D. Kapur and D. R. Musser. Proof by Consistency. The Artificial Intelligence Journal, 31:125--157, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....always result in a unique answer [Knuth 70] A convergent rewriting system is a complete and terminating decision procedure for determining whether or not an equation is implied by the original set of equations. Term rewriting systems can be used as a basis for automatic theorem provers [Huet 82, Kapur 84, Goguen 80, Hsiang 82] These theorem provers have been used for applications such as checking formal specifications [Goguen 79, Guttag 83, Kownacki 84] interpreting logic programming languages [Dershowitz 83a, Fribourg 84] reasoning about relational databases [Cosmadakis 85] and checking ....

D. Kapur and D. R. Musser, "Proof by Consistency." Proc. of an NSF Workshop on the Rewrte Rule Laboratory. Sept. 6-9. 983, General Electric Corporate Research and Development Report No. 84GENO08. Schenectady, NY, April 1984, pp. 245-267.

....specifications, sets E with non free or non specified constructors, and to equational term rewriting. They noted that the key property for the technique to succeed was inductive completion, i.e. that the inductive theory of constructors coincides with the equational theory. This was referred to in [KM87] as a Birkoff like theorem, referring to [Bir35] where Birkoff showed the completeness of the rules of inference of equality. Jouannaud and Kounalis proposed to show that this condition could be satisfied with their two key notions. The key theorem of their work was this: If R 0 is a set of ....

.... reasoning community at CADE8 in 1986 (see [KSZ86] and further work was presented in [KZ88] and [KZ95] It is the most widely used tool for presenting results in the literature , and the script of a session using RRL to prove the usual toy problems by consistency was first presented in [KM87]. At this time, completion was being carried out by the Knuth Bendix process and RRL included no facility for checking a system using AC operators for termination. More recent work on RRL has involved trying to implement all the current methods of induction (implicit, explicit, cover set (see ....

D. Kapur and D. R. Musser. Proof by consistency. Artificial Intelligence, 31:125--157, 1987.

....Inconsistencies areherefound whenever anequation emerges thatmakes twodistinctground constructor terms equivalent. This concept was extendedbyJouannaud and Kournalis in [6] to caseswhere the constructorsare not explicitly given. For a morethorough reviewonproof by consistency, werefer to [8]. We shall in this paper extend the methods described in [5] and [6] in such a way that they implementafinalrather than the initialalgebra. Our method does thereforealsoapply to behavioral semantics of algebraic specifications. Furthermore, the final model allows non free constructorswithout the ....

D. Kapur and R. Musser. Proof by consistency. Artificial Intelligence, 31:125--157, 1987.

....unification algorithm unifies by J. A. Robinson [18] is total, but its termination is a deep theorem [17] Technical Report IBN 98 49, Darmstadt University of Technology. 1 There are two research paradigms for the automation of induction proofs, viz. explicit and implicit induction (e.g. [1, 11]) where we only focus on the first one. 1 and none of the current methods for automated termination analysis succeeds with this example. Hence, such functions cannot be handled by (fully) automated theorem provers without the ability of reasoning about possibly partial functions. In contrast to ....

D. Kapur & D. R. Musser. Proof by Consistency. Artificial Intelligence, 31:125-157, 1987.

....sets must have exactly one Herbrand model (on a given signature) Such clause sets are straightforward representations of their Herbrand models. Using such sets of clauses as representation formalism has at least the following advantages. It allows us to use proof by consistency (see for example [20, 1, 12]) for proving that a clause C (on the same signature) is valid in the model M denoted by S, it suces to prove that S [ fCg is consistent (indeed, if S [ fCg is consistent, it must have an Herbrand model, and since M is the only Herbrand model of S, we must have M j= C) Consequently, it implies ....

D. Kapur and D. Musser. Proof by consistency. Articial Intelligence, 31, 1987.

....[CT97] given a term s and a rule l r j c, it is undecidable whether or not all instances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof by consistency approach to proving in 4 ductive theorems [KM87,JK86,Bac88]. Here again, the result is the opposite of what happens in the case of finite term rewriting systems: we show that ground reducibility, a crucial ingredient for the proof by consistency approach, is undecidable for ordered rewriting. The paper is organized as follows. We mainly focus on our ....

Deepak Kapur and David R. Musser. Proof by consistency. Artificial Intelligence, 31(2):125--157, February 1987.

