# Download Rewriting Logic and Its Applications: 8th International by Natarajan Shankar (auth.), Peter Csaba Ölveczky (eds.) PDF

By Natarajan Shankar (auth.), Peter Csaba Ölveczky (eds.)

This e-book constitutes the refereed complaints of the eighth foreign Workshop on Rewriting good judgment and its functions, WRLA 2010, held as a satellite tv for pc occasion of ETAPS 2010, Paphos, Cyprus, in March 2010. The thirteen revised complete papers awarded have been conscientiously reviewed and chosen from 29 submissions. The papers are geared up in topical sections on termination and narrowing; instruments; the okay framework; functions and semantics; maude version checking and debugging; and rewrite engines.

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**Additional info for Rewriting Logic and Its Applications: 8th International Workshop, WRLA 2010, Held as a Satellite Event of ETAPS 2010, Paphos, Cyprus, March 20-21, 2010, Revised Selected Papers**

**Example text**

We carefully analyze the structure of inﬁnite rewrite sequences for A∨C-rewrite theories. This leads to appropriate deﬁnitions of A∨C-dependency pair and minimal chain. After some technical preliminaries, in Section 3 we investigate the drawbacks of previous notions of minimal term when modeling inﬁnite A∨C-rewrite sequences. Then, we introduce the notion of stably minimal non-E-terminating term which is the basis of our development. Section 4 investigates the structure of inﬁnite sequences starting from such stably minimal terms.

Note that, if E is an A∨C-equational theory, then root(t) ∈ D whenever t ∈ T∞,R,E . As remarked by Giesl and Kapur (see also Example 5 below) this is not true for arbitrary equational theories. The problem with Giesl and Kapur’s Deﬁnition 2 is that minimality is not preserved under E-equivalence. Example 2. Consider the following TRS R: f (x, x) → f (0, f (1, 2)) (1) where f ∈ ΣAC . Hence, ExtAC (R) only adds the following rule to R: f (f (x, x), y) → f (f (0, f (1, 2)), y) (2) Note that t = f (f (0, 1), f (0, f (1, 2))) is non-(ExtAC (R), AC)-terminating: f (f (0, 1), f (0, f (1, 2))) ∼A f (0, f (1, f (0, f (1, 2)))) ∼A f (0, f (f (1, 0), f (1, 2))) ∼C Λ f (0, f (f (0, 1), f (1, 2))) ∼A f (0, f (0, f (1, f (1, 2)))) ∼A f (f (0, 0), f (1, f (1, 2)))→ExtAC (R) f (f (0, f (1, 2)), f (1, f (1, 2))) →ExtAC (R),AC · · · Since f (0, 1) and f (0, f (1, 2)) are in (ExtAC (R), AC)-normal form, we have that t ∈ T∞,R,AC .

Given u → v, u → v ∈ P, there is an arc from u → v to u → v if u → v, u → v is a minimal (P, R, S, μ)-chain for some substitution σ. In termination proofs, we are concerned with the so-called strongly connected components (SCCs) of the dependency graph, rather than with the cycles themselves (which are exponentially many) [15]. The following result formalizes the use of SCCs for dealing with CS problems. Theorem 6 (SCC Processor). Let τ = (P, R, S, μ) be a CS problem. Then, the processor ProcSCC given by ProcSCC (P, R, S, μ) = {(Q, R, SQ , μ) | Q are the pairs of an SCC in G(P, R, S, μ)} Proving Termination in the CSDP Framework 27 (where SQ are the rules from S involving a possible (Q, R, S, μ)-chain) is sound and complete.