## Theorem Proving Modulo

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Venue: | Journal of Automated Reasoning |

Citations: | 76 - 14 self |

### BibTeX

@ARTICLE{Dowek_theoremproving,

author = {Gilles Dowek and Thérèse Hardin and Claude Kirchner},

title = {Theorem Proving Modulo},

journal = {Journal of Automated Reasoning},

year = {},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higher-order logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higher-order logic subsumes full higher-order resolution.

### Citations

785 | Constraint Logic Programming
- Jaffar, Lassez
- 1987
(Show Context)
Citation Context ... use of constraints. Starting from the seminal work of G. Huet on higher-order resolution (Huet, 1972; Huet, 1973), the notion of deduction with constraints spread, with constraint logic programming (=-=Jaffar and Lassez, 1987-=-), and then constraint programming. The counterpart in theorem proving is deduction with constraints (Kirchner et al., 1990) and complete constraint saturation processes (Nieuwenhuis and Rubio, 1994; ... |

751 | Rewrite systems
- Dershowitz, Jouannaud
- 1992
(Show Context)
Citation Context ...ut extension of the methods proposed here to many-sorted first-order logic can be done as usual. We assume that the reader is familiar with the basic notions of rewriting as defined, for instance in (=-=Dershowitz and Jouannaud, 1990-=-). Since we consider rules rewriting propositions that contain binders (quantifiers), we also use some notions of combinatory reduction systems (Klop et al., 1993). The usual notions such as that of o... |

489 |
Simple word problems in universal algebras
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ... (see also (Colata, 1996)). Besides this idea of building-in part of the equality in general refutation processes, the same concern of building-in part of the equality in equational reasoning itself (=-=Knuth and Bendix, 1970-=-) has led to the study of equational reasoning modulo, whose main landmarks are the study of associative-commutative completion (Peterson and Stickel, 1981), the general study of coherence of an equat... |

166 |
Canonical forms and unification
- Hullot
- 1980
(Show Context)
Citation Context ... . . = ? E Rp} is an abuse of notation for {P1 = ? E P2, . . . , P1 = ? E Pn, P1 = ? E R1, . . . , P1 = ? E Rp}. The Extended Narrowing rule is close to the one used in basic or constraint narrowing (=-=Hullot, 1980-=-; Nutt et al., 1989) but the difference here is that narrowing is applied to atomic propositions and not to terms. Because atomic propositions can be rewritten into non-atomic ones, the narrowed claus... |

161 |
Solving equations in abstract algebras: A rulebased survey of unification
- Jouannaud, Kirchner
- 1991
(Show Context)
Citation Context ...ists of equations of the form t1 = t2 and a solution of such an equation is a substitution σ such that σt1 and σt2 are equivalent modulo E. Equational unification may be undecidable or quite complex (=-=Jouannaud and Kirchner, 1991-=-). Fortunately deciding unifiability can be postponed and modularized by the use of constraints. Starting from the seminal work of G. Huet on higher-order resolution (Huet, 1972; Huet, 1973), the noti... |

151 |
Logic for Computer Science. Foundation of Automatic Theorem Proving
- Gallier
- 1986
(Show Context)
Citation Context ...alled constants. We assume that there is at least one constant. The set of terms, atomic propositions, propositions and sentences (i.e. closed propositions) are defined as usual in first-order logic (=-=Gallier, 1986-=-). In particular remember that a closed (also called ground) term is a term without variable. We consider single-sorted first-order logic, but extension of the methods proposed here to many-sorted fir... |

145 |
Completion of a set of rules modulo a set of equations
- Jouannaud, Kirchner
- 1986
(Show Context)
Citation Context ...modulo, whose main landmarks are the study of associative-commutative completion (Peterson and Stickel, 1981), the general study of coherence of an equational theory with respect to a rewrite system (=-=Jouannaud and Kirchner, 1986-=-), its unified presentation in (Bachmair, 1987) and its extension in (Marché, 1994; Viry, 1995). Extending narrowing and resolution Equational resolution permits to integrate smoothly, and in a comple... |

