Guarded Dependent Type Theory with Coinductive Types

Bizjak, Aleš; Grathwohl, Hans Bugge; Clouston, Ranald; Møgelberg, Rasmus Ejlers; Birkedal, Lars

doi:10.1007/978-3-662-49630-5_2

Cited by 43 publications

(81 citation statements)

References 20 publications

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“…One first direction is to enhance the expressiveness of the underlying simply typed language. For instance, it would be interesting to introduce clock variables and some type dependency as in [13], and extend the proof system accordingly. This would allow us, for example, to type the function taking the n-th element of a guarded stream, which cannot be done in the current system.…”

Section: Resultsmentioning

confidence: 99%

“…This example, taken from [13], proves a property about the ZipWith function, which takes two streams of type A, a function on pairs of elements, and "zips" the two streams by applying that function to the elements that are at the same position on the two streams. We want to show that if the function on the elements is commutative, zipping two streams with that function is commutative as well.…”

Section: E1 Proof Of Zipwithmentioning

confidence: 90%

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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: Resultsmentioning

confidence: 99%

Section: E1 Proof Of Zipwithmentioning

confidence: 90%

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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“…Core λ * enjoys decidable type-checking, but not type inference; in contrast, type inference for the later modality has been studied by Severi [31]. Finally, Core λ * might be difficult to extend to dependent types, since it is inherently call-by-value, whereas several dependent type theories with later have been proposed [6,8].…”

Section: Guarded Type Theoriesmentioning

confidence: 99%

A Generalized Modality for Recursion

Guatto

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Nakano's later modality allows types to express that the output of a function does not immediately depend on its input, and thus that computing its fixpoint is safe. This idea, guarded recursion, has proved useful in various contexts, from functional programming with infinite data structures to formulations of step-indexing internal to type theory. Categorical models have revealed that the later modality corresponds in essence to a simple reindexing of the discrete time scale.Unfortunately, existing guarded type theories suffer from significant limitations for programming purposes. These limitations stem from the fact that the later modality is not expressive enough to capture precise input-output dependencies of functions. As a consequence, guarded type theories reject many productive definitions.Combining insights from guarded type theories and synchronous programming languages, we propose a new modality for guarded recursion. This modality can apply any well-behaved reindexing of the time scale to a type. We call such reindexings time warps. Several modalities from the literature, including later, correspond to fixed time warps, and thus arise as special cases of ours.

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“…Therefore our use of the modality seems the simpler choice, especially as it allows us to adapt previously published work on term calculus for the modal logic Intuitionistic S4 [5]. However in our work on extending guarded type theory to dependent types [11] the explicit substitutions become more burdensome, resulting in our adoption of clock quantifiers for that work.…”

Section: Related Workmentioning

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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Guarded Dependent Type Theory with Coinductive Types

Cited by 43 publications

References 20 publications

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

A Generalized Modality for Recursion

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

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