#### DMCA

## Canonical sequent proofs via multifocusing (2008)

### Cached

### Download Links

Venue: | Fifth IFIP International Conference on Theoretical Computer Science, volume 273 of IFIP International Federation for Information Processing |

Citations: | 27 - 11 self |

### Citations

414 | Logic Programming with Focusing Proofs in Linear Logic
- Andreoli
- 1992
(Show Context)
Citation Context ...s for multiplicativeadditive linear logic (MALL), including units and literals. Motivating the construction of canonical representatives is as follows. A first step is to consider only focused proofs =-=[2]-=-, with a strict alternation of negative (invertible) and positive (focused) phases. Focused proofs systems can be used to distinguish between micro rules, i.e., introduction rules in the ordinary sequ... |

125 | Proof-nets: the parallel syntax for proof-theory
- Girard
- 1996
(Show Context)
Citation Context ...ence) are in one-to-one correspondence with MLL proof nets [9]: we show how to uniquely associate a maximally multi-focused proof to an MLL proof net. We also discuss proof nets in MALL without units =-=[10, 12]-=- and for other fragments of linear logic: maximal multi-focusing proofs should also be applicable in various other richer logics where the nature of proof nets is less well developed or satisfying, su... |

53 | Proof nets for unit-free multiplicative-additive linear logic
- Hughes, Glabbeek
- 2003
(Show Context)
Citation Context ...ence) are in one-to-one correspondence with MLL proof nets [9]: we show how to uniquely associate a maximally multi-focused proof to an MLL proof net. We also discuss proof nets in MALL without units =-=[10, 12]-=- and for other fragments of linear logic: maximal multi-focusing proofs should also be applicable in various other richer logics where the nature of proof nets is less well developed or satisfying, su... |

48 | A logical characterization of forward and backward chaining in the inverse method
- Chaudhuri, Pfenning, et al.
(Show Context)
Citation Context ...veral other open questions remain about multi-focused proofs. Firstly, we lack a cut-elimination theorem for multi-focused proofs that generalizes similar theorems for singly focused proofs (see, eg. =-=[4, 15]-=-). Moreover, it is considerably unclear how maximality interacts with cut-elimination, for the standard procedure would not preserve maximality. In terms of larger fragments of linear logic, multi-foc... |

34 |
Permutation of inferences in Gentzen’s calculi LK and LJ
- Kleene
- 1951
(Show Context)
Citation Context ...l-known that the following cut and (non-atomic) initial rules are admissible. ⊢Γ, A ⊢∆, A⊥ ⊢Γ,∆ C and ⊥ I* ⊢ A, A Local permutations of inference rules form a natural relation between cut-free proofs =-=[13]-=-. For example, in a proof of the form E⊢∆, B, C F⊢∆, B, D & D⊢Γ, A ⊢∆, B, C & D ⊗, ⊢Γ,∆, A⊗ B, C & D the order of the⊗and & rules may be locally switched to yield the proof (1) D⊢Γ, A E⊢∆, B, C D⊢Γ, A... |

27 | From proof nets to the free *-autonomous category
- Lamarche, Straßburger
(Show Context)
Citation Context ...ally multi-focused proofs. MLL − proof nets are certainly the most concise canonical structures for this fragment. There are candidates to extend MLL − proof nets to broader fragments (MLL with units =-=[14]-=-, MALL [12] or MELL) but they are not as satisfactory as for MLL − . The analysis we just made could be carried to MALL proof nets as introduced by Hughes and van Glabbeek [12] for the appropriate ext... |

24 | From proofs to focused proofs: a modular proof of focalization in linear logic
- Miller, Saurin
- 2007
(Show Context)
Citation Context ...e the negative connectives are decomposed; this phase is indicated in sequents of the form⊢Γ⇑∆. This phase separation is complete for cut-free proofs, i.e., every provable sequent has a focused proof =-=[2, 16]-=-. In this paper, we generalize this usual focusing strategy further in the following way: when deciding to focus, we may focus on more than one positive formula at a time, i.e., our positive sequents ... |