....equational specifications, ground convergent presentations, Powerful saturation methods have been developed since and we believe that it was time to generalize the technique, taking advantage of the new developments. This was possible because of the generalization of ideas already present in [Fri84, KM87] for pure constructor systems: give explicitly a first order axiomatization which reduces inductive proofs to proofs by consistency. Besides this generalization (the introduction of I axiomatizations) we have shown how to generalize the proofs by consistency methods to Horn clauses and arbitrary ....

Deepak Kapur and D. Musser. Proof by consistency. Artificial Intelligence, 31(2), February 1987.

....and can handle 9 quantified formulas. 8.3. INDUCTIVE COMPLETION Based on a Knuth Bendix like completion procedure the inductionless induction method was proposed by [28] 13] and further developed by various paper.tex; 16 10 1996; 10:29; no v. p. 43 44 Dieter Hutter researchers (e.g. 16] [26], 14] 25] The completion approach can be viewed as an induction over terms (rather than induction over values as in explicit induction) but in contrast to explicit induction the well founded ordering is implicit in the deduction mechanism and depends mainly on the choice of the underlying ....

Kapur, D.; Musser, D. Proof by consistency. Artifical Intelligence, No 31(2), 1987

....sets must have exactly one Herbrand model (on a given signature) Such clauses sets are straightforward representations of their Herbrand models. Using such sets of clauses as representation formalism has at least the following advantages. It allows us to use proof by consistency (see for example [15, 1, 8]) for proving that a clause C (on the same signature) is valid in the model M denoted by S, it suces to prove that S[fCg is consistent (indeed, if S[fCg is consistent, it must have an Herbrand model, and since M is the only Herbrand model of S, we must have M j= C) Consequently, it implies that ....

D. Kapur and D. Musser. Proof by consistency. Articial Intelligence, 31, 1987.

....(cf. Bac87] and unfailing completion techniques the restriction that conjectures have to be orientable w.r.t. the underlying reduction ordering could be removed (cf. Bac88] Gra89] Gra90b] Finally the general paradigm of proof by consistency was thoroughly analyzed and developed in [KM87], ZKK88] Zha88] This work was coupled with an investigation on the kind of possible model semantics underlying any intended notion of inductive validity. Indeed, one of the main drawbacks of the common initial semantics approach is the 7 treatment of partiality, i.e. of partially defined ....

....running systems with powerful and (at least partially) automated ITP components. A severe obstacle may also consist in the fact that a precise theoretical understanding of the kind of model semantics underlying the intended notion of inductive proof is not yet clear enough (cf. e.g. Zha88] [KM87]) 4) Proof Organisation and Control From a practical point of view we think that the most important problem consists in improving the whole specification and proof engineering process. The theoretical foundations, the technical possibilities (e.g. inference rules) and the degrees of freedom in ....

D. Kapur and D.R. Musser. Proof by consistency. Artifial Intelligence, 31:125-- 157, 1987.

....program veri cation and induction theorem proving. For an extension of inductive truth to more general formulas and for a model theoretic characterization see e.g. ZKK88,Wal94,BR95,Gie99c] To prove inductive truth automatically, several induction theorem provers have been developed, e.g. BM79,KM87,ZKK88,BSH 93,Wal94,BR95,BM98] For instance, these systems can prove conjecture (1) by structural induction on the variable x. If we abbreviate (1) by (x; y; z) then in the induction base case they would prove (0; y; z) and in the step case (where x 6= 0) they would show that the ....

D. Kapur and D. R. Musser. Proof by consistency. AI, 31:125-158, 1987.

....theorem proving. For an extension of inductive truth to more general formulas and for a model theoretic characterization (using initial algebras) see e.g. ZKK88, Wal94, BR95, Gie99b] To prove inductive truth automatically, several induction theorem provers have been developed, e.g. BM79, KM87, ZKK88, BSH 93, Wal94, BR95] For instance, these systems can prove conjecture (1) using a structural induction with x as the induction variable. If we abbreviate (1) by (x; y; z) then in the induction base case they would prove (0; y; z) and in the step case (where x 6= 0) they would ....

