J. Jaffar. Minimal and complete word unification. Journal of the ACM, 37(1):47-- 85, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....called Markov s problem by mathematicians in eastern countries. It is called Lob s problem by mathematicians in western countries, for example by A. Lentin and M.P. Schutzenberger [11] A solution to the string unification problem was found by Makanin [13] in 1977. Subsequent papers on this topic [18, 9, 23, 10] were concerned with finding a better description of Makanin s algorithm, closing small gaps in the proof of correctness, and studying its complexity. Context Unification. Context unification is a subproblem of linear second order unification (see below) and a generalization of string ....

J. Jaffar. Minimal and complete word unification. Journal of the ACM, 37(1):47-- 85, 1990.

....a given k. Remark 4.1.4 The special case k = 1 of the Open Problem corresponds to word equations. Indeed, in this case, by Remark 4.1.1 there exists a solution only if t 1 and t 2 are linear trees, i.e. they are words. The solvability of word equations has been proved by Makanin [29] cf. also [24]) Notice that, for k 2, once solved Step 1. that is, the arity function over the variables is given) the problem of solvability of a tree equation becomes trivially decidable. In fact, as we remarked before, when we know the arity of a variable, we can retrieve the size of its image by . In ....

J. Jaffar. Minimal and complete word unification. J. ACM vol. 37, pagg. 47--85, 1990.

....given by the variable names. Vice versa, one can view a tuple in relation r as a ground substitution for the variables formed by the attributes of r. This correspondence is key to the semantics of RA(S) Unification of strings with variables is decidable [17] but has no efficient algorithm [2, 1, 15]. In fact string unification has particular problems because it is infinitary [16] i.e. it can have more than one maximal unifier and possibly an infinite number of them. This implies that the substitutions from the unification of string expressions can also be infinitary. But, under certain ....

J. Jaffar. Minimal and Complete Word Unification. Jour. ACM, 37(1):67--85, 1990.

....procedure is one of the most complicated algorithms existing in the literature. There were several attempts to simplify it [2, 12] The algorithm has been implemented [1] During last 20 years its complexity has been improved several times: 4 NEXPTIME (composition of four exponential functions) [7, 17], 3NEXPTIME [9] 2 EXPSPACE [4] EXPSPACE [8] The exact complexity of the algorithm is still not known. Current version of the algorithm, the full version of which can be found in [5] is still very complicated. Recently, another algorithm has been proposed in [15] It works nondeterministically ....

Jaffar J., Minimal and complete word unification, Journal of the ACM 37(1), 47-85, 1990.

....For free SGA (without any further condition) the decidability of the problem of satisfiability of equations is still open, although we conjecture it is decidable. Both of Makanin s algorithms have received very much attention. The enumeration of all unifiers was done by Jaffar for semigroups [6] and by Razborov 2 Claudio Guti errez for groups [15] Then, the complexity has become the main issue. Several authors have analyzed the complexity of Makanin s algorithm for semigroups [6] 16] 1] being EXPSPACE the best upper bound so far [3] Very recently Plandowski, without using ....

....have received very much attention. The enumeration of all unifiers was done by Jaffar for semigroups [6] and by Razborov 2 Claudio Guti errez for groups [15] Then, the complexity has become the main issue. Several authors have analyzed the complexity of Makanin s algorithm for semigroups [6], 16] 1] being EXPSPACE the best upper bound so far [3] Very recently Plandowski, without using Makanin s algorithm, presented an upper bound of PSPACE for the problem of satisfiability of equations in free semigroups [14] On the other hand, the analysis of the complexity of Makanin s ....

J. Jaffar, Minimal and Complete Word Unification, Journal of the ACM, Vol. 37, No.1, January 1990, pp. 47-85.

....instance of CUP is solvable iff its translation to a WE has a solution satisfying the associated LIC. As a result, we get the decision procedure for the class of CUP instances that translate into WEs possessing finite Minimal Complete Sets of Unifiers (MCSU) In fact, there is an algorithm due to (Jaffar 1990, Section 4) cf. also (Schulz 1993, Section 6) which, given a WE enumerates the corresponding MCSU, and terminates whenever this set is finite 1 . It remains to check, for each unifier from MCSU whether the associated LIC is satisfiable, which can be done effectively by using any known ....

