F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. Symbolic Computation, 21(2):211--244, 1996.  Home/Search   Document Details and Download   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 ....

....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 (as in Sect. 2.1.1) ....

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F. Baader and K. U. Schulz. Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures. J. of Symbolic Computation, 21(2):211--243, 1996.

....constraint if and only if the following holds: for every x V ar(T ) #(x) does not contain any of the function symbols below x in C. In other words, if x h j , then #(x) does not contain any occurrence of h j . These linear constraints are similar to the linear constant restrictions of [8]. The following two theorems relate a unification problem S over ACUID l to its h image T over ACUIH , where H denotes the set of homomorphisms H(S) Theorem 5. If a simple ACUID l unification problem S has a discriminating unifier, then its h image T is solvable as an ACUIH unification ....

F. Baader and K.U. Schultz. Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures. Proc. 11th Conference on Automated Deduction (CADE-11), Saratoga Springs, NY, Springer LNAI 607, 1992, 50--65.

....studied by Tiwari [Tiw00] Barrett, Dill, and Stump [BDS02] and Ganzinger [Gan02] but none of these methods includes a general canonization procedure as is required for a Shostak combination. 16 Variable abstraction is also used in the combination unification procedure of Baader and Schulz [BS96], which addresses a similar problem to that of combining Shostak solvers. In our case, there is no need to ensure that solutions are compatible across distinct theories. Furthermore, variable dependencies can be cyclic across theories so that it is possible to have y vars(S i (x) and vars(S ....

F. Baader and K. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. Symbolic Computation, 21:211-- 243, 1996.

....between fl and a is obvious. String unification has been discovered and investigated by several independent research communities (for an overview see [2] The notion of string unification stems from the field of automated deduction [21, 24] where it is also called A unification [1] with a single associative function symbol. String unification has first been presented by Markov [15] in 1954 and is 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 ....

F. Baader and K. Schulz. Unification in the union of disjoint equational theories. Int. Conference on Automated Deduction, volume 607 of LNCS, pages 50--65, 1992.

....symbol of arity n 1 can be expressed in the positive # fragment of the first order theory of word equations with regular constraints over the alphabet . n . Unfortunately, even the positive fragment of a single word equation is undecidable [26] except if the alphabet is infinite [27] or a singleton [28] Therefore, it remains open whether NSSE is decidable or not. But it becomes clear that the di#culty is raised by word equations hidden behind cap set expressions R # , i.e. the equation ##=# in Lemma 1. Theorem 1 constitutes a promising starting point to further investigate ....

F. Baader, K. Schulz, Unification in the union of disjoint equational theories: Combining decision procedures, Journal of Symbolic Computation 21 (1996) 211--243.

....symbol of arity n 1 can be expressed in the positive # fragment of the first order theory of word equations with regular constraints over the alphabet . n . Unfortunately, even the positive fragment of a single word equation is undecidable [3] except if the alphabet is infinite [2] or a singleton [24] Therefore, it remains open whether NSSE is decidable or not. But it becomes clear that the difficulty is raised by word equations hidden behind cap set expressions R # , i.e. the equation ##=# in Lemma 3. Theorem 1 constitutes a promising starting point to further ....

F. Baader and K. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. In Journal of Symbolic Computation, volume 21, pages 211--243, 1996.

.... Introduction Nelson and Oppen were among the first to provide a fairly general method to combine logical theories and relative satisfiability procedures ( 17] Since then, almost all the effort in the field of combination has been concentrated on equational theories and unification algorithms ([2, 3, 5, 6, 7, 9, 10, 14, 19, 20, 21, 25, 28, 27]) Others have worked on combinations of more general theories as well (see [15, 23, 24] for instance) but to date the Nelson Oppen method appears to be still one of the most general in the field. The need of extending the focus from unification to more general satisfiability problems is well ....

..... If is not already in separate form, we can apply a conversion procedure that, given , returns an equivalent separate form. The separation procedure (and its correctness proof) is straightforward but to describe it we need some definitions and notation first. We have adapted these from those in [2], among others, which appear to be well established in the field. Consider the theories described above. For i 1, n, a member of Z is an i symbol. A Z term t is an i term if it is a variable or it has the form f5 and f is an symbol. An predicate is defined analogously. A sub term of ....

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Franz Baader and Klaus U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. In Proceedings of the 11th International Conference on Automated Deduction, volume 607 of Lecture Notes in Artificial Intelligence, pages 50-65. Springer-Verlag, 1992.