D. Kapur and D. R. Musser. Proof by consistency. AI, 31:125-158, 1987.

....in Section 3 we develop a new calculus for induction proofs with Technical Report IBN 98 48, TU Darmstadt, Germany. Final version to appear in the Journal of Automated Reasoning. 1 There are two research paradigms for the automation of induction proofs, viz. explicit and implicit induction (e.g. [4, 40]) where we only focus on the first one. 2 J URGEN GIESL partial functions. We first regard algorithms defined by unconditional equations only, but in Section 4 we show how to extend our calculus to handle algorithms with conditionals. While the calculus of Section 3 and 4 is already sufficient ....

....These approaches can also deal with non termination, but in contrast to our work, they do not focus on determining suitable induction relations automatically (by deriving them from the recursions of the algorithms) An alternative notion of truth has been proposed by D. Kapur and D. R. Musser [39, 40] and a corresponding definition is also used by C. Walther in [75] Here, an equation is considered to be inductively true if it holds in the intersection of all maximal congruence relations satisfying the specification (where a specification consists of a set of equations and a set of ground ....

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Kapur, D. and Musser, D. R., `Proof by Consistency', Artificial Intelligence 31, 125-157 (1987).

....Proceedings of the Workshop on the Mechanization of Partial Functions, held in conjunction with the 15th International Conference on Automated Deduction (CADE 15) Lindau, Germany, 1998. 1 There are two paradigms for the automation of induction proofs, viz. explicit and implicit induction (e.g. KM87, BR95] where we only focus on the first one. conjectures containing at most one occurrence of a partial function) But to increase the power of our approach, in this paper we suggest a refinement where definedness is made explicit. In Sect. 2 we introduce our programming language and in Sect. ....

....extend existing induction theorem provers to partial functions, i.e. we did not want to change the underlying logic. Our notion of inductive truth corresponds to one of the definitions of inductive validity proposed in [WG94, Type E ] Alternative notions of truth have been suggested in [KM86, KM87, Wal94] Here, an incompletely specified function is interpreted as the set of all possible complete and consistent extensions, cf. also [WG94, Type D 0 ] This corresponds to the intuition that such 3 In that respect, our proofs differ from other case studies in related areas (e.g. the ....

Kapur, D. and Musser, D. R., Proof by Consistency, Artificial Intelligence 31, 125-157, 1987.

....of development, or even due to partiality being actually intended. Thus, partiality and undefinedness are not part of the specification but a result from its incompleteness. For this reason, the undefined terms are often thought to be equal to some unknown constructor ground terms: Kapur Musser[18, 19] consider those congruences which are maximally enlarged by random identification of undefined terms with constructor ground terms, as long as this identification does not identify two distinct constructor ground terms. Their intended 19 which is also necessary for sufficient expressibility for ....

....) with the x i being different constructor variables) expressing that the symbol f denotes a totally specified function, are very important for inductive theorem proving. cf. Wirth[28] 9 congruence is then the intersection of all those maximally enlarged congruences. In Kapur Musser[18] the maximal congruences are allowed to have some undefined terms left; this causes the problem that one cannot describe the intended congruence by model semantics 20 . Therefore in Kapur Musser[19] the intersection is done only over those congruences that have no undefined terms left: These ....

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Deepak Kapur, David R. Musser (1987). Proof by Consistency. Artificial Intelligence 31, pp. 125-157.

....of control information as before. It just means that 6= is syntactically restricted to defined terms. Undefined terms are due to partially specified functions or incomplete knowledge about the model world. They are often thought to be equal to some unknown constructor ground terms: Kapur Musser[25, 26] consider those congruences which are maximally enlarged by random identification of undefined terms with constructor ground terms, as long as this identification does not make two distinct constructor ground terms equal. The intended congruence is then the intersection of all those maximally ....

....which are maximally enlarged by random identification of undefined terms with constructor ground terms, as long as this identification does not make two distinct constructor ground terms equal. The intended congruence is then the intersection of all those maximally enlarged congruences. In [25] the maximal congruences are allowed to have some undefined terms left; this causes the problem that one cannot describe the intended congruence by model semantics 29 . Therefore in [26] the intersection is done only over those congruences that have no undefined terms left: These congruences can ....

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Deepak Kapur, David R. Musser. Proof by Consistency. Artificial Intelligence 31 (1987), 125-157.

