A Decision Algorithm for Stratified Context Unification

Abstract

Context uni cation is a variant of second-order uni cation and also a generalization of string uni cation. Current l y i t i s n o t k n o wn whether context uni cation is decidable. An expressive fragment o f c o ntext uni cation is strati ed context uni cation. Recently, i t t u r n e d o u t that strati ed context uni cation and one-step rewrite constraints are equivalent. This paper contains a description of a decision algorithm SCU for strati ed context uni cation together with a proof of its correctness, which shows decidability of strati ed context uni cation as well as of satis ability of one-step rewrite constraints.

Context Unification and Traversal Equations

“…There it is proved that context unification is decidable when, for any context variable, all its occurrences have the same argument. Later, it was proved [SS96, SS98,SS99b] that the problem is also decidable when context variables are stratified, i.e. when, for any variable, the list of context variables we find going from the root of the term to any occurrence of this variable is always the same.…”

Dominance constraints in stratified context unification

“…Stratified CU is the restriction of CU to stratified CU-equations [20,13]. We extend the usual setting by SO-equations, in order to account for once-only SO-nesting.…”

“…Whether context unification is decidable is a prominent open question. Only some fragments of CU could be shown decidable so far [2,20,21,12,14], among them stratified CU [20]. It is of particular interest, since it can express one-step rewriting constraints [18] for which decidability could not be shown by more direct methods.…”

Simplifying the signature in second-order unification

“…Roughly speaking the variant consists of constraining unifiers to instantiate second-order variables by linear terms. The problem has also several decidable fragments [16,20,28,30]. One of them is Word Unification [12,25,27] where function symbols are at most unary.…”