.... argument for Generalized Context Equations (that such equations with large number of 1 A WE may have empty, finite, or infinite MCSU, cf. Baader Siekmann 1994, Section 5) an early procedure of (Plotkin 1972) for enumeration of MCSU was not guaranteed to terminate when MCSU is finite; (Jaffar 1990, Schulz 1993) provide complete enumeration procedures, which terminate when MCSUs are finite. 3 boundary constraints may only have solutions with high EP) Thus the hardest part of Makanin s proof should be passed through once again. Our program is different. We first translate CUP to word ....

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Jaffar, J. (1990), `Minimal and complete word unification', J. ACM 37(1), 47--85.

....theses [1] 10] studying this algorithm, possible simplifications and implementations. In 1977 Makanin [8] solved the problem in its complete generality giving us the first (and still the only known) algorithm to find solutions for arbitrary string equations. It was later extended by Jaffar [5] to give all possible solutions to an equation as well. In the meantime, there has been some work simplifying various aspects of the algorithm and even some implementations [10] 1] 14] 13] The problem of solving equations in (equationally defined free) algebras is a well established area ....

....better complexity bounds. First, we introduce a substantially simpler datatype for the concept of generalized equation which considerably simplifies the algorithm, making it more understandable and allowing shorter and simpler proofs of the correctness and termination of the algorithm (compare [5], 13] Secondly, we introduce the associated diophantine equations for an equation, which prune the search tree significantly, and by itself could possibly give another approach to solve string equations. Third, we give a thorough analysis of the complexity of the algorithm, obtaining smaller ....

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J. Jaffar, 1990. Minimal and Complete Word Unification, Journal ACM, Vol. 37, No.1, January 1990, pp.47-85.

.... In 1977 Makanin [12] solved the problem in its complete generality giving us an algorithm to decide if arbitrary systems of word equations have solutions (the case of systems of equations reduces easily to the case of only one equation) This decision procedure was later extended by Jaffar [9] to give all possible solutions to an equation as well. In the meantime, there has been some work simplifying various aspects of the algorithm and even some implementations [14] 1] 19] 18] Also, Schulz [19] generalized the result for the case of variables with regular constraints. Jaffar ....

....to give all possible solutions to an equation as well. In the meantime, there has been some work simplifying various aspects of the algorithm and even some implementations [14] 1] 19] 18] Also, Schulz [19] generalized the result for the case of variables with regular constraints. Jaffar in [9] calculated an upper bound for the running time of Makanin s algorithm which was four times exponential in the length of the equation. Later Ko scielski and Pacholski [10] improved it to nondeterministic triple exponential time. But a more detailed analysis, see [7] 6] shows an upper bound of ....

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J. Jaffar, 1990. Minimal and Complete Word Unification. Journal ACM, Vol. 37, No.1, January 1990, pp.47-85.

....of second order variables may have an arbitrary number of bound variables, each having exactly one occurrence. 2 at least some significant difference between context unification and second order unification. Context unification can also be considered as a generalization of word unification [15, 1, 11, 25, 26, 12, 5]. Decidability of word unification had been an open problem for many years. The problem was raised by A. A. Markov in the late 1950 s who hoped to prove the undecidability of Hilbert s tenth problem by showing undecidability of the word unification problem. In this context, Y. Matiyasevich [17] ....

J. Jaffar. Minimal and complete word unification. J. of the ACM, 37(1):47--85, 1990.

.... (i.e. contain at most one occurrence of the same variable) In such a case there exists a complete set of minimal unifiers that is finite: roughly speaking, 5 it suffices to consider all the manners in which t and overlap depending on the possible instantiations of their variables (see, e.g. [21]) Assume given a minimal complete set of unifiers 1 ; k (t i = i for i = 1; k) each instance t i of t reduces to r i (t i r i ) However, we disregard unifiers j which instantiate t at a variable position (replacing variable W of t with a subword of the form #uW 0 v# with u; ....

J. Jaffar. "Minimal and Complete Word Unification". J. ACM 37:1, 1990, pp. 4785.

.... and are linear (i.e. contain at most one occurrence of the same variable) In such a case the number of most general unifiers is finite: roughly speaking, it suffices to consider all the manners in which t and overlap depending on the possible instantiations of their variables (see, e.g. [20]) Suppose now that t and are unifiable via a set of most general unifiers 1 ; k (t i = i for i = 1; k) so that each instance t i of t reduces to r i . We will disregard unifiers j which instantiate t at a variable position . We then say that t is minimally reducible at ....

