A Condensed Goal-Independent Fixpoint Semantics Modeling the Small-Step Behavior of Rewriting

Abstract

In this paper we present a novel condensed narrowing-like semantics that contains the minimal information which is needed to describe compositionally all possible rewritings of a term rewriting system. We provide its goal-dependent top-down definition and, more importantly, an equivalent goal-independent bottom-up fixpoint characterization.We prove soundness and completeness w.r.t. the small-step behavior of rewriting for the full class of term rewriting systems.

A program analysis framework for tccp based on abstract interpretation

“…Now we prove that α τ (r↓ c ) α τ (r)↓ τ (c) by induction on the structure of r. Again, we first recall the definition for the concrete operator from [CTV13]. Definition A.3 (Concrete weak propagation operator) Let r ∈ CT and c ∈ C. We define the weak propagation of c in r, denoted r↓ c , as ↓ c ∶= , ⊠↓ c ∶= ⊠, and…”

“…Let us first recall the definition for the concrete operator, technically adapted from [CTV13]. where, for all c ∈ C, ∃ {x1,...,xn} c ∶= ∃x 1 ⋯ ∃x n c and, for all C ∈ ℘(C), ∃V C ∶= {∃ V c c ∈ C}.…”

“…In this section, we briefly recall the concrete denotational domain and semantics of [CTV13], which is fully abstract 5 w.r.t. the small-step operational behavior of tccp.…”

“…the small-step operational behavior of tccp. Fully detailed version of all definitions and proof of full abstraction, as well as of all results here summarized can be found in [CTV13].…”

“…However, it does not allow one to analyze existential properties, for instance to verify that there exists a suspension trace. In our proposal, we follow such approach starting from the concrete semantics for tccp defined in [CTV13]. This semantics addresses all thorniest difficulties of tccp (i.e., infinite computations, use of negative information and non-determinism).…”