Context Unification and Traversal Equations

Abstract

Abstract. Context unification was originally defined by H. Comon in ICALP'92, as the problem of finding a unifier for a set of equations containing first-order variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are second-order variables that are restricted to be instantiated by linear terms (a linear term is a λ-expression λx 1 · · · λxn . t where every xi occurs exactly once in t). In this paper, we prove that, if the so called rank-bound conjecture is true, then the context unification problem is decidable. This is done reducing context unification to solvability of traversal equations (a kind of word unification modulo certain permutations) and then, reducing traversal equations to word equations with regular constraints.

Sequence Unification Through Currying

“…The previous lemma proves the rank-bound conjecture of [20] for a variant of context unification. Therefore, we can conclude decidability of LHCU from a small modification of the results of that paper.…”

“…Therefore, this reduction proves decidability of LHCU. The reduction is based on some ideas from [20]. There, it is proved (see Corollary 21) that if the rank-bound conjecture is true, then CU is decidable.…”

“…The conjecture (see Conjecture 15) claims that there exists a computable upper bound for the Strahler number of some unifier of every solvable CU problem. Like in [20], the reduction will be done via the traversals of the terms that allows us to encode LHCU equations into WU equations. We need the regular constraints to make this encoding sound and ensure that the solutions of the WU equations really encode solutions of the corresponding LHCU problem.…”

“…These regular expressions define postorder traversals of terms with a bounded rab and allows us to avoid the use of traversal equations which can be replaced by simple word equations with regular constraints. What follows is then an alternative proof for the decidability of LHCU based on some ideas of [20]. Definition 9.…”

Simplifying the signature in second-order unification

“…The reduction also works for Context Unification [5,24,31] and for Bounded Second-Order Unification [21,29]. Context Unification is a variant of Second-Order Unification.…”