## Equational Inference, Canonical Proofs, And Proof Orderings (1992)

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Venue: | Journal of the ACM |

Citations: | 30 - 11 self |

### BibTeX

@ARTICLE{Bachmair92equationalinference,,

author = {Leo Bachmair and Nachum Dershowitz},

title = {Equational Inference, Canonical Proofs, And Proof Orderings},

journal = {Journal of the ACM},

year = {1992},

volume = {41},

pages = {236--276}

}

### Years of Citing Articles

### OpenURL

### Abstract

We describe the application of proof orderings---a technique for reasoning about inference systems---to various rewrite-based theorem-proving methods, including re#nements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," co-authored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR-89-01322, CCR-90-07195, and CCR-90-24271. 1 ...

### Citations

510 |
Simple word problems in universal algebra
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ...des can be rewritten to the identical normal form. The rst rewriting-based decision procedure (for loops) was given by Evans [22], in 1951. See [19] for a survey of the eld. In 1970, Knuth and Bendix =-=[42] prop-=-osed a procedure that attempts to construct a convergent rewrite system from a nite axiomatization of an equational theory. This \completion" procedure must be supplied with an ordering that is u... |

466 | Termination of rewriting
- Dershowitz
- 1987
(Show Context)
Citation Context ...-headed arrows are used only in the direction that results in a decrease in a speci c ordering (in this case, the lexicographic path ordering with a total ordering of the operators and constants; see =-=[17]-=-). Ordered completion, starting with this system and (the Skolemized negation of) a theorem in rst-order predicate calculus with equality (taking + to be exclusive-or and to be conjunction), can be us... |

375 |
Confluent reductions: Abstract properties and applications to term rewriting systems
- Huet
- 1980
(Show Context)
Citation Context ...arded rules). In the past, Church-Rosser properties for rewrite systems have been proved by induction with respect to various orderings on terms. Some particularly elegant proofs are given by Huet in =-=[32]-=-. An intricate and rather complicated proof of correctness of a speci c completion procedure can be found in [33]. We believe that simpli cation and deletion mechanisms are best discussed in a proof-t... |

202 | Rewrite-based equational theorem proving with selection and simplification - Bachmair, Ganzinger - 1994 |

137 |
Complete sets of reductions for some equational theories
- Peterson, Stickel
- 1981
(Show Context)
Citation Context ...laps can be e ectively dealt with if a nite, complete A-uni cation algorithm is given. An associativecommutative completion procedure based on this approach has been described by Peterson and Stickel =-=[53]-=-; for a formulation within the inference rule cum proof normalization approach, see [7] 6. Ordered Completion Standard completion fails whenever an equation s t between persistently irreducible, yet i... |

125 | Completion without failure
- Bachmair, Dershowitz, et al.
- 1989
(Show Context)
Citation Context ...n the above theory. It can be shown that, under certain reasonable assumptions, ordered completion actually succeeds in constructing a convergent rewrite system, if such asystemexists.For details see =-=[8]-=-. Ordered completion is a refutationally complete theorem proving method for equational theories. Let E be a set of equations and s t be an equation provable in E, that is, s$ E t. Let bs bt be a Skol... |

117 |
On proving term rewriting systems are Noetherian
- Lankford
- 1979
(Show Context)
Citation Context ...gs that are total on equivalent ground terms. All general-purpose term orderings used in practice can be extended to complete orderings. For instance, any ordering based on polynomial interpretations =-=[46, 47]-=- can be extended to a complete ordering by combining it with a well-founded ordering to distinguish ground terms having the same interpretations. As an example of ordered completion consider the equat... |

82 |
Proofs by induction in equational theories with constructors
- Huet, Hullot
- 1982
(Show Context)
Citation Context ...]. In theories with free constructors ground reducibility is trivially decidable: a term is ground reducible if and only if it contains a non-constructor symbol (cf. the discussion of constructors in =-=[34, 36, 24]-=-). In the example above, the function symbols 0 and 0 are constructors: all ground normal-form terms are built from these twosymbols only. The constructors are also free: no two ground terms built sol... |

