Optimization modulo the theories of signed bit-vectors and floating-point numbers. (English) Zbl 07433028

Summary: 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 \((\mathcal{BV})\) for the integers and that of floating-point numbers \((\mathcal{FP})\) for the reals respectively. Whereas an approach for OMT with (unsigned) \(\mathcal{BV}\) objectives has been proposed by Nadel & Ryvchin, unfortunately we are not aware of any existing approach for OMT with \(\mathcal{FP}\) objectives. In this paper we fill this gap, and we address for the first time OMT with \(\mathcal{FP}\) objectives. We present a novel OMT approach, based on the novel concept of attractor and dynamic attractor, which extends the work of Nadel and Ryvchin to work with signed-\(\mathcal{BV}\) objectives and, most importantly, with \(\mathcal{FP}\) objectives. We have implemented some novel OMT procedures on top of OptiMath-SAT and tested them on modified problems from the SMT-LIB repository. The empirical results support the validity and feasibility of our novel approach.


68V15 Theorem proving (automated and interactive theorem provers, deduction, resolution, etc.)
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[1] http://disi.unitn.it/trentin/resources/floatingpoint_test.tar.gz
[2] https://hub.docker.com/repository/docker/patricktrentin88/jar2020_floatingpoint_test
[3] nuXmv. https://nuxmv.fbk.eu
[4] SmtLibv2. www.smtlib.cs.uiowa.edu/
[5] IEEE standard 754, 2008. http://grouper.ieee.org/groups/754/
[6] Albuquerque, H.F., Araujo, R.F., de Bessa, I.V., Cordeiro, L.C., de Lima Filho, E.B.: OptCE: a counterexample-guided inductive optimization solver. In: SBMF, volume 10623 of Lecture Notes in Computer Science, pp. 125-141. Springer (2017)
[7] Araujo, RF; Albuquerque, HF; de Bessa, IV; Cordeiro, LC; Filho, JEC, Counterexample guided inductive optimization based on satisfiability modulo theories, Sci. Comput. Program., 165, 3-23 (2018)
[8] Araújo, R., Bessa, I., Cordeiro, L.C., Filho, J.E.C.: SMT-based verification applied to non-convex optimization problems. In: 2016 VI Brazilian Symposium on Computing Systems Engineering (SBESC), pp. 1-8 (2016)
[9] Bjorner, N., Phan, A.-D.: \( \nu{}Z\)-Maximal satisfaction with Z3. In: Proceedings of International Symposium on Symbolic Computation in Software Science, Gammart, Tunisia, December 2014. EasyChair Proceedings in Computing (EPiC)
[10] Bjorner, N., Phan, A.-D, Fleckenstein, L.: \( \nu Z\)—an optimizing SMT solver. In: Proceedings of TACAS, volume 9035 of LNCS. Springer (2015)
[11] Bozzano, M., Bruttomesso, R., Cimatti, A., Franzèn, A., Hanna, Z., Khasidashvili, Z., Palti, A. Sebastiani, R.: Encoding RTL constructs for MathSAT: a preliminary report. In: Proceedings of 3rd Workshop of Pragmatics on Decision Procedure in Automated Reasoning, PDPAR’05, ENTCS. Elsevier (2005) · Zbl 1272.68230
[12] Brain, M., D’Silva, V., Griggio, A., Haller, L., Kroening, D:. Interpolation-based verification of floating-point programs with abstract CDCL. In: SAS, pp. 412-432 (2013)
[13] Brain, M., D’Silva, V., Griggio, A., Haller, L., Kroening, D.: Deciding floating-point logic with abstract conflict driven clause learning. Formal Methods Syst. Des. 45(2), 213-245 (2014) · Zbl 1317.68110
[14] Brain, M., Tinelli, C., Rümmer, P., Wahl, T:. An automatable formal semantics for IEEE-754 floating-point arithmetic. In: ARITH, pp. 160-167. IEEE (2015)
[15] Brillout, A., Kroening, D., Wahl, T.: Mixed abstractions for floating-point arithmetic. In: 2009 Formal Methods in Computer-Aided Design, pp. 69-76 (2009)
[16] Brinkmann, R., Drechsler, R.: RTL-datapath verification using integer linear programming. In: Proceedings of ASP-DAC 2002, pp. 741-746. IEEE (2002)
[17] Brummayer, R.: Efficient SMT Solving for Bit-Vectors and the Extensional Theory of Arrays. PhD thesis, Informatik, Johannes Kepler University Linz (2009) · Zbl 1187.68168
[18] Brummayer, R., Biere, A.: Boolector: An efficient SMT solver for bit-vectors and arrays. In: TACAS, pp. 174-177. Springer, Berlin (2009)
[19] Bruttomesso, R., Cimatti, A., Franzén, A., Griggio, A., Hanna, Z., Nadel, A., Palti, A., Sebastiani, R.: A lazy and layered SMT \(( \cal{BV} )\) solver for hard industrial verification problems. In CAV, volume 4590 of LNCS, pp. 547-560. Springer (2007)
[20] Cimatti, A., Franzén, A., Griggio, A., Sebastiani, R., Stenico, C.: Satisfiability modulo the theory of costs: foundations and applications. In: TACAS, volume 6015 of LNCS, pp. 99-113. Springer (2010) · Zbl 1284.68388
[21] Cimatti, A., Griggio, A., Schaafsma, B.J., Sebastiani, R.: A modular approach to MaxSAT modulo theories. In: International Conference on Theory and Applications of Satisfiability Testing, SAT, volume 7962 of LNCS, July 2013 · Zbl 1390.68572
