Martelli, A. and Montanari, U. (1982). An efficient unification algorithm. ACM Transactions on Programming Languages and Systems, 4, 258--282.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= oet . A unifier oe of t and t is called most general unifier iff oe [Var(t) Var (t ) for all unifiers of t and t . The definitions above can be extended to atoms, equations, and sets of equations in the obvious way. For a unification algorithm we use the operations suggested in [MM82]. 3 Cycle Unification C = fL Rg is called a cyclic theory , or cycle for short, if the atoms L and R are weakly unifiable, i.e. there exist two substitutions oe and oe such that oeL = oe R [Ede85] Let G and F be two atoms such that Var(G) Var(F ) A cycle unification problem hG ....

....= j Var(G) Throughout the paper will denote the most general unifier of G and F restricted to Var(G) if it exists. Similarly, k will denote the most general unifier of C restricted to Var(G) if it exists. The solutions k will be computed by applying the Martelli Montanari operations [MM82] to the set C . Let C = hG Gamma F i be a cycle unification problem. A set Sigma of substitutions is a complete set of solutions for C iff each substitution in Sigma is a solution for C and for each solution for C we find a substitution oe in Sigma such that oe [Var(G) A ....

A. Martelli and U. Montanari. An efficient unification algorithm. ACM TOPLAS, 4(2):258--282, 1982.

....extended to domain variables) now taking into account the extension to HiLog terms. Instead, we will give an efficient unification algorithm through term equations solving, obtained by extending the algorithm in [CKW,1993] which is in turn an adaptation of the efficient unification algorithm in [MM,1982]. With respect to terms, we will use the terms of unifier (or: unifying substitution) and most general unifier for a set of terms in the same relationship to the substitution notion as they are in the predicate calculus. We will see in the subsequent that it will not be the same in the case of ....

.... We can use the encoding function encode t used by [CKW,1993] to reduce HiLog terms (containing finite domain variables) to predicate calculus terms (with such variables) Proving the algorithm for such terms implies only a simple adaptation of the proof for the Martelli and Montanari algorithm in [MM,1982]. Here is the definition for encode t : 7 encode t (X) X, for each (either simple or domain) variable; encode t (s) s, for each parameter symbol s in the language alphabet; encode t ( t(t 1 ,t 2 , t n ) apply n 1 ( encode t (t) encode t (t 1 ) encode t (t n ) where apply ....

A. Martelli, U. Montanari, An Efficient Unification algorithm, ACM Trans. on Progr. Lang. and Systems 4(2):258-282, 1982.

.... e n and every common instance of the e i s is an instance of e 1 . An algorithm is said to solve the unification problem, iff it decides whether two expressions are unifiable and, if so, computes the most general unifier. It is well known that this problem can be solved in linear time (cf. [27] ) An important prerequisite for an efficient unification algorithm is the appropriate representation of terms, namely as directed acyclic graphs (dags) Likewise, for our results in this paper, it is important to represent equational problems as dags such that each subterm of the problem occurs ....

Martelli, A. and U. Montanari: 1982, `An Efficient Unification Algorithm'. ACM Transactions on Programming Languages and Systems 4(2), 258--282.

.... Our main result is that solvability is decidable and that solvable problems possess most general solutions (for a reasonably obvious notion of most general ) The proof is via a unification algorithm which is very similar to the first order algorithm given in the now common transformational style [17]. See [16, Sect. 2.6] or [1, Sect. 4.6] for expositions of this. Section 4 considers the relationship of our version of unification modulo # equivalence to existing approaches. Section 5 assesses what has been achieved and the prospects for applications. To appreciate the kind of problem that ....

....problem, decides whether or not it has a solution and if it does, returns a most general solution. Proof. We describe an algorithm using labelled transformations directly generalising the presentation of first order unification in [16, Sect. 2. 6] which in turn is based upon the approach in [17]. See also [1, Sect. 4.6] for a detailed exposition, but not using labels. We use two types of labelled transformation between unification problems, namely =# P # and P # =# P # where the substitution # is either the identity #, or a single replacement [X : t] and where the freshness ....

A. Martelli and U. Montanari. An efficient unification algorithm. ACM Trans. Programming Languages and Systems, 4(2):258--282, 1982.

....theory, the first algorithm was described in [Robinson 71] and is exponential in the size of the input. It has been modified by representing terms as directed acyclic graphs rather than trees [Corbin 83] to give an n 2 algorithm. The algorithm of [Paterson 78] runs in linear time and those of [Martelli 82] and [Baxter 73] run in nearly linear time. Martelli 82] while theoretically slower than the linear algorithms, runs faster on some typical examples. Also, the modified algorithm of [Corbin 83] is fast in practice and has the additional advantage that the structure of the algorithm is simple ....

