Mark Stickel. A unification algorithm for associative-commutative functions. Journal of the ACM, 28(3):423--434, July 1981. 7  Home/Search   Document Not in Database   Summary   Related Articles   Check

....available, he was unable to prove that any of these conditions follow from the axioms. Then, in 1996, with the development of a new theorem prover called EQP [McCune, 1997b] the problem was cracked. The 133 step solution, which relied on a technique known as associativecommutative unification [Stickel, 1981], required 20 hours using 18 megabytes on a Unix workstation. The answer all Robbins algebras are Boolean took the world by storm [McCune, 1997a] It was covered by the New York Times [Kolata, 1996] and was cited as one of five major accomplishments in artificial intelligence by the NEC ....

....little experimentation in that period accounts for this fact the past two decades have seen impressive advances, in theory, in implementation, and in application. Included among the significant advances are the research and implementation of associative commutative unification algorithms [Stickel, 1981; Burckert et al. 1988; Siekmann, 1989; McCune, 1997b] the development of new strategies [Wos and McCune 1988; Wos, 1995; Wos and Pieper, 1999b; Veroff, 1996, 2001b] the formulation and use of linked inference rules [Veroff and Wos, 1990; Veroff, 2001a] and the design and implementation of ....

M. Stickel. A unification algorithm for associative-commutative functions. J. ACM, 28:423--34, 1981.

....of combining decision procedures for universally quantified formulae, i.e. arbitrary Boolean combinations of equations that are universally quantified, was solved by Nelson and Oppen [8] in 1979. Work on combining unification algorithms started also in the seventies with Stickel s investigation [12] of uni fication of terms containing several associative commutative and free symbols. The first general result on how to combine decision procedures for unification was published by Baader and Schulz [1] in 1992. It turned out that decision procedures for unification (with constants) are not ....

M. E. Stickel. A unification algorithm for associative commutative functions. J. ACM 28, 1981.

....82, Baxter 73] and linear [Paterson 78] Equational unification procedures, and in most cases terminating algorithms, are currently known for a number of equational theories. Some examples include procedures for the associative [Plotkin 72] commutative [Siekmann 79] associative commutative [Stickel 81, Livesey 76] identity [Arnborg 85] and one sided distributive [Arnborg 85] theories. The unification problem in some theories, e.g. the associative distributive theory, has been proven undecidable [Szab o 78] In others, such as the associative theory, no terminating procedure exists [Makanin ....

M. E. Stickel. A Unification Algorithm for AssociativeCommutative Theories. Journal of the ACM, 28(3):423--434, July 1981.

....and intuitive; one disadvantage of the [Corbin 83] approach is that it depends heavily on a data structure for terms that may or may not be appropriate within an application. Some of the currently known complete E unification algorithms are for commutative operators [Siekmann 79] AC operators [Stickel 81, Livesey 76] with termination in the multiple instance case proved in [Fages 84] signed trees [Kirchner 81] one sided distributivity [Arnborg 85] and transitivity [Kirchner 85] There are variations on the AC algorithm [Livesey 76, Fages 84] for AC with idem potence, and AC with a unit ....

.... presented abelian group finitely presented boolean ring 1 3: Some Common Equational Theories Theory A Variable Only Free Symbols [Robinson 71] and others [Plotkin 72] Plotkin 72] procedure to enumerate unifiers Multiple Instance open C [Siekmann 79] Fages 83b] Fages 83b] AC [Stickel 81] Fages 83b] Fages 83b] I [Raulefs 78] Hullot 80] Hullot 80] Cl [Raulefs 78] Jouannaud 83] Jouannaud 83] Jouannaud 83] AI decidable [Szabo 78] open open ACI [Livesey 76] Jouannaud 83] Jouannaud 83] ACU [Livesey 76] Jouannaud 83] Jouannaud 83] D I (or Dr) Arnborg 85] open open D ....

[Article contains additional citation context not shown here]

M.E. Stickel, "A Unification Algorithm for Associative-Commutative Theories," Journal of the ACM 28(3):423.434, July 1981. Preliminary version in Proc. 4th Intl. Joint Conf. on Artificial Intelligence, Tbilissi, 1975.

