Sarah Winkler

Ordinals and knuth-bendix orders

In this paper we consider a hierarchy of three versions of Knuth-Bendix orders. (1) We show that the standard definition can be (slightly) simplified without affecting the ordering relation. (2) For the extension of transfinite Knuth-Bendix orders we show that transfinite ordinals are not needed as weights, as far as termination of finite rewrite systems is concerned. (3) Nevertheless termination proving benefits from transfinite ordinals when used in the setting of general Knuth-Bendix orders defined over a weakly monotone algebra. We investigate the relationship to polynomial interpretations and present experimental results for both termination analysis and ordered completion. For the latter it is essential that the order is totalizable on ground terms.

Termination tools in ordered completion

Ordered completion is one of the most frequently used calculi in equational theorem proving. The performance of an ordered completion run strongly depends on the reduction order supplied as input. This paper describes how termination tools can replace fixed reduction orders in ordered completion procedures, thus allowing for a novel degree of automation. Our method can be combined with the multi-completion approach proposed by Kondo and Kurihara. We present experimental results obtained with our ordered completion tool omkbTT for both ordered completion and equational theorem proving.

Beyond Peano Arithmetic - Automatically Proving Termination of the Goodstein Sequence.

Kirby and Paris (1982) proved in a celebrated paper that a theorem of Goodstein (1944) cannot be established in Peano (1889) arithmetic. We present an encoding of Goodsteins theorem as a termination problem of a finite rewrite system. Using a novel implementation of ordinal interpretations, we are able to automatically prove termination of this system, resulting in the first automatic termination proof for a system whose derivational complexity is not multiple recursive. Our method can also cope with the encoding by Touzet (1998) of the battle of Hercules and Hydra, yet another system which has been out of reach for automated tools, until now.

Encoding Dependency Pair Techniques and Control Strategies for Maximal Completion

This paper describes two advancements of SAT-based Knuth-Bendix completion as implemented in Maxcomp. (1) Termination techniques using the dependency pair framework are encoded as satisfiability problems, including dependency graph and reduction pair processors. (2) Instead of relying on pure maximal completion, different SAT-encoded control strategies are exploited.

Recording Completion for Certificates in Equational Reasoning

We introduce recording completion, a variant of Knuth-Bendix completion which facilitates the construction of certificates for various equational logic proofs (completion proofs, entailment proofs and dis-proofs). The approach generalizes to more powerful variants of completion such as ordered completion and AC completion. We implemented recording completion in the tools KBCV and MKBTT. Both tools allow to choose among different formats of proof certificates, namely conversions, proof trees, and conversions with history. We report on experimental results in which all generated certificates have been verified by the trustable checker CeTA.

Beyond polynomials and Peano arithmetic-automation of elementary and ordinal interpretations

Kirby and Paris (1982) proved in a celebrated paper that a theorem of Goodstein (1944) cannot be established in Peano arithmetic. We present an encoding of Goodsteins theorem as a termination problem of a finite rewrite system. Using a novel implementation of algebras based on ordinal interpretations, we are able to automatically prove termination of this system, resulting in the first automatic termination proof for a system whose derivational complexity is not multiple recursive. Our method can also cope with the encoding by Touzet (1998) of the battle of Hercules and Hydra as well as a (corrected) encoding by Beklemishev (2006) of the Worm battle, two further systems which have been out of reach for automatic tools, until now. Based on our ideas of implementing ordinal algebras we also present a new approach for the automation of elementary interpretations for termination analysis.

Formalizing Soundness and Completeness of Unravelings

Unravelings constitute a class of program transformations to model conditional rewrite systems as standard term rewrite systems. Key properties of unravelings are soundness and completeness with respect to reductions, in the sense that rewrite sequences in the unraveled system correspond to rewrite sequences in the conditional system and vice versa. While the latter is easily satisfied, the former holds only under certain conditions and is notoriously difficult to prove. This paper describes an Isabelle formalization of both properties. The soundness proof is based on the approach by Nishida, Sakai, and Sakabe 2012 but we also contribute to the theory by showing it applicable to a larger class of unravelings. Based on our formalization we developed the first certifier to check output of conditional rewrite tools. In particular, quasi-decreasingness proofs by AProVE and conditional confluence proofs by ConCon can be certified.

AC-KBO Revisited

We consider various definitions of AC-compatible Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are revisited. The former is enhanced to a more powerful AC-compatible order and we modify the latter to amend its lack of monotonicity on non-ground terms. We further present new complexity results. An extension reflecting the recent proposal of subterm coefficients in standard Knuth-Bendix orders is also given. The various orders are compared on problems in termination and completion.

Normalized Completion Revisited

Normalized completion (Marche 1996) is a widely applicable and efficient technique for com- pletion modulo theories. If successful, a normalized completion procedure computes a rewrite system that allows to decide the validity problem using normalized rewriting. In this paper we consider a slightly simplified inference system for finite normalized completion runs. We prove correctness, show faithfulness of critical pair criteria in our setting, and propose a different notion of normalizing pairs. We then show how normalized completion procedures can benefit from AC- termination tools instead of relying on a fixed AC-compatible reduction order. We outline our implementation of this approach in the completion tool mkbtt and present experimental results, including new completions.

AC completion with termination tools

We present mascott, a tool for Knuth-Bendix completion modulo the theory of associative and commutative operators. In contrast to classical completion tools, mascott does not rely on a fixed ACcompatible reduction order. Instead, a suitable order is implicitly constructed during a deduction by collecting all oriented rules in a similar fashion as done in the tool Slothrop. This allows for convergent systems which cannot be completed using standard orders. We outline the underlying inference system and comment on implementation details such as the use of multi-completion, term indexing techniques, and critical pair criteria.