## Normalisation control in deep inference via atomic flows (2008)

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Citations: | 23 - 11 self |

### BibTeX

@MISC{Guglielmi08normalisationcontrol,

author = {Alessio Guglielmi and Tom Gundersen},

title = {Normalisation control in deep inference via atomic flows},

year = {2008}

}

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### Abstract

Abstract. We introduce ‘atomic flows’: they are graphs obtained from derivations by tracing atom occurrences and forgetting the logical structure. We study simple manipulations of atomic flows that correspond to complex reductions on derivations. This allows us to prove, for propositional logic, a new and very general normalisation theorem, which contains cut elimination as a special case. We operate in deep inference, which is more general than other syntactic paradigms, and where normalisation is more difficult to control. We argue that atomic flows are a significant technical advance for normalisation theory, because 1) the technique they support is largely independent of syntax; 2) indeed, it is largely independent of logical inference rules; 3) they constitute a powerful geometric formalism, which is more intuitive than syntax. 1.

### Citations

338 | The relative efficiency of propositional proof systems
- Cook, Reckhow
- 1979
(Show Context)
Citation Context ... above. In classical propositional logic, the situation is more complicated. There are proof nets for classical propositional logic, but they are not a proof system (as defined by Cook and Reckhow in =-=[CR79]-=-), because checking their correctness requires exponential time in their size. There is not much hope for improvement, because obtaining a proof system for classical propositional logic based on proof... |

265 |
Investigations into logical deductions
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- 1935
(Show Context)
Citation Context ...formalism, which is more intuitive than syntax. 1. Introduction We are interested in normalising derivations in proof systems of propositional logic. As for natural deduction and the sequent calculus =-=[Gen69]-=-, we intend normalisation as eliminating cuts, or, more in general, ‘detours’, from derivations. Normalisation is performed by algorithms that, given a non-normal derivation, produce a normal derivati... |

117 | Theorem Proving via General Matings
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- 1981
(Show Context)
Citation Context ...d similar constructions in his paper [Gui06], where he analyses this way the structural bureaucracy of SKS derivations. Finally, there might be connections with the theory of matings, as in the works =-=[And81]-=- by Peter Andrews and [Bib81] by Wolfgang Bibel; that theory also can be considered an abstract characterisation of proofs. Acknowledgements We thank Michel Parigot for feedback during the early devel... |

98 |
On Matrices with Connections
- Bibel
- 1981
(Show Context)
Citation Context ...s paper [Gui06], where he analyses this way the structural bureaucracy of SKS derivations. Finally, there might be connections with the theory of matings, as in the works [And81] by Peter Andrews and =-=[Bib81]-=- by Wolfgang Bibel; that theory also can be considered an abstract characterisation of proofs. Acknowledgements We thank Michel Parigot for feedback during the early development of atomic flows, and f... |

85 | A system of interaction and structure
- Guglielmi
(Show Context)
Citation Context ...without atomic flows is, in our opinion, inconceivable. We consider derivations in the proof system SKS of the calculus of structures [BT01], which is a very general formalism based on deep inference =-=[Gug07]-=-. Because of their comparatively larger expressive power, normalisation in pure, unconstrained proof systems in the calculus of structures is much more challenging than in more disciplined formalisms,... |

55 | Non-commutativity and MELL in the calculus of structures - Guglielmi, Straßburger - 2001 |

47 |
Brünnler and Alwen Fernanto Tiu. A local system for classical logic
- Kai
- 2001
(Show Context)
Citation Context ...is conceivable, but perverse; finding the normalisation algorithm without atomic flows is, in our opinion, inconceivable. We consider derivations in the proof system SKS of the calculus of structures =-=[BT01]-=-, which is a very general formalism based on deep inference [Gug07]. Because of their comparatively larger expressive power, normalisation in pure, unconstrained proof systems in the calculus of struc... |

38 | Higher-dimensional word problems with applications to equational logic
- Burroni
- 1993
(Show Context)
Citation Context ...roofs [Hug06]; we are especially interested in generating combinatorial proofs by manipulating atomic flows. Atomic flows could be given an algebraic characterisation with Albert Burroni’s polygraphs =-=[Bur93]-=-; Yves Guiraud investigated similar constructions in his paper [Gui06], where he analyses this way the structural bureaucracy of SKS derivations. Finally, there might be connections with the theory of... |

31 | On the proof complexity of deep inference
- Bruscoli, Guglielmi
(Show Context)
Citation Context ...T07, Sto07] and commutative/non-commutative logics [Gug07, Tiu06b, Str03a, Bru02, DG04, GS01, GS02, GS07, Kah06, Kah07b]. The basic proof complexity properties of the calculus of structures are known =-=[BG08]-=-. The calculus of structures promoted the discovery of a new class of proof nets for classical and linear logic [LS05a, LS05b, LS06, SL04] (see also [Gui06]). There exist implementations in Maude of d... |

29 | Atomic cut elimination for classical logic
- Brünnler
- 2003
(Show Context)
Citation Context ...and cut, and plugging one copy of the derivation into itself. In principle, this is very similar to normalisation in natural deduction. This idea is due to Alwen Tiu and then used by Kai Brünnler, in =-=[1]-=-, to simplify his previous proof of cut elimination in the calculus of structures. In our case, the idea allows us to dispense entirely with the usual case analysis and permutation of rules, which is ... |

27 | Deep sequent systems for modal logic - BRÜNNLER |

27 | A non-commutative extension of MELL - Guglielmi, Straßburger - 2002 |

26 | The undecidability of k-provability
- Buss
- 1991
(Show Context)
Citation Context ...c flow is a directed graph obtained from a derivation by only retaining information about the creation and destruction of atom occurrences. Atomic flows are, essentially, specialised Buss flow graphs =-=[Bus91]-=-. Notably, the atomic flow of a derivation completely disregards all the logical relations and associated inference steps; so, an atomic flow is not a derivation, but only a very abstract representati... |

24 | A purely logical account of sequentiality in proof search - Bruscoli - 2002 |

20 | Deep Inference and Symmetry - Brünnler - 2004 |

16 | Reducing nondeterminism in the calculus of structures - Kahramano˘gulları - 2006 |

14 | Deep inference and its normal form of derivations
- Brünnler
- 2006
(Show Context)
Citation Context ...plicable to other normalisation procedures and categorical axiomatisations of classical logic. Our methods solve the problem of finding an internal normalisation procedure for system SKS mentioned in =-=[3]-=- (Problem 8; our solution actually provides a much finer normal form than required by the problem). We have exploited the possibility, peculiar to deep inference, of designing proof systems whose stru... |

12 | Structures for multiplicative cyclic linear logic: Deepness vs cyclicity - Gianantonio - 2004 |

11 | Locality for classical logic’, Notre Dame - Brünnler |

11 | A system of interaction and structure IV: The exponentials and decomposition
- Guglielmi, Straßburger
- 2010
(Show Context)
Citation Context ...eamlined forms of derivations, of course, take care of this. For an instructive example of this phenomenon, in a more complicated logic to which we hope to extend atomic flows, we refer the reader to =-=[GS07]-=-. The complexity of Str and HStr is exponential on the size of the input derivation, mainly due to the recursive duplication of derivations (see Remark 5.15). This is in line with the expectations for... |

9 | Cut elimination inside a deep inference system for classical predicate logic - Brünnler |

7 | The three dimensions of proofs
- Guiraud
- 2006
(Show Context)
Citation Context ...ties of the calculus of structures are known [BG08]. The calculus of structures promoted the discovery of a new class of proof nets for classical and linear logic [LS05a, LS05b, LS06, SL04] (see also =-=[Gui06]-=-). There exist implementations in Maude of deep-inference proof systems [Kah07a]. For a better introduction than this, we refer the reader to [Brü03]. Definition 2.1. Formulae, α, β, γ, δ are freely b... |

7 | Maude as a platform for designing and implementing deep inference systems
- Kahramano˘gulları
- 2008
(Show Context)
Citation Context ... promoted the discovery of a new class of proof nets for classical and linear logic [LS05a, LS05b, LS06, SL04] (see also [Gui06]). There exist implementations in Maude of deep-inference proof systems =-=[Kah07a]-=-. For a better introduction than this, we refer the reader to [Brü03]. Definition 2.1. Formulae, α, β, γ, δ are freely built from: units, f (false), t (true); atoms, a, b, c, d, e; disjunction and con... |

6 | Classical modal display logic in the calculus of structures and minimal cut-free deep inference calculi for S5 - Goré, Tiu |

6 | Proofs without syntax
- Hughes
- 2006
(Show Context)
Citation Context ... the size of interpolants (see Alessandra Carbone’s work [Car97] for a possibly related approach to ours). There appear to be connections between atomic flows and Dominic Hughes’ combinatorial proofs =-=[Hug06]-=-; we are especially interested in generating combinatorial proofs by manipulating atomic flows. Atomic flows could be given an algebraic characterisation with Albert Burroni’s polygraphs [Bur93]; Yves... |

4 |
cut elimination and flow graphs for the propositional calculus
- Interpolants
- 1997
(Show Context)
Citation Context ...n is known. We are also interested in exploring the connections between our work and Craig’s interpolation theorem, especially in relation with the size of interpolants (see Alessandra Carbone’s work =-=[Car97]-=- for a possibly related approach to ours). There appear to be connections between atomic flows and Dominic Hughes’ combinatorial proofs [Hug06]; we are especially interested in generating combinatoria... |

1 |
Annals of Pure and Applied Logic, 83:249–299
- Cook, Reckhow
- 1997
(Show Context)
Citation Context ... above. In classical propositional logic, the situation is more complicated. There are proof nets for classical propositional logic, but they are not a proof system (as defined by Cook and Reckhow in =-=[CR79]-=-), because checking their correctness requires exponential time in their size. There is not much hope for improvement, because obtaining a proof system for classical propositional logic based on proof... |