Deepak Kapur and Paliath Narendran. Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning, To appear, 199?  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to these problems, still using the [ based representation, is obtained by assuming that a unary (free) operator f Deltag is present in Sigma. The set fs 1 ; s m g can be described as fs 1 g[ fs m g. To our knowledge, the only proposals that have addressed this problem so far are [20, 16, 3]. In particular, 3] shows how to combine unification algorithms for equational theories with disjoint signatures and theories. A general ACI1 unification algorithm (namely, ACI1 dealing also with free symbols, such as f Deltag) can be obtained by combining ACI1 unification for ; and constants ....

....Like in CLP (Set) 7] in order to check satisfiability of a given constraint C, we transform C to an equivalent disjunction of constraints in solved form, guaranteed to be satisfiable. The problem we tackle here extends the satisfiability problem for set unification, shown to be NP complete in [16]. Thus, it is an NP hard problem. NP completeness ensues from [23] The algorithm we propose here clearly does not belong to NP since it is non deterministic and it applies syntactic substitutions. However, one could devise implementations of the algorithm adopting standard techniques (e.g. ....

D. Kapur and P. Narendran. Complexity of Unification Problems with Associative-Commutative Operators. J. of Automated Reasoning, 9:261--288, 1992.

....(ii) this problem is NP hard. Basically, all algorithms for unification under associativity are based on Makanin s algorithm for word equations [96] The 3 NEXPTIME upper bound is obtained in [86] The following theorem characterizes other popular kinds of equational theories. Theorem 8. 2 ([82, 83]) Let E be an equational theory defining some symbols as AC symbols or ACI symbols or AC1 symbol or ACI1 symbols (there can be one or more of these kinds of symbols) The theory E is assumed to contain no other equations. Then the E unification problem is NP complete. 8.3. Complexity of ....

D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operators. J. Automated Reasoning, 9(2):261--288, 1992.

....The drawback is that most general unifiers need not be unique any more, and that each unification problem may generate a (potentially large) number of unifiers. In the case of associativity and commutativity, there are doubly exponentially many unifiers in the size of the terms to be unified (Kapur and Narendran, 1992). Fortunately, the explicit computation of unifiers can be avoided, if constraints are used to specify the requisite unification problems for an inference. Then it suffices to check whether the constraints are satisfiable, which in the case of associativity and commutativity is only an ....

....the explicit computation of unifiers can be avoided, if constraints are used to specify the requisite unification problems for an inference. Then it suffices to check whether the constraints are satisfiable, which in the case of associativity and commutativity is only an NP complete problem, see (Kapur and Narendran, 1992). For a detailed description of associative commutative superposition calculi with equality constraints see Nieuwenhuis and Rubio (1994) and Vigneron (1994) Constraints can also be used to represent implicit or explicit types of variables such that type inference is integrated into proof search, ....

Kapur, D. and P. Narendran: 1992, `Complexity of Unification Problems with Associative-Commutative Operators'. J. Automated Reasoning 9(2), 261--288.

....unification which is unitary , i.e. admits at most one most general unifier) In the general case the decision problem for ACI1 unification whether two terms 1 and 2 are unifiable is NPcomplete. This can be shown by reducing the ACI matching problem (which is shown to be NP complete in [23]) to ACI1 unification as shown in [24] In our domain we consider a restricted alphabet for ACI1 expressions and consequently, ACI1 unification is far simpler. In our domain there is only one binary function symbol and only one constant. As a consequence, two abstract terms are always unifiable ....

D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning, 9(2):261--288, October 1992.

....For example, the syntactic unification problem (empty theory) is unitary a minimal complete set al..ways exists and has cardinality 1. For ACI unification, a minimal 5 complete set al..ways exists and is finitary [4] The underlying decision problem of ACI unification is in general, NP complete [30]. The algorithms for ACI unification which are described in the literature compute a complete set of ACI unifiers which is in general not minimal. For efficiency in applications which use these unifiers, such a set should be close to minimal in practice. Abstract Interpretation: For semantic ....

....in Section 2, ACI unification is an instance of the more general notion of E unification. The ACI unification of two terms 1 and 2 consists in finding a substitution satisfying 1 =ACI 2 . The underlying decision problem deciding whether two terms are ACI unifiable is NP complete [30]. It is interesting to note that the restricted ACI unification problem for monomorphic types is P time decidable. The algorithm is also simpler an implementation is illustrated in Appendix A. However, even for monomorphic types, there may be exponentially many most general unifiers for a ....

D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning, 9(2):261--288, October 1992.

