H. Comon, "Disunification: a survey". In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, MIT Press, pp 322-359, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in [43] Decidability of an extension of term algebras with membership tests is presented in [10] in the form of a terminating term rewriting system. Unification and disunification problems are special cases of decision problem for first order theory of term algebras, for a survey see e.g. [45, 9]. We believe that our proof provides some insight into di#erent variations of quantifier elimination procedures for term algebras. Like [22] we use selector language symbols, but retain the usual constructor symbols as well. The advantage of the selector language is that #y. z = f(x, y) is ....

Hubert Comon. Disunification: A survey. In JeanLouis Lassez and Gordon Plotnik, editors, Computational Logic: Essays in Honor of Alan Robinson. The MIT Press, Cambridge, Mass., 1991. 3.4

....to the work in this paper. At this point there are a number of questions raised by this approach which we hope to address. Can a significant portion of our relational machine be captured with a Church Rosser, strongly normalizing set of rewrite rules Comon and Jouannaud and Kirchner s work [Comon, CoHaJo] on rewriting systems for unification and disunification suggests that this is quite feasible, as does the work of Bellia and Occhiuto, op. cit. We would also like to exploit the rich semantics of relational formalisms to obtain new notions of observables, and abstract interpretation, as well as ....

Comon, H. "Disunification, A Survey", in Logic and Automation, Lassez and Plotkin, eds., MIT, 1992.

....add C to B end The constraints passed into the put methods are then purely conjunctive. Finally, the set di#erence operation can be modeled either by taking negation into the constraint language, or by introducing subsumption checks at various places. We will not elaborate this here. See e.g. [Com91] for a survey on syntactic constraint solving methods. 4 Refinements The Incremental Closure approach has the desirable property that it can easily be refined in a number of ways. We stress this point, because incremental closure is surely not the answer to all problems in automated theorem ....

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, chapter 9, pages 322--359. MIT Press, Cambridge, MA, USA, 1991.

....under which a formula could have been discarded in the constraint. To do this, we have to require the constraint language to be closed under negation (denoted ) as well as conjunction. The resulting constraint satisfiability problems are known as dis unification problems, see e.g. [5], so I will talk of dis unification or DU constraints. 4 This might be enforced through a regularity condition that forbids the application of a rule that would lead to variant formulae on one of the extended branches. A little care has to be taken with the semantics of DU constraints: Some DU ....

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, chapter 9, pages 322--359. MIT Press, Cambridge, MA, USA, 1991.

.... x are the eliminable variables and y are the parameters. Colmerauer [4] gives an algorithm that will place a set of equations in rational solved form, or report unsatisfiability. Decision procedures for the theory of rational trees are given in [26, 5] This work and others is surveyed in [6]. The following equality theory ERT (dependent on Sigma) is given in [26] see also [14] as an axiomatization of the algebra of rational trees. In addition to the usual axioms of equality (i.e. reflexivity, symmetry etc) ERT contains axioms defined by the following schemes For every f 2 Sigma ....

H. Comon, Disunification: A Survey, in: Computational Logic: Essays in Honor of Alan Robinson, J-L. Lassez and G. Plotkin (Eds), MIT Press, 1991, 322--359.

....models and the free product in section 3 and the work on model completeness in section 4. The final two sections discuss solution compactness and the solution compact models of Clark s axioms respectively. Although an attempt has been made to keep this paper self contained, the references [3, 5, 17] should contain any missing definitions. 2 Preliminary Definitions Throughout this paper Sigma will denote a set of function symbols, each with a fixed associated arity. X denotes an infinite set of variables. Function symbols of arity 0 are also called constants. We will use a; b; c; d to ....

