H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. Symbolic Computation, 8(5):499--522, 1990. 46  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is decidable. Based on the decision procedure, a constraint based approach to theorem proving with sequence variables and flexible arity symbols can be developed (compare [22] 25] Particular instances of unification with sequence variables and flexible arity symbols are word equations ([1], 15] 26] equations over free semigroups ( 19] equations over lists of atoms with concatenation ( 7] pattern matching. We have implemented the unification procedure (without decision algorithm) as a Mathematica package and incorporated it into the Theorema system [6] which aims at ....

....and individual variables, free flexible arity function symbols, free constants and free fixed arity function symbols. We denote it as GUP . The unification procedure is a tree generation process based on two basic steps: projection and transformation. 4. 1 Projection The idea of projection ([1]) is to eliminate some sequence variables from the given unification problem UP . Let #(UP ) be the following set of substitutions: x # S S vars(UP ) # SV , where vars(UP ) is a set of variables of UP . #(UP ) is called the set of projecting substitutions for UP . Each # # ....

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H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. Symbolic Computation, 8(5):499--522, 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 ....

H. Abdulrab and J-P. Pecuchet. Solving Word Equations. Jour. Symbolic Computation, 8:499--521, 1989.

....equations has a simple formulation: Find out whether or not an input word equation has a solution. The decidability of the problem was proved by Makanin [11] His decision 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 ....

Abdulrab H., Pecuchet J.P., Solving word equations, Journal of Symbolic Computation 8(1989), 499-521.

....1 summarizes part of our current knowledge about uni cation in equational theories. 11 Name Type Deci Main references ; 1 yes [Her30, Rob65, Rob71, Hue76, MM82, PW78, CB83, Fag83] Subject of Section 3.7. A(f) 1 yes The decidability has been proved in [Mak77] Related works are [Abd87, AP90, Jaf90, Plo72] P ec81, Sie75, LS75] C(f) yes [Sie79, Her87, Kir86] Subject of Section 3.8. I(f) yes First studied in [RS78] Hullot [Hul80a] derives an algorithm by narrowing. A(f) C(f) yes [Sti76, Sti75, Sti81, LS76, Fag84, HS85, Kir89, Kir85] A(f) I(f) 0 yes Studied in ....

H. Abdulrab and J.-P. Pecuchet. Solving word equations. Journal of Symbolic Computation, 8(5):499-522, 1990.

....J 0 ) if this combined substitution is J admissible and fails otherwise or if either of the two unifications fails. 3 Prefix Unification. Computing the most general unifiers of a set of prefix equations EQ is by no means trivial. General string unification [Matiyasevic, 1968, Manakin, 1977, Abdulrab and Pecuchet, 1990] although decidable, is already complicated, because two arbitrary strings may have infinitely many most general unifiers. Fortunately, prefixes are a very restricted class of strings. They do not contain duplicates and the same character cannot occur in two prefixes p and q of atoms in the same ....

H. Abdulrab and J.-P. Pecuchet. Solving word equations. In C. Kirchner, ed., Unification, pages 353--375. Academic Press, 1990.

....In the case of string unification (the monoid problem or resolution problem for word equations) the only axiom is the associativity of string concatenation. An algorithm for enumerating the most general unifiers of a set of string equations has first been presented by Plotkin [16] whereas Makanin [9, 10, 1] showed that it is possible to decide whether a set of string equations has a unifier or not. In general, however, the number of most general unifiers of a set of string equations is infinite. For Gamma = faX=Xag, for instance, the set of most general unifiers is foe j 9i 2 IN: oe(X) a i g. ....

....important non classical logics. For these logics, however, a characterization for validity and the notion of a prefix still has to be developed. Our algorithm is much simpler and considerably more efficient than other string unification algorithms developed so far. The algorithms described in [9, 10, 1] are developed for general string unification and do not take advantage of the special properties of prefix strings. Ohlbach s algorithm [13] does not compute a minimal set of unifiers and thus wastes computation time. Besides this one of the main advantages of our algorithm is that it generates ....

H. Abdulrab and J.-P. Pecuchet. Solving word equations. In C. Kirchner, editor, Unification, pages 353--375. Academic Press, London, 1990.

