C. Kirchner. Methodes et Outils de Conception Systematique d'Algorithms d'Unification dans les Th'eories Equationalles. PhD thesis, Centre de Recherche en Informatique de Nancy, UER de Mathematiques, Universit'e de Nancy I, Nancy, France, June 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....approach produces a modally complete unification procedure whenever the equational presentation upon which narrowing operates is composed entirely of moded equations. No more general condition guaranteeing modal completeness is known. 6.1. 3 Combining algorithms Three researchers [Yelick 85, Kirchner 85, Tid en 86] have independently developed algorithms that address the problem of combining complete unification algorithms for disjoint theories to produce a complete unification algorithm for their union. Below, we characterize these approaches, which are designed to cope with unsorted theories ....

C. Kirchner. Methodes et Outils de Conception Systematique d'Algorithms d'Unification dans les Th'eories Equationalles. PhD thesis, Centre de Recherche en Informatique de Nancy, UER de Mathematiques, Universit'e de Nancy I, Nancy, France, June 1985.

....to Martelli and Montanari s rule based approach to syntactic unification, where a given unification problem is transformed into a problem in solved form using syntactic transformations of terms. Usually, these approaches apply to a whole class of equational theories (e.g. syntactic theories [9, 10] or theories axiomatized by a canonical term rewriting system [7, 8] where this class is defined by syntactic restrictions on the form of the identities defining the equational theories. In contrast, semantic approaches depend on properties of the (free) algebras defined by the equational ....

C. Kirchner. Methodes et Outils de Conception Systematique d'Algorithmes d'Unification dans les Theories Equationelles. These d' Etat, Universite de Nancy I, France, 1985.

....AC Unification Alexandre Boudet and Evelyne Contejean LRI, CNRS URA 410 Bat 490, Universit e Paris Sud, Centre d Orsay 91405 Orsay Cedex, France Abstract. The rules for unification in a simple syntactic theory, using Kirchner s mutation [15, 16] do not terminate in the case of associativecommutative theories. We show that in the case of a linear equation, these rules terminate, yielding a complete set of solved forms, each variable introduced by the unifiers corresponding to a (trivial) minimal solution of the (trivial) Diophantine ....

....equations. 1 Introduction The syntactic theories were defined in 85 by Kirchner as those collapse free equational theories which admit a finite resolvent presentation, this being a presentation where any equality proof can be performed with at most one application of an axiom at the root [15, 16]. At that time, the unification community was not aware that there were many syntactic theories on top of commutativity and its variants. The main advantage of syntacticness is that one can nondeterministically guess which axiom of a resolvent presentation will possibly apply at the root. This ....

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Claude Kirchner. M'ethodes et outils de conception syst'ematique d'algorithmes d'unification dans les th'eories equationnelles. Th`ese d'Etat, Univ. Nancy, France, 1985.

....presentations orient the equations. E unification was introduced by Plotkin (1972) The explicit use of transformations on systems as an approach to unification is due to Martelli and Montanari (1982) but as Gallier and Snyder point out, it was implicit in Herbrand s (1930) dissertation. Kirchner (1984, 1985, 1986) was apparently the first to investigate the use of transformations in E unification. 1.1. notation Terms (A; B; C; are built from variables (x; y; z; using a set of (one sorted) first order function symbols. Subterms are referenced in the usual way by sequences of positive ....

Kirchner, C. (1985), M'ethodes et Outils de Conception Systematique d'Algorithmes d'Unification dans les Theories Equationnelles, Th`ese d'Etat, Universit'e de Nancy I.

....one meant a procedure which computes a finite complete set of unifiers. The problem was first considered in [St75, St81, Fa84, HS87] for the case where several AC symbols and free symbols may occur in the terms to be unified. More general combination problems were, for example, treated in [Ki85, Ti86, He86, Ye87, BJ89], but the theories considered in these papers always had to satisfy certain restrictions (such as collapse freeness or regularity 1 ) on the syntactic form of their defining identities. The problem was finally solved in its until now most general form by SchmidtSchau [Sc89] His algorithm ....

C. Kirchner, M'ethodes et Outils de Conception Syst'ematique d'Algorithmes d'Unification dans les Th'eories equationnelles, Th`ese d'Etat, Univ. Nancy, France, 1985.