....free product. It now follows immediately from the final part of Theorem 2 that EFT or indeed any incomplete subset of EFT ; DCA is not model complete when Sigma is finite. Corollary 2. If Sigma is finite, EFT is not model complete. From the quantifier elimination procedure of Comon (see [6, 5]) every formula is equivalent in EFT ; DCA to an existential formula. It follows that every formula OE is also equivalent to a universal formula (since :OE is equivalent to an existential formula) and so, using Theorem 2, EFT ; DCA is model complete. Proposition 3. If Sigma is finite, EFT ; DCA ....

H. Comon, Disunification: A Survey, in: Computational Logic, J-L. Lassez & G. Plotkin (Eds.), MIT Press, 1991.

....example, if there is one constant symbol a and one binary function symbol f then the system f8u; v; u 0 ; v 0 f(x; y) 6= f(u; u) f(x; y) 6= f(f(u; v) f(u 0 ; v 0 ) g has an mgss which consists of the substitutions fx a; y f(x 0 ; y 0 )g and fx f(x 0 ; y 0 ) y ag. In [5] Comon gives an authoritative survey of the theory of the algebra of finite trees, which is the appropriate context for the work presented in this section. Elements of an mgss have also been studied under the names basic sets [18] basic formulas [29] and parameterized substitutions [30] and the ....

....cover for the calling patterns must consist only of linear terms, when each sort of the arguments is infinite. This provides a strong justification for the common requirement in functional languages that the patterns be, in fact, linear. A similar application of these results, pointed out by Comon [5], is that every ground convergent constructor based rewrite system is equivalent to a (ground convergent constructor based) rewrite system where all the patterns are linear. Term subtraction and the previous results are also relevant to the compilation of pattern matching in functional languages, ....

H. Comon, Disunification: A Survey, in: Computational Logic, J-L. Lassez & G. Plotkin (Eds.), MIT Press, 1991.

....interpretation I is such that I ; ae j= s t if and only if I [ s] ae = I [ t] ae, i.e. it is an interpretation in which I( is exactly the diagonal set f(v; v) j v 2 S I( g. Now we shall be interested in interpretations that are free, in that they will obey non confusion in the following way [Com91] Definition 1.2 An interpretation I is free if and only if: ffl if I [ c(s 1 ; s m ) ae = I [ d(t 1 ; t n ) ae, then c = d, m = n, and I [ s i ] ae = I [ t i ] ae for every i, 1 i m; ffl and if I [ x] ae = I [ t] ae, then either x = t or x is not free in t. 2 Design of ....

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

.... terms have become a popular tool because they allow to express and encode strategies and to modularise deduction processes [19] We call symbolic, constraints over terms [11] There are plenty of symbolic constraint systems, some examples are unification (see [16] for a survey) disunification [10], ordering [5] membership [9, 13] and feature constraints [1] The most well known example is equational unification. Equational unification is nothing but solving equations between terms when the function symbols of the terms satisfy a certain equational theory. Within programming languages ....

H. Comon. Disunification: a survey. In Jean-Louis Lassez and G. Plotkin, editors, Computational Logic. Essays in honor of Alan Robinson, chapter 9, pages 322--359. The MIT press, Cambridge (MA, USA), 1991.

....presentable, i.e. their generating set of equational axioms is infinite [McN92] If we consider theories also with other predicates than equality, we can think of theories generated by an infinite set of disequations or inequations. The former appear in complement problems and disunification [CL89, Com91], the latter appear in the presence of symbolic ordering constraints [Com90, Com93, KKR90] The Knuth Bendix completion procedure [DJ90] often generates an infinite family of rewrite rules in an attempt to produce a confluent and terminating rewrite system. Infinite sets of objects are not ....

H. Comon. Disunification: a survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic. Essays in honor of Alan Robinson, chapter 9, pages 322-- 359. MIT Press, Cambridge (MA, USA), 1991.

