A modality for recursion

Nakano, Hiroshi

doi:10.1109/lics.2000.855774

Cited by 116 publications

(134 citation statements)

References 19 publications

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“…The step-indexing aspect of CMRAs is internalized into the logic by adding a new modality: the later modality, P [23,3]. Intuitively, P asserts that P holds "at the next step-index" (or "one step later").…”

Section: The Later Modality and Guarded Fixed-pointsmentioning

confidence: 99%

The Essence of Higher-Order Concurrent Separation Logic

Krebbers

Jung

Bizjak

et al. 2017

Programming Languages and Systems

102

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Abstract. Concurrent separation logics (CSLs) have come of age, and with age they have accumulated a great deal of complexity. Previous work on the Iris logic attempted to reduce the complex logical mechanisms of modern CSLs to two orthogonal concepts: partial commutative monoids (PCMs) and invariants. However, the realization of these concepts in Iris still bakes in several complex mechanisms-such as weakest preconditions and mask-changing view shifts-as primitive notions.In this paper, we take the Iris story to its (so to speak) logical conclusion, applying the reductionist methodology of Iris to Iris itself. Specifically, we define a small, resourceful base logic, which distills the essence of Iris: it comprises only the assertion layer of vanilla separation logic, plus a handful of simple modalities. We then show how the much fancier logical mechanisms of Iris-in particular, its entire program specification layer-can be understood as merely derived forms in our base logic. This approach helps to explain the meaning of Iris's program specifications at a much higher level of abstraction than was previously possible. We also show that the step-indexed "later" modality of Iris is an essential source of complexity, in that removing it leads to a logical inconsistency. All our results are fully formalized in the Coq proof assistant. IntroductionIn his paper The Next 700 Separation Logics, Parkinson [26] observed that "separation logic has brought great advances in the world of verification. However, there is a disturbing trend for each new library or concurrency primitive to require a new separation logic." He argued that what is needed is a general logic for concurrent reasoning, into which a variety of useful specifications can be encoded via the abstraction facilities of the logic. "By finding the right core logic," he wrote, "we can concentrate on the difficult problems."The logic he suggested as a potential candidate for such a core concurrency logic was deny-guarantee [12]. Deny-guarantee was indeed groundbreaking in its support for "fictional separation"-the idea that even if threads are concurrently manipulating the same shared piece of physical state, one can view them as operating on logically disjoint pieces of it and use separation logic to reason modularly about those pieces. It was, however, far from the last word on the subject. Rather, 2 Krebbers, Jung, Bizjak, Jourdan, Dreyer, Birkedal it spawned a new breed of logics with ever more powerful fictional-separation mechanisms for reasoning modularly about interference [11,16,29,9,30,27]. Several of these also incorporated support for impredicative invariants [28,18,17,4], which are needed if one aims to verify code in languages with semantically cyclic features (such as ML or Rust, which support higher-order state).Although exciting, the progress in this area has come at a cost: as these new separation logics become ever more expressive, each one accumulates increasingly baroque and bespoke proof rules, which are primitive in the sense that their sou...

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Section: The Later Modality and Guarded Fixed-pointsmentioning

confidence: 99%

The Essence of Higher-Order Concurrent Separation Logic

Krebbers

Jung

Bizjak

et al. 2017

Programming Languages and Systems

102

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“…Related work. Nakano presented a simple type theory with guarded recursive types [30] which can be modelled using complete bounded ultrametric spaces [6]. We show in Section 5 that the category BiCBUlt of bisected, complete bounded ultrametric spaces is a co-reflective subcategory of S. Thus, our present work can be seen as an extension of the work of Nakano to include the full internal language of a topos, in particular dependent types, and an associated higher-order logic.…”

mentioning

confidence: 95%

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

Birkedal¹,

Møgelberg²,

Schwinghammer³

et al. 2012

Logical Methods in Computer Science

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Abstract. We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a wellfounded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S.

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“…What then is the guarded version of this combinator? Following the need for the recursion variable to be guarded, and the original observation of Nakano [38] that guarded fixed point combinators should have type (◮A → A) → A, we reconstruct the type Rec A by the addition of later modalities in the appropriate places. The terms θ and fix can then be constructed by adding next term-formers, and replacing function application with ⊛, to the original terms so that they type-check:…”

Section: Examplesmentioning

confidence: 99%

“…A more flexible approach, first suggested by Nakano [38], is to guarantee productivity via types. A new modality, for which we follow Appel et al [3] by writing ◮ and using the name 'later', allows us to distinguish between data we have access to now, and data which we have only later.…”

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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A modality for recursion

Cited by 116 publications

References 19 publications

The Essence of Higher-Order Concurrent Separation Logic

The Essence of Higher-Order Concurrent Separation Logic

First steps in synthetic guarded domain theory: step-indexing in the topos of trees

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

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