63 |
On proving inductive properties of abstract data types
- Musser
- 1980
(Show Context)
Citation Context ...is of a refutationally complete prover (cf. [30]). Finally, in Section 7, we present, within the same proof-transformation framework, a method for proving inductive theorems, due originally to Musser =-=[49], bas-=-ed on the concept of \proof by consistency." With this method, it is easy to prove automatically from the de nition of multiplication given at the outset that 0 x 0. We end with a brief conclusio... |

59 |
On word problems in equational theories
- Hsiang, Rusinowitch
- 1987
(Show Context)
Citation Context ...any reduction ordering. Standard completion will fail for this set of equations, whereas with ordered completion we can obtain a set of equations (x y) z (x w) z (x y) x!x x (y z)!x z ((x y) z) w!x w =-=[31]-=-, which is ground convergent with respect to the lexicographic path ordering, and therefore provides a decision procedure for the word problem in the above theory. It can be shown that, under certain ... |

52 | On restrictions of ordered paramodulation with simplification - Bachmair, Ganzinger - 1990 |

47 |
A complete proof of correctness of the Knuth-Bendix completion algorithm
- Huet
- 1981
(Show Context)
Citation Context ...) The following version of completion [16] takes as input a set of equations Eo and a well-founded ordering ~. It generates all new critical pairs at once; more practical versions (notably the one in =-=[33]-=-) do this incrementally. Let E be Eo and R be the empty set. Then repeat the following six steps as long as equations are left in E; if none remain, terminate successfully: 21. Remove an equation s ~... |

45 |
Term rewriting induction
- Reddy
- 1990
(Show Context)
Citation Context ... Gramlich [29] describes a more re ned proof by consistency system along lines similar to ordered completion. The relation of completion-based approaches to well-founded induction has been studied in =-=[56]-=-. 8. Conclusion We have represented completion and related rewrite methods as equational inference systems and have described techniques, based on proof reduction orderings, for reasoning about such p... |

43 |
A strong restriction of the inductive completion procedure
- Fribourg
- 1989
(Show Context)
Citation Context ...]. In theories with free constructors ground reducibility is trivially decidable: a term is ground reducible if and only if it contains a non-constructor symbol (cf. the discussion of constructors in =-=[34, 36, 24]-=-). In the example above, the function symbols 0 and 0 are constructors: all ground normal-form terms are built from these twosymbols only. The constructors are also free: no two ground terms built sol... |

35 |
A completion procedure for conditional equations
- Ganzinger
- 1987
(Show Context)
Citation Context ... cation. For a formalization of such procedures within the proof ordering framework, see [7]. The inference system cum proof complexity approachisby no means limited to purely equational theories. In =-=[27, 9, 18]-=-, completion procedures for conditional equations (Horn clauses) have been described in this framework. Applications to rst-order theorem proving (with or without equality) are described in [6]. The t... |

35 |
Satisfiability of systems of ordinal notations with the subterm property is decidable
- Jouannaud, Okada
- 1991
(Show Context)
Citation Context ...erms s, t, u, and v, whether there exists a ground substitution oe, such that soe �� toe and uoe �� voe. This question is decidable, for instance, if �� is a path ordering based on a total=-= precedence [15, 37]. If-=- inequations can not be solved in general for the given ordering ��, or if the decision procedure is prohibitively expensive, then completion may have to deduce more equations than are actually ne... |

35 |
Semantic confluence tests and completion methods
- Plaisted
- 1985
(Show Context)
Citation Context ...und instance 0 00 �� 0 0 , in which both terms are irreducible, yet distinct; and hence 0 00 #0 0 is not ground reducible by R [ fx#x ! ?g. Ground reducibility is decidable for finite rewrite syst=-=ems [54, 39]-=-, but, according to [38] is in exponential time even for left-linear rewrite systems. Algorithms for deciding ground-reducibility with respect to left-linear rewrite systems have been described in [41... |