....and Section applies the result to the transformational approach to negation in higher order logic programs. We conclude in 9 with some speculation on future research. 2 Related Wok 2. 1 Complement Problems Complement problems have some relevant applications in theoretic computer science (see [Com91] for a list of references) for example in functional programming to produce non ambiguous function definitions by patterns and to improve 2 its compilation, in rewriting systems, to check whether an algebraic specification is sufficiently completeness,or to analyze communicating processes ....

....Consider a finite language L and a linear term t. NotL (x) NotL (f(t n ) fg(x) j for all g 2 Sigma distinct from fg [ ff(z 1 ; z i Gamma1 ; s; z i 1 ; z n ) j s 2 NotL (t i ) i 2 [1; n]g An alternative solution to the relative complement problem is disunification (see [Com91] for a survey and [Lug95] for an extension to the simply typed calculus) here operations on sets of terms are translated into conjunctions or disjunctions of equations and disequation under explicit quantifiers. Nondeterministic application of a few dozen rules eventually turns a given problem ....

H. Comon. Disunification: a survey. In J-L. Lassez and G.Plotkin, editors, Computational Logic. MIT Press, Cambridge,MA, 1991.

....solution in A Sigma and ( Gamma 3; Delta ; L) has a restrictive solution in B Delta . 6 Independence Properties of Equational Theories In equational theories, the only negative constraints are of course disequations. For a general introduction to disunification, we refer the reader to [8]. Unitary Theories Theorem 6.1 Let E be a unitary equational theory. Then E has the independence of negative constraints property. The reason for this theorem is that every solution oe i of Gamma C Gamma i is an instance of , the most general unifier of Gamma , and is also a ....

Hubert Comon. Disunification: A Survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic, pages 322--359. MIT Press, 1991.

....a solution in the initial algebra. For disunification, solvability in the initial algebra (called ground solvability in the following) implies solvability in the free algebra (simply called solvability below) but not vice versa. Both types of solvability are considered in the literature (see [Com91, Bur88]) but ground solvability seems to be more interesting for most applications. For solvability, the adaptation of the combination method to disunification problems is relatively straightforward. The main tool of the method is a decomposition algorithm which transforms every disunification problem ....

....solution. It should be noted that the notion of a disunification problem does not always refer to the same kind of problem in the literature. Our definition coincides with the one of Burckert [Bur88] who considers existentially quantified equational formulae, but other authors (e.g. Comon [Com91]) allow for arbitrary quantification. As in the case of unification, one has to distinguish several types of disunification problems. The (E; Sigma) disunification problem is called elementary, if Sigma = sig(E) it is a disunification problem with constants, if Sigma n sig(E) is a finite set ....

H. Comon, "Disunification: a Survey," in J.-L. Lassez, G. Plotkin (editors) , Computational Logic, MIT Press, 1991.

....to the work in this paper. At this point there are a number of questions raised by this approach which we hope to address. Can a significant portion of our relational machine be captured with a ChurchRosser, strongly normalizing set of rewrite rules Comon and Jouannaud and Kirchner s work [Comon, CoHaJo] on rewriting systems for unification and disunification suggests that this is quite feasible, as does the work of Bellia and Occhiuto, op. cit. We would also like to exploit the rich semantics of relational formalisms to obtain new notions of observables, and abstract interpretation, as well as ....

Comon, H. "Disunification, A Survey", in Logic and Automation, Lassez and Plotkin, eds., MIT, 1992.

....our fragment. We conclude in Section 8 with some applications and speculation on future research. For reasons of space, a number of lemmas and proofs are omitted here and can be found in [11] 2 Related Work Complement problems have a number of applications in theoretical computer science (see [4] for a list of references) For example, they are used in functional programming to produce non ambiguous function definitions by patterns and to improve their compilation, and in rewriting systems to check whether an algebraic specification is sufficiently complete. They can also be employed to ....

