F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theoretical Computer Science, 43:189--200, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....V is a denumerable set of logical variables disjoint from Sigma. If t is a term, then vars(t) denote the set of variables occurring in t. Capital letters X;Y; Z, etc. are used to represent variables; f , g, etc. stand for function symbols (i.e. elements of F) We assume the usual notions (e.g. [11]) of T unifier, complete set of T unifiers, and related notions involved in the general theory of unification w.r.t. an equational theory T. There are three main ways of representing sets as terms: ffl using the binary union symbol [ as the set constructor, as done, for instance, in [2, 5, ....

F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theoretical Computer Science, 43:189--200, 1986.

....is infinite. If there is a finite complete set then a minimal complete set al..ways exists and can be found by filtering out non minimal unifiers through matching. For some infinite cases, the properties of minimality and completeness may conflict, so that no minimal and complete set exists [Fages 83a] For completeness, it may be necessary for an E unification algorithm to use more variables in the range of the unifiers than occur in the terms being unified. Because unification procedures are often used within a larger system containing variables of its own, it is useful to require an ....

F. Fages and G. Huet, "Complete Sets of Unifiers and Matchers in Equational Theories," Trees in Algebra and Programming, CAAP '83, Proceedings of the 8th Colloquium, L'Aquila, Italy, Lecture Notes in Computer Science, Springer-Verlag, March 1983, pp. 205-220.

....of solutions should be as small as possible and complete, i.e. for each solution there should be a solution oe in Sigma such that oe is equal to under E or can be instantiated to a substitution which is equal to under E. Unfortunately, minimal complete sets of solutions do not always exist [Fages and Huet, 1986]. One possibility to compute a complete set of solutions for a given E unification problem is to add to the equational theory E its axioms of equality and apply resolution (see RESOLUTION) However, such an E unification algorithm would compute many irrelevant and redundant solutions since the ....

F. Fages and G Huet. Complete sets of unifiers and matchers in equational theories. Journal of Theoretical Computer Science, 43:189--200, 1986.

....then (7) is an SLDE resolvent of (5) and (6) As an immediate consequence of the results of Jaffar et al. Jaffar et al. 1984; Jaffar et al. 1986] we find that the success set with respect to SLDE resolution is equal to the least Herbrand E model of an equational logic program. As Fages Huet [Fages and Huet, 1986] showed, the concept of a most general unifier can only be generalized to a complete set of EP unifiers if EP 6= Such a complete set of EP unifiers for P (s 1 ; s n ) and P (t 1 ; t n ) is nothing else as a complete set of solutions for the problem of whether there exists a ....

Fages, F. and Huet, G. (1986). Complete sets of unifiers and matchers in equational theories. Journal of Theoretical Computer Science, 43:189--200.

....procedure can be obtained from the results of [BS96] 8 If T is the equational theory constituted by the permutativity and absorption axioms (E p ) E a ) see Section 2. 3) this algorithm computes, for each given Herbrand system E , a complete set of T unifiers for E (refer to, e.g. [FH86] for the notions of T unifier and complete set of T unifiers, as well as related notions involved in the general theory of unification with respect to an equational theory T) In this context, is used to express equality with respect to the theory T. We briefly recall the main ideas underlying ....

F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theoretical Computer Science, 43:189--200, 1986. 46

....omit the superscript X if X is the set of all variables. 2.2 Unification When you do equational unification, you are trying to find values for variables so that terms become equal in some equational theory; thus, equational unification is just equation solving in various algebras. Fages and Huet [4] give crisp definitions of some basic concepts; Siekmann [25] and Jouannaud and Kirchner [10] give broad surveys. Definition 1 An E unification problem is a finite set of pairs of terms. Each pair hs; ti is called an equation and is usually written s = E t. A substitution S is an E unifier of ....

F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theor. Comput. Sci. 43:189--200, 1986.

