Abstract
This paper poses the following basic question: Given a quantified Boolean formula ∃ x. ϕ, what should a function/formula f be such that substituting f for x in ϕ yields a logically equivalent quantifier-free formula? Its answer leads to a solution to quantifier elimination in the Boolean domain, alternative to the conventional approach based on formula expansion. Such a composite function can be effectively derived using symbolic techniques and further simplified for practical applications. In particular, we explore Craig interpolation for scalable computation. This compositional approach to quantifier elimination is analyzably superior to the conventional one under certain practical assumptions. Experiments demonstrate the scalability of the approach. Several large problem instances unsolvable before can now be resolved effectively. A generalization to first-order logic characterizes a composite function’s complete flexibility, which awaits further exploitation to simplify quantifier elimination beyond the propositional case.
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References
Berkeley Logic Synthesis and Verification Group. ABC: A System for Sequential Synthesis and Verification (2005), http://www.eecs.berkeley.edu/~alanmi/abc/
Caviness, B.F., Johnson, J.R. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer, Heidelberg (1998)
Coudert, O., Madre, J.C.: A unified framework for the formal verification of sequential circuits. In: Proc. Int’l. Conf. Computer-Aided Design, pp. 126–129 (1990)
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
Craig, W.: Linear reasoning: A new form of the Herbrand-Gentzen theorem. J. Symbolic Logic 22(3), 250–268 (1957)
Enderton, H.B.: A Mathematical Introduction to Logic, 2nd edn. Academic Press, London (2000)
Ganai, M., Gupta, A., Ashar, P.: Efficient SAT-based unbounded symbolic model checking using circuit cofactoring. In: Proc. Int’l. Conf. Computer-Aided Design, pp. 510–517 (2004)
Jhala, R., McMillan, K.: Interpolant-based transition relation approximation. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 39–51. Springer, Heidelberg (2005)
Lee, R.-R., Jiang, J.-H.R., Hung, W.-L.: Bi-decomposing large Boolean functions via interpolation and satisfiability solving. In: Proc. Design Automation Conf., pp. 636–641 (2008)
Lee, C.-C., Jiang, J.-H.R., Huang, C.-Y., Mishchenko, A.: Scalable exploration of functional dependency by interpolation and incremental SAT solving. In: Proc. Int’l. Conf. on Computer-Aided Design, pp. 227–233 (2007)
Lin, H.-P., Jiang, J.-H.R., Lee, R.-R.: To SAT or not to SAT: Ashenhurst decomposition in a large scale. In: Proc. Int’l. Conf. Computer-Aided Design, pp. 32–37 (2008)
Mishchenko, A., Brayton, R.K., Jiang, J.-H.R., Jang, S.: Scalable don’t-care-based logic optimization and resynthesis. In: Proc. Int’l. Symp. on Field Programmable Gate Arrays, pp. 151–160 (2009)
Mishchenko, A., Chatterjee, S., Brayton, R.K.: DAG-aware AIG rewriting: A fresh look at combinational logic synthesis. In: Proc. Design Automation Conference, pp. 532–536 (2006)
Mishchenko, A., Chatterjee, S., Jiang, J.-H.R., Brayton, R.K.: FRAIGs: A unifying representation for logic synthesis and verification. Technical Report, EECS Dept., UC Berkeley (2005)
McMillan, K.: Applying SAT methods in unbounded symbolic model checking. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 250–264. Springer, Heidelberg (2002)
McMillan, K.: Interpolation and SAT-based model checking. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003)
Pigorsch, F., Scholl, C., Disch, S.: Advanced unbounded model checking based on AIGs, BDD sweeping, and quantifier scheduling. In: Proc. Formal Methods on Computer Aided Design, pp. 89–96 (2006)
Skolem, T.: Uber die mathematische Logik. Norsk. Mat. Tidsk. 10, 125–142 (1928); Translation in From Frege to Gödel, A Source Book in Mathematical Logic, J. van Heijenoort. Harvard Univ. Press (1967)
Seidl, A., Sturm, T.: Boolean quantification in a first-order context. In: Proc. Int’l. Workshop on Computer Algebra in Scientific Computing, pp. 329–345 (2003)
Sturm, T.: New domains for applied quantifier elimination. In: Proc. Int’l. Workshop on Computer Algebra in Scientific Computing, pp. 295–301 (2006)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry. University of California Press, Berkeley (1951)
Touati, H., Savoj, H., Lin, B., Brayton, R.K., Sangiovanni-Vincentelli, A.: Implicit enumeration of finite state machines using BDDs. In: Proc. Int’l. Conf. on Computer-Aided Design, pp. 130–133 (1990)
Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5, 3–27 (1988)
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Jiang, JH.R. (2009). Quantifier Elimination via Functional Composition. In: Bouajjani, A., Maler, O. (eds) Computer Aided Verification. CAV 2009. Lecture Notes in Computer Science, vol 5643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02658-4_30
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DOI: https://doi.org/10.1007/978-3-642-02658-4_30
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