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2018 | OriginalPaper | Buchkapitel

Focussing, \(\mathsf {MALL}\) and the Polynomial Hierarchy

verfasst von : Anupam Das

Erschienen in: Automated Reasoning

Verlag: Springer International Publishing

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Abstract

We investigate how to extract alternating time bounds from ‘focussed’ proofs, treating non-invertible rule phases as nondeterministic computation and invertible rule phases as co-nondeterministic computation. We refine the usual presentation of focussing to account for deterministic computations in proof search, which correspond to invertible rules that do not branch, more faithfully associating phases of focussed proof search to their alternating time complexity.

As our main result, we give a focussed system for \(\mathsf {MALL}\mathsf {w}\) (\(\mathsf {MALL}\) with weakening) with encodings to and from true quantified Boolean formulas (QBFs): in one direction we encode QBF satisfiability and in the other we encode focussed proof search. Moreover we show that the composition of the two encodings preserves quantifier alternation, yielding natural fragments of \(\mathsf {MALL}\mathsf {w}\) complete for each level of the polynomial hierarchy. This refines the well-known result that \(\mathsf {MALL}\mathsf {w}\) is \(\mathbf {PSPACE}\)-complete.

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Vorheriges Kapitel Efficient Model Construction for Horn Logic with VLog

Nächstes Kapitel Checking Array Bounds by Abstract Interpretation and Symbolic Expressions

\(\mathsf {MALL}\mathsf {w}\) is also \(\mathbf {PSPACE}\)-complete, a folklore result subsumed by this work.

In fact the same phenomenon presents in this work, cf. Fig. 3.

Notice that, by definition of satisfaction these two notions coincide for closed QBFs.

In fact, quantifier-free Boolean formula evaluation is known to be \(\mathbf {NC}^{1}\)-complete [2].

Note that, for the derivations for the innermost quantifier (\(\exists \) or \(\forall \)), the topmost \(R\) or \(\bar{R}\) step of Figs. 3 or 4 (resp.) does not occur.

This is actually exemplary of the more general phenomenon that invertible phases of rules are ‘confluent’.

Andreoli, J.: Logic programming with focusing proofs in linear logic. J. Log. Comput. 2(3), 297–347 (1992)MathSciNetCrossRef

Buss, S.R.: The Boolean formula value problem is in ALOGTIME. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC) (1987)

Buss, S.R., Iemhoff, R.: The depth of intuitionistic cut free proofs (2003). http://www.phil.uu.nl/~iemhoff/Mijn/Papers/dpthIPC.pdf

Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. J. ACM 28(1), 114–133 (1981)MathSciNetCrossRef

Chaudhuri, K., Miller, D., Saurin, A.: Canonical sequent proofs via multi-focusing. In: Ausiello, G., Karhumäki, J., Mauri, G., Ong, L. (eds.) TCS 2008. IIFIP, vol. 273, pp. 383–396. Springer, Boston, MA (2008). https://doi.org/10.1007/978-0-387-09680-3_26CrossRef

Cook, S., Nguyen, P.: Logical Foundations of Proof Complexity, 1st edn. Cambridge University Press, New York (2010)CrossRef

Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44(1), 36–50 (1979)MathSciNetCrossRef

Das, A.: Alternating time bounds from variants of focussed proof systems (2017, submitted). http://anupamdas.com/alt-time-bnds-var-foc-sys.pdf

Delande, O., Miller, D., Saurin, A.: Proof and refutation in MALL as a game. Ann. Pure Appl. Logic 161(5), 654–672 (2010)MathSciNetCrossRef

10.

Girard, J.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)MathSciNetCrossRef

11.

Heuerding, A., Seyfried, M., Zimmermann, H.: Efficient loop-check for backward proof search in some non-classical propositional logics. In: Miglioli, P., Moscato, U., Mundici, D., Ornaghi, M. (eds.) TABLEAUX 1996. LNCS, vol. 1071, pp. 210–225. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61208-4_14CrossRef

12.

Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press, New York (1995)CrossRef

13.

Laurent, O.: A study of polarization in logic. Theses, Université de la Méditerranée - Aix-Marseille II (2002)

14.

Letz, R.: Lemma and model caching in decision procedures for quantified Boolean formulas. In: Automated Reasoning with Analytic Tableaux and Related Methods, International Conference (TABLEAUX) (2002)MATH

15.

Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theor. Comput. Sci. 410(46), 4747–4768 (2009)MathSciNetCrossRef

16.

Lincoln, P., Mitchell, J.C., Scedrov, A., Shankar, N.: Decision problems for propositional linear logic. In: 31st Annual Symposium on Foundations of Computer Science (FOCS) (1990)

17.

Lincoln, P., Mitchell, J.C., Scedrov, A., Shankar, N.: Decision problems for propositional linear logic. Ann. Pure Appl. Log. 56(1–3), 239–311 (1992)MathSciNetCrossRef

18.

Marin, S., Miller, D., Volpe, M.: A focused framework for emulating modal proof systems. In: 11th Conference on Advances in Modal Logic, pp. 469–488 (2016)

19.

Miller, D., Volpe, M.: Focused labeled proof systems for modal logic. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 266–280. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48899-7_19CrossRefMATH

20.

Negri, S., Von Plato, J., Ranta, A.: Structural Proof Theory. Cambridge University Press, Cambridge (2008)

21.

Nigam, V.: Investigating the use of lemmas (2007, preprint)

22.

Ono, H.: Proof-Theoretic Methods in Nonclassical Logic – An Introduction. MSJ Memoirs, vol. 2, pp. 207–254. The Mathematical Society of Japan, Tokyo (1998)

23.

Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Boston (1994)MATH

24.

Statman, R.: Intuitionistic propositional logic is polynomial-space complete. Theor. Comput. Sci. 9, 67–72 (1979)MathSciNetCrossRef

25.

Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 1–22 (1976)MathSciNetCrossRef

26.

Troelstra, A., Schwichtenberg, H.: Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science, vol. 43. Cambridge University Press, Cambridge (1996)

Titel: Focussing, and the Polynomial Hierarchy
verfasst von: Anupam Das
Verlag: Springer International Publishing
Buch: Automated Reasoning
Print ISBN: 978-3-319-94204-9

Electronic ISBN: 978-3-319-94205-6

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-94205-6_45

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