34 |
Proof by induction in equational theories without constructors
- Jouannaud, Kounalis
- 1989
(Show Context)
Citation Context ...r it. This case of inconsistency is indeed decidable. The key here is the notion of \ground reducibility." A term t is called ground reducible by R if all its ground instances are reducible by R =-=(see [36]-=-). For example, every term in which the function symbol + or occurs is ground reducible by the system R 0, since in any ground term the subterm rooted at the rightmost occurrence of + or can be rewrit... |

32 |
A technique for establishing completeness results in theorem proving with equality
- Peterson
- 1983
(Show Context)
Citation Context ...iting). Computation of equational consequences from unorientable equations already appears in the workofBrown [11] and Lankford [46] on integrating resolution and simpli cation by rewriting. Peterson =-=[52]-=- proved the refutation completeness of an inference system combining resolution, paramodulation, and simpli cation with respect to orderings on ground terms that are order-isomorphic to the natural nu... |

27 |
Proof methods for equational theories
- Bachmair
- 1987
(Show Context)
Citation Context ... of equations is viewed as a process of proof simpli cation, the goal of which is the derivation of rewrite proofs. A detailed exposition of this approach iscontained in the rst author's dissertation =-=[2]-=-;seealso[3]. What distinguishes completion from ordinary logical inference is the incorporation of rules for removing redundant items from the set of inferred equations. In this context, an inference ... |

27 |
Canonical inference
- Lankford
- 1975
(Show Context)
Citation Context ...s a convergent system. This notion 3sof correctness is crucial for the use of completion as a (relatively e cient) semi-decision procedure for validity in equational theories, as proposed by Lankford =-=[46]-=- and Huet [33]. Any such correctness proof has to address three problems: rst, that the deduction mechanism is general enough for construction of convergent systems (which essentially amounts to provi... |

27 |
Rigid E-unification is NP-complete
- Gallier, Narendran, et al.
(Show Context)
Citation Context ...out equality) are described in [6]. The techniques underlying ordered completion have also been applied to other forms of unification of importance in theorem provers based on equational matings; see =-=[26, 25]-=-. For another interesting application, to unification in Boolean rings and Abelian groups, see [10]. Our approach was found to be of advantage in describing rewrite techniques for program synthesis [5... |

22 |
Critical pair criteria for completion
- Bachmair, Dershowitz
- 1988
(Show Context)
Citation Context ...ns is viewed as a process of proof simpli cation, the goal of which is the derivation of rewrite proofs. A detailed exposition of this approach iscontained in the rst author's dissertation [2];seealso=-=[3]-=-. What distinguishes completion from ordinary logical inference is the incorporation of rules for removing redundant items from the set of inferred equations. In this context, an inference rule is a b... |

22 |
Completion-Time Optimization of Rewrite-Time Goal Solving
- Bertling, Ganzinger
- 1989
(Show Context)
Citation Context ... cation. For a formalization of such procedures within the proof ordering framework, see [7]. The inference system cum proof complexity approachisby no means limited to purely equational theories. In =-=[27, 9, 18]-=-, completion procedures for conditional equations (Horn clauses) have been described in this framework. Applications to rst-order theorem proving (with or without equality) are described in [6]. The t... |

22 | Rewrite methods for clausal and nonclausal theorem proving
- Hsiang, Dershowitz
- 1983
(Show Context)
Citation Context ... a theorem in rst-order predicate calculus with equality (taking + to be exclusive-or and to be conjunction), can be used (by Herbrand's Theorem) as the basis of a refutationally complete prover (cf. =-=[30]-=-). Finally, in Section 7, we present, within the same proof-transformation framework, a method for proving inductive theorems, due originally to Musser [49], based on the concept of \proof by consiste... |

20 |
How to prove algebraic inductive hypotheses without induction
- Goguen
- 1980
(Show Context)
Citation Context ...nt rewrite system R if and only if all equations in C are inductive theorems of R. 2 If R is ground convergent, then the algebra de ned on the set of ground normal-form terms is an initial model of R =-=[28]-=-. It is isomorphic to the quotient of the set of ground terms by the congruence $ R. 28sConsistency in this sense is not decidable (not even semi-decidable). However, we will show that any inconsisten... |

