Francois Fages. Associative-commutative unification. Journal of Symbolic Computation, 3:257--275, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....appendices are included. Appendix A gives some additional technical details of the proof of correctness, Appendix B gives a glossary of some terms used in this thesis, and Appendix C gives a list of special symbols and there uses. The following definitions are consistent with the definitions of [Fages 84] and [Huet 80a] begin with basic definitions of terms and functions on terms. We Let V be a countable set of variables and F be a family of function symbols with associated arity such that V and F are disjoint. We recursively define the set of terms, T(F, V) as either a variable or a function ....

....to be complete if and only if it generates all unifiers: VO(UE(t,s ) o Z O V Definition. Let : be a set of unifiers of t and s and V = t)Us) is minimal if and only if no substitution in T is redundant: v E When it exists, a minimal complete set of unifiers is unique up to = for any [Fages 84] The size E of the minimal complete set is bounded for certain values of E. If E = 0, there is always a singleton complete set for any two unifiable terms. If E contains only the associative and commutative axioms (the AC theory) then the complete set is always finite. If E contains only the ....

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F. Fages, "Associative-Commutative Unification," Proc. 7th CADE, Napa Valley, Springer-Verlag, 1984, pp. 194-208.

.... extensively cf. 11,23] simulation is called path inclusion in [23] Several unification methods have been proposed that, like simulation unification, process flexible terms or structures, notably feature unification [24,25] and associative commutative unification, short AC unification, [26]. Simulation 16 unification di#ers from feature unification in several aspects: First, it accepts terms with several identically rooted terms corresponding to a feature with more than one value. In contrast, feature logic languages and feature unification usually preclude this: A term (in ....

Fages, F.: Associative-Commutative Unification. In: Proc. 7th Int. Conf. on Automated Deduction (Napa, CA). Volume 170., Berlin, Springer (1984) 194--208

.... extensively cf. 9,20] simulation is called path inclusion in [20] Several unification methods have been proposed that, like simulation unification, process flexible terms or structures, notably feature unification [21,22] and associative commutative unification, short AC unification, [23]. Simulation unification di#ers from feature unification in several aspects (discussed in [8] Simulation unification might remind of theory unification [24] The significant di#erence between both is that simulation unification is based upon an order relation, while theory unification refers to ....

Fages, F.: Associative-Commutative Unification. In: Proc. 7th Int. Conf. on Automated Deduction (Napa, CA). Volume 170., Berlin, Springer (1984) 194--208

....3.4. In the transformation of an AC1 unification problem, atomic AC1 unifi cation problems of various forms occur. The most difficult case consists of a problem with the same AC symbol on both sides. The decomposition step for this case forms the heart of most AC unification procedures. In [Fag 84] an algorithm to decompose these atomic AC1 unification problems is presented. Based on this algorithm, we define a simplifier for AC unification problems. Definition 3.3 A simplifier for AC unification problems is a partial function dio AC : CAC Theta 2 V 2 CAC such that for all f 2 FAC ....

....=s k;i ; 9 = CHAPTER 3. UNIFICATION 24 where Sol(9 z: f(X) f(Y ) h2[1: k] Sol(9 z h : i2[1: n h ] r h;i =s h;i ; and for all h 2 [1: k] for all i; i 0 2 [1: m h ] 1. z z h , 2. r h;i 2 X [ Y (see the algorithm trans in [Fag 84] 3. r h;i 6= r h;i 0 (see the algorithm elimcom in [Fag 84] 4. if r h;i 2 V then s h;i 2 X [ Y or s h;i = f(Z) Z (V Gamma W ) X [ Y (see the algorithm trans in [Fag 84] 5. if r h;i 62 V , then r h;i and s h;i 2 X [ Y (see Proposition 1 in [Fag 84] Table 1 shows the set AC Unify of ....

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F. Fages. Associative-Commutative Unification. In: Proceedings 7 th International Conference on Automated Deduction, Napa Valley (California, USA), R. Shostak, editor, Lecture Notes in Computer Science 170, pp. 194--208, Springer-Verlag, 1984.