....Improved General E Unification Method 5 We cannot hope for a set of transformations for general E unification which performs as well as those for syntactic unification. E unification is undecidable even under stringent conditions on E, most general E unifiers do not necessarily exist, and in fact Fages and Huet (1986) have shown that there are equational theories E and systems S which do not possess E minimal unifiers. Consequently we say that an E unification procedure is complete (for E) if for every system S and every substitution which E unifies S , there is a computation on S yielding an E unifying ....

Fages, F., and Huet, G. (1986), Complete sets of unifiers and matchers in equational theories, Theoretical Computer Science 43, 189--200.

....T unifiable if and only if there is a substitution # such that s# = T t#; such a # is called a T unifier. The set of all T unifiers of two terms s and t is denoted by U T (s, t) 5 The Journal of Functional and Logic Programming 1997 7 Arenas et al. Minimality Study 2 Following [FH86] given a congruence = T on T# , we write, for any set of variables W # V , # = W T # if and only if #x # W x# = T x#, for any two substitutions # and #. In the same way, # is more general than # in T over W (# # W T #) if and only if ## # = W T # # #. 1 The ....

.... is any complete set of T unifiers CSU T (s, t) satisfying the condition (### # CSU T (s, t) # #= # # # ## W T #) where W = Var(s) # Var(t) Note that if CSU T (s, t) exists and is finite for any s, t # T# (V) then CSU T (s, t) exists and is unique up to # V T ( FH86] Definition 5 (T Unifiable Herbrand Systems) A Herbrand system E t 1 . s 1 , t n . s n is T unifiable if and only if there is a substitution # such that t i # = T s i #, for all 1 # i # n. In such a case, # is called a T unifier for E. The set of all T ....

F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories.<F4.877e+05> Theoretical Computer<F5.38e+05> Science, 43:189--200, 1986.

....to find an appropriate reduction [Val79b, Gal74] Moreover, not all NP decision problems have counting counterparts that belong to the class #P. For example, AC unification as a decision problem is NP complete [KN92] but there are AC unification problems whose minimal complete set of unifiers (see [FH86] for the definition) has double exponential cardinality [Dom92] This situation is due to the fact that the decision problem asks for the existence of a unifier, not necessarily a member of the minimal complete set. On the other hand, computing the (minimal) complete set of unifiers is a crucial ....

F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theoretical Computer Science, 43(1):189--200, 1986.

....unifiability is decidable; some partial results have been obtained in [NPS89] see Section 6. 2 Unification Matching in Equational Theories Unification theory is a young discipline, and notation and definitions vary between different authors. I have picked my notation from several sources, mostly [HO80, FH86, Tid86, Kir86, Sie89]. When the definitions vary, which they do only in details, I have picked one that seemed appropriate, though I stay close to [FH86] This section is sparse in motivations and is not intended as an introduction to the subject. I have found the introductory parts of [Tid86] readable for a beginner, ....

....is a young discipline, and notation and definitions vary between different authors. I have picked my notation from several sources, mostly [HO80, FH86, Tid86, Kir86, Sie89] When the definitions vary, which they do only in details, I have picked one that seemed appropriate, though I stay close to [FH86]. This section is sparse in motivations and is not intended as an introduction to the subject. I have found the introductory parts of [Tid86] readable for a beginner, and [Sie89] is a survey article that contains an extensive bibliography. Let F be a set of function symbols, and let V be a ....

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F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theoretical Comput. Sci., 43:189--200, 1986.

....member of Th(F ; E) An E unifier of s and t is a substitution ae such that sae =E tae holds; equivalently, an E unifier of s and t is a solution of the equation s : E t in the algebra T (F ; X ) E . If a minimal complete set of E unifiers of s and t exists, then it is unique up to j V E (cf. [FH86]) In this case, we let CSUE (s; t) denote the minimal complete set of E unifiers of s and t, if s and t are unifiable, or the empty set, otherwise. A theory E is said to be unitary if for every pair of terms (s; t) the set CSUE (s; t) exists and j CSUE (s; t)j 1. Similarly, E is said to be ....

F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theoretical Computer Science, 43(1):189--200, 1986.