....theory E cannot always be shown by normalisation, so the generalisation rules need extending, though where possible normalisation appears to be a better solution. Unification modulo E has been studied for many theories. Stickel s algorithm for associative commutative functions is a good example [15]. Generalisation relative to a background theory for atomic formulas has also been studied [11] Baader [1] defines E generalisation in the same framework as E unification. Promoting embedding equations to be more than passive suppliers of substitutions could give smaller generalisations. Yet ....

M E Stickel. A unification algorithm for associative-commutative functions. JACM, 28(3):423--434, July 1981.

....if the syntactic approach is a very elegant way to design unification matching algorithms or procedures, it does not lead in general to efficient implementations. This is obviously true for the AC theory. 3. 3 Diophantine Constraint Solving The existing efficient implementation of AC unification [47, 48, 12, 11, 2] needs to solve two hard subproblems: first, generate a basis of solutions for linear homogeneous Diophantine equations and second, search for a AC unifier through all combinations of solutions in this basis. Given an AC( equation t 1 = AC t 2 such that Var(t 1 ) fx 1 ; xm g and ....

M. E. Stickel. A unification algorithm for associative-commutative functions. Journal of the ACM, 28:423--434, 1981.

....most of the paramodulation strategies we considered, we ran experiments with ratios 1, 4, 8, and 1 (i.e. purely best first) We used a value of 4 for almost all of the experiments reported in [11] 4.5. 2 Associative Commutative (AC) Operations EQP has AC unification (Stickel s AC algorithm [16], with Huet s algorithm [4] for finding basis solutions) and AC matching (our own algorithm) The AC unification algorithm is quite complicated, but our implementation is straightforward. AC terms are stored as binary terms in canonical form, and they are flattened into arrays during the ....

M. Stickel. A unification algorithm for associative-commutative functions. J. ACM, 28(3):423--434, 1981.

....linear Diophantine equations on the nonnegative integers, or systems of such equations, had a significant increase in the recent past. This may be explained by their use in term rewriting techniques, namely in unification algorithms of terms with associative and commutative function symbols ([Stickel 1981]; Huet 1978] Guckenbiehl and Herold 1985] and also by the surge of programming languages and systems based on constraint solvers examples are Constraint Logic Programming languages under the CLP scheme ( Jaffar et al. 1986] Jaffar and Lassez 1987] and problem solvers emerging from ....

M. E. Stickel, A unification algorithm for associativecommutative functions. JACM, 28(3), 1981. 78

....most of the paramodulation strategies we considered, we ran experiments with ratios 1, 4, 8, and 1 (i.e. purely best first) We used a value of 4 for almost all of the experiments reported in [12] 5.5. 2 Associative Commutative (AC) Operations EQP has AC unification (Stickel s AC algorithm [18], with Huet s algorithm [5] for finding basis solutions) and AC matching (our own algorithm) The AC unification algorithm is quite complicated, but our implementation is straightforward. AC terms are stored as binary terms in canonical form, and they are flattened into arrays during the ....

M. Stickel. A unification algorithm for associative-commutative functions. JACM, 28(3):423--434, 1981.

....is a theorem prover written in CAML Light (18000 lines) a functional language of the ML family; it has a graphical interface written in Tcl Tk, X11 Toolkit based on the language Tcl. It runs on SUN, HP and IBM PC workstations. It uses an AC unification algorithm based on the algorithm of Stickel [27] and the technique for solving Diophantine equations of Fortenbacher [9] The algorithm for AC matching is inspired by the algorithm of Hullot [11] The ordering for comparing terms is the APO of Bachmair and Plaisted [5] with the improvements of Delor and Puel [7] Let us detail an example of ....

M. E. Stickel. A Unification Algorithm for Associative-Commutative Functions. Journal of the ACM, 28:423--434, 1981.

....3 5 (26) where e y denotes the variables which occur in oe = but not in e x . 18 As shown in [ Holldobler and Thielscher, 1995 ] a unification complete theory for our axioms AC1 can be obtained by computing, for each two terms s; t , some complete set cUAC1 (s; t) of AC1unifiers (see, e.g. Stickel, 1981; Buttner, 1986 ] and taking the corresponding equational formula which is to the right of the entailment symbol in (26) In what follows, this theory will be called extended unique name assumption, abbreviated EUNA. As an example, consider the terms heavy(x)ffiz and ....