....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] ....

H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. Symbolic Computation, 8:499--521, 1989.

....series dedicated to word equations and related topics: 23] and [2] Makanin s Algorithm is the construction of a nite search graph. It s niteness proof is probably among the most complex proofs in theoretical computer science. The algorithm was implemented in 1987 at Rouen by Abdulrab, see [1]. In 1990 Schulz showed an important generalization: Makanin s result remains true when adding regular constraints, 22] Thus, we may specify for each word variable x a regular language L x and we are only looking for solutions, where the value of each variable x is in L x . Having this form it ....

Habib Abdulrab and Jean-Pierre Pecuchet. Solving word equations. J. Symbolic Computation, 8(5):499-521, 1990.

....and in fact there may be an 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, ....

H. Abdulrab and J-P. Pecuchet. Solving Word Equations. Jour. Symbolic Computation, 8:499--521, 1989.

....steps at assignments. Intuitively, this specifies a flow insensitive data flow analysis, refined by data flow sensitivity at assignments. The unification theory described in Section 3. 2 is a special and feasible case of associative unification (word unification, solving of word equations) [AP89] that, however, does not seem to have been treated in the literature before. Independently of us, Ramalingam, Field and Tip have developed 3 Year 2000 unsafe years in the range 1900 1999 are considered a special case of windowed years with pivot 00. prog : dec stmt dec : ffl j dec dn: elem ....

Habib Abdulrab and Jean-Pierre P'ecuchet. Solving word equations. Journal of Symbolic Computation, 8(5):499--521, November 1989.

....In the case of string unification (the monoid problem or resolution problem for word equations) the only axiom is the associativity of string concatenation. An algorithm for enumerating the most general unifiers of a set of string equations has first been presented by Plotkin [16] whereas Makanin [9, 10, 1] showed that it is possible to decide whether a set of string equations has a unifier or not. In general, however, the number of most general unifiers of a set of string equations is infinite. For Gamma = faX=Xag, for instance, the set of most general unifiers is foe j oe(X) a i ; i 2 IN 0 ....

....important non classical logics. For these logics, however, a characterization for validity and the notion of a prefix still has to be developed. Our algorithm is much simpler and considerably more efficient than other string unification algorithms developed so far. The algorithms described in [9, 10, 1] are developed for general string unification and do not take advantage of the special properties of prefix strings. Ohlbach s algorithm [13] does not compute a minimal set of unifiers and thus wastes computation time. Besides this one of the main advantages of our algorithm is that it generates ....

H. Abdulrab and J.-P. Pecuchet. Solving word equations. In C. Kirchner, editor, Unification, pages 353--375. Academic Press, London, 1990.

....J ( A) It returns (oe Q 0 ; oe J 0 ) if this combined substitution is J admissible and fails otherwise or if either of the two unifications fails. 3 Prefix Unification. Computing the most general unifiers of a set of prefix equations EQ is by no means trivial. General string unification [31, 27, 1], although decidable, is already complicated, because two arbitrary strings may have infinitely many most general unifiers. Fortunately, prefixes are a very restricted class of strings. They do not contain duplicates and the same character cannot occur in two prefixes p and q of atoms in the same ....

H. Abdulrab and J.-P. Pecuchet. Solving word equations. In C. Kirchner, ed., Unification, pages 353--375. Academic Press, 1990.

....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 [Benanav, Kapur and Narendran ....

Abdulrab H. and P' ecuchet J.-P. [1989], `Solving word equations', Journal of Symbolic Computation 8(5), 499--521.

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H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. Symbolic Computation, 8(5):499--522, 1990. 46

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H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. Symbolic Computation, 8(5):499--522, 1990.

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H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. of Symbolic Computation, 8(5):499--522, 1990.

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H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. Symbolic Computation, 8(5):499--522, 1990.

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H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. Symbolic Computation, 8(5):499--522, 1990. 46

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H. Abdulrab and J.-P. Pecuchet. Solving word equations. J. of Symbolic Computation, 8(5):499--522, 1990.

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H. Abdulrab, J.-P. P'ecuchet, "Solving Word Equations," J. Symbolic Computation 8 (1989), pp. 499-521.

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