Critical Peaks Redefined - Φ⊔Ψ=⊤

6th International Workshop on Confluence6th International Workshop on ConfluenceLet a cluster be a term with a number of patterns occurring in it. We give two accounts of clusters, a geometric one as sets of (node and edge) positions, and an inductive one as pairs of terms with gaps (2nd order variables) and pattern-substitutions for the gaps. We show both notions of cluster and the corresponding refinement/coarsening orders on them, to be isomorphic. This equips clusters with a lattice structure which we lift to (parallel/multi) steps to yield an alternative account of the notion of critical peak

A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems

We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating them as equational term rewriting systems and considering E-critical pairs and/or termination modulo E. In contrast, our method is based solely on usual critical pairs and it also (partially) works even if the system is not terminating modulo E. We first present confluence criteria for term rewriting systems whose rewrite rules can be partitioned into a terminating part and a possibly non-terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system

The reproductive ecology of Leptodactylus fuscus (Anura, Leptodactylidae): new data from natural temporary ponds in the Brazilian Cerrado and a review throughout its distribution

Lucas, Elaine M., Brasileiro, Cínthia A., Oyamaguchi, Hilton M., Martins, Marcio (2008): The reproductive ecology of Leptodactylus fuscus (Anura, Leptodactylidae): new data from natural temporary ponds in the Brazilian Cerrado and a review throughout its distribution. Journal of Natural History 42 (35-36): 2305-2320, DOI: 10.1080/00222930802254698, URL: http://dx.doi.org/10.1080/0022293080225469