InforLorV4, PascalFrancis, Corpus, bibRecord, 000232

Regaining cut admissibility in deduction modulo using abstract completion

Identifieur interne : 000232 ( PascalFrancis/Corpus ); précédent : 000231; suivant : 000233

Regaining cut admissibility in deduction modulo using abstract completion

Auteurs : Guillaume Burel ; Claude Kirchner

Source :

Information and computation : (Print) [ 0890-5401 ] ; 2010.

RBID : Pascal:10-0105560

Descripteurs français

Pascal (Inist)
- Admissibilité, Complétion, Preuve, Règle inférence, Calcul séquent, Congruence, Déploiement, Procédure, Puissance, Démonstration théorème, Réécriture, Relation ordre, Informatique théorique, Déduction naturelle, 68T15, 68Q42, Système deductif.

English descriptors

KwdEn :
- Admissibility, Completion, Computer theory, Congruence, Deductive system, Inference rule, Ordering, Power, Procedure, Proof, Rewriting, Sequent calculus, Theorem proving, Unfolding.

Abstract

Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore. We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w.r.t. cut-free proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system. Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the Knuth-Bendix completion in a non-trivial way, using the framework of abstract canonical systems. These results enlighten the entanglement between computation and deduction, and the power of abstract completion procedures. They also provide an effective way to obtain systems admitting cuts, therefore extending the applicability of deduction modulo in automated theorem proving.

Notice en format standard (ISO 2709)

Pour connaître la documentation sur le format Inist Standard.

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Format Inist (serveur)

NO :	PASCAL 10-0105560 INIST
ET :	Regaining cut admissibility in deduction modulo using abstract completion
AU :	BUREL (Guillaume); KIRCHNER (Claude)
AF :	Nancy-Université, Université Henri Poincaré, Faculté des sciences et techniques, Campus Victor Grignard, BP 70239/54506 Vandœuvre-lés-Nancy/France (1 aut.); INRIA, Centre de Recherche INRIA Bordeaux - Sud-Ouest, 351 cours de la Libération, Bâtiment A29/33405 Talence/France (2 aut.); LORIA, Équipe Pareo, Bâtiment B, Campus Scientifique, BP239/54506 Vandœuvre-lés-Nancy/France (1 aut., 2 aut.)
DT :	Publication en série; Niveau analytique
SO :	Information and computation : (Print); ISSN 0890-5401; Coden INFCEC; Etats-Unis; Da. 2010; Vol. 208; No. 2; Pp. 140-164; Bibl. 39 ref.
LA :	Anglais
EA :	Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore. We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w.r.t. cut-free proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system. Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the Knuth-Bendix completion in a non-trivial way, using the framework of abstract canonical systems. These results enlighten the entanglement between computation and deduction, and the power of abstract completion procedures. They also provide an effective way to obtain systems admitting cuts, therefore extending the applicability of deduction modulo in automated theorem proving.
CC :	001D02A08; 001D02C02; 001D02A05
FD :	Admissibilité; Complétion; Preuve; Règle inférence; Calcul séquent; Congruence; Déploiement; Procédure; Puissance; Démonstration théorème; Réécriture; Relation ordre; Informatique théorique; Déduction naturelle; 68T15; 68Q42; Système deductif
ED :	Admissibility; Completion; Proof; Inference rule; Sequent calculus; Congruence; Unfolding; Procedure; Power; Theorem proving; Rewriting; Ordering; Computer theory; Deductive system
SD :	Admisibilidad; Compleción; Prueba; Regla inferencia; Càlculo sequente; Congruencia; Despliegue; Procedimiento; Potencia; Demostración teorema; Reescritura; Relación orden; Informática teórica
LO :	INIST-8341.354000180885010030
ID :	10-0105560