.... modules (see [GPT96] for a generalized approach) In recent years, considerable research has focused on both domain and solver combinations (see, for instance, BS95a, BS95b, KS96, NO79, Sho84] although most of the efforts have been concentrated on unification problems and equational theories ( BS92, Bou93, DKR94, Her86, KR92, SS89, Yel87] among others) The current results of these investigations are limited in scope, and a deep understanding of many model and proof theoretic issues involved is still out of reach. Despite that, in this paper we attempt to show the e#ectiveness of ....

....# i is a (possibly empty) formula of sCNF(# i ) If # is not already in separate form, we can apply a conversion procedure that, given #, returns an equivalent separate form of #. To describe such a procedure, we need to introduce some definitions and notation that we have adapted from those in [BS92] among others. Consider the signatures introduced above. For i =1, 2, a member of # i is an i symbol . A term t is an i term if it is a variable or if its root symbol is an i symbol. An i atomic formula (i atom for short) is defined analogously. A subterm of an i term t is an alien subterm of t ....

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Franz Baader and Klaus U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. In Deduction, volume 607 of Lecture Notes in Artificial Intelligence, pages 50--65, Berlin, 1992. Springer-Verlag. 46

....on theorem proving has focused on decomposition for parallel implementations [5, 14, 50] and has followed decomposition methods guided by lookahead and subgoals, neglecting the types of structural properties we used here. Another related line of work focuses on combining logical systems (e.g. [38, 45, 3]) Contrasted with this work, we focus on interactions between theories with overlapping signatures, the efficiency of reasoning, and automatic decomposition. Decomposition for propositional SAT has followed differ ent tracks. Some work focused on heuristics for clause weighting or symbol ....

F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. In 11th Intl. conf on automated deduction, volume 607 of LNAI, pages 50 65. Springer-Verlag, 1992.

....on theorem proving has focused on decomposition for parallel implementations [8, 5, 15, 43] and has followed decomposition methods guided by lookahead and subgoals, neglecting the types of structural properties we used here. Another related line of work focuses on combining logical systems (e.g. [32, 40, 3, 36, 44]) Contrasted with this work, we focus on interactions between theories with overlapping signatures, the efficiency of reasoning, and automatic decomposition. Decomposition for propositional SAT has followed different tracks. Perhaps the most relevant work to ours is [19] which presented ....

F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. In 1 lth Intl. conf on automated deduction, volume 607 of LNAI, pages 512L65. Springer-Verlag, 1992.

....if and only if Sol encodes a most general E 1 [ E 2 unifier of PH . Another normal form true:P 1 :P 2 :Sol means that P 1 P 2 is not E 1 [ E 2 unifiable. 4.1. 3 Combination rules for arbitrary theories We describe in this section a rule based specification of the unification algorithm [5] for a union of disjoint arbitrary equational theories. In the general case, we are faced with the problem of finding two solved forms Sol 1 and Sol 2 of two pure unification problems P 1 and P 2 such that their conjunction Sol 1 Sol 2 leads directly (after variable replacement) to a compound ....

F. Baader and K. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. Journal of Symbolic Computation, 21(2):211--243, February 1996.

.... for such popularity lies in the fact that almost all systems are enhanced by the use of a small amount of deduction, and unification is fundamental to this process [JJ98, BM99, Bod99, IS99] 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 ....

F. Baader and K.U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. Journal of Symbolic Computation, 21(2):211--244, 1998.

....and E contains x Delta x = x, we call Delta an ACI symbol (I stands for idempotence) Also, Delta is called an AC1 symbol (or an ACI1 symbol) if Delta is an AC symbol (an ACI symbol respectively) and E contains the equation x Delta 1 = x where 1 is a constant belonging to L. Theorem 8. 1 ([96, 11, 17, 86]) Let E be an equational theory defining a function symbol Delta in L as an associative symbol (E contains all logical consequences of x Delta (y Delta z) x Delta y) Delta z and no other equations) The following upper and lower bounds on the complexity of the E unification problem hold: ....

F. Baader and K. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. In D. Kapur, editor, 11th CADE, LNCS/LNAI 607, pp. 50-- 65, 1992.