.... of this problem have been exhaustively studied in the areas of automated theorem proving and constraint programming, such as combination of decision procedures [NO79, Ri96, TH96] combination of algorithms to solve the word problem [DKR94, BT96] combination of unification and matching algorithms [Ki85, He86, Ti86, Ye87, Ni89, Sc89, Bo93, BS92, DKR94], combination of disunification algorithms [BS95a] and combination of constraints over non free solution domains such as rational trees, feature structures, non wellfounded sets and lists [Col90, BS95b, KS96] In the meantime, general combination methods have been developed that solve most of ....

C. Kirchner, "M'ethodes et outils de conception syst'ematique d'algorithms d`unification dans les th'eories 'equationelles," Th'ese d'Etat, Universit'e de Nancy 1, France, 1985.

....E unification can be considered as a combination of elementary E unification with free (Robinson) unification. Hence it is one instance of the more general problem of combining unification algorithms for disjoint equational theories. The latter problem has been considered by many authors ([Ki85, He86, Ti86, Ye87, Sc89, Bo93, BS92]) general solutions have been given in [Sc89, Bo93, BS92] On the basis of [BS92] it is possible to obtain an algorithm for general E unification problems by combining a given algorithm for E unification with linear constant restriction with an algorithm for free (Robinson) unification with ....

C. Kirchner, "M'ethodes et outils de conception syst'ematique d'algorithmes d'unification dans les th'eories 'equationelles," Th`ese d'Etat, Universit'e de Nancy 1, France, 1985.

....axioms [Tre90] Therefore, we have to impose strong restrictions on the set of axioms E in order to derive decidability results. Actually, we show in this paper the decidability of the first order theory of T (F ) E when =E is quasi free. E is quasi free if E is collapse free resolvent [Kir85, Jou90], quasi strict (this notion is a new notion introduced in this paper) and if the equations in E have a depth 0 or 1. Typically, commutativity axioms are quasi free (and more generally, permutative axioms of Mal cev) We also show that, if E is a set of ground axioms (i.e. equations without ....

....theories In order to design finite terminating and complete sets of transformation rules for equational theories, a natural idea is to restrict our attention to those theories where it is possible to use a top down strategy of paramodulation. More precisely, C. Kirchner introduced in his thesis [Kir85] syntactic (equational) theories. In such equational theories, every equality proof can be done with at most one inference step at the root of the tree. For such theories it is possible to design simple unification rules [Kir85, KK90, JK91, Jou90] It is very easy to give the negative counterpart ....

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Claude Kirchner. M'ethodes et outils de conception syst'ematique d'algorithmes d'unification dans les th'eories equationnelles. Th`ese d'Etat, Univ. Nancy, France, 1985.

.... 1984 ] a summary of complexity results for some of the decidable cases is [ Kapur Narendran, 1987 ] A popular exposition on the undecidability of the existence of solutions to Diophantine equations is [ Davis Hersh, 1973 ] For a comprehensive treatment of narrowing and E unification, see [ Kirchner, 1985 ] For the satisfiability problem of arbitrary first order formulae with equality as the only predicate, see [ Maher, 1988 ] Comon Lescanne 1989. 7 CRITICAL PAIRS In this section, we continue our study of the Church Rosser property for rewrite systems. In particular, we will see that ....

C. Kirchner, Methodes et outils de conception systematique d'algorithmes d'unification dans les theories equationnelles, Th`ese d'Etat, Universit'e de Nancy, Nancy, France (June 1985).

....variables on a single variable. In this case, two terms occurring in the same equation become syntactically equal. Hence, the question is not the solvability of such problems, but a full description of their solutions. In [6] Kirchner and Klay have proved that distributivity is not syntactic [5, 6], by giving an infinite set of uncomparable solutions for the very simple equation x y = D u v. We introduce a representation of the solutions which enables us to represent such an infinite set by a single scheme. Actually, we cannot represent all solutions, but only the upper part of ....

Claude Kirchner. M'ethodes et outils de conception syst'ematique d'algorithmes d'unification dans les th'eories equationnelles. Th`ese d'Etat, Univ. Nancy, France, 1985.

....by Herold and Siekmann [HS87] More general combination problems were treated by Yelick, Kirchner, Herold, Tid en, Boudet, Jouannaud, and Schmidt Schau , who designed algorithms for combination of equational theories that satisfy certain restrictions on the syntactic form of their axioms. Kirchner [Kir85] requires E 1 and E 2 to be sets of simple axioms. Yelick [Yel87] gives a solution for the combination of regular and collapse free theories. Similar results with the same restriction were obtained by Herold [Her86] Tid en [Tid86] extended Yelick s result to collapse free theories. Boudet, ....