J. Jaffar. "Minimal and Complete Word Unification". J. ACM 37:1, 1990, pp. 47-85.

....the fifties by A. A. Markov, who posed in [5] the problem of satisfiability of equations on the free monoid. This has been an open problem until 1976, when G. S. Makanin proved the decidability of the satisfiability problem for equations on words (cf. 3] More recently, in 1990, J. Jaffar (cf. [2]) designed an algorithm to find the set of all the principal solutions to a word equation (when this set is finite) Partially supported by the Italian Ministry of Universities and Scientific Research MURST 40 Efficienza di Algoritmi e Progetto di Strutture Informative. Partially ....

Jaffar, J.: Minimal and complete word unification. J. ACM 37 (1990) 47--85.

....infinite number of maximally general unifiers. The decidability of the string unification problem (also called as the word problem) was established by Makanin [15] and procedures based on his technique have been developed by other researchers: Abdulrab and Pecuchet [1] Koscielski [14] and Jaffar [10]. But such procedures are not suitable for use in an automated reasoning environment or in a logic programming language because of their generation of multiple (maximally general) unifiers and non termination when there are infinite number of such unifiers. In [18, 16] see also [17] we offer a ....

J. Jaffar. Minimal and Complete Word Unification. Jour. ACM, 37(1):67--85, 1990.

....now been implemented [Ab87] In this paper we shall present a version of Makanin s algorithm which probably cannot be further simplified without fundamentally new insights into the problem insights which essentially exceed Makanin s original ideas. Our presentation will be based on J. Jaffar s [Ja90] modified notion of generalized equations which replaces Makanin s concept of a boundary connection by the concept of a boundary equation. With this step, various important improvements are obtained: the normalization subprocedure which occurs in [Ma77, P e81, Ab87, AbPe89, Sc90] is avoided ....

....in [Ma77] most readers were led to a real misinterpretation of this definition. In particular, P ecuchet s and Abdulrab s definition of a position equation in [P e81, Ab87, AbPe89] follows such a misinterpretation and does not lead to a correct proof. Similar difficulties arise from Jaffar s [Ja90] notion of a proper generalized equation. In the meantime, these points are more or less wellknown among the experts in the field. But, to my knowledge, there is no journal publication which contains a complete and error free proof showing that transformation of generalized equations behaves as it ....

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J. Jaffar, "Minimal and Complete Word Unification", Journal of the ACM 37, No. 1, 1990, pp. 47-85.

....The one used in Prolog III is based on a variable ordering and an equation ordering [140] Each equation in the system is of the form x = t (where t is a linear term) and each l.h.s. member is a variable which appears neither in t nor in the next equations. Conversely, the one used in CLP(R) [106] and CHIP for instance, does not need any ordering. In this last solved form, each equation in the system is of the form x = t (where t is a linear term) and each l.h.s. member is a variable which appears only once in the whole system. This last case implies an overcost to maintain little systems ....

.... on intelligent backtracking both in the sequential and concurrent cases [44, 42, 46] Preliminary results on assumption based constraint languages are reported in [151] Linear constraint solving over numeric domains has been investigated in various CLP systems such as PROLOG III [140] CLP(R) [106], CHIP [84] etc. Linear constraints simplification is mainly based on the method of Fourier [70] Various modification and improvements have been made[31, 32, 117, 67, 122, 120, 103, 104, 102, 121] based on a one at a time variable elimination, and on a simplex method and or optimizations. ....

J. Jaffar. Minimal and complete word unification. Journal of the ACM, 37 (1):47--85, 1990.

....for a given k. Remark 8.4 The special case k = 1 of the Open Problem corresponds to word equations. Indeed, in this case, by Remark 8.1 there exists a solution only if t 1 and t 2 are linear trees, i.e. they are words. The solvability of word equations has been proved by Makanin [17] cf. also [13]) Remark 8.5 By item 4. of Remark 8.2 one can derive that, if there exists a k ary solution to a tree equation for k 2, then there exists a solution for k = 2. So, in order to decide the existence of a k 2 such that (t 1 ; t 2 ) has a k ary solution, it suffices to decide the existence of a ....

J. Jaffar. Minimal and complete word unification. J. ACM vol. 37, pagg. 47--85, 1990.

....the algorithm in the Shostak combination for deciding verification conditions with non fixed bit vector equalities. The unification problem for non fixed bit vectors is also reminiscent of the word unification problem, originally solved by Makanin and later solved using a unification procedure in [Jaf90] The main difference is that variables ranging over words in that problem do not have associated size constraints which bit vectors have. By performing comparisons and arithmetic on these lengths symbolically and allowing admissible answers to be paired with accumulated constraints (as explained ....