Mark E. Stickel. A unification algorithm for associative commutative functions. Journal of the ACM, 28(3):207--274, 1981.

....languages such as Prolog taking advantage of built in backtracking. The algorithm unifies tuples of ACI terms, e.g. h 1 ; n i with h 0 1 ; 0 n i thus solving systems of ACI equations. This algorithm as well as all other published algorithms of general ACI unification [41] computes a complete but not necessary minimal set of unifiers. Of course a minimal complete set of unifiers can be obtained in a second phase by choosing the most general unifiers from the computed set. As far as we know the direct, one phase and efficient computation of a minimal complete set of ....

M. E. Stickel. A unification algorithm for associative-commutative functions. Journal of the ACM, 28(2):423--434, April 1981.

....nontrivial mathematical problems, for instance, see (Kapur and Zhang, 1991) AC unification is also an interesting problem because of the long history in trying to understand the problem. To the best of our knowledge, the first algorithms for ac unification were by Livesey and Siekmann (1976) and Stickel (1975, 1981). Of these Stickel s recursive algorithm is more general and was also shown to be complete. However, due to its complicated nature, termination of this algorithm could be shown only for a simple (restricted) subcase in (Stickel, 1981) Livesey and Siekmann s algorithm dealt only with terms ....

....for ac unification were by Livesey and Siekmann (1976) and Stickel (1975, 1981) Of these Stickel s recursive algorithm is more general and was also shown to be complete. However, due to its complicated nature, termination of this algorithm could be shown only for a simple (restricted) subcase in (Stickel, 1981). Livesey and Siekmann s algorithm dealt only with terms involving a single ac function symbol having variables and constants as arguments. This algorithm was subsequently extended to general terms by Herold and Siekmann (1985) The question of termination of Stickel s ac unification algorithm ....

[Article contains additional citation context not shown here]

Stickel, M.E. (1981). A unification algorithm for associative-commutative functions. J. of Assoc. of Comp. Mach. 28, 423-434.

....results from mathematics. The two most prominent instances of the semantic approach are 1. unification in Boolean algebras and rings [6, 16, 15] and its generalization to finite and to primal algebras [4, 5, 17, 11] 2. unification modulo the theories ACU (associativity commutativity with unit [14, 20]) ACUI (associativity commutativity idempotency with unit [14, 2] and AG (the theory of Abelian groups [12] It has turned out that the approach used in the second case can be generalized to a whole class of equational theories, called commutative theories in [1] and monoidal theories in ....

M.E. Stickel. A unification algorithm for associative commutative functions. J. of the ACM, 28(3):423--434, 1981. 5

....rewriting modulo associativity and commutativity (AC) i.e. certain operators in FAC F 2 are known to be both associative and commutative. We assume the basic algorithms (equalitytest, match, unification) do have built in knowledge of the associativity and commutativity of the operators in FAC [Hul79, Sti81]. Equational Specifications A set E of equations s = t where s and t are terms induces a relation E = E where E = fs t; t s j s = t 2 e.g. A term rewriting system R is equivalent to a set of equations E if R = E . The Knuth Bendix completion procedure [KB70] transforms on success a ....

Mark E. Stickel. A unification algorithm for associative-commutative functions. JACM, 28(3):423--434, July 1981.

....and the solving of an AC1 unification problem in Section 3.3. A computer implementation of the procedures presented in this chapter is described in Chapter 5 and examples are given in Appendix B. AC and AC1 unification problems are inherently complex. Following the intial work in [LS 75] and [Sti 81] in AC unification, considerable research has been done. The reader may consult [JK 91] for an extensive list of references. The approach presented herein is not intended to optimize the running time of the solution of a given AC1 unification problem but rather to modularize the very process of ....

.... (for an example see [Fag 84] One way to guarantee termination is to apply the decomposition step recursively in a sense that once an atomic AC1 unification problem f(X) f(Y ) is simplified, all emerging subproblems are completely solved before another atomic problem is examined ( Sti 81] Fag 84] This method is not satisfactory for our approach to completion modulo AC1, where we want to be able to interrupt the solving of a constraint when a particularly hard atomic AC1 unification problem arises. Other approaches to AC unification are described in [Kir 89] and [Bou 90] We ....