....(that is, multiplication, inverse, and the identity element) My algorithm can probably not improve dimension derivation until these restrictions are removed. 2 Semi unification in equational theories 2. 1 Terms, substitutions, and equivalence I assume familiarity with terms and substitutions [3, 11, 15]. V(t) is the set of variables in the term t, D(oe) is the set of variables changed by the substitution oe, and I(oe) is the set of variables introduced by oe (occurring in some term oe(x) where x 2 D(oe) If X is a variable set, oe X is a substitution such that (oe X ) x) if x 2 X then ....

....set X as parameter: oe = X E iff oe(x) E (x) for all x 2 X; oe X E iff ae ffi oe = X E for some ae; oe j X E iff oe X E and X E oe: 2.2 Semi unification I am not aware of any earlier work on equational semi unification. My definitions combine equational unification [3, 11, 15] with ordinary, or syntactic, semiunification [5, 8, 9] If you let E = in my definitions, you get syntactic semiunification (since =E is then syntactic equality) Definition 1 Let S be a system of inequations fs 1 E t 1 ; s n E t n g; then oe E semi unifies S iff oe(s i ) E ....

[Article contains additional citation context not shown here]

F. Fages and G. Huet, Complete sets of unifiers and matchers in equational theories, Theor. Comput. Sci. 43 (1986) 189--200.

....stating that no refuter is an instance of another can be added to the definition of CSR. However it is not required for correctness or completeness issues; even more it may be advisable to leave minimality away, as there are cases where a complete set of minimal refuters may not exist (See (Fages and Huet, 1986) for a proof in the context of theory unifiers) 3.2.2 Formal Definition of Theory Consolution Being equipped with the definition of the consolution calculus, a generalization to a theory consolution calculus is fairly straightforward now. At first the notion of product has to be generalized: the ....

Fages, F. and Huet, G. (1986). Complete Sets of Unifiers and Matchers in Equational Theories.

....subset is complete. A least element is an element a 2 S such that a a 0 for all a 0 2 S. Obviously, complete subsets of S always exist, since S is a complete subset of itself. Minimal complete subsets, however, need not exist, but if they do exist, any two of them have equal cardinality [FH86]. If S 0 S is a minimal complete set, then every a 2 S 0 is minimal. A minimal complete subset of S exists if and only if the minimal elements of S form a complete subset. In other words, S has a minimal complete subset if and only if for every a 2 S there is a minimal element a 0 2 S such ....

F. Fages and G. Huet. Complete sets of unifiers and matchers in equational theories. Theoretical Computer Science, 43(2,3):189--200, 1986.

....generates a minimal complete set of A unifiers of a given A unification problem over an arbitrary set of function symbols F , which shows that A is in fact infinitary and not of type zero. 3.8. Example (zero) The first example of an equational theory of unification type zero was described by Fages and Huet [1983] and [1986]. In [Baader 1986] it is shown that the theory of idempotent semigroups, i.e. AI : A [ ff(x; x) xg is of unification type zero since the AI unification problem 28 Franz Baader and Wayne Snyder ff(x; f(y; x) AI f(x; f(z; x) g does not have a minimal complete set of AI unifiers. This ....

Fages F. and Huet G. [1986], `Complete sets of unifiers and matchers in equational theories', Theoretical Computer Science 43(1), 189--200.

....generates a minimal complete set of A unifiers of a given A unification problem over an arbitrary set of function symbols F , which shows that A is in fact infinitary and not of type zero. 3.8. Example (zero) The first example of an equational theory of unification type zero was described by Fages and Huet [1983] and [1986] In [Baader 1986] it is shown that the theory of idempotent semigroups, i.e. AI : A [ ff(x; x) xg is of unification type zero since the AI unification problem 28 Franz Baader and Wayne Snyder ff(x; f(y; x) AI f(x; f(z; x) g does not have a minimal complete set of AI ....

....guesses 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 ....

Fages F. and Huet G. [1983], Complete sets of unifiers and matchers in equational theories, in `Proceedings of the 5th Colloquium on Automata, Algebra, and Programming', Vol. 159 of Lecture Notes in Computer Science, Springer-Verlag, pp. 205--220.

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