## Strong normalisation for a gentzen-like cut-elimination procedure (2001)

Venue: | In TLCA |

Citations: | 9 - 0 self |

### BibTeX

@INPROCEEDINGS{Urban01strongnormalisation,

author = {C. Urban},

title = {Strong normalisation for a gentzen-like cut-elimination procedure},

booktitle = {In TLCA},

year = {2001},

pages = {415--429}

}

### OpenURL

### Abstract

Abstract. In this paper we introduce a cut-elimination procedure for classical logic, which is both strongly normalising and consisting of local proof transformations. Traditional cut-elimination procedures, including the one by Gentzen, are formulated so that they only rewrite neighbouring inference rules; that is they use local proof transformations. Unfortunately, such local proof transformation, if defined naïvely, break the strong normalisation property. Inspired by work of Bloo and Geuvers concerning the λx-calculus, we shall show that a simple trick allows us to preserve this property in our cut-elimination procedure. We shall establish this property using the recursive path ordering by Dershowitz.

### Citations

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Untersuchungen uber das Logische Schlieen I und II
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(Show Context)
Citation Context ...roperty using the recursive path ordering by Dershowitz. Keywords. Cut-Elimination, Classical Logic, Explicit Substitution, Recursive Path Ordering. 1 Introduction Gentzen showed in his seminal paper =-=[6]-=- that all cuts can be eliminated from sequent proofs in LK and LJ. He not only proved that cuts can be eliminated, but also gave a simple procedure for doing so. This procedure consists of proof trans... |

281 | Orderings for term-rewriting systems
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(Show Context)
Citation Context ...hall present next is rather involved. 5 Proof of Strong Normalisation In this section we shall give a proof for Theorem 1. In this proof we shall make use of the recursive path ordering by Dershowitz =-=[3]-=-. Definition 3 (Recursive Path Ordering). Let s ≡ f(s1, . . . , sm) and t ≡ g(t1, . . . , tn) be terms, then s> rpo t iff (i) si ≥ rpo t for some i = 1, . . . , m (subterm) or (ii) f ≫ g and s> rpo tj... |

112 | A new deconstructive logic: Linear logic
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- 1997
(Show Context)
Citation Context ...edure fulfilling both demands. The problem with Gentzen-like cut-reductions is that they, if defined naïvely, break the strong normalisation property, as illustrated in the following example given in =-=[2, 5]-=-. Consider the following LK-proof. A A A A A∨A A, A ∨L A∨A A ContrR A A A A A, A A∧A ∧R A A∧A ContrL Cut A∨A A∧A Using Gentzen-like proof transformations there are two possibilities for permuting the ... |

69 | Classical Logic and Computation
- Urban
- 2000
(Show Context)
Citation Context ...g inference rules, possibly by duplicating a subderivation. Most of the traditional cut-elimination procedures, including Gentzen’s original procedure, consist of such local proof transformations. In =-=[11]-=- and [12] three criteria for a cut-elimination procedure were introduced: 1. the cut-elimination procedure should not restrict the collection of normal forms reachable from a given proof in such a way... |

61 | Constructive logics part I: A tutorial on proof systems and typed λ-calculi
- Gallier
- 1993
(Show Context)
Citation Context ...edure fulfilling both demands. The problem with Gentzen-like cut-reductions is that they, if defined naïvely, break the strong normalisation property, as illustrated in the following example given in =-=[2, 5]-=-. Consider the following LK-proof. A A A A A∨A A, A ∨L A∨A A ContrR A A A A A, A A∧A ∧R A A∧A ContrL Cut A∨A A∧A Using Gentzen-like proof transformations there are two possibilities for permuting the ... |

44 | Strong normalisation of cut-elimination in classical logic
- Urban, Bierman
- 2001
(Show Context)
Citation Context ...o be applied until no further −−→-reduction is applicable (later we shall refer to such a term as x-normal form). Here we omit an inductive definition of the proof substitution, which can be found in =-=[11, 12]-=-. Using this proof substitution we can reformulate the reduction for commuting cuts as follows. c ′ 5’. Cut(〈a〉M, (x)N) −−→ M{a := (x)N} if M does not freshly introduce a, or 6’. Cut(〈a〉M, (x)N) c ′ −... |

42 | Cut-Elimination and a Permutation-free Sequent Calculus for Intuitionistic Logic. Studia Logic
- Dyckho, Pinto
- 1998
(Show Context)
Citation Context ...op by constantly applying this reduction. Thus a common restriction is to not allow a cut to pass over another cut in any circumstances. Unfortunately, this has several serious drawbacks, as noted in =-=[4, 7]-=-; it limits, for example, in the intuitionistic case the correspondence between cut-elimination and beta-reduction. In particular, strong normalisation of beta-reduction cannot be inferred from strong... |

34 |
A lambda-calculus structure isomorphic to sequent calculus structure
- Herbelin
- 1994
(Show Context)
Citation Context ...op by constantly applying this reduction. Thus a common restriction is to not allow a cut to pass over another cut in any circumstances. Unfortunately, this has several serious drawbacks, as noted in =-=[4, 7]-=-; it limits, for example, in the intuitionistic case the correspondence between cut-elimination and beta-reduction. In particular, strong normalisation of beta-reduction cannot be inferred from strong... |

33 | Explicit substitution: on the edge of strong normalization
- Bloo, Geuvers
- 1999
(Show Context)
Citation Context ...ivial, the corresponding strong normalisation proof is rather non-trivial (mainly because we allow cuts to pass over other cuts). To prove this property, we shall make use of a technique developed in =-=[1]-=-. This technique appeals to the recursive path ordering theorem by Dershowitz. Our proof is more difficult than the one given in [4], which also appeals to the recursive path ordering theorem, because... |

26 |
Explicit substitution – tutorial & survey
- Rose
- 1996
(Show Context)
Citation Context ...rate in this section. Explicit substitution calculi have been developed to internalise the substitution operation—a meta-level operation on lambda-terms—arising from betareductions. For example in λx =-=[10]-=-, the beta-reduction (λx.M)N β −−→ M[x := N] is replaced by the reduction (λx.M)N b −−→ M〈x := N〉 where the reduct contains a new syntactic constructor. The following reduction rules apply to this con... |

20 |
Typed lambda-calculi with explicit substitutions may not terminate
- Melliès
- 1995
(Show Context)
Citation Context ...he situation with the lambda-calculus and λx: strong normalisation for the explicit substitution calculus does not follow directly from strong normalisation of the lambda-calculus. Indeed as shown in =-=[9]-=- explicit substitution calculi, if defined naïvely, may break the strong normalisation property. So the proof we shall present next is rather involved. 5 Proof of Strong Normalisation In this section ... |