Context-Sensitive Dependency Pairs

Abstract

Termination is one of the most interesting problems when dealing with context-sensitive rewrite systems. Although there is a good number of techniques for proving termination of context-sensitive rewriting (CSR), the dependency pair approach, one of the most powerful techniques for proving termination of rewriting, has not been investigated in connection with proofs of termination of CSR. In this paper, we show how to use dependency pairs in proofs of termination of CSR. The implementation and practical use of the developed techniques yield a novel and powerful framework which improves the current state-of-the-art of methods for proving termination of CSR.

Improving Context-Sensitive Dependency Pairs

“…This was solved first in [1]. The corresponding implementation in the tool mu-term [3] outperformed all previous tools for termination of CS rewriting.…”

“…Nevertheless, the existing results on CS-DPs [1,2,4,19] still have major disadvantages compared to the DP method for ordinary rewriting, since CS-DPs can be collapsing. To handle such DPs, one has to impose strong requirements which make the CS-DP method quite weak and which make it difficult to extend refined termination techniques based on DPs to the CS case.…”

“…5 If all t i σ are terminating, then the chain is minimal [13,16,21]. But due to the collapsing DPs, the notion of "chains" has to be adapted when it is used with CS-DPs [1]. If s i → t i is a collapsing DP (i.e., if t i ∈ V), then instead of t i σ ֒→ * R,µ s i+1 σ (and termination of t i σ for minimality), one requires that there is a term w i with t i σ ¤ µ w i and w ♯ i ֒→ * R,µ s i+1 σ.…”

“…A CS-TRS is terminating iff there is no infinite chain [1]. As in the non-CS case, the above notion of chains can also be adapted to innermost rewriting.…”

“…In the TRS of Ex. 1, minus hides both positions 1 and 2 and p hides position 1 due to Rule (1). So the hiding contexts are P, p(P), minus(P, P), p(p(P)), minus(P, p(P)), .…”