K. Yelick. Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3(1 & 2):153--182, April 1987. 35  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... section, we study the problem of combining two unification algorithms for signature disjoint equational theories E 1 = F 1 ; E 1 ) and E 2 = F 2 ; E 2 ) such that F 1 F 2 = in order to design a new modular unification algorithm for the union of theories E 1 [ E 2 = F 1 [ F 2 ; E 1 [ E 2 ) [46, 9, 53]. This is crucial for solving heterogeneous unification problems involving for instance several free symbols and or several commutative symbols and or several associative commutative symbols. However, designing a modular unification algorithm is not in general an easy task, and we need to make ....

K. Yelick. Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3(1 & 2):153--182, April 1987. 35

....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 ....

K. Yelick, "Unification in Combinations of Collapse Free Regular Theories, " J. Symbolic Computation 3, 1987.

.... 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 ....

K. Yelick, "Unification in Combinations of Collapse Free Regular Theories," J. Symbolic Computation 3, 1987.

....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 ....

K. Yelick, "Unification in Combinations of Collapse Free Regular Theories," J. Symbolic Computation 3, 1987. This article was processed using the L a T E X macro package with LLNCS style

.... problem for unification can be stated as follows: given two unification algorithms in two (consistent) equational theories E 1 on T (F 1 ; X ) and E 2 on T (F 2 ; X ) how to find a unification procedure for E 1 [ E 2 on T (F 1 [ F 2 ; X ) Combining unification algorithms was initiated in [6, 8, 19, 20] where syntactic conditions on the axioms of the theories to be combined were assumed. Combination of arbitrary theories with disjoint sets of symbols is considered in [2, 17] and the case of theories sharing constants is studied in [16] The general idea of unification in a combination of ....

K. Yelick. Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3(1 & 2):153--182, April 1987.

....specialized matching algorithm. For these two reasons, the specific unification problem called matching has attracted interest for itself. The problem addressed in this paper is the modular construction of matching algorithms. The combination problem for unification has been extensively studied in [6, 4, 12, 13, 10, 2] for equational theories built over disjoint signatures. Recently, F. Baader and K. Schulz [1] have shown an improved method for solving the combined unification problem. The greatest interest of this new method is that unification algorithms or decision algorithms for unification can be combined ....

K. Yelick. Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3(1 & 2):153--182, April 1987.

....I denotes the words input into the rule, A denotes the state variable values input, O denotes the words output, and A 0 denotes the state variable values output. Definitions of terms and related ideas are given below. The concepts are standard, but the particular definitions used are from [34]. Definition 3.2: Let Phi be a countable set of variables and F a family of function symbols disjoint from Phi with associated arity. A term is either a variable or a function symbol followed by n terms, where n is the arity of the function symbol. A function symbol of arity zero is called a ....

K. A. Yelick. Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3:153--181, 1987.

....for solving the problem P in E 1 [ E 2 . Almost all results known up to now are restricted to the case where the signatures of E 1 and E 2 are disjoint, in which case, we speak of disjoint theories. Many authors studied the problem of combining unification algorithms for disjoint theories [6, 14, 13] and the best result is due to F. Baader and K. Schulz [1] who described a general technique for combining decision procedures for unifiability and, by the way, solved the problem of combining unification algorithms for non finitary theories. This result was extended in two different ways: F. ....

K. Yelick. Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3(1 & 2):153--182, April 1987.

....1 ; t 2 )g [P ; S) fs 1 = t 2 ; s 2 = t 1 g [ P ; S) More complex cases, involving bounded applications of axioms in E prior to decomposition, can be treated similarly. Methods of combining unification algorithms for well behaved theories that do not share symbols have been given in [ Yelick, 1987; Kirchner, 1989; Boudet etal, 1988 ] The general case was solved in [ SchmidtSchauss, 1988 ] Note that a unification algorithm that generates a complete set of most general unifiers (for terms without free constants) does not automatically work for matching (one cannot just treat the ....

K. A. Yelick, Unification in combinations of collapse-free regular theories, J. of Symbolic Computation 3 (1&2), pp. 153-181 (February/April 1987).

.... Academic Press Limited 2 H el ene Kirchner and Christophe Ringeissen on a set of terms T (F 1 ; X ) and E 2 on T (F 2 ; X ) how to find a unification procedure for E 1 [ E 2 on T (F 1 [ F 2 ; X ) Combining unification algorithms was initiated in Kirchner (1985) Herold (1986) Tid en (1986) Yelick (1987) where syntactic conditions on the axioms of the theories to be combined were assumed. Combination of arbitrary theories with disjoint sets of symbols is considered in Schmidt Schau (1989) Boudet (1990) and in Baader Schulz (1992) The general idea of unification in a combination of theories ....

....are unifiers since =E i is included in =E . Care must be taken that this method is also complete: each unifier must be an instance of at least one of these substitutions. This method has been successively shown complete for the combination of disjoint regular and collapse free equational theories (Yelick (1987)) and, later on, for disjoint equational theories (Schmidt Schau (1989) Boudet (1990) Boudet (1993) We prove next that it is also true when constants are shared. Lemma 4.2. Let s be a i term such that alien subterms are R normalized. Assume that s is R reducible. Then there exists a term t ....

Yelick, K. (1987). Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3(1 & 2):153--182.

....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, Jouannaud Schmidt Schau [BJSS89] gave an algorithm for combining an ....

K. Yelick. Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3(1 & 2):153--182, 1987.

....algorithm for E 1 [ E 2 on T (F 1 [ F 2 ; X) The general idea of the unification in a combination of theories consists in breaking an equational problem into sub problems that are pure in the sense that they can be solved in one component of the combination. This problem was initiated in [6, 13, 15] where syntactic conditions on the axioms of the disjoint theories to be combined were assumed. Then, this problem has been solved by M. SchmidtSchau [12] and A. Boudet [2, 3] in the general case of arbitrary disjoint equational theories: a unification algorithm with free constants and a free ....

....are unifiers since =E i is included in =E . Care must be taken that this method is also complete: each unifier must be an instance of at least one of these substitutions. This method has been successively shown complete for the combination of disjoint regular and collapse free equational theories [15] and, later on, for disjoint equational theories [12, 3, 2] We prove next that it is also true when only constants are shared. Theorem 9. Let (s = t) be a i pure equation and oe a substitution normalized w.r.t. R. Then soe =E toe ( soe) i =E i (toe) i : Note that (soe) i =E i ....

K. Yelick. Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation, 3(1 & 2):153--182, April 1987.

.... provided that they satisfy certain restrictions (such as collapse freeness or regularity 19 ) on the syntactic form of their defining identities, which make sure that the theories behave similarly to associativity commutativity and syntactic equality [Kirchner 1985, Tid en 1986, Herold 1986, Yelick 1987, Boudet et al. 1989] The problem of combining algorithms computing complete sets of unifiers was solved in a very general form by Schmidt Schau [1989] His approach imposes no restriction on the syntactic form of the identities. The only requirements on the component theories E i are of an ....

Yelick K. [1987], `Unification in combinations of collapse-free regular theories', J. Symbolic Computation 3(1,2), 153--182.

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