Programming and Reasoning with Guarded Recursion for Coinductive Types

Clouston, Ranald; Bizjak, Aleš; Grathwohl, Hans Bugge; Birkedal, Lars

doi:10.1007/978-3-662-46678-0_26

Cited by 29 publications

(48 citation statements)

References 23 publications

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“…Our starting point is the Guarded Higher-Order Logic [1] (Guarded HOL) inspired by the topos of trees. In addition to the usual constructs of HOL to reason about lambda terms, this logic features the ⊲ and modalities to reason about infinite terms, in particular streams.…”

Section: Base Logic: Guarded Higher-order Logicmentioning

confidence: 99%

“…Our probabilistic guarded λ-calculus and the associated logic Guarded HOL build on top of the guarded λ-calculus and its internal logic [1]. The guarded λ-calculus has been extended to guarded dependent type theory [13], which can be understood as a theory of guarded refinement types and as a foundation for proof assistants based on guarded type theory.…”

Section: Related Workmentioning

confidence: 99%

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Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Aguirre

Barthe

Birkedal

et al. 2018

Lecture Notes in Computer Science

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We extend the simply-typed guarded λ-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like Markov chains. We demonstrate the logic sound by interpreting its judgements in the topos of trees and by using probabilistic couplings for the semantics of relational assertions over distributions on discrete types. The program logic is designed to support syntax-directed proofs in the style of relational refinement types, but retains the expressiveness of higher-order logic extended with discrete distributions, and the ability to reason relationally about expressions that have different types or syntactic structure. In addition, our proof system leverages a well-known theorem from the coupling literature to justify better proof rules for relational reasoning about probabilistic expressions. We illustrate these benefits with a broad range of examples that were beyond the scope of previous systems, including shift couplings and lump couplings between random walks.The goal of this paper is to develop a programming and reasoning framework for probabilistic computations over infinite objects, such as Markov chains. Although programming and reasoning frameworks for infinite objects and probabilistic computations are well-understood in isolation, their combination is challenging. In particular, one must develop a proof system that is powerful enough for proving interesting properties of probabilistic computations over infinite objects, and practical enough to support effective verification of these properties.Modelling probabilistic infinite objects A first challenge is to model probabilistic infinite objects. We focus on the case of Markov chains, due to its importance. A (discrete-time) Markov chain is a sequence of random variables {X i } over some fixed type T satisfying some independence property. Thus, the straightforward way of modelling a Markov chain is as a stream of distributions over T . Going back to the simple example outlined above, it is natural to think about this kind of discrete-time Markov chain as characterized by the sequence of positions {p i } i∈N , which in turn can be described as an infinite set indexed by the natural numbers. This suggests that a natural way to model such a Markov chain is to use streams in which each element is produced probabilistically from the previous one. However, there are some downsides to this representation. First of all, it requires explicit reasoning about probabilistic dependency, since X i+1 depends on X i . Also, we might be interested in global properties of the executions of the Markov chain, such as "The probability of passing through the initial state infinitely many times is 1". These properties are naturally expressed as properties of the whole stream. For these reasons, we want to represent Markov chains as distributions over streams. Seemingly, one downside of this repre...

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Section: Base Logic: Guarded Higher-order Logicmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Aguirre

Barthe

Birkedal

et al. 2018

Lecture Notes in Computer Science

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“…A more flexible alternative to syntactic criteria is to have users annotate the functions' types with information that controls termination and productivity. Approaches in these category include fair reactive programming [19,24,37], clock variables [8,20], and sized types [2]. Sized types are implemented in MiniAgda [3] and in newer versions of Agda, in conjunction with a destructor-oriented (copattern) syntax for corecursion [5].…”

Section: Related Workmentioning

confidence: 99%

Foundational extensible corecursion: a proof assistant perspective

Blanchette

Popescu

Traytel³

2015

SIGPLAN Not.

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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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“…In Section 5 we conclude with a discussion of related and further work. This paper is based on a previously published conference paper [13], but has been significantly revised and extended. We have improved the presentation of our results and examples throughout the paper, but draw particular attention to the following changes:…”

Section: Introductionmentioning

confidence: 99%

The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

Clouston¹,

Bizjak²,

Grathwohl³

et al. 2017

Logical Methods in Computer Science

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Abstract. We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We introduce a program logic with Löb induction for reasoning about the contextual equivalence of programs. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten's behavioural differential equations.

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Programming and Reasoning with Guarded Recursion for Coinductive Types

Cited by 29 publications

References 23 publications

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Foundational extensible corecursion: a proof assistant perspective

The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

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