135 |
Complete sets of reductions for some equational theories
- Peterson, Stickel
- 1981
(Show Context)
Citation Context ... of the equality in equational reasoning itself (Knuth and Bendix, 1970) has led to the study of equational reasoning modulo, whose main landmarks are the study of associative-commutative completion (=-=Peterson and Stickel, 1981-=-), the general study of coherence of an equational theory with respect to a rewrite system (Jouannaud and Kirchner, 1986), its unified presentation in (Bachmair, 1987) and its extension in (Marché, 19... |

131 | ndergaard. Analysis of constraint logic programs - Marriott, S - 1990 |

129 | Solution of the robbins problem
- Mccune
- 1997
(Show Context)
Citation Context ...surveyed in (Kirchner, 1998) has led to very powerful results and systems. This allowed, in combination with the use of equational constraints, to solve problems considered by mathematicians as hard (=-=McCune, 1997-=-) (see also (Colata, 1996)). Besides this idea of building-in part of the equality in general refutation processes, the same concern of building-in part of the equality in equational reasoning itself ... |

121 | Automated Deduction by Theory Resolution
- Stickel
- 1985
(Show Context)
Citation Context ...s been solved by G. Plotkin (Plotkin, 1972) by the introduction of equational unification. This idea led a decade later to the so-called equational resolution and more generally to theory resolution (=-=Stickel, 1985-=-). This idea has also been exploited in tableaux methods (see, for4 Gilles Dowek, Thérèse Hardin, Claude Kirchner instance, (Gallier et al., 1989)). Rewriting techniques have also been implemented in... |

110 |
Building-in equational theories
- Plotkin
- 1972
(Show Context)
Citation Context ... (b + (y + (d + e)))) z = z since the standard unification algorithm does not “see” that instantiating x by c+d permits a new reduction of the proposition. This problem has been solved by G. Plotkin (=-=Plotkin, 1972-=-) by the introduction of equational unification. This idea led a decade later to the so-called equational resolution and more generally to theory resolution (Stickel, 1985). This idea has also been ex... |

106 | Resolution in Type Theory
- Andrews
- 1971
(Show Context)
Citation Context ..., because there are no bound variables in clauses. For higher-order logic, such an intermediate system with an instantiation rule and an identical resolution rule has been introduced by P.B. Andrews (=-=Andrews, 1971-=-). This system contains also additional rules, for instance a conversion rule. For deduction modulo, we consider a system containing the rules Identical Resolution and Instantiation together with two ... |

102 | Higher-order unification via explicit substitutions - Dowek, Hardin, et al. |

67 | Basic paramodulation
- Bachmair, Ganzinger, et al.
- 1995
(Show Context)
Citation Context ...traint programming. The counterpart in theorem proving is deduction with constraints (Kirchner et al., 1990) and complete constraint saturation processes (Nieuwenhuis and Rubio, 1994; Vigneron, 1995; =-=Bachmair et al., 1995-=-). The integration of rewrite based techniques and orderings in first-order theorem proving surveyed in (Kirchner, 1998) has led to very powerful results and systems. This allowed, in combination with... |

66 |
Deduction with symbolic constraints. Revue Francaise d'Intelligence Artificielle, 4(3):9--52
- Kirchner, Kirchner, et al.
- 1990
(Show Context)
Citation Context ...n of deduction with constraints spread, with constraint logic programming (Jaffar and Lassez, 1987), and then constraint programming. The counterpart in theorem proving is deduction with constraints (=-=Kirchner et al., 1990-=-) and complete constraint saturation processes (Nieuwenhuis and Rubio, 1994; Vigneron, 1995; Bachmair et al., 1995). The integration of rewrite based techniques and orderings in first-order theorem pr... |

56 |
Rksoluiion d'e'quations dans les langages d'ordre 1,2,. . . w
- Huet
- 1976
(Show Context)
Citation Context ...rder logic, P.B. Andrews (Andrews, 1971) proposes to build-in conversion axioms. Unification is then replaced by unification modulo βη-conversion, usually called higher-order unification (Huet, 1975; =-=Huet, 1976-=-), which is the kernel of higher-order resolution (Huet, 1972; Huet, 1973). The ENAR method, extended to many-sorted first-order logic, can be applied to a presentation of higher-order logic in many-s... |