C. Kirchner. M'ethodes et outils de conception syst'ematique d'algorithmes d'unification dans les th'eories 'equationnelles. Th`ese d'Etat, Universit'e de Nancy 1, France, 1985.

....of (non orientable) inclusions. We will see that the characterization of Huet for left linear rules (in terms of ff and fl properties [10, lemma 2. 8] the generalization of Peterson and Stickel [25] for non left linear rules (in terms of E compatibility) and the result of Jouannaud Kirchner [13, 14, 15] (in terms of E coherence or confluence of cliffs) all of them are not valid for inclusions. We present a new characterization of bi rewriting modulo a set of inclusions where stronger properties are required. We have divided section 3 in two subsections, the first devoted to abstract ....

....are sufficient to have a more efficient complete and decidable proof procedure. The solution we will propose comes mainly from the two solutions known for the equational case [13, 25] In the following we consider how they can be adapted to bi rewriting. Huet [10] and Jouannaud Kirchner [14, 13] have proved that given a set of rules R and equations E such that Gamma Gamma E ffi Gamma R is terminating, R is strong Church Rosser modulo E iff all peaks and cliffs are confluent: Gamma R ffi Gamma R Gamma R ffi Gamma Gamma E ffi Gamma R and Gamma Gamma E ffi ....

Claude Kirchner. Methodes et Outils de Conception Systematique d'Algorithmes d'Unification dans les Theories Equationnelles. PhD thesis, Universite de Nancy I, 1985.

.... modulo the axioms of associativity and commutativity (AC for short) and has been given as an open problem in Comon (1988) In this paper the existential fragment has been shown decidable thus extending the results for AC unification (Stickel (1981) Livesey Siekmann (1976) Fages (1987) and Kirchner (1985) for arbitrary additional free function symbols) This example shows that equational problems may be undecidable even in case that unification with free function A New Method for Undecidability Proofs of First Order Theories 3 symbols is decidable. The extension by the axiom of idempotency to ....

Kirchner, C. (1985). M'ethodes et Outils de Conception Syst'ematique d'Algorithmes d'Unification dans les Th'eories ' Equationelles. PhD thesis, Centre de R'echerche en Informatique de Nancy.

....have the most general E unifier oe = fx 7 s tg. Now, as soon as such an equation is AC unifiable, foeg is an AC complete set of E unifiers. 3 The general case One of the major difficulties with AC unification is the combination of the elementary AC unification algorithms for each AC operator [25, 16, 11, 8]. Indeed, it took almost a decade before the termination of Stickel s combination algorithm [25] was proved by Fages 3 [11] Yet AC theories, being simple theories (i.e. theories in which there is no equality between a term and one of its proper subterms) are in the easiest class for the ....

C. Kirchner. M'ethodes et outils de conception syst'ematique d'algorithmes d'unification dans les th'eories equationnelles. Th`ese d'Etat, Univ. Nancy, France, 1985.

....the right one. Unification type: finitary for all three kinds of unification problems [Siekmann 1979] Unification algorithm: In addition to Siekmann s simple (non minimal) unification algorithm for general C f unification [Siekmann 1979] various other methods have been proposed [Fages 1983, Kirchner 1985, Herold 1987] However, none of them directly produces a minimal complete set of C f unifiers. Distributivity The theories D l f;g : ff(x; g(y; z) g(f(x; y) f(x; z) g and D r f;g : ff(g(y; z) x) g(f(y; x) f(z; x) g axiomatize left distributivity and right distributivity of f over ....

.... f unification problem [Kapur and Narendran 1992b] Unification algorithm: Because unification modulo associativity commutativity has many applications in automated deduction, a great variety of unification algorithms has been developed for AC f and ACU f [Stickel 1975, Livesey and Siekmann 1975, Kirchner 1985, Fortenbacher 1985, Buttner 1986a, Herold 1987, Herold and Siekmann 1987, Lincoln and Christian 1989, Boudet, Contejean and Devie 1990] see also subsection 5.1) Associativity commutativity idempotency We consider the theories ACI f : AC f [ ff(x; x) xg, its extension by a unit e, ACUI f : ....

[Article contains additional citation context not shown here]

Kirchner C. [1985], M'ethodes et Outils de Conception Syst'ematique d'Algorithmes d'Unification dans les Th'eories Equationelles., Th`ese d' ' Etat, Universit'e de Nancy I, France.

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