Cut normal forms and proof complexity
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| Title: | Cut normal forms and proof complexity |
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| Authors: | Matthias Baaz, Alexander Leitsch |
| Source: | Annals of Pure and Applied Logic. 97:127-177 |
| Publisher Information: | Elsevier BV, 1999. |
| Publication Year: | 1999 |
| Subject Terms: | Complexity of proofs, cut normal forms, equational sequents, Logic, cut-elimination, non-monotone cuts, Cut-elimination and normal-form theorems, 0102 computer and information sciences, 0101 mathematics, non-elementary complexity, 01 natural sciences |
| Description: | Cut-elimination in first-order logic is known to have a non-elementary complexity. The authors investigate ``reduction classes'' \(K\) for cuts [cuts over formulas \(\notin K\) can be feasibly eliminated or replaced by cut-formulas in \(K]\) and for cut-elimination [if all cut formulas are in \(K\) then all cuts can be feasibly eliminated]. For \(K=\text{NNF}\) (negative normal form: no implication, and negations are applied only to atomic formulas) every cut is easily reduced to a formula in \(K\), and hence cut-elimination for \(K\) is not feasible. For \(K=\) Prenex Normal Forms there is a quadratic transformation of a proof with arbitrary cut-formulas into a proof with cut-formulas in \(K\) sending derivable formulas into \(K\). However, elimination of new cuts introduced in this way can be non-elementary. Monotone formulas are constructed by atoms including \(\perp\) by \(\forall,\exists,\&,\vee\). The authors prove that the elimination of non-monotone cuts has non-elementary complexity in the worst case. First, all monotone cuts are proved to be quickly (almost exponentially) eliminable from proofs of quasi-monotone formulas. Then it is noticed that Statman's sequence of equational sequents with no Kalmar-elementary bound for cut-elimination can be encoded by quasi-monotone formulas. The long proof of quick cut-elimination in the paper can be simplified using two observations. Statman's sequence is encoded by Horn formulas, and pruning transformations (underlying Harrop's theorem) allow drastically ``skolemize'' monotone formulas proved from universal Horn axioms. |
| Document Type: | Article |
| File Description: | application/xml |
| Language: | English |
| ISSN: | 0168-0072 |
| DOI: | 10.1016/s0168-0072(98)00026-8 |
| Access URL: | https://zbmath.org/1340820 https://doi.org/10.1016/s0168-0072(98)00026-8 https://www.sciencedirect.com/science/article/pii/S0168007298000268 https://dblp.uni-trier.de/db/journals/apal/apal97.html#BaazL99 https://philpapers.org/rec/BAACNF |
| Rights: | Elsevier Non-Commercial |
| Accession Number: | edsair.doi.dedup.....f61a4b72efcb0eb844cc67ba5eb1d81d |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | Cut-elimination in first-order logic is known to have a non-elementary complexity. The authors investigate ``reduction classes'' \(K\) for cuts [cuts over formulas \(\notin K\) can be feasibly eliminated or replaced by cut-formulas in \(K]\) and for cut-elimination [if all cut formulas are in \(K\) then all cuts can be feasibly eliminated]. For \(K=\text{NNF}\) (negative normal form: no implication, and negations are applied only to atomic formulas) every cut is easily reduced to a formula in \(K\), and hence cut-elimination for \(K\) is not feasible. For \(K=\) Prenex Normal Forms there is a quadratic transformation of a proof with arbitrary cut-formulas into a proof with cut-formulas in \(K\) sending derivable formulas into \(K\). However, elimination of new cuts introduced in this way can be non-elementary. Monotone formulas are constructed by atoms including \(\perp\) by \(\forall,\exists,\&,\vee\). The authors prove that the elimination of non-monotone cuts has non-elementary complexity in the worst case. First, all monotone cuts are proved to be quickly (almost exponentially) eliminable from proofs of quasi-monotone formulas. Then it is noticed that Statman's sequence of equational sequents with no Kalmar-elementary bound for cut-elimination can be encoded by quasi-monotone formulas. The long proof of quick cut-elimination in the paper can be simplified using two observations. Statman's sequence is encoded by Horn formulas, and pruning transformations (underlying Harrop's theorem) allow drastically ``skolemize'' monotone formulas proved from universal Horn axioms. |
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| ISSN: | 01680072 |
| DOI: | 10.1016/s0168-0072(98)00026-8 |