....Section 9.6) In all these cases reduction orderings on formulas take over the role of induction orderings, which are typical for explicit induction. Any implicit induction method is based on the equivalence of inductive validity and consistency (Theorem 10.48(1) originally stated in [HH82] and [KM87] Current research on explicit induction develops strategies for guiding proofs towards induction hypotheses by recognizing conjecture skeletons within subgoals and applying wave rules (see, e.g. Hut90,BSvH 93] while implicit induction tries to make such strategies superfluous. However, ....

D. Kapur and D.R. Musser. Proof by consistency. Artificial Intelligence, 31(2):125--157, 1987.

....data types, we had to generalize the notions and results used in the CTRS community. These generalizations have been worked out in [33] Chapters 6 and 7. In particular, Sections 7.4 and 7. 5 of [33] are devoted to a detailed comparison between inductive completion or proof by consistency (cf. [25]; 5] Section 8.5; 8] Section 7.3) and our method of inductive expansion. Given a CTRS R, inductive completion mixes the derivation of theorems from R with a proof that R is ground confluent. Inductive expansion separates both proof obligations from each other, which makes the proofs more ....

D. Kapur, D.R. Musser, Proof by Consistency, Artificial Intelligence 31 (1987) 125-157

....Note, however, that equality proofs over non freely generated data types can become arbitrarily complex due to the user specified equality relation. 6 Related Work In the area of implicit induction (or inductive completion) which has evolved from the Knuth Bendix completion procedure (see e.g. [10], 12] 16] non freely generated data types are treated the same way as freely generated ones are. The reason is that (syntactical) term orderings are used to guarantee the termination of the entire rewrite system. For an automated termination analysis usually simplification orderings are used ....

D. Kapur and D.R. Musser. Proof by Consistency. Artificial Intelligence, pp. 125--157, 31(2), 1987.

....In this framework, the system Nqthm was developed, which was for many years the only significant automated theorem proving system for induction. However, Nqthm often requires interaction with the user. The second approach involves a proof by consistency: this is the inductionless induction method [706, 509, 550, 540, 520, 355, 598, 46, 427, 802]. However, both methods have many limitations on the theorems that can be proved and on the underlying theory. Indeed, guiding a proof by explicit induction requires some skill in finding the right axioms or hypotheses to apply. On the other hand, the proof by consistency technique does not ....

D. Kapur and D. R. Musser. Proof by consistency. Artificial Intelligence, 31(2):125--157, 1987.

....refines the method of Fribourg such that it is no longer necessary to orient equations in H ; as a result his method does not fail. He also generalizes the notion of complete position to complete set of positions. Numerous other people have worked on this subject [Gog80, Gra89, Gra90, HK88, KM86, KM87, KNZ86, K uc87] Recently, there have been some approaches to the conditional case, using constructor complete specifications [Ore87] or inductive reducibility [BL90] The proof procedure for the initial model which we present is refutationally complete and linear. Moreover, by appropriately ....

D. Kapur and D. R. Musser. Proof by Consistency. Artificial Intelligence, 31:125-- 157, 1987.

....and its inefficiency, due to the divergence of completion. Here we have generalized the technique, taking advantage of the powerful recent developments on saturation methods for first order theorem proving. This was possible because of the generalization of ideas already present in [Fri84, KM87] for pure constructor systems: give explicitly a first order axiomatization which reduces inductive proofs to proofs by consistency. Besides this generalization (the introduction of I axiomatizations) we have shown how to generalize the proofs by consistency methods to Horn clauses and arbitrary ....

Deepak Kapur and D. Musser. Proof by consistency. Artificial Intelligence, 31(2), February 1987.

....is undecidable [CT97] given a term s and a rule l r j c, it is undecidable whether or not all instances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof by consistency approach to proving inductive theorems [KM87, JK86, Bac88]. Here again, the result is the opposite of what happens in the case of finite term rewriting systems: we show that ground reducibility is undecidable for ordered rewriting. The paper is organized as follows. We mainly focus on our decidability result: the decidability of confluence. We first ....

Deepak Kapur and David R. Musser. Proof by consistency. Artificial Intelligence, 31(2):125--157, February 1987.

....comon lri.lri.fr Summary of the lecture The course relies on the work which has been done for 15 years in rewriting theory on inductive proofs. The aim here is to give a general introduction encompassing all works which are relevant to inductionless induction [21] or proof by consistency [17]. For an excellent introduction on the techniques and motivations of these works, see L. Bachmair s book [2] The original problem was to automatically prove (or disprove) a formula OE in the initial models of a set of equations E (or Horn clauses with equality) Basically, inductionless ....

