🤖 AI Summary
This work addresses the completeness problem for left-linear term rewriting systems under associative-commutative (AC) equational theories. We present the first completion algorithm that simultaneously preserves left-linearity and AC-normalization properties. Unlike conventional approaches relying on AC-unification, our method introduces an AC-aware critical pair generation and reduction mechanism, ensuring that left-linearity and AC-compatibility are maintained throughout the entire completion process, while automatically yielding a confluent, terminating, and decidable rewrite system. The approach integrates rewriting theory, an enhanced AC-normalization strategy, inductive proof techniques, and a symbolic computation framework. Experimental evaluation on standard AC benchmark suites demonstrates that the algorithm efficiently generates correct rewrite systems, thereby unifying theoretical completeness and practical feasibility for left-linear AC completion for the first time.