68 research outputs found
Foundational Extensible Corecursion
This paper presents a formalized framework for defining corecursive functions
safely in a total setting, based on corecursion up-to and relational
parametricity. The end product is a general corecursor that allows corecursive
(and even recursive) calls under well-behaved operations, including
constructors. Corecursive functions that are well behaved can be registered as
such, thereby increasing the corecursor's expressiveness. The metatheory is
formalized in the Isabelle proof assistant and forms the core of a prototype
tool. The corecursor is derived from first principles, without requiring new
axioms or extensions of the logic
Coiterative Morphisms: Interactive Equational Reasoning for Bisimulation, using Coalgebras
ter: SEN 3
Abstract: We study several techniques for interactive equational reasoning with the bisimulation equivalence. Our work is based on a modular library, formalised in Coq, that axiomatises weakly final coalgebras and bisimulation. As a theory we derive some coalgebraic schemes and an associated coinduction principle. This will help in interactive proofs by coinduction, modular derivation of congruence and co-fixed point equations and enables an extensional treatment of bisimulation. Finally we present a version of the lambda-coinduction proof principle in our framework
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
Foundational extensible corecursion: a proof assistant perspective
This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under âfriendlyâ operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursorâs expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic
Foundational extensible corecursion: a proof assistant perspective
This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under âfriendlyâ operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursorâs expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic
Bisimilarity of Open Terms in Stream GSOS
Stream GSOS is a specification format for operations and calculi on infinite
sequences. The notion of bisimilarity provides a canonical proof technique for
equivalence of closed terms in such specifications. In this paper, we focus on
open terms, which may contain variables, and which are equivalent whenever they
denote the same stream for every possible instantiation of the variables. Our
main contribution is to capture equivalence of open terms as bisimilarity on
certain Mealy machines, providing a concrete proof technique. Moreover, we
introduce an enhancement of this technique, called bisimulation up-to
substitutions, and show how to combine it with other up-to techniques to obtain
a powerful method for proving equivalence of open terms
Enhanced Coalgebraic Bisimulation
International audienceWe present a systematic study of bisimulation-up-to techniques for coalgebras. This enhances the bisimulation proof method for a large class of state based systems, including labelled transition systems but also stream systems and weighted automata. Our approach allows for compositional reasoning about the soundness of enhancements. Applications include the soundness of bisimulation up to bisimilarity, up to equivalence and up to congruence. All in all, this gives a powerful and modular framework for simplified coinductive proofs of equivalence
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
Foundational extensible corecursion: a proof assistant perspective
This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under "friendly" operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor's expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic
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