Function Calls at Frozen Positions in Termination of Context-Sensitive Rewriting

Abstract

Abstract. Context-sensitive rewriting (CSR) is a variant of rewriting where only selected arguments of function symbols can be rewritten. Consequently, the subterm positions of a term are classified as either active, i.e., positions of subterms that can be rewritten; or frozen, i.e., positions that cannot. Frozen positions can be used to denote subexpressions whose evaluation is delayed or just forbidden. A typical example is the if-then-else operator whose second and third arguments are not evaluated until the evaluation of the first argument yields either true or false. Imposing replacement restrictions can improve the termination behavior of rewriting-based computational systems. Termination of CSR has been investigated by several authors and a number of automatic tools are able to prove it. In this paper, we analyze how frozen subterms affect termination of CSR. This analysis helps us to improve our ContextSensitive Dependency Pair (CS-DP) framework for automatically proving termination of CSR. We have implemented these improvements in our tool mu-term. The experiments show the power of the improvements in practice.

mu-term: Verify Termination Properties Automatically (System Description)

“…Full explanations of the processors can be found in [4,12,13,19,20,24,25]. The mu-term 6.0 logic-based approach has led to dramatic improvements, as reported here:…”

“…upv.es/muterm/?page id=82 for an historical account. In the CSR subcategory, since 2014 mu-term is able to prove all the examples proved by any other participating tool (thanks to the results in [12]).…”

“…Hiding contexts are contexts where hidden terms can occur at active positions within a context-sensitive rewrite sequence [1,13]. There, hidden terms can restart a delayed recursive call after the application of a rule with migrating variables (see [12] for a detailed analysis). For R and μ in Example 8, the only rule with hidden terms is fact(x) → if(zero(x), s(0), fact(p(x)) × x).…”

“…Symbol p hides no position. The refinements introduced in [12] have led to a more precise notion of hidden terms and contexts, enabling a better analysis of the connections between them. This has greatly improved the ability of mu-term to prove termination of CSR.…”