20 |
A Critical-Pair/Completion Algorithm for Finitely Generated Ideals in Rings
- Buchberger
- 1984
(Show Context)
Citation Context ...ect criteria that have been suggested in the literature can be viewed as applications of compositeness, a concept that is the analogue, on the proof level, of the term-based notion of "connectedn=-=ess" [12]. A peak P o-=-f the form s /E u !E t is called composite with respect to a proof ordering ���� if there exist a sequence of proofs Q 1 ; : : : ; Q n in E, where Q i is a proof of u i\Gamma1 �� u i , suc... |

17 | Automating inductionless induction using test sets
- Kapur, Narendran, et al.
- 1991
(Show Context)
Citation Context ... 39], but, according to [38] isin exponential time even for left-linear rewrite systems. Algorithms for deciding ground-reducibility with respect to left-linear rewrite systems have been described in =-=[41, 36]-=-. In theories with free constructors ground reducibility is trivially decidable: a term is ground reducible if and only if it contains a non-constructor symbol (cf. the discussion of constructors in [... |

16 |
Inductive completion by ground proof transformation
- Küchlin
- 1989
(Show Context)
Citation Context ...ed to sets of positions. A setfp 1;:::;png of non-variable positions in a term t is said to be inductive if every ground instance t is reducible by R at some position pi, whenever is irreducible (cf. =-=[44, 13]-=-). For example, in x +(y + z) the subterm y + z is rooted at an inductive position (the rightmost occurrence of + or is reducible by R, aswehave pointed out above). Suppose now that we wish to prove t... |

15 |
Automated Proofs of the Moufang Identities in Alternative Rings
- Anantharaman, Hsiang
(Show Context)
Citation Context ...Implementations of ordered completion procedures have been reported in [50, 51]. Experiments with a procedure that combines ordered completion with associative-commutative completion are described in =-=[1]-=-. 7. Proof by Consistency In many applications, such as algebraic data type speci cations and equational programming, equations are intended to de ne a certain standard model, called the \initial mode... |

15 |
Only prime superpositions need be considered in the Knuth-Bendix procedure
- Kapur, Musser, et al.
- 1985
(Show Context)
Citation Context ...apsed if s >s 0 , it is su cient to require only q6= in completion procedures that construct reduced rewrite systems. This special case of compositeness was introduced by Kapur, Musser, and Narendran =-=[40]-=-. For example, if R 19scontains rewrite rules (x , y) , ! y , x ,, , x x , ! e, and x ,, ! x, then the critical overlap x ,,, x ,, R (x , x ,, ) , !R e , (between the rst two rules) is composite, beca... |

14 |
Applications of the Knuth-Bendix completion procedure
- Dershowitz
- 1982
(Show Context)
Citation Context ...r rule, the two possible results of rewriting the left-hand side instance form what is called a \critical pair." (Precise de nitions will be given in Section 2.3.) The following version of comple=-=tion [16]-=- takes as input a set of equations E 0 andawell-founded ordering . It generates all new critical pairs at once; more practical versions (notably the one in [33]) do this incrementally. Let E be E 0 an... |

13 |
About the rewriting systems produced by the Knuth-Bendix completion algorithm
- Metivier
- 1983
(Show Context)
Citation Context ... v R . A rewrite system R is reduced if, for each rule l!r in R, the right-hand side r is irreducible by R and no term s less than l in the encompassment ordering is reducible [19]. It has been shown =-=[14, 48]-=- that there is exactly one ( nite or in nite) reduced convergent rewrite system (up to renaming of variables) contained in any given reduction ordering. The contraction rules we have su ce for reducin... |

13 |
On sufficient-completeness and related properties of term rewriting systems
- Kapur, Narendran, et al.
- 1987
(Show Context)
Citation Context ...und instance 0 00 �� 0 0 , in which both terms are irreducible, yet distinct; and hence 0 00 #0 0 is not ground reducible by R [ fx#x ! ?g. Ground reducibility is decidable for finite rewrite syst=-=ems [54, 39]-=-, but, according to [38] is in exponential time even for left-linear rewrite systems. Algorithms for deciding ground-reducibility with respect to left-linear rewrite systems have been described in [41... |