....similar to this algorithm. It does not perform an occur check and it returns x y only if x can be reached from y and y has been labelled by another structure. The Theory ACI In the theory of Abelian monoids, ACI, the binary function symbol is associative, commutative and idempotent. In [11], an algorithm was given that decides solvability of ACI unification with constants. The main idea is to set up Horn clauses which describe the solvability of the equations. The Horn clauses are built from propositional variables P x;a which are true iff the constant a does not occur in a solution ....

....iff the constant a does not occur in a solution for the variable x. A clause P x;a P y;a ) False means that the problem is unsolvable if a does neither appear in x nor in y, or equivalently: if we can deduce that a does not occur in x, then it must appear in y. We extend the algorithm given in [11] for our situation where the set of variables and constants is not fixed in the beginning. By this, we prevent that new Horn clauses have to be set up when a new labelling decision is made. Let VACI be the set of variables in Gamma ACI ; note that there are no constants in Gamma ACI . We ....

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D. Kapur and P. Narendran. Complexity of unification problems with associativecommutative operators. Journal of Automated Reasoning, 9:261--288, 1992.

....a reduction of disunification to unification does not seem to be possible. In addition, even if a theory is finitary, the computation of a complete set of unifiers can be of higher complexity than deciding solvability (associativity and commutativity is an example for this phenomenon; see, e.g. [KN92a, KN92b]) In [BS91a] we have shown how to combine decision procedures for unification, and in the present paper we shall investigate how this method can be generalized to treat solvability of disunification problems. For unification, solvability means having a solution in the free algebra (in ....

D. Kapur, P. Narendran, "Complexity of Unification Problems with Associative-Commutative Operators," J. Automated Reasoning 9, 1992, pp.261-288.

.... Among other consequences we mentioned (1) that the algorithm provides us with a decision procedure for the solvability of general A and AI unification problems and (2) that Kapur and Narendran s result about the NPdecidability of the solvability of general AC and ACI unification problems (see [KN91]) may be obtained from our results. In [BS91] we did not give detailled proofs for these two consequences. In the present paper we will treat these problems in more detail. Moreover, we will use the two examples of general A and AI unification for a case study of possible optimizations of the ....

....general associative unification problems, i.e. associative unification problems with free function symbols. The decidability of general A and AI unification problems was open for a long time (compare Kapur and Narendran s table of known decidability and complexity results in unification theory [KN91]) A positive answer was obtained in our paper [BS91] where we used the fact that general E unification may be regarded as an instance of the combination problem. The combination problem (see e.g, BS91, Sc89] is concerned with the question of how to derive unification algorithms (for the ....

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D. Kapur, P. Narendran, "Complexity of Unification Problems with Associative-Commutative Operators," Preprint, 1991. To appear in J. Automated Reasoning .

....the same variables. minimal complete sets of A unifiers for general A unification problems, and in 1977 Makanin [Ma77] has shown that A unification with constants is decidable. But in 1991, decidability of general A unification was still mentioned as an open problem by Kapur and Narendran [KN91] in their table of known decidability and complexity results for unification. Such a decision procedure could, for example, be useful when building associativity into a theorem prover via constraint resolution; and it could be used to make Plotkin s enumeration procedure terminating for equations ....

....are elements of given regular languages over the constants. It is easy to see that problems with constant restriction are a special case of these more generally restricted problems. 4. General AI unifiability, where AI : A [ ff(x; x) xg, is decidable. This was also stated as an open problem in [KN91]. For AI, decidability of unification problems with constant restriction easily follows from the wellknown fact (see e.g. Ho76] that finitely generated idempotent semigroups are finite (see [BS91] for details) 5. If solvability of the E i unification problems with linear constant restriction ....

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D. Kapur, P. Narendran, "Complexity of Unification Problems with Associative-Commutative Operators," J. Automated Reasoning 9, 1992, pp.261-288.

....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 doubly exponentially many AC unifiers for two terms (Domenjoud, 1992b) and therefore as many conclusions in an inference; e.g. a minimal complete set for x ....

....clause C with an AC equality constraint T of the form s 1 = t 1 : sn = t n can be proved redundant by means of efficient incomplete methods detecting cases of unsatisfiability of T . Only if C is the empty clause one has to actually decide the AC unifiability of T (which is NP complete, cf. Kapur and Narendran, 1992a) in order to know whether an inconsistency has been derived or not. In our completeness proofs we apply an essential ingredient which we gave for this purpose in (Rubio and Nieuwenhuis, 1995) an AC compatible simplification ordering that is total (up to AC equality) on ground terms. This ....

Kapur, D., Narendran, P. (1992). Complexity of unification problems with associative commutative operators. Journal of Automated Reasoning, 9:261--288.