....do actually show that the presentation reduced to the distributivity axiom is resolvent and that in this particular case, the unification process terminates. Surprisingly, while associativity commutativity is doubtless the theory for which unification has been the most extensively investigated [19, 21, 15, 9, 10, 12, 5, 4, 3, 1, 14, 8, 2], it has not been taken advantage of the syntacticness for AC unification. The problem is that the syntactic method, while it constructs all the AC solutions, does not terminate. Actually, an attempt has been made by Franzen and Henschen in 88 who already use a resolvent presentation of AC for ....

Francois Fages. Associative-commutative unification. Journal of Symbolic Computation, 3(3), June 1987.

....all alien subterms. The remaining steps are obvious. Proof of claim 2: AC Unification with Linear Constant Restriction. It is a well known fact that solving AC unification problems with constants can be reduced to solving systems of linear equations over the nonnegative integers (see e.g. [St81, Fa84]) As an easy consequence one can show that solvability of AC unification problems with linear constant restriction can be expressed as an integer programming problem, thus establishing NP decidability. Instead of giving a formal presentation of this reduction, we shall illustrate it by an ....

F. Fages, "Associative-Commutative Unification," Proceedings of the 7th International Conference on Automated Deduction, LNCS 170, 1984.

.... unification is decidable, but unification with constants is undecidable (see [Bu86] ffl From the development of the first algorithm for AC unification with constants [St75, LS75] it took almost a decade until the termination of an algorithm for general AC unification was shown by Fages [Fa84]. The applications of theory unification mentioned above require algorithms for general unification. This fact is illustrated by the following example. Example 1.1 The theory A = ff(f(x; y) z) f(x; f(y; z) g only contains the binary symbol f . When talking about A unification, one first thinks ....

....AC [ AC for computing critical pairs. When considering the combination problem, until now the attention was mostly restricted to finitary unifying theories, and by unification algorithm 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 ....

F. Fages, "Associative-Commutative Unification," Proceedings of the 7th International Conference on Automated Deduction, LNCS 170, 1984.

.... t 0 s t (s[t 0 ] p t)oe 8 oe 2 cU AC (sj p ; s 0 ) i.e. one conclusion is added for each oe in cU AC (sj p ; s 0 ) a minimal complete set of ACunifiers of sj p and s 0 . This has motivated a huge amount of research on computing complete sets of AC unifiers (e.g. Stickel, 1981; Fages, 1987; Fortenbacher, 1987; Burckert et al. 1988; Kirchner, 1989; Lincoln and Christian, 1989; Boudet et al. 1990; Domenjoud, 1992a; Kapur and Narendran, 1992b ; see Baader and Siekmann, 1993 for a recent survey on (AC )unification) One drawback is the complexity of AC unification: there may be ....

Fages, F. (1987). Associative-commutative unification. Journal of Symbolic Computation, 3(3).

....expressions are viewed as functions. 3.3 Associative Commutative Unification The associative commutative unification (or unification under the A, C and A C theory) is the unification problem in the presence of associative and or commutative functions. The improved version of Stickel s algorithm [7] was chosen for implementation in AlgBench. As it is common in such algorithms [4] the problem is reduced to the solution of (in this case linear homogeneous) diophantine equations. For the declaration of associative and commutative functions we provide like in Mathematica the command ....

F. Fages. Associative commutative unification. J. of Symbolic Computation, 3:257--275, 1987.

....together satisfy the conditions of Lemma 3. Procedure Unify Conjunction is justified by the fact that CSU(e1 : en) T , where T = S 2CSU(e 1 ) CSU(e2 : en ) Finally, the completeness of procedure Matrix Solve was already established by Lemma 3. 2 6. 2 Termination Fages work [3] is witness to the difficulty of demonstrating termination in the general case of AC unification. We have discovered that some special mechanism is required in order to assure termination of our algorithm in the case that both terms contain repeated variables. We have come up with two ....

F.Fages. "Associative-Commutative Unification". Journal of Symbolic Computation, Vol. 3, Number 3, June 1987 pp 257-275.