....71] and is exponential in the size of the input. It has been modified by representing terms as directed acyclic graphs rather than trees [Corbin 83] to give an n 2 algorithm. The algorithm of [Paterson 78] runs in linear time and those of [Martelli 82] and [Baxter 73] run in nearly linear time. Martelli 82] while theoretically slower than the linear algorithms, runs faster on some typical examples. Also, the modified algorithm of [Corbin 83] is fast in practice and has the additional advantage that the structure of the algorithm is simple and intuitive; one disadvantage of the [Corbin 83] approach ....

[Article contains additional citation context not shown here]

A. Martelli and U. Montanari, "An Efficient Unification Algorithm," ACM Transactions on Programming Languages and Systems 4(2):258.282, April 1982.

....or the set of inference rules for the equational theory. Hence, it can be instanciated in many different ways. For example, if the equational theory consists only of the reflexivity axiom and if we consider the transformation rules defined by Herbrand [Herbrand, 1930] or Martelli Montanari [Martelli and Montanari, 1982], then we obtain the strong completeness of SLD resolution or the strong completeness of the general procedure plus the axiom of reflexivity [van Emden and Lloyd, 1984] or the strong completeness of BF resolution [Wolfram et al. 1983] depending on the strategy we apply to solve the unification ....

Martelli, A. and Montanari, U. (1982). An efficient unification algorithm. ACM Transactions on Programming Languages and Systems, 4:258--282.

....irreducible terms. A transformation system is a finite set of transformation rules. A transformation system is convergent if all sequences of transformations lead to a unique normal form. Below we sketch a transformation system that extends the Martelli and Montanari s transformation system [52] to take into consideration metavariables and name equations (see [28] for a full treatment of it) Below we indicate with V the set of variables and with M the set of metavariables of HC . Let e be an equation. Equations yet to be solved are written as s = t and terms of the form f(t 1 ; ....

....rather than substitutions (see, e.g. Clark [17] A computation state is a pair hM; Hi, where M is a set of atoms that have to be proved and H is a Herbrand assignment. Unification can in this process be seen as a rewrite system that takes a set of equations to an equivalent Herbrand assignment [52]. The assumption that unification rewrites the whole set of equations to a Herbrand assignment can be relaxed. Let a state instead consist of a triple hM; H;F i, where M is a set of atoms, H is a Herbrand assignment, and F is a set of irreducible name equations. We can see H and F as the solved ....

A. Martelli and U. Montanari. An efficient unification algorithm. ACM TOPLAS, 4:258--282, 1982.

....and k d ff(k d ) j s j j t j O(js j j t j) Table 1: Table of Asymtotic Complexities 1.1 Summary of Results The main contribution of this paper is an efficient subterm unification algorithm that exploits commonality among subterms. Our algorithm, following Martelli and Montanari s approach [6], does unification by solving term equations. The basic operations in this algorithm are computing the common part of terms and substitutions for variables, as described in Section 2. We transform the common part computation into a highly structured string matching problem. This structure enables ....

....that reaches the root of the term or :i ( is a position in the term and i is an integer) which reaches the ith argument of the root of the subterm reached by . We use t= to refer to the subterm of t reached by . The development of our algorithm is based on Martelli and Montanari s algorithm in [6]. We sketch its high level description below. Their algorithm views unification as the problem of solving term equations of the form s = t where s and t are the terms to be unified. It operates by repeatedly transforming the initial equation s = t into an equivalent set of equations until either ....

[Article contains additional citation context not shown here]

A. Martelli and U. Montanari, An Efficient Unification Algorithm, ACM TOPLAS, Vol. 4 No. 2, 1982, pp. 258-282.

....But the constraint satisfaction step, when analyzed in its own right, often turns out to have sharply delineated complexity. For example, in the case of firstorder logic (without equality) a linear time unification algorithm suffices to solve the constraints associated with a deduction see [Martelli and Montanari, 1982]. Hence the lifted calculus is useful not only for carrying out proof search in practice, but also, in many cases, for establishing theoretical bounds on the com 54 plexity of proof search problems in logical fragments [Lincoln and Shankar, 1994, Voronkov, 1996, McAllister and Rosenblitt, ....

.... and certain other primitive constraints (governing the types of values for variables and the occurrence of values of variables as subterms of other terms) This perspective is assumed in algorithmic characterizations of instantiation, both for classical inference (e.g. in work on unification [Martelli and Montanari, 1982]) and FIRST ORDER MULTI MODAL DEDUCTION 65 for equational or constrained reasoning in modal inference [Wallen, 1990, Frisch and Scherl, 1991, Auffray and Enjalbert, 1992, Ohlbach, 1993, Otten and Kreitz, 1996, Schmidt, 1998, Stone, 1999c] The opening of sections 4 and 4.1 indicated that the ....