....when they exist, do not always provide efficient algorithms for solving equality constraints. For instance associativecommutative unifiability is an NP complete problem whereas the computation of a base of associative commutative unifers is complete for the double exponential complexity class [28]. 4.4 CLP( In CLP( we consider the terms of the simply typed calcul, they are defined by the following grammar of types t and typed expressions e : t: t : v j t 1 t 2 e : t : x : t j (x : t 1 :e : t 2 ) t 1 t 2 j (e 1 : t 1 t 2 (e 2 : t 1 ) t 2 The symbol represents the ....

D. Kapur, P. Narendran, Complexity of unification problems with associativecommutative operators, Journal of Automated Reasoning, 9, pp. 261-288 (1992).

.... solvability of associative commutativeidempotent (ACI ) unification problems with free constants is decidable in polynomial time, but the problem of deciding solvability of ACI unification problems with additional free function symbols (called general ACI unification) is NP hard (see [KN91]) The same phenomenon holds for the theory ACUN of an associative commutative and nilpotent function with unit, and for the theory ACUNh which contains in addition a homomorphic unary function symbol (see [GN96] for formal definitions) These examples show that there cannot be a general ....

D. Kapur, P. Narendran, "Complexity of Unification Problems with Associative-Commutative Operators," J. Automated Reasoning 9, 1992, pp. 261-288.

....for the complexity of general E unification. For most of the equational theories E that This work was supported by the EC Working Group CCL, EP6028. have been discussed in the literature the problem of deciding solvability of general E unification problems has turned out to be NP hard (see [KN92, BS94] for surveys) When looking at the proofs of these intractability results it becomes clear that some of the encoding techniques that are used are closely related. Yet, a common background is missing that would explain these results from an abstract point of view. This yields one motivation for the ....

D. Kapur, P. Narendran, "Complexity of Unification Problems with Associative-Commutative Operators," J. Automated Reasoning 9, 1992, pp. 261-288.

.... X ) Psi 5 = Phi A= list Phi X ) B=list ; C= integer Phi X ) Psi 6 = Phi B=list ; C= integer Phi A) Psi 7 = Phi A=list ; C= integer Phi B) Psi 8 = Phi A=list ; B=list ; C=integer Psi The theoretical complexity of the underlying ACIunification problem is NP complete [20]. Our unification algorithm is exponential in the size of the unificands. Our analysis algorithm considers a complete set of unifiers which in general may consist of an exponential number of elements. However, in practice the cost of the unification as well as the number of unifiers is markedly ....

D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operations. Journal of Automated Reasoning, 9:261--288, 1992.

....survey) A clause C with an AC equality constraint T of the form s 1 = AC t 1 : s n = AC t n can be proved redundant by means of efficient incomplete methods detecting cases of unsatisfiability of T . If C is the empty clause one can decide the AC unifiability of T (which is NP complete, cf. [KN92]) to know whether an inconsistency has been derived or not. The first results on (almost basic) constrained deduction modulo AC were reported by Laurent Vigneron. In a recent version of his work [Vig94] he also avoids the computation of AC unifiers (by applying our notion of irreducibility, defin. ....

Deepak Kapur and Paliath Narendran. Complexity of unification problems with associative commutative operators. Journal of Automated Reasoning, 9:261-- 288, 1992.

....Research of this author was partially supported by NSF Grant CCR 9610257. The computational complexity of general ACI matching and general ACIunification (that is, the terms to be unified or matched may contain both free function and free constant symbols) was investigated by Kapur and Narendran [KN86,KN92], who established that these decision problems are NP complete. In contrast, they also proved that elementary ACI unification with a finite number of free constants is solvable in polynomial time [KN92] More recently, Narendran [Nar96] showed that ground elementary ACIU disunification is NPhard, ....

....both free function and free constant symbols) was investigated by Kapur and Narendran [KN86,KN92] who established that these decision problems are NP complete. In contrast, they also proved that elementary ACI unification with a finite number of free constants is solvable in polynomial time [KN92]. More recently, Narendran [Nar96] showed that ground elementary ACIU disunification is NPhard, where ACIU is the extension of ACI with a unit element. In this paper, we investigate further the computational complexity of elementary ACI unification and ACI disunification with a finite number of ....

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D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operators. J. of Autom. Reasoning, 9:261--288, 1992.

....bad behavior for most of the sample problems. 1 Introduction Many papers concerning constraint logic programming with sets (see, e.g. DR93, Ger94, LL91] have pointed out that the complexities of the set unification problem and even of the simplest set matching problem (see, e.g. KN86, BKN87, KN92] are the real bottlenecks of any attempt to extend logic programming with set entities. In [LL91] the problem is avoided using a delay technique: any set unification problem is delayed until it is transformed into a simple ground test. This improves e#ciency; however, if the two terms do not ....

D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operators.<F4.877e+05> Journal of Automated<F5.38e+05> Reasoning, 9:261--288, 1992.

