H.J. Burckert. Solving Disequations in Equational Theories. In E. Lusk and R. Overbeek, editors, , volume 310 of , pages 517--526. Springer-Verlag, Berlin, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which have already been unsuccessfully tried by the constraint solver and that are useful for a heuristic search of other solutions. empty state, terminal states ( labels (5 6) We must show that is an unifier of . If a substitution solves an equation then every instance solves it [12, 39]. Since is an unifier of , also is an unifier of c. Moreover, since is an unifier of then is also an unifier of . So, is an unifier of both and . If is the unique mgu over of then ( since ( and ( Thus the ....

H.J. Burckert. Solving Disequations in Equational Theories. In E. Lusk and R. Overbeek, editors, , volume 310 of , pages 517--526. Springer-Verlag, Berlin, 1988.

....a solution in the initial algebra. For disunification, solvability in the initial algebra (called ground solvability in the following) implies solvability in the free algebra (simply called solvability below) but not vice versa. Both types of solvability are considered in the literature (see [Com91, Bur88]) but ground solvability seems to be more interesting for most applications. For solvability, the adaptation of the combination method to disunification problems is relatively straightforward. The main tool of the method is a decomposition algorithm which transforms every disunification problem ....

....in Gamma to variable free Sigma terms. Gamma is called (ground) solvable iff it has a (ground) solution. It should be noted that the notion of a disunification problem does not always refer to the same kind of problem in the literature. Our definition coincides with the one of Burckert [Bur88], who considers existentially quantified equational formulae, but other authors (e.g. Comon [Com91] allow for arbitrary quantification. As in the case of unification, one has to distinguish several types of disunification problems. The (E; Sigma) disunification problem is called elementary, ....

H.J. Burckert, "Solving Disequations in Equational Theories," Proceedings of the 9th International Conference on Automated Deduction, Argonne, LNCS 310, Springer 1988.

....problems in the obvious way. It should be mentioned that we consider here only one possible semantics for disunification problems. Often these problems are also solved over the ground term algebra modulo E, the initial algebra for E. For a more thourough description of disunification we refer to [Bur88, Com91]. An equational theory E is unitary if every elementary E unification problem fl has a most general unifier, i.e. a unifier such that for every unifier of fl there exists a substitution such that (x) E ( x) for all x 2 Var(fl) Let fl be an elementary E (dis)unification problem over the ....

H.J. Burckert, "Solving Disequations in Equational Theories," Proceedings of the 9th International Conference on Automated Deduction, Argonne, LNCS 310, Springer 1988.

....may involve quantifiers and we may be interested in irreducible solutions. This slightly differs from the unification case. Actually, depending on the application at hand, syntaxes as well as semantics may differ. For example, the syntax may allow quantification [KL87, CL89, Mah88a] or not [Col84, Bur88]. The semantics may consider one particular algebra (e.g. the Herbrand Universe [LM87] or the algebra of rational trees [Col84] or irreducible trees [Com89] or a class of algebras [Mal71, MSK90] The definition of a solved form may also differ, depending on the application at hand. Indeed, we may ....

....in U is mapped to an element of A. 2.3 Examples 2.3.1 A = T (F ) or A = T (F; X) This is the interpretation used for complement problems (for example [Sch87, Sch88b] Thi84] BMPT90] LM87] Com86] and others) This interpretation is also considered in e. g [Ven87] CL89] Mah88a] [Bur88], Kun87b, Kun87a, Nic87] It appears that the main point for such interpretations is to know whether F is finite or infinite: the rules for solving equational formulas will be different in the two cases. In particular, as shown in e.g. Mah88a] see also section 4) the axiomatizations of T (F ) ....

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H. J. Burckert. Solving disequations in equational theories. In Proc. 9th Conf. on Automated Deduction, Argonne, LNCS 310. Springer-Verlag, May 1988.

....u) v) is 2 ffi [ Q(v; v) 9x; y: v = f(y) u 6= f(y) x = g(u; u) simplified to: 3 ffi [ Q(v; v) u 6= v] c 2 can then be removed from the set of c clauses and is replaced by 1 ffi and 3 ffi (see Lemma 3.6) 3.4.3. the distautology generation rule Definition 3.8. see also [Burckert, 1988]) Let c : l(t) l c (s) c 0 : X ] be a c clause. The rule of Distautology generation is defined as follows: l(t) l c (s) c 0 : X ] l(t) l c (s) c 0 : X s 6= t] Sigma Lemma 3.7. Let I be a partial Herbrand interpretation. I j= l(t) l c (s) c 0 : X ] iff I j= ....

Burckert, 1988 Burckert, H.-J. (1988). Solving disequations in equational theories. In Proceedings of the 9 th Conference on Automated Deduction (Lusk, E. and Overbeek, R., eds), pp. 517--526. Springer Lecture Notes in Computer Sciences 310.

....any predicate symbol other than equality. However, we do not assume that the model is freely generated: equality is assumed to be generated by a finite set of equations E. In other words, we are interested in the quotient T (F ) E where =E is a finitely generated congruence. This is related to [Bur88], but we consider here the full first order theory (not only a fragment) and we assume a finite alphabet F (the alphabet is assumed to be infinite in [Bur88] which is simpler in some respects) For an arbitrary E, the first order theory of T (F ) E is of course undecidable. At least the word ....

....of equations E. In other words, we are interested in the quotient T (F ) E where =E is a finitely generated congruence. This is related to [Bur88] but we consider here the full first order theory (not only a fragment) and we assume a finite alphabet F (the alphabet is assumed to be infinite in [Bur88], which is simpler in some respects) For an arbitrary E, the first order theory of T (F ) E is of course undecidable. At least the word problem (a subset of the Pi 1 fragment) and unification problem (a subset of the Sigma 1 fragment) should be decidable for the congruence =E . Unfortunately, ....

[Article contains additional citation context not shown here]

H. J. Burckert. Solving disequations in equational theories. In Proc. 9th Int. Conf. on Automated Deduction, Argonne, IL, LNCS 310. Springer-Verlag, May 1988.

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