12 | Ordering-based strategies for Horn clauses
- Dershowitz
- 1991
(Show Context)
Citation Context ... cation. For a formalization of such procedures within the proof ordering framework, see [7]. The inference system cum proof complexity approachisby no means limited to purely equational theories. In =-=[27, 9, 18]-=-, completion procedures for conditional equations (Horn clauses) have been described in this framework. Applications to rst-order theorem proving (with or without equality) are described in [6]. The t... |

11 |
Solving inequations in term algebras
- Comon
- 1990
(Show Context)
Citation Context ... to decide, given terms s, t, u, andv, whether there exists a ground substitution , such that s t and u v . This question is decidable, for instance, if is a path ordering based on a total precedence =-=[15, 37]-=-. If inequations can not be solved in general for the given ordering , or if the decision procedure is prohibitively expensive, then completion may have to deduce more equations than are actually nece... |

10 |
Rewriting Techniques for Program Synthesis
- Reddy
- 1989
(Show Context)
Citation Context ...25]. For another interesting application, to uni cation in Boolean rings and Abelian groups, see [10]. Our approach was found to be of advantage in describing rewrite techniques for program synthesis =-=[55]-=-. Many of the rewrite techniques, which wehave discussed in the context of equational theories, can also be applied in the more general setting of rst-order clausal theorem proving, as described in [4... |

10 |
On multiplicative systems defined by generators and relations
- Evans
- 1953
(Show Context)
Citation Context ...n identity holds (in all models of the axioms) if and only if both its sides can be rewritten to the identical normal form. The first rewriting-based decision procedure (for loops) was given by Evans =-=[22]-=-, in 1951. See [19] for a survey of the field. In 1970, Knuth and Bendix [42] proposed a procedure that attempts to construct a convergent rewrite system from a finite axiomatization of an equational ... |

9 |
A Structured Design-Method for Specialized Proof Procedures
- Brown
- 1975
(Show Context)
Citation Context ... version of paramodulation, enriched by contraction rules (including simpli cation by rewriting). Computation of equational consequences from unorientable equations already appears in the workofBrown =-=[11]-=- and Lankford [46] on integrating resolution and simpli cation by rewriting. Peterson [52] proved the refutation completeness of an inference system combining resolution, paramodulation, and simpli ca... |

8 |
Computing ground reducibility and inductively complete positions
- Bundgen, Kuchlin
- 1989
(Show Context)
Citation Context ...ed to sets of positions. A setfp 1;:::;png of non-variable positions in a term t is said to be inductive if every ground instance t is reducible by R at some position pi, whenever is irreducible (cf. =-=[44, 13]-=-). For example, in x +(y + z) the subterm y + z is rooted at an inductive position (the rightmost occurrence of + or is reducible by R, aswehave pointed out above). Suppose now that we wish to prove t... |

8 |
On su cient-completeness and related properties of term rewriting systems
- Kapur, Narendran, et al.
- 1987
(Show Context)
Citation Context ... is a ground instance 0 00 0 0 , in which both terms are irreducible, yet distinct; and hence 0 00 #0 0 is not ground reducible by R[fx#x!>g. Ground reducibility is decidable for nite rewrite systems =-=[54, 39]-=-, but, according to [38] isin exponential time even for left-linear rewrite systems. Algorithms for deciding ground-reducibility with respect to left-linear rewrite systems have been described in [41,... |

7 |
Semantic con uence tests and completion methods
- Plaisted
- 1985
(Show Context)
Citation Context ... is a ground instance 0 00 0 0 , in which both terms are irreducible, yet distinct; and hence 0 00 #0 0 is not ground reducible by R[fx#x!>g. Ground reducibility is decidable for nite rewrite systems =-=[54, 39]-=-, but, according to [38] isin exponential time even for left-linear rewrite systems. Algorithms for deciding ground-reducibility with respect to left-linear rewrite systems have been described in [41,... |