.... and a linear term t we define: Not Sigma (x) Not Sigma (f(t n ) fg( x) j for all g 2 Sigma distinct from fg [ ff(z 1 ; z i Gamma1 ; s; z i 1 ; z n ) j s 2 Not Sigma (t i ) i 2 [1; n]g An alternative solution to the relative complement problem is disunification (see [4] for a survey and [8] for an extension to the simply typed calculus) here operations on sets of terms are translated into conjunctions or disjunctions of equations and disequation under explicit quantifiers. Non deterministic application of a few dozen rules eventually turns a given problem ....

H. Comon. Disunification: A survey. In J.-L. Lassez and G.Plotkin, editors, Computational Logic. MIT Press, Cambridge,MA, 1991.

....problems in the obvious way. It should be mentioned that we consider here only one possible semantics for disunification problems. Often these problems are also solved over the ground term algebra modulo E, the initial algebra for E. For a more thourough description of disunification we refer to [Bur88, Com91]. An equational theory E is unitary if every elementary E unification problem fl has a most general unifier, i.e. a unifier such that for every unifier of fl there exists a substitution such that (x) E ( x) for all x 2 Var(fl) Let fl be an elementary E (dis)unification problem over the ....

H. Comon, "Disunification: a Survey," in J.-L. Lassez, G. Plotkin (editors), Computational Logic, MIT Press, 1991.

....a practical way to find efficient algorithms that solve the problem. Another interesting problem consists in solving the negation of equations, called disequations, i.e. in finding the solutions of s 6 = t modulo a theory. This problem is a particular case of the equational disunification (see [2] for a survey) when formulas to be solved are just disequations, instead of general first order formulas. Disunification has a large interest in the domain of automated deduction. In symbolic constraint solving, disunification problems appear as soon as negation is handled. From functional logic ....

....has to be solved modulo a theory. Few authors have studied equational disunification. Decidability results have been established for quasi free theories and compact theories [3] and for shallow theories [4] Other work gives algorithms that enumerate the solutions, as in syntactic theories [2], or modulo theories presented by a convergent set of rewrite rules [6] These enumerations do not terminate in general, therefore these works give semi decision procedures of existence of solutions. In this paper our goal is both to decide existence of solutions of disequations and to give a ....

H. Comon. Disunification: a Survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

....how convenient computational systems can be for expressing inference systems of common use in the algebraic and logic programming community. In fact, the rulebased approach of general deduction presented in [MOM93] as well as specific ones as advocated for unification [JK91] disunification [Com91] constraint solving [CDJK95] or theorem proving, can be realistically used in order to directly implement these concepts. This is made concrete using ELAN [BKK 96b] our implementation of computational systems, which is used for running all the examples presented now. ELAN, considered as a ....

H. Comon. Disunification: a survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic. Essays in honor of Alan Robinson, chapter 9, pages 322--359. The MIT press, Cambridge (MA, USA), 1991.

....with which some constraints are accompanied. Widely known studies among them are; unification problems with membership constraints[2] in which each variable in goal terms is associated with a regular tree language from which a term to substitute for the variable is chosen; disunification problems[3] in which some variables in goal terms must be substituted by different terms; unification problems with ordering constraints [4] in which terms to substitute for variables must satisfy given term ordering. SUPCS and TDunification problems can be regarded as two of such extensions of unification ....

H. Comon, "Disunification: a survey," in Computational Logic: Essays in Honor of Alan Robinson, pp.322-359, MIT Press, Boston, (1991).

....algebra. The previous deduction rules are actually parameterized by the constraint solving process. The next step is to describe it also as a computational system. 3 Constraint solvers as specific computational systems The description of constraint solving using rule based algorithms as in [Com91, JK91], allows easier correctness and completeness proofs of constraint solvers, partly thanks to the explicit distinction made in this approach between deduction rules and control. This is also the starting point for incorporating constraint solving in our framework. A constraint solver for symbolic ....

