## Quasipolynomial normalisation in deep inference via atomic flows and threshold formulae (2009)

Citations: | 8 - 4 self |

### BibTeX

@MISC{Bruscoli09quasipolynomialnormalisation,

author = {Paola Bruscoli and Alessio Guglielmi and Tom Gundersen and Michel Parigot},

title = {Quasipolynomial normalisation in deep inference via atomic flows and threshold formulae},

year = {2009}

}

### OpenURL

### Abstract

ABSTRACT. Jeˇrábek showed that analytic propositional-logic deep-inference proofs can be constructed in quasipolynomial time from nonanalytic proofs. In this work, we improve on that as follows: 1) we significantly simplify the technique; 2) our normalisation procedure is direct, i.e., it is internal to deep inference. The paper is self-contained, and provides a starting point and a good deal of information for tackling the problem of whether a polynomial-time normalisation procedure exists. 1.

### Citations

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Citation Context ...ch is the main result of this section. Threshold formulae realise boolean threshold functions, which are defined as boolean functions that are true if and only if at least k of n inputs are true (see =-=[Weg87]-=- for a thorough reference on threshold functions). In the following, ⌊x⌋ denotes the maximum integer n such that n � x. There are several ways of encoding threshold functions into formulae, and the pr... |

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Citation Context ...and a good deal of information for tackling the problem of whether a polynomial-time normalisation procedure exists. 1. INTRODUCTION Deep inference is a new methodology in proof theory, introduced in =-=[Gug07]-=- and subsequently developed to the point that all major logics can be expressed with analytic deep-inference proof systems (see [Gug] for a complete overview). Deep inference is more general than trad... |

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Citation Context ...analytic Gentzen proof systems, but the converse does not hold: in fact, Statman’s tautologies admit polynomial proofs in analytic SKS but, as is well known, only exponential ones in analytic Gentzen =-=[Sta78]-=-. In this paper, we work in CoS and SKS, but we introduce a new notation for CoS. We do so to conveniently describe certain derivations related to threshold formulae, which would seem very cumbersome ... |

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Citation Context ...t be exportable to other proof-theoretic realms. Two recent research threads involving SKS are particularly relevant to this work: • The basic proof-complexity properties of SKS have been studied, in =-=[BG09]-=-. • System SKS enjoys normal forms, which include analytic proofs, that can be defined and obtained in a largely syntax-independent way, by resorting to special graphs associated with SKS derivations,... |

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Citation Context ...tems of deep inference are strictly more powerful than Gentzen analytic proof systems because they exhibit exponential speed-ups. Atomic flows, which can be considered as specialised Buss flow graphs =-=[Bus91]-=-, play a major role in designing and controlling the normalisation procedure presented in this paper. They contribute to the overall clarification of this highly technical matter, by reducing our depe... |

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Citation Context ...ormal forms, which include analytic proofs, that can be defined and obtained in a largely syntax-independent way, by resorting to special graphs associated with SKS derivations, called ‘atomic flows’ =-=[GG08]-=-. Contrary to Frege proof systems, deep-inference proof systems have a proof theory, in the sense of a normalisation theory with a notion of analyticity, like the Gentzen formalisms have. The work [BG... |

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Citation Context ...which generalises CoS and formally allows for the new notation. In this section, we quickly introduce all the necessary notions. The standard reference for SKS in CoS and its typical constructions is =-=[Brü04]-=-; an introduction to SKS with an emphasis on proof complexity is in [BG09]. Formulae, denoted by A, B, C and D are freely built from: units, f (false), t (true); atoms, denoted by a, b, c, d and e; di... |

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Citation Context ...alisation, as is the case in Gentzen proof systems. Jeˇrábek obtained his result by relying on a construction over threshold functions by Atserias, Galesi and Pudlák, in the monotone sequent calculus =-=[AGP02]-=-. We note that the monotone sequent calculus specifies a weaker logic than propositional logic because negation is not freely applicable. The technique that Jeˇrábek adopts is indirect because normali... |

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Citation Context ...Jeˇrábek adopts is indirect because normalisation is performed over proofs in the sequent calculus, which are, in turn, related to deep-inference ones by polynomial simulations, originally studied in =-=[Brü06b]-=-. In this work, we demonstrate again Jeˇrábek’s result, still by adopting, essentially, the Atserias-Galesi-Pudlák technique, and we improve on that as follows: Date: January 25, 2010. Bruscoli and Gu... |

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Citation Context ..., formalism that has been defined in deep inference is called the calculus of structures, or CoS [Gug07]. CoS is now well developed for classical [Brü03, Brü06a, Brü06d, BT01, Brü06b], intuitionistic =-=[Tiu06a]-=-, linear [Str02, Str03b], modal [Brü06c, GT07, Sto07] and commutative/non-commutative logics [Gug07, Tiu06b, Str03a, Bru02, DG04, GS01, GS02, SG09, GS09, Kah06, Kah07]. The standard proof system of pr... |

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Citation Context ...ep inference is a new methodology in proof theory, introduced in [Gug07] and subsequently developed to the point that all major logics can be expressed with analytic deep-inference proof systems (see =-=[Gug]-=- for a complete overview). Deep inference is more general than traditional Gentzen proof theory because proofs can be freely composed by the logical operators, instead of having a rigid formula-direct... |

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Citation Context ...ep inference is a new methodology in proof theory, introduced in [Gug07] and subsequently developed to the point that all major logics can be expressed with analytic deep-inference proof systems (see =-=[Gug]-=- for a complete overview). Deep inference is more general than traditional Gentzen proof theory because proofs can be freely composed by the logical operators, instead of having a rigid formula-direct... |

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