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

Nagele, Julian; Middeldorp, Aart

doi:10.1007/978-3-319-43144-4_18

Cited by 8 publications

(10 citation statements)

References 20 publications

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“…We gave a lattice-theoretic characterisation of overlap. Since many results in term rewriting, and beyond, are based on reasoning about overlap, which is notoriously hard [24], we expect that formalising our characterisation could simplify or even enable formalising them. We expect that both methods generalise to commutation, extend to HRSs [21], and can be strengthened by considering rule specialisations.…”

Section: Discussionmentioning

confidence: 99%

“…Huet's parallel closedness result relies on the notion of overlap whose geometric intuition is subtle [1,24], and reasoning becomes intricate for development closedness as covered by Theorem 3. We factor the classical theory of overlaps and critical pairs through the encompassment lattice in which overlapping redexpatterns is taking their join and the amount of overlap between redex-patterns is computed via their meet, thus allowing to reason algebraically about overlaps.…”

Section: Supported By Jsps Kakenhi Grant Number 17k00011mentioning

confidence: 99%

“…This fails for, e.g., connected graphs; these may fall apart into non-connected ones 8. For the amount of overlap for redexes in parallel reduction −→, see e.g [16,1,24]9.…”

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confidence: 99%

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Confluence by Critical Pair Analysis Revisited

Hirokawa

Nagele

Oostrom

et al. 2019

Lecture Notes in Computer Science

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Section: Discussionmentioning

confidence: 99%

Section: Supported By Jsps Kakenhi Grant Number 17k00011mentioning

confidence: 99%

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Confluence by Critical Pair Analysis Revisited

Hirokawa

Nagele

Oostrom

et al. 2019

Lecture Notes in Computer Science

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“…8 Since 2012 CeTA supports checking (non-)confluence certificates. CSI supports certifiable output for the following criteria checkable by CeTA: Knuth and Bendix' criterion [17,39], (weak) orthogonality [35,25], Huet's results on strongly closed and parallel closed critical pairs and Toyama's extenson of the latter [12,42,24], the rule labeling heuristic for decreasing diagrams [23,45], and transformations based on redundant rules [22]. For non-confluence CeTA can check that, given derivations s → * t 1 and s → * t 2 , t 1 and t 2 cannot be joined.…”

Section: Certificationmentioning

confidence: 93%

CSI: New Evidence – A Progress Report

Nagele

Felgenhauer

Middeldorp

2017

Automated Deduction – CADE 26

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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 CSIˆho, a tool built on the same framework and similar ideas as CSI that automatically checks confluence of higher-order rewrite systems.

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“…The code generator produces these proof checkers from their verified HOL formalizations. Today, IsaFoR/CeTA can also check proofs of a term rewriting system's complexity [DJK + 18] and confluence[NM16].…”

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confidence: 99%

From LCF to Isabelle/HOL

Paulson

Nipkow

Wenzel³

2019

Form. Asp. Comput.

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Interactive theorem provers have developed dramatically over the past four decades, from primitive beginnings to today's powerful systems. Here, we focus on Isabelle/HOL and its distinctive strengths. They include automatic proof search, borrowing techniques from the world of first order theorem proving, but also the automatic search for counterexamples. They include a highly readable structured language of proofs and a unique interactive development environment for editing live proof documents. Everything rests on the foundation conceived by Robin Milner for Edinburgh LCF: a proof kernel, using abstract types to ensure soundness and eliminate the need to store proofs. Compared with the research prototypes of the 1970s, Isabelle is a practical and versatile tool. It is used by system designers, mathematicians and many others.Isabelle is a leading interactive theorem prover. It is generic, supporting a number of different formal calculi, but by far the most important of these is its instantiation to higher-order logic: Isabelle/HOL. Already during the 1990s, Isabelle/HOL was being applied with great success to the task of verifying cryptographic protocols [Pau98]. Turning to mathematics, it played a critical role in Hales's Flyspeck project, which verified his proof of the Kepler conjecture [HAB + 17]. It is the basis for the seL4 project, under which an entire operating system kernel was verified, proving full functional correctness [KAE + 10]. It was adopted by researchers outside the verification milieu for specifying and verifying algorithms for replicated datatypes that provide "eventual consistency" [GKMB17]. Numerous other projects are underway around the world. Like other proof assistants, Isabelle is not directly concerned with program verification, i.e. with verifying code written in a programming language, but it can be used as a back end to prove verification conditions. Isabelle can also be used as a verified programming environment, where mathematical functions can be proved correct and then automatically translated to executable code in one of several different programming languages. The translation process itself is currently unverified, but even this is likely to change in the near future.This essay focuses on Isabelle/HOL and therefore has little to say about techniques common to most systems. For example, simplification by rewriting-coupled with recursive simplification to handle conditional rewrite rules-was already realised in both the Boyer/Moore theorem prover [BM79] and Edinburgh LCF by the end of the 1970s. Recursive datatype and function definitions, as well as inductive definitions, were commonplace by around 1995. Linear arithmetic decision procedures were also widely available by then.Our title echoes Mike Gordon's paper "From LCF to HOL: a Short History" [Gor00]. Like Mike, we begin with LCF, the source of the most fundamental ideas. But we pass over this material quickly in order to focus on Isabelle. There is no way to surpass Mike's account of the early years. He starts in 1969, wi...

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

Cited by 8 publications

References 20 publications

Confluence by Critical Pair Analysis Revisited

Confluence by Critical Pair Analysis Revisited

CSI: New Evidence – A Progress Report

From LCF to Isabelle/HOL

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