....language and a constraint solver. We are interested here in constraint solving processes that can be described with transition rules that compute solved forms of constraints. This includes for now constraint languages built from elementary constraints that may be equations (as above) disequations [Com91], inequations [Com90] on terms expressed with simplification orderings, membership constraints [CD91] This view has several advantages over constraint solving systems where solvers are encapsulated in black boxes. 1. Although completeness may be in some cases, like syntactic unification, proved ....

H. Comon. Disunification: a survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic. Essays in honor of Alan Robinson, chapter 9. MIT Press, Cambridge (MA, USA), 1991.

....P system is regular, Ax(P ) Nax(P ) allows to search for answers of open negative goals, basically using the computation model of SLD resolution. In [22] and later in [5] more targeted to logic programs) algorithms to compute the (relative) complement of first order terms are described (see also [11]) Those algorithms can be adapted to compute regular splitting, as informally described in the examples below. Example 40 Let us consider the program EVEN (3) over the signature Sigma EVEN = f0; sg of natural numbers. The failure axiom 8x(even(x) 9u(x = 0 true x = s(u) even(u) can be ....

H. Comon. Disunification: A Survey. In Computational Logic, J-L. Lassez and G. Plotkin, eds., pp. 322--359, MIT Press, 1991.

....a given equational theory. ffl Sorted unification, either many sorted or order sorted [111, 112, 99, 80, 104, 53] where type constraints are added to variables in equations. ffl Higher order unification [49, 82] which corresponds to solving equations between expressions. ffl Disunification [22], which corresponds to solving not only equalities but also negated equalities. ffl Solution of equalities and inequalities in a theory, as for example the solution of numerical constraints built into the constraint logic programming language CLP(R) 50] and in other languages. A remarkable ....

....one of applying transformations to a set or multiset of constraints. Furthermore, many authors have realized that the most elegant and simple way to specify, prove correct, or even implement many constraint solving problems is by expressing those transformations as rewrite rules (see for example [34, 53, 21, 22, 90]) In particular, the survey by Jouannaud and Kirchner [53] makes this viewpoint the cornerstone of a unified conceptual approach to unification. For example, the so called decomposition transformation present in syntactic unification and in a number of other unification algorithms can be ....

H. Comon, Disunification: A survey, in: J.-L. Lassez and G. Plotkin (eds.), Computational Logic: Essays in Honor of Alan Robinson, The MIT Press, 1991, pages 322--359.

....our fragment. We conclude in Section 8 with some applications and speculation on future research. For reasons of space, a number of lemmas and proofs are omitted here and can be found in [13] 2 Related Work Complement problems have a number of applications in theoretical computer science (see [5] for a list of references) For example, they are used in functional programming to produce non ambiguous function definitions by patterns and to improve their compilation and in rewriting systems to check whether an algebraic specification is sufficiently complete. They can also be employed to ....

.... Sigma and a linear term t we define: Sigma (x) Sigma (f(t n ) fg( x) j for all g 2 Sigma distinct from fg [ ff(z 1 ; z i Gamma1 ; s; z i 1 ; z n ) j s 2 : Sigma (t i ) i 2 [1; n]g An alternative solution to the relative complement problem is disunification (see [5] for a survey and [10] for an extension to the simply typed calculus) here operations on sets of terms are translated into conjunctions or disjunctions of equations 1 This notion of linearity should not be confused with the eponymous concept in linear logic and calculus. and disequation ....

H. Comon. Disunification: A survey. In J-L. Lassez and G.Plotkin, editors, Computational Logic. MIT Press, Cambridge,MA, 1991.

....convincingly the operations of generalisation (also called antiunification or disunification) and specialisation (unification) on such first order terms. This trend of ideas was followed by more work done in the direction of Machine Learning, where disjunction and negation were studied in [9] and [22] These ideas were studied in an unsorted world (corresponding to PROLOG terms for instance) althought these results usually extend to many sorted signatures. Nevertheless adding sorts is a reasonable step when wanting to describe the real world : any given term will have a meaning, ....

