Up-to techniques using sized types

Danielsson, Nils Anders

doi:10.1145/3158131

Cited by 13 publications

(8 citation statements)

References 40 publications

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“…Future work also includes investigating more advanced proofs of bisimulation e.g., using upto-techniques [Danielsson 2018;Milner 1983;Pous and Sangiorgi 2012] and weak bisimulation, which is generally challenging for guarded recursion [Mùgelberg and Paviotti 2016]. It would also be desirable to have an implementation of guarded recursion in a proof assistant, which would allow proofs such as those presented in this paper to be formally checked by a computer.…”

Section: Discussionmentioning

confidence: 99%

Bisimulation as path type for guarded recursive types

Møgelberg

Veltri

2019

Proc. ACM Program. Lang.

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In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, allowing this to be encoded in types. Indeed, coinductive types can be encoded using a combination of guarded recursion and universal quantification over clocks. This paper studies the notion of bisimilarity for guarded recursive types in Ticked Cubical Type Theory, an extension of Cubical Type Theory with guarded recursion. We prove that, for any functor, an abstract, category theoretic notion of bisimilarity for the final guarded coalgebra is equivalent (in the sense of homotopy type theory) to path equality (the primitive notion of equality in cubical type theory). As a worked example we study a guarded notion of labelled transition systems, and show that, as a special case of the general theorem, path equality coincides with an adaptation of the usual notion of bisimulation for processes. In particular, this implies that guarded recursion can be used to give simple equational reasoning proofs of bisimilarity. This work should be seen as a step towards obtaining bisimilarity as path equality for coinductive types using the encodings mentioned above.

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Section: Discussionmentioning

confidence: 99%

Bisimulation as path type for guarded recursive types

Møgelberg

Veltri

2019

Proc. ACM Program. Lang.

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“…In Agda, coinductive types are encoded as coinductive records which can optionally be parametrized by a size to ease the productivity checking of corecursive definitions [2,8]. For example, the type of trees given above is implemented in Agda as the following pair of mutually defined types: (2) The type Tree I A i is inductive, while Tree ′ I A i is a coinductive record type.…”

Section: Programs As Treesmentioning

confidence: 99%

Inductive and Coinductive Predicate Liftings for Effectful Programs

Veltri

Voorneveld

2021

Electron. Proc. Theor. Comput. Sci.

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We formulate a framework for describing behaviour of effectful higher-order recursive programs. Examples of effects are implemented using effect operations, and include: execution cost, nondeterminism, global store and interaction with a user. The denotational semantics of a program is given by a coinductive tree in a monad, which combines potential return values of the program in terms of effect operations.Using a simple test logic, we construct two sorts of predicate liftings, which lift predicates on a result type to predicates on computations that produce results of that type, each capturing a facet of program behaviour. Firstly, we study inductive predicate liftings which can be used to describe effectful notions of total correctness. Secondly, we study coinductive predicate liftings, which describe effectful notions of partial correctness. The two constructions are dual in the sense that one can be used to disprove the other.The predicate liftings are used as a basis for an endogenous logic of behavioural properties for higher-order programs. The program logic has a derivable notion of negation, arising from the duality of the two sorts of predicate liftings, and it generates a program equivalence which subsumes a notion of bisimilarity. Appropriate definitions of inductive and coinductive predicate liftings are given for a multitude of effect examples.The whole development has been fully formalized in the Agda proof assistant.

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“…A function is below the companion if and only if it preserves all the elements of this sequence of approximations. This approach to enhancements as 'approx-imation preserving' functions has been implemented in Agda [Dan18], where sized types (intuitively, types indexed with a bound on the size their elements) give an explicit access to the sequence of approximations: approximation preserving functions become size-preserving functions (at least for coinductive datatypes that are obtained by a sequence of approximations that converges at the first infinite ordinal ω: it remains unclear whether one can go beyond this ordinal with sized types -moreover the development of a dependent type theory with sized types is still an ongoing research program [Dan18, end of p.4]).…”

Section: Compositions and Algebras Of Enhancementsmentioning

confidence: 99%

“…Operations defined using the GSOS format [Plo04b] typically satisfy this condition: lookaheads are not permitted in this format. So do size-preserving functions from [Dan18].…”

Section: Enhanced Corecursion Schemesmentioning

confidence: 99%

Bisimulation and Coinduction Enhancements: A Historical Perspective

Pous

Sangiorgi

2019

Form. Asp. Comput.

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Bisimulation is an instance of coinduction. Both bisimulation and coinduction are today widely used, in many areas of Computer Science, as well as outside Computer Science. Over, roughly, the last 25 years, enhancements of the principles and methods related to bisimulation and coinduction (i.e., techniques to make proofs shorter and simpler) have become a research topic on its own. In the paper the origins and the developments of the topic are reviewed.

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Up-to techniques using sized types

Cited by 13 publications

References 40 publications

Bisimulation as path type for guarded recursive types

Bisimulation as path type for guarded recursive types

Inductive and Coinductive Predicate Liftings for Effectful Programs

Bisimulation and Coinduction Enhancements: A Historical Perspective

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