is used some times for unit propagation/conflicts [14,12].…”

“…For example, 1799972218749879 2251799813685248 is a sample weight value from problems in[35].3 One could run a MAXSAT-based search separately on each objective, but doing this he/she would loose the benefits of boxed optimization, see[19,13,32].…”

Multi-objective reasoning with constrained goal models

“…The tool offers functionalities to create CGM models as graphical diagrams and to explore alternatives scenarios running automated reasoning techniques. CGM-Tool uses the SMT/OMT solver OptiMathSAT [38,40,39], which is built on top of the SMT solver MATHSAT5 [8], as automated reasoning backend. 1 The structure of the paper is as follows: §2 provides a succinct account of necessary background on goal modelling and on SMT/OMT; §3 introduces the notion of CGM through an example; §4 introduces the syntax and semantics of CGMs; §5 presents the set of automated reasoning functionalities for CGMs; §6 gives a quick overview of our tool based on the presented approach; §7 provides an experimental evaluation of the performances of our tool on large CGMs, showing that the approach scales well with respect to CGM size; §8 gives overview of related work, while in §9 we draw conclusions and present future research challenges.…”