....We will study in this paper the negation of a term and of a disjunction of terms. For instance not (on top (red, ball, W) empty, on top (blue, W, W) on top (green, W, W) on top (W, cube, W) In the case of first order terms, and also using the closed world assumption, it is proved (in [9] or [22] that the explicit (positive) negation of a term may not be finite. The negation of a term f(x, x) is indeed infinite as soon as the set of all terms is so. A first way to make sure that negation introduces finite sets, is to consider constrained terms [9] where constraints concern the ....

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H. Comon, "Disunification: a survey". In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, MIT Press, pp 322-359, 1991.

....yields very simple reduction proofs. A first set of applications illustrating the method proposed is concerned with equational problems, that is validity of formulas with equality as the only predicate symbol in the initial, respectively the free algebra of an equational specification (see Comon (1991) and Burckert Schmidt Schau (1989) The first example (A) treats the decision problem for the theory of ground term algebras 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 ....

....holds. Comon (1988) remarks that the case of one AC function symbol plus one constant is equivalent to Presburger arithmetic and is therefore decidable. Using this idea the case of one AC function symbol plus a finite set of constants (called the theory of finitely generated multisets in Comon (1991)) can as well be reduced to the theory of Presburger arithmetic. Furthermore the theory of a ground term algebra modulo an empty set of equations has been shown to be decidable in Comon Lescanne (1989) and Maher (1988) 2 The decidability of the Sigma 1 fragment of the theory of ground term ....

Comon, H. (1991). Disunification: A survey. In Lassez, J.-L., Plotkin, G., editors, Computational Logic, chapter 9, pages 322--359. MIT Press.

....head is P (t 1 ; t n ) In the least fixed point of the set of clauses, the converse implication holds, hence, in this model, we have :P (x 1 ; x n ) OE Now, we may use a quantifier elimination procedure for the theory of finite trees and get a definition of :P . See e.g. [Com91] for more details on the quantifier elimination procedures) The only weakness is that, if some clauses contain variables in the body which do not appear in the head, then universal quantifiers cannot always be eliminated. However, if this is not the case, then the result is an I axiomatization. ....

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

....head is P (t 1 ; t n ) In the least fixed point of the set of clauses, the converse implication holds, hence, in this model, we have :P (x 1 ; x n ) OE Now, we may use a quantifier elimination procedure for the theory of finite trees and get a definition of :P . See e.g. [Com91] for more details on the quantifier elimination procedures) The only weakness is that, if some clauses contain variables in the body which do not appear in the head, then universal quantifiers cannot always be eliminated. However, if this is not the case, then the result is an I axiomatization. ....

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

....equations in the (free) term algebra, is known to be a fundamental operation in many areas of computer science and, in particular, in logic programming. Disunification, which consists in solving more complex formulae in the (free) term algebra, also revealed to be a fundamental operation (see [29, 13] for surveys on unification and disunification respectively) Recently, these computations have been seen as constraint solving in term algebras and this point of view is actually fruitful. 1 Constraints: a definition A constraint system is defined by a logical language C (which is in practice a ....

.... depending on the finiteness of F : when F is finite, the complete axiomatization consists of what is known as Clark s axioms of equality plus the domain closure axiom 8x; f2F 9 x f :x = f( x f ) Such formulae have been used for solving various problems in rewriting theory (see [13]) and as a constraint system in automated theorem proving, searching simultaneously for a proof and for a counter model [7] Equational formulae can be generalized in various directions. One of them consists in adding sort constraints, which is studied in the next section. Another generalization ....

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H. Comon. Disunification: a survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

....in this context. A typical example is the most well known problem of unification of terms, as it appears, e.g. in classical logic programming. Unification is nothing but solving conjunctions of equations over the set of terms, as we explain in section 2. We call symbolic, constraints over terms [4, 16]. Symbolic constraints have been used for years in computer science, with the following applications: ffl to represent sets of formulae. A constrained formula is a pair OE j C which stands for the set of instances of OE by the solutions of C. Consider arithmetic expressions built over the ....

