A Quasipolynomial Cut-Elimination Procedure in Deep Inference via Atomic Flows and Threshold Formulae

Jeřabek showed in 2008 that cuts in propositional-logic deepinference proofs can be eliminated in quasipolynomial time. The proof is an indirect one relying on a result of Atserias, Galesi and Pudlak about monotone sequent calculus and a correspondence between this system and cut-free deep-inference proofs. In this paper we give a direct proof of Jeřabek's result: we give a quasipolynomial-time cut-elimination procedure in propositional-logic deep inference. The main new ingredient is the use of a computational trace of deep-inference proofs called atomic flows, which are both very simple (they trace only structural rules and forget logical rules) and strong enough to faithfully represent the cutelimination procedure.

[1]  Lutz Straßburger,et al.  A system of interaction and structure V: the exponentials and splitting , 2011, Math. Struct. Comput. Sci..

[2]  Alessio Guglielmi,et al.  Deep Inference and the Calculus of Structures , 2006 .

[3]  Lutz Straßburger,et al.  Breaking Paths in Atomic Flows for Classical Logic , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[4]  Charles A. Stewart,et al.  Purity through Unravelling , 2005 .

[5]  Kai Brünnler.  Cut Elimination inside a Deep Inference System for Classical Predicate Logic , 2006, Stud Logica.

[6]  Lutz Straßburger,et al.  Linear logic and noncommutativity in the calculus of structures , 2003 .

[7]  Paola Bruscoli.  A Purely Logical Account of Sequentiality in Proof Search , 2002, ICLP.

[8]  Anupam Das,et al.  Complexity of Deep Inference via Atomic Flows , 2012, CiE.

[9]  Phiniki Stouppa.  A Deep Inference System for the Modal Logic S5 , 2007, Stud Logica.

[10]  Samuel R. Buss,et al.  The Undecidability of k-Provability , 1991, Ann. Pure Appl. Log..

[11]  Rajeev Goré,et al.  Classical Modal Display Logic in the Calculus of Structures and Minimal Cut-free Deep Inference Calculi for S5 , 2007, J. Log. Comput..

[12]  Ozan Kahramanogullari.  System BV is NP-complete , 2008, Ann. Pure Appl. Log..

[13]  Alexander Artikis,et al.  Specifying norm-governed computational societies , 2009, TOCL.

[14]  Kai Br.  Deep Sequent Systems for Modal Logic , 2006 .

[15]  Pietro Di Gianantonio.  Structures for Multiplicative Cyclic Linear Logic: Deepness vs Cyclicity , 2004, CSL.

[16]  Ingo Wegener,et al.  The complexity of Boolean functions , 1987 .

[17]  Lutz Straßburger,et al.  MELL in the calculus of structures , 2003, Theor. Comput. Sci..

[18]  Lutz Straßburger,et al.  Non-commutativity and MELL in the Calculus of Structures , 2001, CSL.

[19]  Alessio Guglielmi,et al.  A system of interaction and structure , 1999, TOCL.

[20]  Tom Gundersen,et al.  Normalisation Control in Deep Inference via Atomic Flows , 2007, Log. Methods Comput. Sci..

[21]  Lutz Straßburger,et al.  A Non-commutative Extension of MELL , 2002, LPAR.

[22]  Charles A. Stewart,et al.  A Systematic Proof Theory for Several Modal Logics , 2004, Advances in Modal Logic.

[23]  Kai Brünnler,et al.  Deep sequent systems for modal logic , 2009, Arch. Math. Log..

[24]  Pavel Pudlák,et al.  Monotone simulations of non-monotone proofs , 2001, J. Comput. Syst. Sci..

[25]  H. Reichel,et al.  Deep Inference and Symmetry in Classical Proofs , 2003 .

[26]  Kai Brünnler,et al.  Atomic Cut Elimination for classical Logic , 2003, CSL.

[27]  Alwen Tiu,et al.  A Local System for Intuitionistic Logic , 2006, LPAR.

[28]  Tom Gundersen,et al.  A Proof Calculus Which Reduces Syntactic Bureaucracy , 2010, RTA.

[29]  Alwen Tiu,et al.  A Local System for Classical Logic , 2001, LPAR.

[30]  R. Statman.  Bounds for proof-search and speed-up in the predicate calculus , 1978 .

[31]  Alessandra Carbone.  Review: Samuel R. Buss, The Undecidability of $k$-Provability , 1997 .

[32]  Tom Gundersen.  A General View of Normalisation through Atomic Flows , 2009 .

[33]  Ozan Kahramanogullari.  Reducing Nondeterminism in the Calculus of Structures , 2006, LPAR.

[34]  Lutz Straßburger,et al.  A Local System for Linear Logic , 2002, LPAR.

[35]  Kai Brünnler.  Locality for Classical Logic , 2006, Notre Dame J. Formal Log..

[36]  Emil Jerábek,et al.  Proofs with monotone cuts , 2012, Math. Log. Q..

[37]  Emil Jerábek,et al.  Proof Complexity of the Cut-free Calculus of Structures , 2009, J. Log. Comput..

[38]  Alwen Tiu,et al.  A System of Interaction and Structure II: The Need for Deep Inference , 2005, Log. Methods Comput. Sci..

[39]  Lutz Straßburger,et al.  A system of interaction and structure IV: The exponentials and decomposition , 2009, TOCL.

[40]  Alessio Guglielmi,et al.  On the proof complexity of deep inference , 2009, TOCL.

[41]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[42]  Kai Brünnler.  Deep Inference and Its Normal Form of Derivations , 2006, CiE.