Julian Nagele

On the Formalization of Termination Techniques based on Multiset Orderings

Multiset orderings are a key ingredient in certain termination techniques like the recursive path ordering and a variant of size-change termination. In order to integrate these techniques in a certifier for termination proofs, we have added them to the Isabelle Formalization of Rewriting. To this end, it was required to extend the existing formalization on multiset orderings towards a generalized multiset ordering. Afterwards, the soundness proofs of both techniques have been established, although only after fixing some definitions. Concerning efficiency, it is known that the search for suitable parameters for both techniques is NP-hard. We show that checking the correct application of the techniques--where all parameters are provided--is also NP-hard, since the problem of deciding the generalized multiset ordering is NP-hard.

CSI: New Evidence – A Progress Report

CSI is a strong automated confluence prover for rewrite systems which has been in development since 2010. In this paper we report on recent extensions that make CSI more powerful, secure, and useful. These extensions include improved confluence criteria but also support for uniqueness of normal forms. Most of the implemented techniques produce machine-readable proof output that can be independently verified by an external tool, thus increasing the trust in CSI. We also report on CSI\(\mathbf {\hat{~}}\)oho, a tool built on the same framework and similar ideas as CSI that automatically checks confluence of higher-order rewrite systems.

Confluence Competition 2015

Confluence is one of the central properties of rewriting. Our competition aims to foster the development of techniques for proving/disproving confluence of various formalisms of rewriting automatically. We explain the background and setup of the 4th Confluence Competition.

Improving Automatic Confluence Analysis of Rewrite Systems by Redundant Rules.

We describe how to utilize redundant rewrite rules, i.e., rules that can be simulated by other rules, when (dis)proving confluence of term rewrite systems. We demonstrate how automatic confluence provers benefit from the addition as well as the removal of redundant rules. Due to their simplicity, our transformations were easy to formalize in a proof assistant and are thus amenable to certification. Experimental results show the surprising gain in power.

Certification of Classical Confluence Results for Left-Linear Term Rewrite Systems

This paper presents the first formalization of three classic confluence criteria for first-order term rewrite systems by Huet and Toyama. We have formalized proofs, showing that (1) linear strongly closed systems, (2) left-linear parallel closed systems, and (3) left-linear almost parallel closed systems are confluent. The third result is extended to commutation. The proofs were carried out in the proof assistant Isabelle/HOL as part of the library IsaFoR and integrated into the certifier CeTA, significantly increasing the number of certifiable proofs produced by automatic confluence tools.

Certified Rule Labeling

The rule labeling heuristic aims to establish confluence of (left-)linear term rewrite systems via decreasing diagrams. We present a formalization of a confluence criterion based on the interplay of relative termination and the rule labeling in the theorem prover Isabelle. Moreover, we report on the integration of this result into the certifier CeTA, facilitating the checking of confluence certificates based on decreasing diagrams for the first time. The power of the method is illustrated by an experimental evaluation on a (standard) collection of confluence problems.

Certification of Nontermination Proofs Using Strategies and Nonlooping Derivations

The development of sophisticated termination criteria for term rewrite systems has led to powerful and complex tools that produce (non)termination proofs automatically. While many techniques to establish termination have already been formalized—thereby allowing to certify such proofs—this is not the case for nontermination. In particular, the proof checker Open image in new window was so far limited to (innermost) loops. In this paper we present an Isabelle/HOL formalization of an extended repertoire of nontermination techniques. First, we formalized techniques for nonlooping nontermination. Second, the available strategies include (an extended version of) forbidden patterns, which cover in particular outermost and context-sensitive rewriting. Finally, a mechanism to support partial nontermination proofs further extends the applicability of our proof checker.