....into a set al..l the parts supplied by every supplier. Bounded set terms are of the form fe 1 ; e n g, where e 1 ; e n are (not necessarily ground) terms corresponding to the elements of the set. Bounded set terms as defined above are special cases of set terms as known in the literature [8, 10, 11, 12, 16, 18, 19, 21, 25, 26, 32, 33, 34, 37]. Indeed, the classical definition of set term assumes that enumeration of the elements in a set term can be partial, thus a set term may have the form feg [ s so that it contains the element e in addition to those in the set s moreover, to recursively decompose a set into strictly smaller ....

....a tight time complexity lower bound for the problem and we present an optimal algorithm that performs weak unification of bounded simple set terms. The problem of unifying commutative, idempotent terms has been deeply investigated by taking into account an additional property, associativity, [8, 12, 18, 25, 21, 32]. An excellent review of significant results in this area has been provided by Siekmann [32] see also [21] various complexity results are presented by Kapur and Narendran in [18] In particular they prove that both unification and matching of terms with commutativity and idempotency are ....

F. Fages, Associative-Commutative Unification, 8th Int. Conf. on Automated Deduction, 1986, 416-430.

....database languages. This is because in set term unification it is no longer true that a most general unifier (mgu) of two set terms is unique modulo renaming of variables. Unification algorithms for functions with the associativity, commutativity, and idempotency properties have been proposed [Sti81, Fag84, LC88]; however, these algorithms cannot be adopted to handle set term unification [STZ92] since functions and set terms have different semantics. In [DOPR91, Sto93] set term unification algorithms that compute unifiers for a given pair of set terms with commutative and idempotent properties were ....

F. Fages. Associative-Commutative Unification. In Proceedings of 7th International Conference on Automated Deduction, pages 194--208, 1984.

....f(x; x) x Table 2. 5: Common equational axioms Name Decidable Some References Phi yes [Robinson, 1965; Paterson and Wegman, 1978; Martelli and Montanari, 1982] A(f) yes [Plotkin, 1972; Makanin, 1977] C(f) yes [Siekmann, 1979; Kirchner, 1986] I(f) yes [Hullot, 1980] A(f) C(f) yes [Stickel, 1981; Fages, 1984; Kirchner, 1989] A(f ) I(f) yes [Siekmann and Szab o, 1984; Baader, 1986] C(f ) I(f) yes [Jouannaud et al. 1983] A(f) C(f ) I(f) yes [Baader and Buttner, 1988] Dr(f; g) yes [Arnborg and Tid en, 1985] Dl(f; g) yes [Arnborg and Tid en, 1985] D(f; g) A(f) no Szabo D(f; g) A(f) C(f) no Szabo D(f; g) ....

F. Fages. Associative-commutative unification. In Proceedings of the Seventh International Conference on Automated Deduction, 1984. Volume 170, pages 194--208, of Lecture Notes in Computer Science, Springer Verlag.

....assumes that s is not a variable and its size is less than or equal to that of t. Can this condition be improved by replacing it with the condition that the rule Check does not apply (In other words, is Check complete for finding cycles when Merge is modified as above ) Problem 40. Fages [32] proved that associative commutative unification terminates when variable replacement is made after each step. Boudet, et al. 16] have proven that it terminates when variable replacement is postponed to the end. Does the same (or similar) set of transformation rules terminate with more flexible ....

F. Fages. Associative-commutative unification. J. Symbolic Computation, 3(3):257--275, June 1987.

.... 1981; Herold Siekmann, 1987 ] alternative algorithms with better performance are [ Kirchner, 1989; Boudet, 1989 ] Other theories for which algorithms are available that compute finite, complete sets of most general E unifiers include commutativity, AC with identity and or idempotency (see [ Fages, 1987 ] as well as Boolean rings (see [ Boudet etal, 1988 ] For many of these theories, unification is believed intractable from the time complexity point of view [ Kapur Narendran, 1986 ] Of course, E unifiability is semi decidable for recursively enumerable E. Paramodulation (without the ....