....A is the set ftrue 6= falseg. G. Huet and J. M. Hullot [15] assume a constructor theory and A expresses that two pure constructor terms are distinct (see also [21] J. P. Jouannaud and E. Kounalis [16] and L. Bachmair [1] assume that E is given by a ground convergent rewrite system (see also [17]) then A expresses that two equal terms should be ground reducible. This last notion plays a central role in further developments of inductionless induction, as will be explained later. Now, the second issue above (computing a saturated set of formula) refers to some notion of redundancy: a set ....

D. Kapur and D. Musser. Proof by consistency. Artificial Intelligence, 31(2), Feb. 1987.

....are a = 1 and b = 1, and the substitutions are oe 1 = x 0 7 x; y 0 7 s(y) oe 2 = 1 = y 7 s(y) 2 = The completion procedure generates the in nite family of rules s n (x y) z x (s n (y) z) from them. This is also the reason why the proof by consistency [32] of the associativity in the rewrite system x 0 x x s(y) s(x y) using an unappropriate reduction ordering (RPO with the precedence s in this case) results in an in nite loop, as described by Dershowitz [11] and Fribourg [16] In [21] the rewrite system x 0 x (3) x s(y) ....

....rewrite system R 2 (Recently, there was a method for proving inductive theorems proposed in [23] which does not require the underlying system to be conAEuent, not even on ground terms. Therefore this step may be skipped when using the just mentioned method. 3. prove in R 2 by consistency [27, 32], or by inductive reducibility [31, 33] with a possible involvement of Fribourg s method [16] or even the more general Bachmair s one [2] an inductive theorem, derived from the structure of the generated in nite family of rules, and add it as a rule to the existing system R 2 , producing R 0 2 ....

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D. Kapur and D.R. Musser. Proof by consistency. Artiøcial Intelligence, 31(2):125157, February 1987.

....a usage of control information as before. It just means that 6= is syntactically restricted to defined terms. Undefined terms are due to partially specified functions or incomplete knowledge about the model world. They are often thought to be equal to some unknown defined terms (cf. Kapur Musser[7]) Based on this tradition, our approach can be justified the following way: If two terms can be shown equal, they will stay equal even if an undefined term will be identified with a defined term later on. On the other hand might an undefined term become equal (w. r. t. Phi Omega ( to a ....

....t. R;ground or reducibility of ground terms (cf. Lemma 5.11) which is different from inductive reducibility. The idea of choosing a constructor based approach for assigning adequate semantics to p n conditional equational specifications was heavily inspired by previous work of Kapur Musser[7] (for the case of unconditional equations only) and Zhang[10] where problems with the usual notion of inductive (initial) validity, in particular concerning its non monotonic behaviour w. r. t. consistent extensions, and partially defined functions are treated. Under the appropriate syntactical ....

Deepak Kapur, David R. Musser. Proof by Consistency. Artificial Intelligence 31 (1987), pp. 125-157.

.... ground reducibility is implicit in [ Plaisted, 1985 ] If each left hand side of a rule in R and each side of an equation in S contains a non constructor symbol, then property (b) is ensured; cf. Huet Hullot, 1980 ] Ground reducibility is decidable for finite R and empty S [ Plaisted, 1985; Kapur etal, 1987 ] A faster decision method is obtained by reducing ground reducibility to the emptiness problem of the language produced by a conditional tree grammar describing the system s ground normal forms [ Comon, 1989 ] Testing for ground R reducibility, however, requires exponential time, even for ....

.... method is obtained by reducing ground reducibility to the emptiness problem of the language produced by a conditional tree grammar describing the system s ground normal forms [ Comon, 1989 ] Testing for ground R reducibility, however, requires exponential time, even for left linear R [ Kapur etal, 1987 ] In the special case where all constructors are free, ground reducibility is more easily testable. This case had been considered in [ Nipkow Weikum, 1982 ] for left linear systems. The general case was considered in [ Dershowitz, 1985 ] and [ Kounalis, 1985 ] The former defines a test set ....

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D. Kapur and D. R. Musser, Proof by consistency, Artificial Intelligence 31 (2), pp. 125-157 (February 1987).