....last 10 years, and resulted in a complete constraint solver mechanism for the first order theory of features on one hand, and in many prototype implementations of feature based languages on the other hand. Finally let us note that most of the efforts in symbolic constraint solving are surveyed in [16, 4] to which the reader is referred for more details. See also [15] for more recent work. 7 3 Semantic Methods Semantic methods, as opposed to syntactic ones, do not simply rely on a translation of the constraint syntax. They rather use another representation of the solutions of the constraint and ....

H. Comon. Disunification: a survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic. Essays in honor of Alan Robinson, chapter 9, pages 322--359. The MIT press, Cambridge (MA, USA), 1991.

....ffl disunification problems in which the formulae are conjunctions of equations and negated equations (called disequations) or more generally, arbitrary formulae involving no other predicate symbol than equality. Such formulae are interpreted in the free or quotient algebras of T (F ) See [6] for a survey) ffl membership constraints in which the formulae involve membership constraints of the form t 2 i where i belongs to an infinite set of sort expressions, generally built from a finite set of sort symbols, logical connectives and applications of function symbols. The membership ....

H. Comon. Disunification: a survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, pages 322--359. MIT Press, 1991.

.... Formulae with Membership Constraints Hubert Comon y and Catherine Delor Laboratoire de Recherche en Informatique et CNRS Bat. 490, Universit e de Paris Sud 91405 ORSAY cedex, France. E mail fcomon,delorg lri.lri.fr. Abstract We propose a set of transformation rules for first order formulae whose atoms are either equations between terms or ....

Hubert Comon, 1991. Disunification: a Survey. In Jean-Louis Lassez, Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press.

....function symbols, since the corresponding equational theory is indeed compact. Our results show that it is possible to use arbitrary first order formulas, interpreted in a quotient T (F ) E , as a constraint language, provided that =E is quasi free. Other applications are described in [Com91b]. For example, complement problems which express instances of finitely many terms which are not instances of another finite set of terms, are useful in many computational problems such as compiling pattern matching, automatic inductive proofs, logic program synthesis . We start in section 1 by ....

....every sentence OE, either T j= OE or T j= OE. Let us recall that a complete and recursively enumerable set of formulas has a decidable theory (see e.g. Sho67] for more details) Finally, if A is an F algebra, the (first order) theory 1 We use actually a set of normalization rules (such as in [CL89, Com91b, Com90a]) and assume that the formulas are kept in normal form with respect to these rules. Th(A) of A is the set of sentences over the alphabet F of function symbols and the only predicate symbol = which are true in A. Phi is an axiomatization of A if Th( Phi) Th(A) 1.5 Transformation Rules We ....

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

....of E [ E 0 . In the case of Horn clauses without equality, if, in each clause, the body of the clause does not contain variables which do not occur in the head, then it is possible to compute a normal axiomatization, using the quantifier elimination for the firstorder theory of finite trees [Comon 1991]. Let us show it on a very simple example: 5.14. Example. Let E be E = E(0) E(s(s(x) E(x) The program is rewritten as E(x) x = 0 (9y:x = s(s(y) E(y) In the least fixed point of the definition, the converse implication also holds, which gives by negating the two members: E(x) ....

Comon H. [1991], Disunification: a survey, in J.-L. Lassez and G. Plotkin, eds, `Computational Logic: Essays in Honor of Alan Robinson', MIT Press.

.... disunification problems in which the formulae are conjunctions of equations and negated equations (called disequations) or more generally, arbitrary formulae involving no other predicate symbol than equality. Such formulae are interpreted in the free or quotient algebras of T (F) See [6] for a survey) membership constraints in which the formulae involve membership constraints of the form t 2 i where i belongs to an infinite set of sort expressions, generally built from a finite set of sort symbols, logical connectives and applications of function symbols. The membership ....