45 | Proof Normalization Modulo
- Dowek, Werner
- 2003
(Show Context)
Citation Context ... that is proved once and for all, cut elimination for sequent calculus modulo depends on the considered congruence. Cut elimination for sequent calculus modulo several congruences has been proved in (=-=Dowek and Werner, 1999-=-), including all the congruences presented by a confluent and terminating quantifier-free rewrite system, those presented by a confluent and terminating positive rewrite system and that of a first-ord... |

40 |
A unification algorithm for typed lambda-calculus
- Huet
- 1975
(Show Context)
Citation Context ...for higher-order logic, P.B. Andrews (Andrews, 1971) proposes to build-in conversion axioms. Unification is then replaced by unification modulo βη-conversion, usually called higher-order unification (=-=Huet, 1975-=-; Huet, 1976), which is the kernel of higher-order resolution (Huet, 1972; Huet, 1973). The ENAR method, extended to many-sorted first-order logic, can be applied to a presentation of higher-order log... |

34 |
A Resolution Principle for a Logic with Restricted Quantifiers
- Burckert
- 1991
(Show Context)
Citation Context ... prop. and U ′ ∈ cℓ({U[r]ω}) Figure 2. Extended narrowing and resolution (ENAR) first in (Huet, 1972; Huet, 1973) (where unification is higher-order unification), then in (Kirchner et al., 1990) and (=-=Bürckert, 1991-=-). Notice that applying this rule can introduce new variables. The set of constraints {P1 = ? E . . . =? E Pn = ? E R1 . . . = ? E Rp} is an abuse of notation for {P1 = ? E P2, . . . , P1 = ? E Pn, P1... |

32 |
A technique for establishing completeness results in theorem proving with equality
- Peterson
- 1983
(Show Context)
Citation Context ... R is empty the rule Extended Narrowing never applies and we get back equational resolution. When R and E are both empty we get back resolution. Notice also that ENAR does not subsume paramodulation (=-=Peterson, 1983-=-) as Extended Narrowing is always performed with the same built-in congruence. REMARK 2.2. The rules of the ENAR method may be very prolific since all the unification problems are delayed as constrain... |

32 |
Basic Narrowing Revisited
- Nutt, R'ety, et al.
- 1989
(Show Context)
Citation Context ... is an abuse of notation for {P1 = ? E P2, . . . , P1 = ? E Pn, P1 = ? E R1, . . . , P1 = ? E Rp}. The Extended Narrowing rule is close to the one used in basic or constraint narrowing (Hullot, 1980; =-=Nutt et al., 1989-=-) but the difference here is that narrowing is applied to atomic propositions and not to terms. Because atomic propositions can be rewritten into non-atomic ones, the narrowed clause may be not in cla... |

30 | HOL-λσ an intentional first-order expression of higher-order logic
- Dowek, Hardin, et al.
(Show Context)
Citation Context ...d to many-sorted first-order logic, can be applied to a presentation of higher-order logic in many-sorted firstorder logic. This way, it subsumes higher-order resolution. This result is explained in (=-=Dowek et al., 1999-=-) where we develop a first-order presentation of higher-order logic based on the calculus of explicit substitutions that is intentionally equivalent to the usual presentation based on λ-calculus. In t... |

29 | AC-supperposition with constraints: No ACunifiers needed
- Nieuwenhuis, Rubio
- 1990
(Show Context)
Citation Context ...ming (Jaffar and Lassez, 1987), and then constraint programming. The counterpart in theorem proving is deduction with constraints (Kirchner et al., 1990) and complete constraint saturation processes (=-=Nieuwenhuis and Rubio, 1994-=-; Vigneron, 1995; Bachmair et al., 1995). The integration of rewrite based techniques and orderings in first-order theorem proving surveyed in (Kirchner, 1998) has led to very powerful results and sys... |

28 |
M'ethodes et outils de conception syst'ematique d'algorithmes d'unification dans les th'eories 'equationnelles. Th`ese de doctorat d"etat en informatique, Universit'e de Nancy 1
- Kirchner
- 1985
(Show Context)
Citation Context ...gly believe that the method above is complete. The assumptions above are52 Gilles Dowek, Thérèse Hardin, Claude Kirchner similar to those necessary to establish the completeness of narrowing modulo (=-=Kirchner, 1985-=-), which is indeed an instance of the method developed here. Conclusion We have presented a sequent calculus that operates in the quotient of the set of propositions modulo a congruence which can equa... |

