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IMPLICATIONAL RELEVANCE LOGIC IS 2-EXPTIME-COMPLETE

Published online by Cambridge University Press:  22 January 2016

SYLVAIN SCHMITZ
SYLVAIN SCHMITZ*
Affiliation:
ENS CACHAN & INRIAFRANCEE-mail:[email protected]: http://www.lsv.ens-cachan.fr/∼schmitz
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Abstract

We show that provability in the implicational fragment of relevance logic is complete for doubly exponential time, using reductions to and from coverability in branching vector addition systems.

Keywords

Relevance logicbranching VASSfocusing proofscomplexity
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 641 - 661
Copyright
Copyright © The Association for Symbolic Logic 2016 

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References

REFERENCES

Anderson, Alan Ross and Belnap, Nuel D. Jr., Entailment: The logic of relevance and necessity, vol. I, Princeton University Press, Princeton, 1975.Google Scholar
Andreoli, Jean-Marc, Logic programming with focusing proofs in linear logic . Journal of Logic and Computation, vol. 2 (1992), no. 3, pp. 297–347.CrossRefGoogle Scholar
Bimbó, Katalin and Michael Dunn, J., On the decidability of implicational ticket entailment , this Journal, vol. 78 (2013), no. 1, pp. 214–236.Google Scholar
Bojańczyk, Mikołaj, Muscholl, Anca, Schwentick, Thomas, and Segoufin, Luc, Two-variable logic on data trees and XML reasoning . Journal of the ACM, vol. 56 (2009), no. 3, pp. 1–48.CrossRefGoogle Scholar
Bouajjani, Ahmed and Emmi, Michael, Analysis of recursively parallel programs . ACM Transactions on Programming Languages and Systems, vol. 35 (2013), no. 3, pp. 10:1–10:49.Google Scholar
Church, Alonzo, The weak theory of implication , Kontrolliertes Denken. Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften (Menne, A., Wilhelmy, A., and Angsil, H., editors), Kommissions-Verlag Karl Alber, Munich, 1951, pp. 22–37.Google Scholar
Curry, Haskell H. and Feys, Robert, Combinatory logic, vol. 1, Studies in Logic and the Foundations of Mathematics, vol. 22, North-Holland Publishing Company, Amsterdam, 1958.Google Scholar
Groote, Philippe de, Towards abstract categorial grammars , Proceedings of ACL 2001, ACL Press, Toulouse, 2001, pp. 252–259.Google Scholar
de Groote, Philippe, Guillaume, Bruno, and Salvati, Sylvain, Vector addition tree automata, Proceedings of LICS 2004 , IEEE Computer Society, 2004, pp. 64–73.CrossRefGoogle Scholar
Demri, Stéphane, Jurdziński, Marcin, Lachish, Oded, and Lazić, Ranko, The covering and boundedness problems for branching vector addition systems . Journal of Computer and System Sciences, vol. 79 (2013), no. 1, pp. 23–38.Google Scholar
Dimino, Jérémie, Jacquemard, Florent, and Segoufin, Luc, FO 2(<,+1,∼) on data trees, data tree automata and branching vector addition systems, preprint, hal.inria.fr:hal-00769249, 2015.Google Scholar
Michael Dunn, J. and Restall, Greg, Relevance logic , Handbook of philosophical logic (Gabbay, Dov M. and Guenthner, Franz, editors), vol. 6, Kluwer Academic Publishers, Dordrecht, 2002, pp. 1–128.Google Scholar
Figueira, Diego, Figueira, Santiago, Schmitz, Sylvain, and Schnoebelen, Philippe, Ackermannian and primitive-recursive bounds with Dickson’s Lemma, Proceedings of LICS 2011 , IEEE Computer Society, 2011, pp. 269–278.CrossRefGoogle Scholar
Kripke, Saul A., The problem of entailment , Proceedings of ASL 1959, vol. 24, this Journal, no. 4, 1959, (Abstract), p. 324.Google Scholar
Lazić, Ranko and Schmitz, Sylvain, Non-elementary complexities for branching VASS, MELL, and extensions, Proceedings of CSL-LICS 2014 , ACM, 2014.CrossRefGoogle Scholar
Lincoln, Patrick, Mitchell, John, Scedrov, Andre, and Shankar, Natarajan, Decision problems for propositional linear logic . Annals of Pure and Applied Logic, vol. 56 (1992), no. 1–3, pp. 239–311.Google Scholar
Majumdar, Rupak and Wang, Zilong, Expand, enlarge, and check for branching vector addition systems , Proceedings of Concur 2013 (D’Argenio, Pedro R. and Melgratti, Hernán, editors), Lecture Notes in Computer Science, vol. 8052, Springer, Heidelberg, 2013, pp. 152–166.Google Scholar
Mayr, Ernst W. and Meyer, Albert R., The complexity of the word problems for commutative semigroups and polynomial ideals , Advances in Mathematics, vol. 46 (1982), no. 3, pp. 305–329.CrossRefGoogle Scholar
Moh, Shaw Kei, The deduction theorems and two new logical systems . Methodos, vol. 2 (1950), pp. 56–75.Google Scholar
Padovani, Vincent, Ticket entailment is decidable . Mathematical Structures in Computer Science, vol. 23 (2013), no. 3, pp. 568–607.CrossRefGoogle Scholar
Rambow, Owen, Multiset-valued linear index grammars: imposing dominance constraints on derivations , Proceedings of ACL’94, ACL Press, Las Cruces, 1994, pp. 263–270.Google Scholar
Riche, Jacques and Meyer, Robert K., Kripke, Belnap, Urquhart and relevant decidability & complexity , Proceedings of CSL ’98 (Gottlob, Georg, Grandjean, Étienne, and Seyr, Katrin, editors), Lecture Notes in Computer Science, vol. 1584, Springer, Heidelberg, 1999, pp. 224–240.Google Scholar
Salvati, Sylvain, Minimalist grammars in the light of logic , Logic and grammar (Pogodalla, Sylvain, Quatrini, Myriam, and Retoré, Christian, editors), Lecture Notes in Computer Science, vol. 6700, Springer, Heidelberg, 2011, pp. 81–117.CrossRefGoogle Scholar
Schmitz, Sylvain, On the computational complexity of dominance links in grammatical formalisms , Proceedings of ACL 2010, ACL Press, Uppsala, 2010, pp. 514–524.Google Scholar
Statman, Richard, Intuitionistic propositional logic is polynomial-space complete . Theoretical Computer Science, vol. 9 (1979), no. 1, pp. 67–72.Google Scholar
Urquhart, Alasdair, The undecidability of entailment and relevant implication , this Journal, vol. 49 (1984), no. 4, pp. 1059–1073.Google Scholar
Urquhart, Alasdair, The complexity of decision procedures in relevance logic , Truth or consequences: Essays in honour of Nuel Belnap (Michael Dunn, J. and Gupta, Anil, editors), Kluwer Academic Publishers, Dordrecht, 1990, pp. 61–76.CrossRefGoogle Scholar
Urquhart, Alasdair, The complexity of decision procedures in relevance logic II , this Journal, vol. 64 (1999), no. 4, pp. 1774–1802.Google Scholar
Verma, Kumar Neeraj and Goubault-Larrecq, Jean, Karp-Miller trees for a branching extension of VASS . Discrete Mathematics and Theoretical Computer Science, vol. 7 (2005), no. 1, pp. 217–230.CrossRefGoogle Scholar