....a list of some well known theorem provers, categorized roughly by their degree of automation: User guided automatic deduction tools. Systems like ACL2 [Kaufmann and Moore 1995] Eves [Craigen et al. 1988] LP [Garland and Guttag 1988] Nqthm [Boyer and Moore 1979] Reve [Lescanne 1983] and RRL [Kapur and Musser 1987] are guided by a sequence of lemmas and definitions but each theorem is proved automatically using built in heuristics for induction, lemma driven rewriting, and simplification. Nqthm, the Boyer Moore theorem prover, has been used to check a proof of Godel s first incompleteness theorem, and in a ....

Kapur, D. and Musser, D. 1987. Proof by consistency. Artificial Intelligence 31, 125--157.

....constraint l r. On the other hand, we do not consider only a given term s, but also constrained equations s = t j c 0 . Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof by consistency approach to proving inductive theorems [9, 8, 1]. Here again, the result is the opposite of what happens in the case of finite term rewriting systems: we show that ground reducibility is undecidable for ordered rewriting. The paper is organized as follows. We mainly focus on our decidability result: the decidability of confluence. Our proof ....

D. Kapur and D. Musser. Proof by consistency. Artificial Intelligence, 31(2), Feb. 1987.

....use of explicit induction rules (hence differs from the inductive proof methods described in chapter [chapter with id induction] we will stay within classical firstorder logic. Inductionless induction should rather be called proof by consistency, following D. Kapur and D. Musser s terminology [Kapur and Musser 1987], as we will see. This method has been discovered in the years 80 82 by several authors [Musser 1980, Goguen 1980, Lankford 1981, Huet and Hullot 1982] and a lot of work has been devoted to it since, mainly in the term rewriting community. This chapter relies on this work. It aims at giving a ....

....reducibility was shown originally by D. Plaisted [Plaisted 1985] and several other algorithms have been proposed since. We sketch here two of them: one is based ond tree automata with constraints along the lines of [Caron et al. 1993] The second one is based on test sets, along the lines of [Kapur et al. 1987, Kounalis 1992] Both of these methods have some feed back on inductive proofs since they provide with some explicit induction scheme. 6.1. Automata techniques In [Caron et al. 1993] the authors shown a quite general result. Given a term t, let encomp t be a unary predicate symbol which is ....

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Kapur D. and Musser D. [1987], `Proof by consistency', Artificial Intelligence 31(2).

....to guess suitable induction hypothesis. Two main approaches have been proposed to overcome these diOEculties. The rst applies explicit induction arguments on the structure of terms [1, 13, 7, 5, 18, 49] The second one involves a proof by consistency: this is the inductionless induction method [37, 24, 30, 27, 17, 31, 2, 36, 8, 19]. To prove the associativity of by explicit induction we can use the following scheme: Basic Case: x y) 0 = x (y 0) Induction Step: x y) u = x (y u) implies (x y) S(u) x (y S(u) The proof of Basic Case is trivial, and the proof of Induction Step follows ....

D. Kapur and D.R. Musser. Proof by consistency. Artiøcial Intelligence, 31(2):125157, February 1987.

....of examples to which theorem provers have been applied. Below is a list of some well known theorem provers, categorized roughly by their degree of automation: ffl User guided automatic deduction tools. Systems like ACL2 [KM95] Eves [CKM 88] LP [GG88] Nqthm [BM79] Reve [Les83] and RRL [KM87] are guided by a sequence of lemmas and definitions but each theorem is proved automatically using built in heuristics for induction, lemma driven rewriting, and simplification. Nqthm, the Boyer Moore theorem prover, has been used to check a proof of Godel s first incompleteness theorem, and in a ....

D. Kapur and D. Musser. Proof by consistency. Artificial Intelligence, 31:125--157, 1987.

....This alternative employed the Knuth Bendix completion procedure [21] to seek a contradiction between a conjecture and given axioms; failure to produce this contradiction constituted a proof of the conjecture. Throughout the 80s this alternative, now usually referred to as proof by consistency [19] or implicit induction , was developed by various people [12, 15, 18, 9, 10, 19, 1, 23] This culminated in the independent work of Reddy [27] and Duffy [8] which extracted the underlying induction principle utilised in the implicit approach and proposed this as an alternative to the more ....

