Incremental Pattern-Based Coinduction for Process Algebra and Its Isabelle Formalization

Popescu, Andrei; Gunter, Elsa L.

doi:10.1007/978-3-642-12032-9_9

Cited by 12 publications

(8 citation statements)

References 27 publications

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“…Specifically, if (given x) we instantiate the reduction lemma's f with f (x −), it yields Incremental Coinduction. In 2010, Popescu and Gunter [16] proposed a proof system for incremental coinduction, tailored towards bisimilarity in a process calculus, and they established the soundness of their system by a global, monolithic argument. Their judgment θ θ corresponds precisely to θ θ G f (θ) (where f is the generating function of their process bisimilarity).…”

Section: Discussion and Related Workmentioning

confidence: 99%

The power of parameterization in coinductive proof

Hur

Neis

Dreyer

et al. 2013

Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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Coinduction is one of the most basic concepts in computer science. It is therefore surprising that the commonly-known lattice-theoretic accounts of the principles underlying coinductive proofs are lacking in two key respects: they do not support compositional reasoning (i.e., breaking proofs into separate pieces that can be developed in isolation), and they do not support incremental reasoning (i.e., developing proofs interactively by starting from the goal and generalizing the coinduction hypothesis repeatedly as necessary). In this paper, we show how to support coinductive proofs that are both compositional and incremental, using a dead simple construction we call the parameterized greatest fixed point. The basic idea is to parameterize the greatest fixed point of interest over the accumulated knowledge of "the proof so far". While this idea has been proposed before, by Winskel in 1989 and by Moss in 2001, neither of the previous accounts suggests its general applicability to improving the state of the art in interactive coinductive proof. In addition to presenting the lattice-theoretic foundations of parameterized coinduction, demonstrating its utility on representative examples, and studying its composition with "up-to" techniques, we also explore its mechanization in proof assistants like Coq and Isabelle. Unlike traditional approaches to mechanizing coinduction (e.g., Coq's cofix), which employ syntactic "guardedness checking", parameterized coinduction offers a semantic account of guardedness. This leads to faster and more robust proof development, as we demonstrate using our new Coq library, Paco.

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Section: Discussion and Related Workmentioning

confidence: 99%

The power of parameterization in coinductive proof

Hur

Neis

Dreyer

et al. 2013

Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

View full text Add to dashboard Cite

show abstract

“…In 2010, Popescu and Gunter [16] proposed a proof system for incremental coinduction, tailored towards bisimilarity in a process calculus, and they established the soundness of their system by a global, monolithic argument. Their judgment θ θ corresponds precisely to θ θ G f (θ) (where f is the generating function of their process bisimilarity).…”

Section: Discussion and Related Workmentioning

confidence: 99%

The power of parameterization in coinductive proof

HurChung-Kil¹,

NeisGeorg²,

DreyerDerek³

et al. 2013

SIGPLAN Not.

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Coinduction is one of the most basic concepts in computer science. It is therefore surprising that the commonly-known lattice-theoretic accounts of the principles underlying coinductive proofs are lacking in two key respects: they do not support compositional reasoning (i.e., breaking proofs into separate pieces that can be developed in isolation), and they do not support incremental reasoning (i.e., developing proofs interactively by starting from the goal and generalizing the coinduction hypothesis repeatedly as necessary).In this paper, we show how to support coinductive proofs that are both compositional and incremental, using a dead simple construction we call the parameterized greatest fixed point. The basic idea is to parameterize the greatest fixed point of interest over the accumulated knowledge of "the proof so far". While this idea has been proposed before, by Winskel in 1989 and by Moss in 2001, neither of the previous accounts suggests its general applicability to improving the state of the art in interactive coinductive proof.In addition to presenting the lattice-theoretic foundations of parameterized coinduction, demonstrating its utility on representative examples, and studying its composition with "up-to" techniques, we also explore its mechanization in proof assistants like Coq and Isabelle. Unlike traditional approaches to mechanizing coinduction (e.g., Coq's cofix), which employ syntactic "guardedness checking", parameterized coinduction offers a semantic account of guardedness. This leads to faster and more robust proof development, as we demonstrate using our new Coq library, Paco.

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“…We have chosen to build our approach on top of paco, but other incremental coinductive techniques exist: incremental pattern-based coinduction [Popescu and Gunter 2010], circular coinduction [Hausmann et al 2005], parametric coinduction [Moss 2001]. We refer to Hur et al 's related work [Hur et al 2013] for a thorough comparison.…”

Section: Distinguishing Internal and Visiblementioning

confidence: 99%

An equational theory for weak bisimulation via generalized parameterized coinduction

Zakowski

Hur

et al. 2020

Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs

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Coinductive reasoning about infinitary structures such as streams is widely applicable. However, practical frameworks for developing coinductive proofs and finding reasoning principles that help structure such proofs remain a challenge, especially in the context of machine-checked formalization.This paper gives a novel presentation of an equational theory for reasoning about structures up to weak bisimulation. The theory is both compositional, making it suitable for defining general-purpose lemmas, and also incremental, meaning that the bisimulation can be created interactively.To prove the theory's soundness, this paper also introduces generalized parameterized coinduction, which addresses expressivity problems of earlier works and provides a practical framework for coinductive reasoning. The paper presents the resulting equational theory for streams, but the technique applies to other structures too.All of the results in this paper have been proved in Coq, and the generalized parameterized coinduction framework is available as a Coq library.CCS Concepts • Software and its engineering → Formal software verification; • Theory of computation → Program verification; Logic and verification; Equational logic and rewriting.

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Incremental Pattern-Based Coinduction for Process Algebra and Its Isabelle Formalization

Cited by 12 publications

References 27 publications

The power of parameterization in coinductive proof

The power of parameterization in coinductive proof

The power of parameterization in coinductive proof

An equational theory for weak bisimulation via generalized parameterized coinduction

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