....484] but the explicite proofs are omitted. In fact, it is sometimes not straightforward 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 ....

....class #P, since each counting problem belonging to #P contains, by definition, only a simple exponential number of different solutions. A polynomial reduction from 1 in 3Sat is the NP hardness proof of many interesting problems, especially in automated deduction. Unification modulo idempotence [KN92] and modulo unit [TA87] are proved NP complete by a reduction from 1 in 3Sat. Within the unification theory, we are interested not only in decision problems (whether two terms are unifiable in a given theory) but also in counting problems (how many substitutions contains the minimal complete set ....

D. Kapur and P. Narendran. Complexity of unification problems with associativecommutative operators. Journal of Automated Reasoning, 9:261--288, 1992.

....as a representation in the first place. TREE CONSTRAINTS FOR NECESSITY 25 It is also noteworthy that the complexity of resolving scopes by unification cannot be established by the usual encodings of hard problems using string unification, such as those presented in (Kapur and Narendran, 1986; Kapur and Narendran, 1992). These encodings repeat variables in different contexts to enforce constraints. Such repetitions are unavailable because of the unique prefix property on occurrences of variables in equations between modal scopes (cf. Wallen, 1990; Auffray and Enjalbert, 1992) For each scope variable or ....

Kapur, D. and Narendran, P. (1992). Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning, 9:261-- 288.

....even nonprimitive recursive or super exponential. In our attempts to study the ac unification problem, we proved in 1986 that the ac unifiability check can be done in non deterministic polynomial time, thus showing the ac unifiability check to be NP complete (Benanav, Kapur and Narendran, 1985; Kapur and Narendran, 1986b 1 ; Kapur and Narendran, 1987) In this paper, we design a deterministic algorithm for computing a complete set of ac unifiers; the algorithm is based on the non deterministic algorithm presented in our earlier paper (Kapur and Narendran, 1986b) The key ideas of the algorithm are: 1. A ....

.... be NP complete (Benanav, Kapur and Narendran, 1985; Kapur and Narendran, 1986b 1 ; Kapur and Narendran, 1987) In this paper, we design a deterministic algorithm for computing a complete set of ac unifiers; the algorithm is based on the non deterministic algorithm presented in our earlier paper (Kapur and Narendran, 1986b) The key ideas of the algorithm are: 1. A decision tree is built by 1 A final version of this paper is to appear in the Journal of Automated Reasoning sometime in 1992. ffl considering whether non variable subterms with the same outermost ac function symbol can be unified or not. This ....

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Kapur, D., and Narendran, P. (1986b). Complexity of Unification Problems with AssociativeCommutative Operators, Unpublished Manuscript, G.E. R&D Center, December 1986. A revised version to appear in J. Automated Reasoning.

....ideas, concepts, and algorithms that could be implemented and tested for their effectiveness. Experimentation using RRL has led, among other things, to development of criteria for identifying redundant inferences [22, 71] complexity studies of primitive operations such as matching, unification [25, 4, 26], efficient algorithms for primitive operations [27] approaches for first order theorem proving [20, 70] methods for proving formulas by induction [30, 72] algorithms for checking the sufficient completeness property of specifications [28] and specialized completion procedures for equational ....

Kapur, D., and Narendran, P., Complexity of unification problems with associativecommutative operators, J. Automated Reasoning 9 (1992) 261--288.

....and there has been work on unification algorithms that build in these theories. There are many other properties that can be treated similarly. Several equational theories have been considered in the literature [5, 13] Decidability and complexity results have been obtained for many of them [15, 23]. Some of the results reported here are in partial fulfillment of Qing Guo s Ph.D. requirements, and will form part of his dissertation. Paliath Narendran was partially supported by NSF grant CCR 9404930. In this paper we consider nilpotence: the simple theory f(x; x) 0 where 0 is a ....

D. Kapur and P. Narendran. Complexity of unification problems with associative- -commutative operators. Journal of Automated Reasoning 9 (2) (1992) 261--288.

....from the ability to build in many proof steps into the pattern matching algorithm, possibly shortening the search for a proof. Several equational theories have been considered in the literature (see the surveys by [4, 10] and decidability complexity results have been obtained for many of them [12, 17]. In this paper we consider linear equations over semirings, in particular N[x 1 ; xn ] and present an undecidability result that is relevant to several unification problems. In other words, we show that the problem of checking whether there is a solution to a set of linear equations over ....

D. Kapur and P. Narendran. Complexity of Unification Problems with Associative-Commutative Operators. Journal of Automated Reasoning 9 (2) (1992) 261-288.

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Deepak Kapur and Paliath Narendran. Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning, To appear, 199?

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D. Kapur and P. Narendran. Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning, 9(2):261-- 288, October 1992.

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