## Cut-elimination and redundancy-elimination by resolution (2000)

Venue: | Journal of Symbolic Computation |

Citations: | 30 - 10 self |

### BibTeX

@ARTICLE{Baaz00cut-eliminationand,

author = {Matthias Baaz and Alexander Leitsch},

title = {Cut-elimination and redundancy-elimination by resolution},

journal = {Journal of Symbolic Computation},

year = {2000},

volume = {29},

pages = {149--176}

}

### OpenURL

### Abstract

A new cut-elimination method for Gentzen’s LK is defined. First cut-elimination is generalized to the problem of redundancy-elimination. Then the elimination of redundancy in LK-proofs is performed by a resolution method in the following way: A set of clauses C is assigned to an LK-proof ψ and it is shown that C is always unsatisfiable. A resolution refutation of C then serves as a skeleton of an LK-proof ψ ′ with atomic cuts; ψ ′ can be constructed from the resolution proof and ψ by a projection method. In the last step the atomic cuts are eliminated and a cut-free proof is obtained. The complexity of the method is analyzed and it is shown that a nonelementary speed-up over Gentzen’s method can be achieved. Finally an application to automated deduction is presented: it is demonstrated how informal proofs (containing pseudo-cuts) can be transformed into formal ones by the method of redundancy-elimination; moreover, the method can even be used to transform incorrect proofs into correct ones. 1.

### Citations

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Citation Context ...nsequent of sequents. The principal formula mostly is identifiable by the context. Thus the rule above will be written as Γ1 ⊢ ∆1,A Γ2 ⊢ ∆2,B Γ1, Γ2 ⊢ ∆1, ∆2,A∧ B Unlike Gentzen’s version of LK (see (=-=Gentzen, 1934-=-)) ours does not contain any “automatic” contractions (we are not interested in the intuitionistic calculus LJ in this paper). Instead we use the additive version of LK as in the book of Girard (Girar... |

171 |
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- 1965
(Show Context)
Citation Context ... A resolution derivation of ⊢ out of C is called a resolution refutation of C. It is easy to verify that resolution, as defined above, simulates resolution defined on clauses in the usual sense (see (=-=Robinson, 1965-=-) and (Leitsch, 1997)). Therefore resolution is complete, i.e. for every unsatisfiable set of clauses there exists a resolution refutation of C. Resolution proofs (in the form of trees) can be transfo... |

70 |
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Citation Context ...3.1 there exists a proof ˆγ[ψ] of∆⊢ Γ with only atomic cuts and l(ˆγ[ψ]) ≤ 2 ·l(ψ)l(γ)(2�γ� + 1). But the elimination of atomic cuts is at most exponential in the length of proofs (see (Tait, 1968), (=-=Schwichtenberg, 1977-=-)). ✷ The bound in Theorem 3.2 can be improved to 2 d·l(ψ)l(γ) . For this purpose we have to replace the LK-proofs in Definition 3.5 by LK-proofs with the mix rule (see (Takeuti, 1987)). In fact the m... |

56 |
The Resolution Calculus
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(Show Context)
Citation Context ...ntials, integrals etc. Therefore it is useful to concentrate on cut-elimination procedures which eliminate cuts by analyzing these explicit definitions and reducing cuts from inside out. In (Baaz and =-=Leitsch, 1997-=-) we defined a projection method for cuts, which rather than decomposing the cut formulas reduces them w.r.t. to arbitrary positions in the cut formula. On a class of proofs called QMON the projection... |

40 |
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Citation Context ...) = 1 and s(n +1)=2 s(n) for n ∈ IN. (b) CERES constructs a cut-free proof out of ψn in ≤ c2 dn steps, where c and d are constants independent of n. Proof. We choose Statman’s sequence (Sn)n∈IN (see (=-=Statman, 1979-=-) and (Baaz and Leitsch, 1994)) where all cut-free LK-proofs of Sn have length > s(n) , but there are 2 proofs πn with cuts of length linear in n. These proofs can be transformed into proofs ρn with c... |

24 | Herbrand-Analysen zweier Beweise des Satzes von Roth: polynomiale Anzahlschranken - Luckhardt - 1989 |

16 |
Cut normal forms and proof complexity
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Citation Context ...fs ρn with cuts and atomic initial sequents s.t. l(ρn) ≤ a2bn for constants a, b independent of n (the double exponential bound in (Baaz and Leitsch, 1994) has been improved to simply exponential in (=-=Baaz and Leitsch, 1999-=-)). ρn contains 2n + 1 cuts with closed cut formulas A1,...,A2n+1. As the end-sequents of ρn are those of πn, every cut-elimination method is nonelementary on ρn, i.e. the number of sequents in a cut-... |

9 |
Proof Theory, North-Holland
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Citation Context ...∆ ¬ : l A{x ← t}, Γ ⊢ ∆ (∀x)A, Γ ⊢ ∆ ∀ : r must fulfil the eigenvariable condition, i.e. the free variable u does not occur in Γ ⊢ ∆,A.In∀ : ltmay be an arbitrary term (w.r.t. the term definition in (=-=Takeuti, 1987-=-) admitting only free variables). ∀ : r is called a strong, ∀ : l aweak quantifier introduction. The conditions for ∃ : r are the same as for ∀ : l and similarly for ∃ : l versus ∀ : r: Γ ⊢ ∆,A{x ← t}... |

5 |
Complexity of resolution proofs and function introduction
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(Show Context)
Citation Context ...e produced proof is too long and unstructured, where cut-introduction should be applied as postprocessing, or it may speed up proof search itself if the minimal length of a cutfree proof is too high (=-=Baaz and Leitsch, 1992-=-). Thus, at the first glimpse, there seems to be no gain in using cut-elimination in automated deduction. But if we widen the scope and consider automated deduction also as a discipline of proof trans... |

5 |
Fast Cut-Elimination by Projection
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(Show Context)
Citation Context .... differentials, integrals etc. Therefore it is useful to concentrate on cut-elimination procedures which eliminate cuts by analyzing these explicit definitions and reducing cuts from inside out. In (=-=Baaz and Leitsch, 1997-=-) we defined a projection method for cuts, which rather than decomposing the cut formulas reduces them w.r.t. to arbitrary positions in the cut formula. On a class of proofs called QMON the projection... |

4 |
Normal derivability in classical logic, The Syntax and Semantics of Infinitary Languages
- Tait
- 1968
(Show Context)
Citation Context ...f. By Theorem 3.1 there exists a proof ˆγ[ψ] of∆⊢ Γ with only atomic cuts and l(ˆγ[ψ]) ≤ 2 ·l(ψ)l(γ)(2�γ� + 1). But the elimination of atomic cuts is at most exponential in the length of proofs (see (=-=Tait, 1968-=-), (Schwichtenberg, 1977)). ✷ The bound in Theorem 3.2 can be improved to 2 d·l(ψ)l(γ) . For this purpose we have to replace the LK-proofs in Definition 3.5 by LK-proofs with the mix rule (see (Takeut... |

3 | Partial Matching for Analogy Discovery in Proofs and Counter Examples
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- 1997
(Show Context)
Citation Context ...the scope and consider automated deduction also as a discipline of proof transformation, then cutelimination becomes more interesting. E.g. consider some newly developed methods of proofs by analogy (=-=Défourneaux and Peltier, 1997-=-). There the raw material does not only consist of the theorem to be proven, but also of several other mathematical proof (schemas) available in a data base; these proofs are generalized versions of r... |

1 | Cut-Elimination by Resolution 29 - Baaz, Leitsch - 1994 |