27 |
Proof methods for equational theories
- Bachmair
- 1987
(Show Context)
Citation Context ...mmutative completion (Peterson and Stickel, 1981), the general study of coherence of an equational theory with respect to a rewrite system (Jouannaud and Kirchner, 1986), its unified presentation in (=-=Bachmair, 1987-=-) and its extension in (Marché, 1994; Viry, 1995). Extending narrowing and resolution Equational resolution permits to integrate smoothly, and in a complete way, term rewriting steps and deduction ste... |

22 | Higher-order Uni via Explicit Substitutions - Dowek, Hardin, et al. - 2000 |

19 | Normalised rewriting and normalised completion - Marché - 1994 |

8 | Positive Deduction modulo Regular Theories
- Vigneron
- 1995
(Show Context)
Citation Context ...), and then constraint programming. The counterpart in theorem proving is deduction with constraints (Kirchner et al., 1990) and complete constraint saturation processes (Nieuwenhuis and Rubio, 1994; =-=Vigneron, 1995-=-; Bachmair et al., 1995). The integration of rewrite based techniques and orderings in first-order theorem proving surveyed in (Kirchner, 1998) has led to very powerful results and systems. This allow... |

8 | Term rewriting: Some experimental results
- Plaisted, Potter
- 1991
(Show Context)
Citation Context ...es the job of all the mentioned refutation derivations at once. In the example above, computation rules apply to the terms of the language. As already remarked by S.J. Lee, D. Plaisted and R. Potter (=-=Plaisted and Potter, 1991-=-; Lee and Plaisted, 1994) such rules canTheorem Proving Modulo 3 also apply to the propositions of the language. For instance, if we want to refute the theory: P ⇔ (Q ∨ R), Q ⇔ S, R ⇔ S, P , ¬S, the ... |

6 |
Rigid E-unification and its application to equational matings
- Gallier, Snyder, et al.
- 1989
(Show Context)
Citation Context ...uational resolution and more generally to theory resolution (Stickel, 1985). This idea has also been exploited in tableaux methods (see, for4 Gilles Dowek, Thérèse Hardin, Claude Kirchner instance, (=-=Gallier et al., 1989-=-)). Rewriting techniques have also been implemented in many proof assistants. Although they are not unifiable in the ordinary sense, these propositions are equationally unifiable. Informally, for an e... |

5 | Higher-Order Equational Unification via Explicit Substitutions
- Kirchner, Ringeissen
- 1997
(Show Context)
Citation Context ...lution. A first step towards such a result, the expression of higher-order equational unification as first-order equational unification in the calculus of explicit substitutions, as been achieved in (=-=Kirchner and Ringeissen, 1997-=-). We also believe that theorem proving modulo is a general framework allowing to backup the integration of decision procedures (as computational part) and theorem provers (the reasoning part). For im... |

4 | Proof normalization modulo. Rapport de Recherche 3542, INRIA - Dowek, Werner - 1998 |

4 |
Constrained Resolution: A Complete Method for Type Theory
- Huet
- 1972
(Show Context)
Citation Context ...lex (Jouannaud and Kirchner, 1991). Fortunately deciding unifiability can be postponed and modularized by the use of constraints. Starting from the seminal work of G. Huet on higher-order resolution (=-=Huet, 1972-=-; Huet, 1973), the notion of deduction with constraints spread, with constraint logic programming (Jaffar and Lassez, 1987), and then constraint programming. The counterpart in theorem proving is dedu... |

4 | Proof normalization for a first-order formulation of higher-order logic
- Dowek
- 1997
(Show Context)
Citation Context ...x)) → ¬ε(x) R = ε(α(α( ˙∨, x), y)) → ε(x) ∨ ε(y) ⎪⎩ ε(α( ˙ ∀T , x)) → ∀y ε(α(x, y)) { α(α(α(S, x), y), z) → α(α(x, z), (y, z)) E = α(α(K, x), y) → x This system is confluent and strongly terminating (=-=Dowek, 1997-=-) and it enjoys cut elimination (Dowek and Werner, 1999).50 Gilles Dowek, Thérèse Hardin, Claude Kirchner Let us see, on an example, the correspondence between a proof in the sequent calculus modulo ... |