Links to Exploration step

Pascal:10-0105560

Le document en format XML

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<front><div type="abstract" xml:lang="en">Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore. We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w.r.t. cut-free proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system. Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the Knuth-Bendix completion in a non-trivial way, using the framework of abstract canonical systems. These results enlighten the entanglement between computation and deduction, and the power of abstract completion procedures. They also provide an effective way to obtain systems admitting cuts, therefore extending the applicability of deduction modulo in automated theorem proving.</div>
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<s5>23</s5>
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<fC03 i1="07" i2="X" l="SPA"><s0>Despliegue</s0>
<s5>23</s5>
</fC03>
<fC03 i1="08" i2="X" l="FRE"><s0>Procédure</s0>
<s5>24</s5>
</fC03>
<fC03 i1="08" i2="X" l="ENG"><s0>Procedure</s0>
<s5>24</s5>
</fC03>
<fC03 i1="08" i2="X" l="SPA"><s0>Procedimiento</s0>
<s5>24</s5>
</fC03>
<fC03 i1="09" i2="X" l="FRE"><s0>Puissance</s0>
<s5>25</s5>
</fC03>
<fC03 i1="09" i2="X" l="ENG"><s0>Power</s0>
<s5>25</s5>
</fC03>
<fC03 i1="09" i2="X" l="SPA"><s0>Potencia</s0>
<s5>25</s5>
</fC03>
<fC03 i1="10" i2="X" l="FRE"><s0>Démonstration théorème</s0>
<s5>26</s5>
</fC03>
<fC03 i1="10" i2="X" l="ENG"><s0>Theorem proving</s0>
<s5>26</s5>
</fC03>
<fC03 i1="10" i2="X" l="SPA"><s0>Demostración teorema</s0>
<s5>26</s5>
</fC03>
<fC03 i1="11" i2="X" l="FRE"><s0>Réécriture</s0>
<s5>27</s5>
</fC03>
<fC03 i1="11" i2="X" l="ENG"><s0>Rewriting</s0>
<s5>27</s5>
</fC03>
<fC03 i1="11" i2="X" l="SPA"><s0>Reescritura</s0>
<s5>27</s5>
</fC03>
<fC03 i1="12" i2="X" l="FRE"><s0>Relation ordre</s0>
<s5>28</s5>
</fC03>
<fC03 i1="12" i2="X" l="ENG"><s0>Ordering</s0>
<s5>28</s5>
</fC03>
<fC03 i1="12" i2="X" l="SPA"><s0>Relación orden</s0>
<s5>28</s5>
</fC03>
<fC03 i1="13" i2="X" l="FRE"><s0>Informatique théorique</s0>
<s5>29</s5>
</fC03>
<fC03 i1="13" i2="X" l="ENG"><s0>Computer theory</s0>
<s5>29</s5>
</fC03>
<fC03 i1="13" i2="X" l="SPA"><s0>Informática teórica</s0>
<s5>29</s5>
</fC03>
<fC03 i1="14" i2="X" l="FRE"><s0>Déduction naturelle</s0>
<s4>INC</s4>
<s5>70</s5>
</fC03>
<fC03 i1="15" i2="X" l="FRE"><s0>68T15</s0>
<s4>INC</s4>
<s5>71</s5>
</fC03>
<fC03 i1="16" i2="X" l="FRE"><s0>68Q42</s0>
<s4>INC</s4>
<s5>72</s5>
</fC03>
<fC03 i1="17" i2="X" l="FRE"><s0>Système deductif</s0>
<s4>CD</s4>
<s5>96</s5>
</fC03>
<fC03 i1="17" i2="X" l="ENG"><s0>Deductive system</s0>
<s4>CD</s4>
<s5>96</s5>
</fC03>
<fN21><s1>067</s1>
</fN21>
<fN44 i1="01"><s1>OTO</s1>
</fN44>
<fN82><s1>OTO</s1>
</fN82>
</pA>
</standard>
<server><NO>PASCAL 10-0105560 INIST</NO>
<ET>Regaining cut admissibility in deduction modulo using abstract completion</ET>
<AU>BUREL (Guillaume); KIRCHNER (Claude)</AU>
<AF>Nancy-Université, Université Henri Poincaré, Faculté des sciences et techniques, Campus Victor Grignard, BP 70239/54506 Vandœuvre-lés-Nancy/France (1 aut.); INRIA, Centre de Recherche INRIA Bordeaux - Sud-Ouest, 351 cours de la Libération, Bâtiment A29/33405 Talence/France (2 aut.); LORIA, Équipe Pareo, Bâtiment B, Campus Scientifique, BP239/54506 Vandœuvre-lés-Nancy/France (1 aut., 2 aut.)</AF>
<DT>Publication en série; Niveau analytique</DT>
<SO>Information and computation : (Print); ISSN 0890-5401; Coden INFCEC; Etats-Unis; Da. 2010; Vol. 208; No. 2; Pp. 140-164; Bibl. 39 ref.</SO>
<LA>Anglais</LA>
<EA>Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e.g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore. We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w.r.t. cut-free proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system. Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the Knuth-Bendix completion in a non-trivial way, using the framework of abstract canonical systems. These results enlighten the entanglement between computation and deduction, and the power of abstract completion procedures. They also provide an effective way to obtain systems admitting cuts, therefore extending the applicability of deduction modulo in automated theorem proving.</EA>
<CC>001D02A08; 001D02C02; 001D02A05</CC>
<FD>Admissibilité; Complétion; Preuve; Règle inférence; Calcul séquent; Congruence; Déploiement; Procédure; Puissance; Démonstration théorème; Réécriture; Relation ordre; Informatique théorique; Déduction naturelle; 68T15; 68Q42; Système deductif</FD>
<ED>Admissibility; Completion; Proof; Inference rule; Sequent calculus; Congruence; Unfolding; Procedure; Power; Theorem proving; Rewriting; Ordering; Computer theory; Deductive system</ED>
<SD>Admisibilidad; Compleción; Prueba; Regla inferencia; Càlculo sequente; Congruencia; Despliegue; Procedimiento; Potencia; Demostración teorema; Reescritura; Relación orden; Informática teórica</SD>
<LO>INIST-8341.354000180885010030</LO>
<ID>10-0105560</ID>
</server>
</inist>
</record>

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Regaining cut admissibility in deduction modulo using abstract completion

Regaining cut admissibility in deduction modulo using abstract completion

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