Skew and ω-Skew Confluence and Abstract Böhm Semantics

Skew confluence was introduced as a characterization of nonconfluent term rewriting systems that had unique infinite normal forms or Böhm like trees. This notion however is not expressive enough to deal with all possible sources of non-confluence in the context of infinite terms or terms extended with letrec. We present a new notion called ωskew confluence which constitutes a sufficient and necessary condition for uniqueness. We also present a theory that can lift uniqueness results from term rewriting systems to rewriting systems on terms with letrec. We present our results in the setting of Abstract Böhm Semantics, which is a generalization of Böhm like trees to abstract reduction systems.