Combination techniques and decision problems for disunification
Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E 1,…, E n in order to obtain a unification algorithm for the union E 1 ∪ ⋯ ∪ E n of the theories. Here we want to show that variants of this method may be used t...
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| Published in: | Theoretical computer science Vol. 142; no. 2; pp. 229 - 255 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article Conference Proceeding |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
15.05.1995
Elsevier |
| Subjects: | |
| ISSN: | 0304-3975, 1879-2294 |
| Online Access: | Get full text |
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| Summary: | Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories
E
1,…,
E
n
in order to obtain a unification algorithm for the union
E
1 ∪ ⋯ ∪
E
n
of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in
E
1 ∪ ⋯ ∪
E
n
. Our first result says that solvability of disunification problems in the free algebra of the combined theory
E
1 ∪ ⋯ ∪
E
n
is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories
E
i
(
i = 1,…,
n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in
E
1 ∪ ⋯ ∪
E
n
we have to consider a new kind of subproblem for the particular theories
E
i
, namely solvability (in the free algebra) of disunification problems with linear constant restriction under the additional constraint that values of variables are not
E
i
-equivalent to variables. The correspondence between ground solvability and this new kind of solvability holds, (1) if one theory
E
i
is the free theory with at least one function symbol and one constant, or (2) if the initial algebras of all theories
E
i
are infinite. Our results can be used to show that the existential fragment of the theory of the (ground) term algebra modulo associativity of a finite number of function symbols is decidable; the same result follows for function symbols which are associative and commutative, or associative, commutative and idempotent. |
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| ISSN: | 0304-3975 1879-2294 |
| DOI: | 10.1016/0304-3975(94)00277-0 |