3 | HOL-lambda-sigma: an intentional expression of higher-order logic - Dowek, Hardin, et al. - 2001 |

3 |
Use of replace rules in theorem proving
- Lee, Plaisted
- 1994
(Show Context)
Citation Context ...oned refutation derivations at once. In the example above, computation rules apply to the terms of the language. As already remarked by S.J. Lee, D. Plaisted and R. Potter (Plaisted and Potter, 1991; =-=Lee and Plaisted, 1994-=-) such rules canTheorem Proving Modulo 3 also apply to the propositions of the language. For instance, if we want to refute the theory: P ⇔ (Q ∨ R), Q ⇔ S, R ⇔ S, P , ¬S, the resolution method starts... |

3 |
With Major Math Proof, Brute Computers Show Flash of Reasoning Power
- COLATA
- 1996
(Show Context)
Citation Context ...98) has led to very powerful results and systems. This allowed, in combination with the use of equational constraints, to solve problems considered by mathematicians as hard (McCune, 1997) (see also (=-=Colata, 1996-=-)). Besides this idea of building-in part of the equality in general refutation processes, the same concern of building-in part of the equality in equational reasoning itself (Knuth and Bendix, 1970) ... |

2 | 1973b, `The Undecidability of Uni in Third Order Logic - Huet - 1973 |

2 | combinators and the comprehension scheme
- Dowek, Lambda-calculus
- 1995
(Show Context)
Citation Context ...uivalent to higher-order logic, because combinators equivalence is weaker than βη-equivalence. However we retrieve equivalence if we add the extensionality axiom to both theories (see, for instance, (=-=Dowek, 1995-=-)). Expressing higher-order logic as a first-order theory and using a firstorder proof search method can be an efficient way to implement higherorder proof search, provided we use the right automated ... |

2 | Axioms vs. rewrite rules: from completeness to cut elimination - Dowek |

2 | Ringeissen (Eds.) Frontiers of Combining Systems - Kirchner, C |

1 | Tuesday December 10 - Colata - 1996 |

1 | Proof normalization for a formulation of higher-order logic - Dowek - 1997 |

1 |
La part du Calcul'. Memoire d'habilitation, Universite de Paris 7
- Dowek
- 1999
(Show Context)
Citation Context ...f l ⇔ r, for each proposition rewrite rule l → r. When we do not have an equality predicate in the theory, me may add this predicate and the corresponding axioms in a conservative way as detailed in (=-=Dowek, 1999-=-). LEMMA 1.4. Let RE be a class rewrite system and T be a theory such that T and RE are compatible. Then we have: T , Γ ⊢ ∆ if and only if T , Γ ⊢RE ∆. Proof: The “only if” part is obvious since a der... |

1 |
A Mechanization of Type Theory’. In: Proceeding of the third international joint conference on artificial intelligence
- Huet
- 1973
(Show Context)
Citation Context ...ay. When we apply this method to the first-order presentation of higherorder logic above, the rule Extended Narrowing specializes exactly to the rule Splitting of higher-order resolution (Huet, 1972; =-=Huet, 1973-=-). The only difference with higher-order resolution is that we are using the combinators S and K and not λ-calculus. Using combinators let unification be only equational unification modulo the axioms ... |

1 | Raamsdonk: 1993, `Combinatory reduction systems: introduction and survey - Klop, Oostrom, et al. |

1 | Theorem Proving Modulo 33 - Peterson, Stickel - 1981 |

1 |
Orderings in Automated Theorem Proving
- Kirchner
- 1998
(Show Context)
Citation Context ...traint saturation processes (Nieuwenhuis and Rubio, 1994; Vigneron, 1995; Bachmair et al., 1995). The integration of rewrite based techniques and orderings in first-order theorem proving surveyed in (=-=Kirchner, 1998-=-) has led to very powerful results and systems. This allowed, in combination with the use of equational constraints, to solve problems considered by mathematicians as hard (McCune, 1997) (see also (Co... |