F. Fages, Associative-commutative unification, J. of Symbolic Computation 3 (3), pp. 257-275 (June 1987).

....and then introduced a special purpose combination algorithm for general AC unification (actually, with several AC symbols) that used the algorithm for elementary AC unification and the algorithm for syntactic unification as subroutines. The termination of Stickel s algorithm was proved by Fages [Fag87]. Similar work was carried out 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 ....

F. Fages. Associative commutative unification. Journal of Symbolic Computation, 3(3):257--275, 1987.

....When they have to be unified with an expression, the specialized unifier for the corresponding class is called. As a result the algorithm works also for new coming classes of pattern objects. All other composite expressions are viewed as functions. The improved version of the Stickel s algorithm [5] was chosen for implementation of associative commutative unification. For the declaration of the ac functions serves the command SetAttributes[symbol, attribute] The attribute is called like in Mathematica Flat for the associative case and Orderless for the commutative case. A mechanism ....

....the user the opportunity to define inheritance hierarchies of types via a subtype command: Subtype[list1 , list2 ] where list1 , list2 are lists of object types. Consider the following example: In[1] Subtype[f, g, h, i] In[2] Subtype[h, j] In[3] Subtype[ g, i, k] In[4] f[x] x In[5]: g[y] x y The type extension is expressed in the definition of g, where the arguments of f are inherited implicitly. With the subtype command we allow user defined type hierarchies within composite expressions, a user defined lattice. Definition: Two object types F,G unify iff there exists a ....

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F. Fages. Associative Commutative Unification. Journal of Symbolic Computation, 3, 257-275, 1987.

.... 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 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 ....

Fages, F. (1987). Associative-commutative unification. Journal of Symbolic Computation, 3(3):257--275.

....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 ....

.... 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 combination problem. What we intend to do here is not to re design a combination procedure for our specific purpose but to reuse ....

F. Fages. Associative-commutative unification. Journal of Symbolic Computation, 3(3), June 1987.

.... polynomial time using linear programming [Domenjoud 1991] Unification theory 41 Unification type: ACU f is unitary for elementary and finitary for all other kinds of unification problems, and AC f is finitary for all three kinds of unification problems [Livesey and Siekmann 1975, Stickel 1981, Fages 1987]. The number of unifiers in a minimal complete set of AC f unifiers may be doubly exponential in the size of a given elementary AC f unification problem [Kapur and Narendran 1992b] Unification algorithm: Because unification modulo associativity commutativity has many applications in automated ....

.... unification (which treats the free function symbols) The research on the combination problem was triggered by the search for a unification algorithm that can deal with terms containing several associativecommutative function symbols and free symbols [Stickel 1975, Stickel 1981, Fages 1984, Fages 1987, Herold and Siekmann 1987] It turned out that the methods used in this particular instance of the combination problem can easily be generalized to other equational theories, provided that they satisfy certain restrictions (such as collapse freeness or regularity 19 ) on the syntactic form of ....

Fages F. [1987], `Associative-commutative unification', J. Symbolic Computation 3, 257--275.

.... for syntactic unification (which treats the free function symbols) The research on the combination problem was triggered by the search for a unification algorithm that can deal with terms containing several associativecommutative function symbols and free symbols [Stickel 1975, Stickel 1981, Fages 1984, Fages 1987, Herold and Siekmann 1987] It turned out that the methods used in this particular instance of the combination problem can easily be generalized to other equational theories, provided that they satisfy certain restrictions (such as collapse freeness or regularity 19 ) on the ....

Fages F. [1984], Associative-commutative unification, in R. Shostak, ed., `Proceedings of the 7th International Conference on Automated Deduction', Vol. 170 of Lecture Notes in Computer Science, Springer-Verlag, New York, pp. 194--208.

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Francois Fages. Associative-commutative unification. Journal of Symbolic Computation, 3:257--275, 1987.

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Fages, F., "Associative-Commutative Unification," Journal of Symbolic Computation 3 (1987) 257--275.

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