Joxan Jaffar. Minimal and complete word unification. J. ACM, 37(1):47--85, 1990.

.... String concatenations j X Variables (Terms) t : s (Constraints) c : s v s Prefixes j s = s Equalities (1.4) The interpretation of these operations and relations is standard. Note that equality can be expressed using prefix. The processing of these constraints is well known (Jaffar, 1987). Basic word unification is undecidable. However, Concurrent constraint programming 13 as is standard in constraint programming, we delay the processing of complex constraints until they can be simplified to a tractable form. In this case, we delay processing constraints of the form X = Y ffi Z ....

Jaffar, Joxan. March 1987. Minimal and complete word unification. Submitted for publication.

....[Mak77] Since Makanin s paper appeared, his algorithm has been the object of many research activities. The objectives have been to simplify the proof of the termination and correctness of his algorithm [Pec81, Sch93] to develop simpler algorithms for deciding the solvability of word equations [Jaf90, Sch90], and to compute a description for the set of all solutions of a solvable word equation [MaAb94] Observe that a word equation can have a minimal complete set of most general unifiers that is infinite, that is, the theory of associativity is of unification type infinitary. Partially supported ....

J. Jaffar. Minimal and complete word unification. Journal Association Computing Machinery 37 (1990) 47--85.

....theory of tree equations, we need that the arity of the variables is unknown. This explains why we define a tree equation as a pair of ordered trees. A general problem in this theory is to decide the existence of a solution of a given equation. This problem has been solved by Makanin [9] cf. also [5]) in the special case of word equations, i.e. in the case of unary trees. It is not known whether this result can be extended to general k ary trees. Another related research line is to develop methods to describe the set (in general infinite) of all solutions of a given tree equation. A step in ....

J. Jaffar. "Minimal and complete word unification". J. ACM vol. 37 ; pagg. 47--85, 1990.

....under (A) is related to unification under standard associativity. Plotkin (1972) shows unification in free semi groups is infinitary and he gives a unification algorithm that is sound and complete, but it is not guaranteed to terminate. There are decision procedures by Makanin (1977) and Jaffar (1990), for example, but these are far too complex for our purposes. Fortunately, though Plotkin s algorithm is non terminating in the general case, it decides unification problems of one linear equation, or one equation in which no variable occurs more than twice (Schulz 1992) This implies, ....

Jaffar, J. (1990), Minimal and complete word unification, J. ACM 37(1), 47--85.

....[Mak77] Abdulrab and Pecuchet [AP89] give an algorithm that produces a description of a set of most general unifiers in the form of a graph. This graph may describe an infinite set of most general unifiers. The graph is not always finite if the set of most general unifiers is finite. Jaffar [Jaf90] gives a unification algorithm that terminates if the complete set of most general unifiers is finite. A unification algorithm for the process algebra with nondeterministic choice and sequential composition cannot be constructed by combining the unification algorithms for each operator separately, ....

Jaffar, J.: Minimal and Complete Word Unification. JACM37,1, pp. 47-85. 1990.

....y) z) f(x; f(y; z) g axiomatizes associativity of the binary function symbol f . Decision problem: This problem, which is very hard and had been open for a long time, was finally solved by Makanin [1977] who proves decidability of A f unification with constants (see also [P ecuchet 1981, Jaffar 1990, Abdulrab and P ecuchet 1989, Schulz 1993] Using general combination techniques and an extension of Makanin s algorithm [Schulz 1992] decidability of general A f unification was shown in [Baader and Schulz 1992, Baader and Schulz 1996] The decision problem for A f unification is NP hard ....

Jaffar J. [1990], `Minimal and complete word unification', J. of the ACM 37(1), 47--85.

.... based on a unitary equality theory [224] the standard representation is the mgu, as in the case of FT (which corresponds to the most elementary equality theory) Word equations over WE , however, are associated with an infinitary theory, and thus a unification algorithm for these equations [127] may not terminate. A solved form for word equations, or any closed form solution for that matter, is not known. The first two kinds of solved form above are also examples of solution forms, that is, a format in which the set of all solutions of the constraints is evident. Here, any instance of ....

J. Jaffar, Minimal and Complete Word Unification, Journal of the ACM, 37(1), 47--85, 1990.

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Jaffar, J., Minimal and complete word unification, J. Assoc. for Comput. Mach. 37(1) (1990) 47--85.

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