2D Dependency Pairs for Proving Operational Termination of CTRSs

Abstract

Abstract. The notion of operational termination captures nonterminating computations due to subsidiary processes that are necessary to issue a single 'main' step but which often remain 'hidden' when the main computation sequence is observed. This highlights two dimensions of nontermination: one for the infinite sequencing of computation steps, and the other that concerns the proof of some single steps. For conditional term rewriting systems (CTRSs), we introduce a new dependency pair framework which exploits the bidimensional nature of conditional rewriting (rewriting steps + satisfaction of the conditions as reachability problems) to obtain a powerful and more expressive framework for proving operational termination of CTRSs.

Models for Logics and Conditional Constraints in Automated Proofs of Termination

“…According to [13], R in Example 1 is operationally terminating if there is a relation on terms such that → * ⊆ , and a wellfounded ordering satisfying • ⊆ such that, for all substitutions σ, if σ(implies (implies(x, implies(x, 0) Now, consider the ordering > 1 over R given by x > 1 y if and only if x−y ≥ 1; it is a well-founded relation on [0, 1] (see [10]). We let be the (well-founded) relation on T (F, X ) induced by > 1 as before.…”

“…According to [13], R is operationally terminating if there is a relation such that → * ⊆ , and is a well-founded ordering such that • ⊆ and for the dependency pair a → a ⇐ b → x, c → x (for a a new symbol), we have that, for all substitutions σ, if b → * σ(x) and c → * σ(x), then a a . With …”

“…We failed to prove operational termination of R by using the ordering-based techniques introduced in [13] and also with the more advanced techniques in [14]. However, below we provide a very simple proof of operational termination based on the use (within [13]!)…”

“…An interpreter for a logic L (for instance the logic for Conditional Term Rewriting Systems (CTRSs) with inference system in Figure 1) is an inference machine that, given a theory S (e.g., the CTRS R in Example 1) and a goal formula ϕ (e.g., a one-step rewriting s → t for terms s and t) tries to incrementally build a proof tree for ϕ by using (instances of) the inference rules B1,...,Bn A ∈ I(L) of the inference system I(L) of L. Then, S is operationally terminating if for any ϕ the interpreter either finds a proof, or fails in all possible attempts (always in finite time). In this setting, practical methods for proving operational termination involve two main issues (see [13] and also [17] for CTRSs): (1) the simulation of the (one-step) rewrite relations → and → * associated to a CTRS R and defined by means of the inference system in Figure 1; and (2) the use of (automatically generated) wellfounded relations to abstract rewrite computations and guarantee the absence (Refl) t → * t (Cong)…”

“…However, below we provide a very simple proof of operational termination based on the use (within [13]!) of a bounded domain [0, 1] which can be easily implemented by using conditional constraints.…”