Martelli, A. and Montanari, U. (1982). An efficient unification algorithm. ACM Transactions on Programming Languages and Systems, 4(2):258--282.

....conjuncts will severely affect precision. 16 Even in the absence of redundancy, one can obtain a more precise result when starting from the solved form, as will be illustrated below. For the Herbrand domain, the solved form can be obtained by applying the Martelli Montanari unification algorithm [Martelli and Montanari 1982]; for generalized linear constraints, a solved form can be obtained by the algorithm of Lassez and McAloon [1992] Before discussing abstract conjunction, let us first further develop the abstract domain. Definition 7.1.7 (Order Relation) Let AC 1 , AC 2 2 Cons F m . Then AC 1 F m AC 2 ....

Martelli, A. and Montanari, U. 1982. An efficient unification algorithm. ACM Trans. Program. Lang. Syst. 4, 3, 258--282.

....Because of both its practical and theoretical interests, extensive research [Rob79, Hue76, Kir90, Bau93, BS98, BO99] has been focused on the definition of good unification algorithms. Linear and almost linear, in both space and time, sequential algorithms were defined [Rob76, Hue76, Bax76, PW78, MM82, Col82, CB83, Muk83, Jaf84, RP90] The advent of parallel processing led to the study of unification on parallel machines [Rob85, MO84] In [Yas84, DKM84] unification was proved to be LogSpace complete for P, which means that there are inputs where parallel unification runs not significantly ....

....an equivalence that is both homogeneous and acyclic. Section 6 introduces a unification algorithm (see Figure 1) which is based on the computation of the unification closure. The algorithms, defined in the literature, for the computation of mgu by sequential machines [Rob76, Hue76, Bax76, PW78, MM82, CB83, Muk83, Jaf84, MMR86, RP90, BO99] compute in a similar way but exploit different algebraic frameworks and use slightly different presentations of substitutions and hence, of the unifier. Definition 2 (Unification problem) The unification problem MGU is the following decision problem: MGU ....

[Article contains additional citation context not shown here]

A. Martelli and U. Montanari. An efficient unification algorithm. ACM Trans. on Progr. Lang. and Sys, 4(2):259--282, 1982.

.... very simple, it is in essence a generate and test algorithm, hence less appealing, from an implementer s point of view, than a specification a la Robinson like the one given in [DOPR92] or (to cope with the new problems arising from the circularity of membership) a la Martelli Montanari (cf. [MM82]) 2 A result generalizing Corollary 10 can be proved in set hyperset theories comprising, in addition to the axioms used so far (cf. Def.8) the binary union and difference axioms: 9 z 8 x(x 2 z (x 2 Y 1 x 2 Y 2 ) 9 z 8 x(x 2 z (x 2 Y 1 x = 2 Y 2 ) The collection of restrictedly ....

A. Martelli and U. Montanari. An Efficient Unification Algorithm. ACM-TOPLAS, 4(2):258--282, 1982.

....p and q: no character is repeated either in p nor in q and equal characters only occur within a common substring at the beginning of p and q. This restriction allows us to give an efficient algorithm computing a minimal set of most general unifiers. Similar to the ideas of Martelli and Montanari [15] rather than by giving a recursive procedure we consider the process of unification as a sequence of transformations. We start with a given equation Gamma = fp=qg and an empty substitution oe= and stop with an empty set Gamma = and a substitution oe representing an idempotent most general ....

A. Martelli, U. Montanari. An efficient unification algorithm.ACM TOPLAS, 4:258--282, 1982.

....for which S is a solution. This set contains S, is trivially closed under ins, ref, sym and cons, is closed under trans because of the confluence of S, is closed under dec and sep because of lemma 3: it contains all s = t for which S j= s = t. QED. In the context of conventional term unification [4], we say that a substitution ff is more general than a substitution fi if there is a fl such that fi j fffl. If ff and fi are idempotent substitutions, as unifiers normally are, and if they are expressed as sets of equations of the form X = t, then the existence of such fl is equivalent to fi j= ....

A. Martelli und U. Montanari, "An Efficient Unification Algorithm", ACM Transactions on Programming Languages and Systems 4, 2 (April 1982), pages 258-282.

....we will assume throughout the rest of this paper that all atoms in any S interpretation SI are standardized apart, i.e. they share no variable symbols. 2. 3 Substitutions We assume that all substitutions are represented as a set of equations in solved form, as described by Martelli and Montanari [12]. A set of equations (or substitutions) is said to be in solved form iff it satisfies the following conditions: 4 1. the equations are x j = t j , j = 1; k; 2. every variable which is the left member of some equation occurs only there. The domain of a substitution oe = fx j = t j j j ....