M. E. Stickel. A Unification Algorithm for Associative-Commutative Functions. Journal of the Association for Computing Machinery 1981 28, pp. 423--434.

....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 ....

....1 ; yn ) P if x = 2 V (s) V (P ) Fig. 1. The set of rules S for unification in simple syntactic theories Before we show how the rules of S can be used, after all, for AC unification, we introduce a convenient notation for AC unification problems and we recall the well known result [19, 21], about the semantic method for AC unification. Definition 6. Consider the table: x1 Delta Delta Delta x i Delta Delta Delta xn t 1 a 11 Delta Delta Delta a 1i Delta Delta Delta a 1n t 2 a21 Delta Delta Delta a2i Delta Delta Delta a2n Delta Delta Delta Delta Delta ....

[Article contains additional citation context not shown here]

M. Stickel. A unification algorithm for associative-commutative functions. Journal of the ACM, 28(3):423--434, 1981.

....rule. However in the ND system this approach is taken even further, allowing free variables to be used also in the # rule. Simple label inequations involving only the # operator generated by the closure rule can be solved in the LKE system using algorithms based on the AC unification technique[Sti81]. See [BF95] for further details. For more complex inequations involving the operator no algorithm has, to the authors knowledge, yet been reported. In the ND approach, the solving process is much simpler. ND proofs are more structured. This structural feature facilitates the definition of an ....

M. Stickel. A Unification Algorithm for Associative-Commutative Functions. Journal of the ACM, 28(3), 1981.

....matching, 2) applies to full first order statements, 3) has fewer paramodulation options, and (4) always uses the given clause algorithm to derive the search. paper.tex; 18 12 1996; 11:32; no v. p.3 4 WILLIAM McCUNE 3.1. AC Unification and Matching Associative commutative (AC) unification [12] builds the properties of associativity and commutativity of a binary operation into the inference process so that the corresponding equations need not be present as explicit axioms. Two terms are AC identical if they can be made identical by reassociating and commuting subterms. An AC unifier of ....

....is always finite) for a pair of terms. AC matching is a special case of AC unification in which only one of the two given terms is instantiated; AC matching is used to simplify derived equations and to determine whether one equation subsumes another. EQP uses Stickel s AC unification algorithm [12], A pair of terms can have a great number of most general AC unifiers, and we have an optional heuristic, the super 0 strategy, that eliminates the more complicated unifiers. Stickel s algorithm constructs a linear homogeneous Diophantine equation that represents identity of the two terms to be ....

M. Stickel. A unification algorithm for associative-commutative functions. J. ACM, 28(3):423--434, 1981.

....algorithm embodying the axioms of the theory itself. Unfortunately, known extended unification algorithms described in the literature fail to capture the properties of with described so far. In particular, ACI unification (i.e. unification under associativity, commutativity, and idempotency [3, 10, 37, 51]) cannot be directly applied in our case. In fact, the identity fZjfY jXgg = ffZjY gjXg, representing associativity, does not hold, for instance, under the substitution fX fcg; Y fbg; Z ag, because the two sets fa; fbg; cg and ffa; bg; cg are distinct in our interpretation. Similarly, the ....

Stickel, M. E., A Unification Algorithm for Associative-Commutative Functions. J. of the ACM, 28(3):423--434 (1981).

....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 ....

M. Stickel, "A Unification Algorithm for Associative-Commutative Functions," J. ACM 28, 1981. This article was processed using the L a T E X macro package with LLNCS style

....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 ....

M. Stickel, "A Unification Algorithm for Associative-Commutative Functions, " J. ACM 28, 1981.

No context found.

Mark Stickel. A unification algorithm for associative-commutative functions. Journal of the ACM, 28(3):423--434, July 1981. 7

No context found.

Mark E. Stickel. A unification algorithm for associative-commutative functions. Journal of the ACM, 28(3):423--434, July 1981.

No context found.

Stickel, M.E. (1981). A unification algorithm for associative-commutative unification. Journal ACM, 28, 3, 423-434. 13

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC