insight - Logic and Formal Methods - # Confluence Analysis of Logically Constrained Rewrite Systems

Undecidability of Local Confluence for Terminating Logically Constrained Rewrite Systems

Q: Are there any decidable fragments of LCTRSs for which local confluence is decidable

In the context provided, it is stated that local confluence of terminating LCTRSs is undecidable, even for a decidable fragment of the theory of integers. This implies that there are no decidable fragments of LCTRSs for which local confluence is decidable. The undecidability of local confluence in LCTRSs poses a significant challenge in automated analysis and verification processes.

Q: What other confluence criteria for TRSs could be lifted to the LCTRS setting using the presented transformation

The transformation presented in the context allows for the lifting of various confluence criteria for TRSs to the LCTRS setting. One such criteria that can be lifted is the concept of (almost) development closed critical pairs. By leveraging the transformation from LCTRSs to TRSs, advanced confluence criteria based on critical pairs in the TRS domain can be extended to the constrained setting of LCTRSs. This enables the application of established confluence techniques for TRSs to the analysis of logically constrained rewrite systems.

Q: How can the insights from this work be applied to improve automated confluence analysis tools for programs that involve complex constraints

The insights from this work can be applied to enhance automated confluence analysis tools for programs that involve complex constraints. By utilizing the transformation from LCTRSs to TRSs, these tools can leverage existing confluence criteria and techniques developed for TRSs. This can lead to more efficient and effective confluence analysis for programs with logical constraints, improving the accuracy and reliability of automated verification processes. Additionally, the ability to relate critical pairs in TRSs to constrained critical pairs in LCTRSs allows for a more seamless integration of advanced confluence analysis methods into tools designed for program analysis and verification.

Core Concepts

Local confluence is undecidable for terminating logically constrained rewrite systems, even when the underlying theory is decidable.

Abstract

The paper investigates the confluence properties of logically constrained rewrite systems (LCTRSs), which extend plain term rewrite systems with native support for constraints handled by SMT solvers. The main results are:

Local confluence of terminating LCTRSs is undecidable, even when the underlying theory is decidable. This is shown by a reduction from the Post Correspondence Problem.
A simple transformation is presented that maps an LCTRS to a (possibly infinite) term rewrite system (TRS). This allows lifting confluence results for TRSs to the LCTRS setting.
Using the transformation, the authors extend two advanced confluence criteria based on (almost) development closed critical pairs and parallel critical pairs from TRSs to LCTRSs. These results have been formally verified in the Isabelle proof assistant.

The paper highlights the challenges in extending confluence analysis techniques from plain term rewrite systems to the more expressive LCTRS setting, and provides a practical approach to leverage existing TRS confluence results for LCTRSs.

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Confluence of Logically Constrained Rewrite Systems Revisited

by Jona... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2402.13552.pdf

Confluence of Logically Constrained Rewrite Systems Revisited

Deeper Inquiries

Are there any decidable fragments of LCTRSs for which local confluence is decidable

In the context provided, it is stated that local confluence of terminating LCTRSs is undecidable, even for a decidable fragment of the theory of integers. This implies that there are no decidable fragments of LCTRSs for which local confluence is decidable. The undecidability of local confluence in LCTRSs poses a significant challenge in automated analysis and verification processes.

What other confluence criteria for TRSs could be lifted to the LCTRS setting using the presented transformation

The transformation presented in the context allows for the lifting of various confluence criteria for TRSs to the LCTRS setting. One such criteria that can be lifted is the concept of (almost) development closed critical pairs. By leveraging the transformation from LCTRSs to TRSs, advanced confluence criteria based on critical pairs in the TRS domain can be extended to the constrained setting of LCTRSs. This enables the application of established confluence techniques for TRSs to the analysis of logically constrained rewrite systems.

How can the insights from this work be applied to improve automated confluence analysis tools for programs that involve complex constraints

The insights from this work can be applied to enhance automated confluence analysis tools for programs that involve complex constraints. By utilizing the transformation from LCTRSs to TRSs, these tools can leverage existing confluence criteria and techniques developed for TRSs. This can lead to more efficient and effective confluence analysis for programs with logical constraints, improving the accuracy and reliability of automated verification processes. Additionally, the ability to relate critical pairs in TRSs to constrained critical pairs in LCTRSs allows for a more seamless integration of advanced confluence analysis methods into tools designed for program analysis and verification.