....in some sense, more important the resulting algorithm proves to have a polynomial nondeterministic complexity, which is not the case for the well founded set unification algorithm. An alternative approach could be to use the general (and elegant) methods for combining unification algorithms (see [BS96]) and, more generally, 3 constraint domains even non well founded and solvers (see [BS95] According to these methods, we could concentrate our e#orts on analyzing the pure case of hyperset (domains and) unification and then obtain automatically the algorithm for the general hybrid case. ....

....a goal driven and terminating algorithm for the pure case. Then the combined algorithm could be obtained, in our context, by adding two (deterministic) actions concerning standard term decomposition. Therefore, we face directly the more general hybrid problem without making use of the results of [BS96, BS95]. The rest of the paper is organized as follows. Section 2 introduces the notion of set term and briefly presents the main ideas underlying the set unification algorithm of [DOPR96] Section 3 gives an intuitive presentation of hypersets and briefly discusses the motivations underlying their ....

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F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. Journal of Symbolic Computation, 21(2):211--243, 1996.

....methodology for deciding what messages should be sent from one partition to another, or which partitions should receive messages from which other partitions. Second, there are no clear criteria for decomposing a theory into sub problems. Another line of work focuses on combining logical systems [80,99,7,87,109]. Here, the computational focus has been on treating combinations of signaturedisjoint theories (allowing the queries to include symbols from all signatures) e.g. 7] Recent work introduced sharing function symbols between two theories (e.g. 87] but no algorithm allowed any sharing of ....

....clear criteria for decomposing a theory into sub problems. Another line of work focuses on combining logical systems [80,99,7,87,109] Here, the computational focus has been on treating combinations of signaturedisjoint theories (allowing the queries to include symbols from all signatures) e.g. [7]. Recent work introduced sharing function symbols between two theories (e.g. 87] but no algorithm allowed any sharing of relation symbols. All approaches either nondeterministically instantiate the (newly created) variables connecting the theories (e.g. 109] or restrict the theories to be ....

Franz Baader and Klaus U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. In Proceedings of the 11th International conference on automated deduction, volume 607 of LNAI, pages 50--65. Springer-Verlag, 1992.

....into a decision procedure for the word problem in E [J E2 [10, 14, 13, 8, 7] For the matching and the unification problem, there also exist very general combination Partially supported by the EC Working Group CCL II. results under the disjointness restriction (see [12] for matching, and, e.g. [13, 4, 1] for unification) It is not hard to extend these results to theories sharing constant symbols [11, 7, 2] The only work we are aware of that presents a general combination approach for the union of equational theories having more than con stant symbols in common is [6] where the problem of ....

.... s t between (l U 2) terms s,t can be transformed into an equisatisfiable formula T1 A T2, where Ti is a conjunction of pure equations and disequations (i = 1, 2) This can be achieved by the usual variable abstraction process in which alien subterms are replaced by new variables (see, e.g. [1, 3] for a detailed description of the process) Obviously, if we know that l A is satisfiable in a model 4 of E1 U E, then is satisfiable in the reduct 4 r, which is a model of E (i = 1, 2) However, the converse need not be true, that is, if is satisfiable in a model 4 of Ei (i = 1, 2) then we ....

F. Baader and K.U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. Symbolic Computation 21, 1996.

....To simplify the presentation, we restrict our attention in this paper to the equational case. 2] described a version of their combination procedure that applies to positive theories, i.e. positive Boolean combinations of equations with an arbitrary quan tifier prefix. They also showed [3] that the decidability of the positive theory is equivalent to the decidability of unification with linear constant restrictions. Since then, the main open problem in the area was how to extend these results to the combination of theories having symbols in common. In general, the existence of ....

F. Baader and K.U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. Symbolic Computation 21, 1996.

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F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. Symbolic Computation, 21(2):211--244, 1996.

No context found.

F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. Symbolic Computation, 21(2):211--244, 1996.

No context found.

F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. of Symbolic Computation, 21(2):211--244, 1996.

No context found.

F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. Symbolic Computation, 21(2):211--244, 1996.

No context found.

F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. Symbolic Computation, 21(2):211--244, 1996.

No context found.

F. Baader and K. U. Schulz. Unification in the union of disjoint equational theories: Combining decision procedures. J. of Symbolic Computation, 21(2):211--244, 1996.

No context found.

F. Baader and K. U. Schulz. Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures. Journal of Symbolic Computation, 21:211--243, 1996.

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F. Baader and K. U. Schulz. Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures. JSC, 21:211--243, 1996.

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