17 |
A neutral approach to proof and refutation
- Delande, Miller
- 2008
(Show Context)
Citation Context ...st considered by Saurin and Miller [16] as naturally arising in the structure of focalization graphs to prove the focalization theorem. Multi-focusing was subsequently also used by Delande and Miller =-=[7]-=- as a necessary generalization of Andreoli’s asymmetric treatment of the positive formulas. Andreoli studied focusing in proof nets [1, 3] and defined a notion of “multi-focus” [3] with a different me... |

16 | Focusing and proof-nets in linear and noncommutative logic
- Andreoli, Maieli
- 1999
(Show Context)
Citation Context ..., neg/pos permutations are also valid since the negative rules are invertible and, hence, may be applied arbitrarily early (reading bottom-up). From a 4⊢Γ1⇓ A,∆1 ⊢Γ2⇓B,∆2 [⊗] ⊢Γ1,Γ2⇓ A⊗ B,∆1,∆2 ⊢·⇓1 =-=[1]-=- ⊢Γ⇑A,∆ ⊢Γ⇑B,∆ [&] ⊢Γ⇑A & B,∆ ⊢Γ⇑⊤,∆ [⊤] ⊢ a ⊥ ⇓ a [I] ⊢Γ⇓∆ ⊢Γ,∆⇑· [MF] ⊢Γ⇓Ai,∆ ⊢Γ⇓A1⊕ A2,∆ [⊕i] ⊢Γ⇑A, B,∆ ⊢Γ⇑A� B,∆ [�] ⊢Γ, A⇑∆ ⊢Γ⇑A,∆ [R⇑] ⊢Γ⇑∆ ⊢Γ⇑⊥,∆ [⊥] ⊢Γ⇑∆ ⊢Γ⇓∆ [R⇓] Fig. 2 Multi-focusing sequen... |

14 |
Focussing proof-net construction as a middleware paradigm
- Andreoli
- 2002
(Show Context)
Citation Context ... Multi-focusing was subsequently also used by Delande and Miller [7] as a necessary generalization of Andreoli’s asymmetric treatment of the positive formulas. Andreoli studied focusing in proof nets =-=[1, 3]-=- and defined a notion of “multi-focus” [3] with a different meaning: there, it refers to a part of the context which is needed in order to apply the decide rule. He also investigates the use of focusi... |

12 |
Introduction to linear logic and ludics
- Curien
- 2005
(Show Context)
Citation Context ...between micro rules, i.e., introduction rules in the ordinary sequent calculus, and the macro rules that comprise an entire focusing phases and correspond to the introduction of synthetic connectives =-=[5]-=-. A first abstraction is then to consider proofs as built up from macro rules introducing synthetic connectives. Unfortunately, this layer of abstraction does not yield canonical representatives of eq... |

12 |
Le point aveugle: Cours de logique: Tome 1, vers la perfection', Vision des Sciences
- Girard
- 2006
(Show Context)
Citation Context ... deal with a restricted fragment of MALL proofs, the unit-free multiplicative fragment, MLL−, for which proof nets have been proposed as a natural candidate for the canonical representation of proofs =-=[8]-=-. We shall show that the maximally multi-focused proofs that we established as canonical in the previous section are indeed in bijection with proof nets. We recall in Figure 4 the syntax for proof str... |

2 |
Introduction to linear logic and ludics, part I
- Curien
- 2005
(Show Context)
Citation Context ...sing proofs systems can be used to distinguish between micro rules, i.e., introduction rules in the sequent calculus, from the macro rules that correspond to the introduction of synthetic connectives =-=[4]-=- that correspond to an entire focusing phase. A first abstraction is then to consider proofs as built up from macro rules introducing synthetic connectives. Unfortunately, this layer of abstraction do... |