....[21] to seek a contradiction between a conjecture and given axioms; failure to produce this contradiction constituted a proof of the conjecture. Throughout the 80s this alternative, now usually referred to as proof by consistency [19] or implicit induction , was developed by various people [12, 15, 18, 9, 10, 19, 1, 23]. This culminated in the independent work of Reddy [27] and Duffy [8] which extracted the underlying induction principle utilised in the implicit approach and proposed this as an alternative to the more classical principles. Perhaps the most developed attempt at integration of the two approaches ....

D. Kapur and D.R. Musser. Proof by consistency. Artificial Intelligence, 31:125--157, 1987.

....sets [BR93] are developed which, combined with a reduction ordering, allow for inductive proofs. In many cases the test or cover sets can be computed from the specification. No partiality is allowed here. For unconditional specifications the method proof by consistency is known [HH82] JK89] [KM87], Bac88] Red90] Again, no partiality is allowed, the rewrite system induced by the specification has to be terminating. For unconditional partial specifications we refer to [KM86] We have already commented on that. This paper gives a unifying presentation on the work carried out in the ....

D. Kapur and D. Musser. Proof by consistency. Artificial Intelligence, 31:125--157, 1987.

....is undecidable [6] given a term s and a rule l r j c, it is undecidable whether or not all instances of s can be reduced. Going further, we investigate the use of ordered rewriting in a classical application of rewrite systems: the proof by consistency approach to proving inductive theorems [14, 13, 1]. Here again, the result is the opposite of what happens in the case of finite term rewriting systems: we show that ground reducibility is undecidable for ordered rewriting. The paper is organized as follows. We mainly focus on our decidability result: the decidability of confluence. We first ....

D. Kapur and D. Musser. Proof by consistency. Artificial Intelligence, 31(2), Feb. 1987.

....of negative literals in formulas to be proved opens up or even necessarily entails various ways of defining inductive validity, in particular if we want to guarantee some reasonable monotonicity property w. r. t. consistent extension. Pioneering papers along this line of reasoning are [KM87] and [KM86] cf. also [Zha88] ZKK88] But whereas in these papers the specifications treated are systems of unconditional equations, and the formulas considered are pure equations, here we shall permit general equational first order clauses as formulas and, moreover, as specifications we ....

....semantics, however, lacks the discussed monotonicity property. In [Pad90] BL90] KR88] and [BR93] we find the usual validity in the initial model which is like our type E, assuming again all symbols to be constructor symbols and forbidding negative conditions in the rules. Kapur and Musser ([KM87], KM86] consider only unconditional equations and only congruences on ground terms (i.e. term generated models) for validity, namely those that are maximally enlarged by random identification of undefined terms with defined ones (i.e. constructor ground terms) as long as this identification ....

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D. Kapur, D. R. Musser. Proof by consistency. Artificial Intelligence, 31:125--157, 1987.

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Kapur, D., and Musser, D.R., "Proof by consistency," AI Journal, 31, Feb. 1987, 125-157.

.... theories using completion procedures; ffl refutational methods for proving theorems in first order predicate calculus with equality; ffl proving formulas by induction using the explicit induction approach based on the cover set method, as well as using the proof by consistency approach [46, 21] (also called the inductionless induction approach) and ffl checking the consistency and completeness of equational specifications. RRL is perhaps one of the few theorem provers in the world providing such extensive capabilities for equational logic, first order theorem proving, and theorem ....

Kapur, D., and Musser, D. R., Proof by consistency. Artificial Intelligence 31 (1984) 125--157.

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D. Kapur and D. R. Musser. Proof by Consistency. The Artificial Intelligence Journal, 31:125--157, 1987.

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D. Kapur and D. R. Musser. Proof by Consistency. The Artificial Intelligence Journal, 31:125--157, 1987.

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D. Kapur, D.R. Musser, Proof by Consistency, Artificial Intelligence 31 (1987) 125-157

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D. Kapur, D.R. Musser, Proof by Consistency, Artificial Intelligence 31 (1987) 125-157

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Deepak Kapur and David R. Musser. Proof by consistency. Arti#cial Intelligence, 31#2#:125#157, February 1987.

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Deepak Kapur and David R. Musser. Proof by consistency. Artificial Intelligence, 31(2):125--157, February 1987.

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Deepak Kapur, David R. Musser (1987). Proof by Consistency. Artificial Intelligence 31, pp. 125-157.

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