7 |
Inference rules for rewrite-based first-order theorem proving
- Bachmair, Dershowitz
- 1987
(Show Context)
Citation Context ..., 9, 18], completion procedures for conditional equations (Horn clauses) have been described in this framework. Applications to first-order theorem proving (with or without equality) are described in =-=[6]-=-. The techniques underlying ordered completion have also been applied to other forms of unification of importance in theorem provers based on equational matings; see [26, 25]. For another interesting ... |

6 |
Sufficient-completeness, quasireducibility and their complexity
- Kapur, Narendran, et al.
- 1987
(Show Context)
Citation Context ... 0 , in which both terms are irreducible, yet distinct; and hence 0 00 #0 0 is not ground reducible by R[fx#x!>g. Ground reducibility is decidable for nite rewrite systems [54, 39], but, according to =-=[38]-=- isin exponential time even for left-linear rewrite systems. Algorithms for deciding ground-reducibility with respect to left-linear rewrite systems have been described in [41, 36]. In theories with f... |

6 |
Rigid E-unification and its application to equational matings
- Gallier, Snyder, et al.
- 1989
(Show Context)
Citation Context ...out equality) are described in [6]. The techniques underlying ordered completion have also been applied to other forms of unification of importance in theorem provers based on equational matings; see =-=[26, 25]-=-. For another interesting application, to unification in Boolean rings and Abelian groups, see [10]. Our approach was found to be of advantage in describing rewrite techniques for program synthesis [5... |

5 |
and N achum Dershowitz. Completion for rewriting modulo a congruence
- Bachmair
- 1989
(Show Context)
Citation Context ...n associativecommutative completion procedure based on this approach has been described by Peterson and Stickel [53]; for a formulation within the inference rule cum proof normalization approach, see =-=[7]-=- 6. Ordered Completion Standard completion fails whenever an equation s t between persistently irreducible, yet incomparable terms s and t is generated. Commutativity, x y y x, is an example of such a... |

5 |
Experiments with computer implementations of procedures which often derive decision algorithms for the word problem in abstract algebras," Memo MTP-7
- Butler, Lankford
- 1980
(Show Context)
Citation Context ... v R . A rewrite system R is reduced if, for each rule l!r in R, the right-hand side r is irreducible by R and no term s less than l in the encompassment ordering is reducible [19]. It has been shown =-=[14, 48]-=- that there is exactly one ( nite or in nite) reduced convergent rewrite system (up to renaming of variables) contained in any given reduction ordering. The contraction rules we have su ce for reducin... |

5 |
Andrzej Tarlecki. Existence, uniqueness, and construction of rewrite systems
- Dershowitz, Marcus
(Show Context)
Citation Context ...d y x are incomparable with respect to any reduction ordering. The strategy used by a completion procedure to construct a derivation may determine whether or not an unorientable equation is generated =-=[21]-=-. To avoid failure a procedure may have to systematically enumerate all possible derivations (for example, via backtracking). In some cases, standard completion is bound to fail even with backtracking... |

4 |
On multiplicative systems de ned by generators and relations
- Evans
- 1951
(Show Context)
Citation Context ... an identity holds (in all models of the axioms) if and only if both its sides can be rewritten to the identical normal form. The rst rewriting-based decision procedure (for loops) was given by Evans =-=[22]-=-, in 1951. See [19] for a survey of the eld. In 1970, Knuth and Bendix [42] proposed a procedure that attempts to construct a convergent rewrite system from a nite axiomatization of an equational theo... |

4 |
A superposition oriented theorem prover
- Fribourg
- 1985
(Show Context)
Citation Context ... with respect to orderings on ground terms that are order-isomorphic to the natural numbers. (This class of orderings excludes many important orderings, though, such as most path orderings.) Fribourg =-=[23]-=- proved 27sthe completeness of a restricted version of paramodulation with locking resolution. Hsiang and Rusinowitch [31] used their trans nite semantic tree method to prove the refutation completene... |