H. Comon. Disunification: a survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, pages 322--359. MIT Press, 1991.

....example illustrates the problem of multiple upper bounds : if some term t is embedded in both g(u) and h(v) then it must be embedded in either u or v. 3 A first set of transformation rules The technique we will use for deciding the satisfiability of inequational formulae is now classical (see [2, 5]) it consists of rewriting the formula, using some rules which preserve the satisfiability, until the problem becomes trivially decidable. Our first set of rules is quite straightforward to derive. It is displayed in figure 1. Let us call R 0 this set of rules. All formulae are assumed to be kept ....

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

....defined as those axioms that decrease the size for all instantiations, such as x x x. Solving cycles is then performed by non deterministically guessing a collapsing rule along that path. Another difficulty arose when dealing with negation: the so called lemma of independence of disequations [5] is unsound if the ground term algebra is partitioned into finitely many congruence classes. On one hand, the latter case can be trivially solved. On the other, we must decide whether a shallow theory has finitely or infinitely many congruence classes in order to apply the appropriate technique. ....

H. Comon. Disunification: a survey. In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

No context found.

H. Comon, "Disunification: a survey", in Computational Logic: Essays in Honor of Alan Robinson, eds J.-L. Lassez and G. Plotkin (MIT Press, 1991) to appear.

....and negation do not appear. Quantifiers occur of course naturally in the expression of various problems, hence more general constraints arise in many applications. Arbitrary first order constraints built upon the equality predicate interpreted over terms are called equational constraints [7]. For example, a method for building A METHODOLOGICAL VIEW OF CONSTRAINT SOLVING 9 The unification transformation rules are parameterized by the vocabulary used for building expressions, called terms. We use f and g for arbitrary function symbols, x and y for variables, and s and t for ....

....10 H. COMON, M. DINCBAS, J. P. JOUANNAUD, C. KIRCHNER counter examples in theorem proving uses equational constraints with quantifiers and negations [4] Jouannaud and Kounalis method for inductive theorem proving [30] uses equational constraints with universal quantifiers and negation (see [7]) Equational constraints also appear in learning from examples and counter examples as described by Lassez and Marriott [35] Equational constraints can be reduced to a solved form, as shown independenlty by Comon and Maher [11, 36] These solved forms are purely existential, hence quantifier ....

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H. Comon. Disunification: a survey. In Jean-Louis Lassez and G. Plotkin, editors, Computational Logic. Essays in honor of Alan Robinson, chapter 9, pages 322--359. The MIT press, Cambridge (MA, USA), 1991.

No context found.

H. Comon, "Disunification: a survey". In J.-L. Lassez and G. Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, MIT Press, pp 322-359, 1991.

No context found.

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, chapter 9, pages 322--359. MIT Press, Cambridge, MA, USA, 1991.

No context found.

H. Comon. Disunification: A survey. In Computational Logic: Essays in Honor of Alan Robinson. MIT Press, Cambridge, MA, 1991.

No context found.

Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson, chapter 9, pages 322--359. MIT Press, Cambridge, MA, USA, 1991.

No context found.

H. Comon. Disunification: a Survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. The MIT Press, Cambridge, Mass., 1991.

No context found.

H. Comon. Disunification: a Survey. In Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

No context found.

H. Comon, Disunification: A survey, in: J.-L. Lassez and G. Plotkin (eds.), Computational Logic: Essays in Honor of Alan Robinson (The MIT Press, 1991) 322--359.

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Comon, H., Disunification: A survey, in: J. Lassez and G. Plotkin, eds., Computational Logic: Essays in Honour of Alan Robinson (MIT Press, Cambridge, MA, to appear).

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Hubert Comon. Disunification: a survey. In Jean-Louis Lassez and Gordon Plotkin, editors, Computational Logic: Essays in Honor of Alan Robinson. MIT Press, 1991.

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