[22] Dillig, I., Dillig, T., McMillan, K.L., Aiken, A.: Minimum satisfying assignments for SMT. In: CAV, pp. 394-409 (2012)
[23] Fazekas, K., Bacchus, F., Biere, A.: Implicit hitting set algorithms for maximum satisfiability modulo theories. In: IJCAR, volume 10900 of Lecture Notes in Computer Science, pp. 134-151. Springer (2018)
[24] Ganesh, V., Dill, D.L:. A decision procedure for bit-vectors and arrays. In: CAV (2007) · Zbl 1135.68472
[25] Hadarean, L.: An Efficient and Trustworthy Theory Solver for Bit-vectors in Satisfiability Modulo Theories. PhD thesis, New York University (2015)
[26] Hadarean, L., Bansal, K., Jovanovic, D., Barrett, C., Tinelli, C.: A tale of two solvers: eager and lazy approaches to bit-vectors. In: CAV, volume 8559 of Lecture Notes in Computer Science, pp. 680-695. Springer (2014)
[27] Haller, L., Griggio, A., Brain, M., Kroening, D.: Deciding floating-point logic with systematic abstraction. In: Proceedings of FMCAD (2012) · Zbl 1317.68110
[28] Kovásznai, G., Biró, C., Erdélyi, B.: Puli-a problem-specific OMT solver. EasyChair Preprint no. 371, EasyChair (2018)
[29] Larraz, D., Oliveras, A., Rodríguez-Carbonell, E., Rubio, A.: Minimal-model-guided approaches to solving polynomial constraints and extensions. In: SAT (2014) · Zbl 1423.68457
[30] Li, Y., Albarghouthi, A., Kincad, Z., Gurfinkel, A., Chechik, M.: Symbolic optimization with SMT solvers. In: POPL (2014) · Zbl 1284.68410
[31] Manolios, P., Papavasileiou, V.: Ilp modulo theories. In: CAV, pp. 662-677 (2013)
[32] Nadel, A., Ryvchin, V.: Bit-vector optimization. In: Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2016, volume 9636 of LNCS. Springer (2016)
[33] Niemetz, A.: Bit-Precise Reasoning Beyond Bit-Blasting. PhD thesis, Informatik, Johannes Kepler University Linz (2017) · Zbl 1377.68134
[34] Niemetz, A., Preiner, M., Fröhlich, A., Biere, A.: Improving local search for bit-vector logics in SMT with path propagation. In: Proceedings of 4th International Working on Design and Implementation of Formal Tools and Systems (DIFTS’15), p. 10 (2015)
[35] Nieuwenhuis, R., Oliveras, A.: On SAT modulo theories and optimization problems. In: Proceedings of Theory and Applications of Satisfiability Testing—SAT 2006, volume 4121 of LNCS. Springer (2006) · Zbl 1187.68558
[36] Roc, O.: Optimization Modulo Theories. Master’s thesis, Polytechnic University of Catalonia (2011). http://hdl.handle.net/2099.1/14204
[37] Ruemmer, P., Wahl, T.: An SMT-LIB Theory of Binary Floating-Point Arithmetic. SMT 2010 Workshop, July 2010. http://www.philipp.ruemmer.org/publications/smt-fpa.pdf
[38] Sebastiani, R., Tomasi, S.: Optimization in SMT with LA(Q) Cost Functions. In: IJCAR, volume 7364 of LNAI, pp. 484-498. Springer (2012) · Zbl 1358.68264
[39] Sebastiani, R., Tomasi, S.: Optimization modulo theories with linear rational costs. ACM Trans. Comput. Logics 16(2) (2015) · Zbl 1354.68233
[40] Sebastiani, R., Trentin, P.: OptiMathSAT: A tool for optimization modulo theories. In: Proceedings of International Conference on Computer-Aided Verification, CAV 2015, volume 9206 of LNCS. Springer (2015) · Zbl 1468.68206
[41] Sebastiani, R., Trentin, P.: Pushing the envelope of optimization modulo theories with linear-arithmetic cost functions. In: Proceedings of International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS’15, volume 9035 of LNCS. Springer (2015) · Zbl 1420.68197
[42] Sebastiani, R., Trentin, P.: On optimization modulo theories, MaxSMT and sorting networks. In Proceedings of International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS’17, volume 10205 of LNCS. Springer (2017)
[43] Sebastiani, R., Trentin, P.: OptiMathSAT: A tool for optimization modulo theories. J. Autom. Reason. (2018) · Zbl 1468.68206
[44] Trentin, P., Sebastiani, R.: Optimization modulo the theory of floating-point numbers. In: In proceedings of 27th International Conference on Automated Deduction—CADE-27., LNCS, pp. 550-567. Springer (2019) · Zbl 07178998
[45] Zeljić, A.; Backeman, P.; Wintersteiger, CM; Rümmer, P.; Galmiche, D.; Schulz, S.; Sebastiani, R., Exploring approximations for floating-point arithmetic using uppsat, Automated Reasoning, 246-262 (2018), Cham: Springer, Cham · Zbl 06958103
[46] Zeljić, A.; Wintersteiger, CM; Rümmer, P.; Demri, S.; Kapur, D.; Weidenbach, C., Approximations for model construction, Automated Reasoning, 344-359 (2014), Cham: Springer, Cham · Zbl 1409.68264
[47] Zeljić, A.; Wintersteiger, CM; Rümmer, P., An approximation framework for solvers and decision procedures, J. Autom. Reason., 58, 1, 127-147 (2017) · Zbl 1409.68265
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