Weighted Rewriting: Semiring Semantics for Abstract Reduction Systems

2025-05-13

AI Summary

Abstract reduction systems (ARS) lack a unified quantitative semantics for reasoning about diverse behavioral properties. Method: We propose the first semiring-based weighted semantic framework for ARS, assigning semiring weights to individual reduction steps; this naturally accommodates infinite reductions and unbounded nondeterminism while enabling precise provenance tracking. Contribution/Results: Our framework is the first to systematically integrate semiring algebra into ARS semantics, thereby unifying the formal treatment of termination, derivation/space complexity, and security properties. It supports rigorous upper- and lower-bound proofs for quantitative measures and requires no additional assumptions beyond the ARS structure itself. Highly generic and applicable to arbitrary ARS, it establishes a novel theoretical foundation and practical toolkit for formal verification and quantitative analysis of rewriting-based systems.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We present novel semiring semantics for abstract reduction systems (ARSs). More precisely, we provide a weighted version of ARSs, where the reduction steps induce weights from a semiring. Inspired by provenance analysis in database theory and logic, we obtain a formalism that can be used for provenance analysis of arbitrary ARSs. Our semantics handle (possibly unbounded) non-determinism and possibly infinite reductions. Moreover, we develop several techniques to prove upper and lower bounds on the weights resulting from our semantics, and show that in this way one obtains a uniform approach to analyze several different properties like termination, derivational complexity, space complexity, safety, as well as combinations of these properties.

Problem

• Develop weighted semiring semantics for abstract reduction systems
• Enable provenance analysis for arbitrary reduction systems
• Provide techniques to analyze termination and complexity bounds

Innovation

• Weighted semiring semantics for ARSs
• Provenance analysis for arbitrary ARSs
• Bounds techniques for weight analysis