Modular coinduction up-to for higher-order languages via first-order transition systems

The bisimulation proof method can be enhanced by employing ‘bisimulations up-to’ techniques. A comprehensive theory of such enhancements has been developed for first-order (i.e., CCS-like) labelled transition systems (LTSs) and bisimilarity, based on abstract fixed-point theory and compatible functions. We transport this theory onto languages whose bisimilarity and LTS go beyond those of first-order models. The approach consists in exhibiting fully abstract translations of the more sophisticated LTSs and bisimilarities onto the first-order ones. This allows us to reuse directly the large corpus of up-to techniques that are available on first-order LTSs. The only ingredient that has to be manually supplied is the compatibility of basic up-to techniques that are specific to the new languages. We investigate the method on the π-calculus, the λ-calculus, and a (call-by-value) λ-calculus with references

An Equational Theory for Weak Bisimulation via Generalized Parameterized Coinduction

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.Comment: To be published in CPP 202

Proofs about Network Communication: For Humans and Machines

Many concurrent and distributed systems are safety-critical and therefore have to provide a high degree of assurance. Important properties of such systems are frequently proved on the specification level, but implementations typically deviate from specifications for practical reasons. Machine-checked proofs of bisimilarity statements are often useful for guaranteeing that properties of specifications carry over to implementations. In this paper, we present a way of conducting such proofs with a focus on network communication. The proofs resulting from our approach are not just machine-checked but also intelligible for humans.Comment: In Proceedings ICE 2023, arXiv:2308.0892

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs

The Power of Convex Algebras

Probabilistic automata (PA) combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of the latter semantics, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull.Comment: Full (extended) version of a CONCUR 2017 paper, to be submitted to LMC

Engineering formal systems in constructive type theory

This thesis presents a practical methodology for formalizing the meta-theory of formal systems with binders and coinductive relations in constructive type theory. While constructive type theory offers support for reasoning about formal systems built out of inductive definitions, support for syntax with binders and coinductive relations is lacking. We provide this support. We implement syntax with binders using well-scoped de Bruijn terms and parallel substitutions. We solve substitution lemmas automatically using the rewriting theory of the -calculus. We present the Autosubst library to automate our approach in the proof assistant Coq. Our approach to coinductive relations is based on an inductive tower construction, which is a type-theoretic form of transfinite induction. The tower construction allows us to reduce coinduction to induction. This leads to a symmetric treatment of induction and coinduction and allows us to give a novel construction of the companion of a monotone function on a complete lattice. We demonstrate our methods with a series of case studies. In particular, we present a proof of type preservation for CC!, a proof of weak and strong normalization for System F, a proof that systems of weakly guarded equations have unique solutions in CCS, and a compiler verification for a compiler from a non-deterministic language into a deterministic language. All technical results in the thesis are formalized in Coq.In dieser Dissertation beschreiben wir praktische Techniken um Formale Systeme mit Bindern und koinduktiven Relationen in Konstruktiver Typtheorie zu implementieren. Während Konstruktive Typtheorie bereits gute Unterstützung für Induktive Definition bietet, gibt es momentan kaum Unterstützung für syntaktische Systeme mit Bindern, oder koinduktiven Definitionen. Wir kodieren Syntax mit Bindern in Typtheorie mit einer de Bruijn Darstellung und zeigen alle Substitutionslemmas durch Termersetzung mit dem -Kalkül. Wir präsentieren die Autosubst Bibliothek, die unseren Ansatz im Beweisassistenten Coq implementiert. Für koinduktive Relationen verwenden wir eine induktive Turmkonstruktion, welche das typtheoretische Analog zur Transfiniten Induktion darstellt. Auf diese Art erhalten wir neue Beweisprinzipien für Koinduktion und eine neue Konstruktion von Pous’ “companion” einer monotonen Funktion auf einem vollständigen Verband. Wir validieren unsere Methoden an einer Reihe von Fallstudien. Alle technischen Ergebnisse in dieser Dissertation sind mit Coq formalisiert

Coalgebras and Their Logics

Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics

Infinitary λ\lambda-Calculi from a Linear Perspective (Long Version)

We introduce a linear infinitary $\lambda$-calculus, called $\ell\Lambda_{\infty}$, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applicative $\lambda$-calculus and is universal for computations over infinite strings. What is particularly interesting about $\ell\Lambda_{\infty}$, is that the refinement induced by linear logic allows to restrict both modalities so as to get calculi which are terminating inductively and productive coinductively. We exemplify this idea by analysing a fragment of $\ell\Lambda$ built around the principles of $\mathsf{SLL}$ and $\mathsf{4LL}$. Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary $\lambda$-calculi

Coinduction in Flow: The Later Modality in Fibrations

This paper provides a construction on fibrations that gives access to the so-called later modality, which allows for a controlled form of recursion in coinductive proofs and programs. The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations. As a result, we obtain a framework that allows the addition of a recursion principle for coinduction to rather arbitrary logics and programming languages. The main interest of using recursion is that it allows one to write proofs and programs in a goal-oriented fashion. This enables easily understandable coinductive proofs and programs, and fosters automatic proof search. Part of the framework are also various results that enable a wide range of applications: transportation of (co)limits, exponentials, fibred adjunctions and first-order connectives from the initial fibration to the one constructed through the framework. This means that the framework extends any first-order logic with the later modality. Moreover, we obtain soundness and completeness results, and can use up-to techniques as proof rules. Since the construction works for a wide variety of fibrations, we will be able to use the recursion offered by the later modality in various context. For instance, we will show how recursive proofs can be obtained for arbitrary (syntactic) first-order logics, for coinductive set-predicates, and for the probabilistic modal mu-calculus. Finally, we use the same construction to obtain a novel language for probabilistic productive coinductive programming. These examples demonstrate the flexibility of the framework and its accompanying results