....3) Fortunately, this fragment is quite large and seems adequate for a wide variety of knowledge representation problems. To see how our algorithm works, we first observe that when a substitution = fX 1 = t 1 ; X n = t n g is in the solved form defined by Martelli and Montanari [12], none of the variables symbols X 1 ; X n can occur anywhere in any of the terms t 1 ; t n . Consequently, an anti cover of the singleton set f g can be obtained by finding, for each X i , 1 i n, a set of terms that do not unify with t i , and which cover the entire ....

[Article contains additional citation context not shown here]

A. Martelli and U. Montanari. (1982) An Efficient Unification Algorithm, ACM Trans. on Prog. Lang. and Systems, 4, 2, pps 258--282.

....atomic subformulas of FH . First we consider the set M = Phi (P; P 0 ; u) j P 2 A P 0 2 A Gamma P [u] P 0 [u] Psi where u is a principal solution of the unification problem P P 0 . First order unification is decidable, and has pseudo linear solutions (see for instance [7, 10]) Then we consider nonempty subsets P of M satisfying conditions (b) and (d) and such that the substitution oe = P;P 0 ;u)2P u exists; therefore, conditions (a d) and (1) are already satisfied. See [1] for an efficient algorithm to find such subsets. Then we keep the subsets who also ....

....ones. So, we have to solve the unification problems u = v where A i ( u) and A j ( v) are two atomic subformulas of FH , respectively appearing positively and negatively. The first order unification is decidable. A lot of algorithms are known, and we used Martelli and Montanari s one (see [10], and also [7] pages 224 226) The function all matches compute the set M value allmatches : formula (formula formula unifier) list; Then we look for the pairs (U ; oe) where U is a subset of M and oe a substitution, such that ffl U is not empty; ffl All atomic formulas of U are ....

A. Martelli and U. Montanari. An efficient unification algorithm. ACM Trans. Prog. Lang. Syst., 4(2):258--282, April 1982.

....information we can conclude that A : D; f( 1 ; m ) is a logical consequence of the program where is any substitution such that it is less general than 1 ; m . In this paper, we will consider a substitution to be a set of equations in solved form (cf. Martelli and Montanari [25]) Definition 13 Let 1 and 2 be two substitutions. oe is said to be the most general common denominator (MGCD) of 1 and 2 iff 1. oe is less general than both 1 and 2 , i.e. there exists substitutions 0 and 00 such that oe = 1 0 = 2 00 . 23 2. For any ....

A. Martelli and U. Montanari. (1982) An Efficient Unification Algorithm, ACM Trans. on Prog. Lang. and Systems, 4, 2, pps 258--282.

....canonical form for OE 2 . Further it can be shown that there is an effective procedure to compute the canonical form. Existence and computability of canonical forms mean that equality checking is decidable. A procedure to find the canonical form of a formula can be based on unification algorithms [PW78, AM82, Ram89]. First, we compute the DNF of the given formula. Each conjunction in this formula is of the form V n 1=1 (Y i = t i ) We use the unification algorithms to compute the most general unifiers of the equations Y 1 = t 1 ; Y n = t n . The substitutions obtained for each variable Y 1 ; Y ....

U. Montanari A. Moartelli. An efficient unification algorithm. ACM TOPLAS, 4(2):258-- 282, April 1982.

....is called internal in [BGMV93] in contrast to the external perspective where one relates several trees. Both views have a long tradition in logics. The internal view is taken in modal logic and in second order monadic logic (SnS) Rab69] whereas the external view is popular in unification theory [MM82,Col84,BS93] and for set constraints [HJ90,AW92,TDT00] In feature logics [KR86,Smo92] both perspectives have been employed and compared [BS95,MN00] Dominance versus Subtree Constraints. Dominance constraints contain node valued variables that we write as capital letters X;Y; Z. An atomic dominance ....

Alberton Martelli and Ugo Montanari. An efficient unification algorithm. ACM TOPLAS, 4(2):258--282, 1982.

No context found.

Martelli, A. and Montanari, U. (1982). An efficient unification algorithm. ACM Transactions on Programming Languages and Systems, 4, 258--282.

No context found.

Martelli, A. and Montanari, U. 1982. An efficient unification algorithm. ACM Trans. on Prog. Lang. and Systems 4, 2, 258--282.

No context found.

A. Martelli, U. Montanari, An efficient unification algorithm, ACM Trans. Programming Languages and Systems 4 (2) (1982) 258--282.

No context found.

A. Martelli, U. Montanari, An efficient unification algorithm, ACM Trans. Programming Languages and Systems 4 (2) (1982) 258--282.

No context found.

A. Martelli, U. Montanari, An Efficient Unification Algorithm, ACM TOPLAS 4 (1982) 258-282

No context found.

A. Martelli and U. Montanari, `An efficient unification algorithm', Trans. Programming Languages and Systems, 4(2), 258--282 (1982).

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