A Coinductive Confluence Proof for Infinitary Lambda-Calculus

Czajka, Łukasz

doi:10.1007/978-3-319-08918-8_12

Cited by 10 publications

(14 citation statements)

References 45 publications

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“…Y λ ) strongly converges to the infinitary term f ω := f (f (...)) (resp. λy.λy....) in the sense of [15], [10]. Thus, Ω and Y f are both zero terms (terms of order 0) and Y λ a term of infinite order.…”

Section: Typing Some Notable Terms In System Rmentioning

confidence: 98%

Every λ-Term is Meaningful for the Infinitary Relational Model

Vial

2018

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

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Infinite types and formulas are known to have really curious and unsound behaviors. For instance, they allow to type Ω, the auto-autoapplication and they thus do not ensure any form of normalization/productivity. Moreover, in most infinitary frameworks, it is not difficult to define a type R that can be assigned to every λ-term. However, these observations do not say much about what coinductive (i.e. infinitary) type grammars are able to provide: it is for instance very difficult to know what types (besides R) can be assigned to a term in this setting. We begin with a discussion on the expressivity of different forms of infinite types. Using the resource-awareness of sequential intersection types (system S) and tracking, we then prove that infinite types are able to characterize the order (arity) of every λ-terms and that, in the infinitary extension of the relational model, every term has a "meaning" i.e. a non-empty denotation. From the technical point of view, we must deal with the total lack of productivity guarantee for typable terms: we do so by importing methods inspired by first order model theory.

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Section: Typing Some Notable Terms In System Rmentioning

confidence: 98%

Every λ-Term is Meaningful for the Infinitary Relational Model

Vial

2018

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

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“…. , T k are again trees over Σ (for an introduction to coinductive definitions and proofs see, e.g., Czajka [12]). We employ the usual notions of nodes, children, branches, etc.…”

Section: Preliminariesmentioning

confidence: 99%

Higher-Order Model Checking Step by Step

Parys

2021

Preprint

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We show a new simple algorithm that solves the model-checking problem for recursion schemes: check whether the tree generated by a given higher-order recursion scheme is accepted by a given alternating parity automaton. The algorithm amounts to a procedure that transforms a recursion scheme of order n to a recursion scheme of order n − 1, preserving acceptance, and increasing the size only exponentially. After repeating the procedure n times, we obtain a recursion scheme of order 0, for which the problem boils down to solving a finite parity game. Since the size grows exponentially at each step, the overall complexity is n-EXPTIME, which is known to be optimal. More precisely, the transformation is linear in the size of the recursion scheme, assuming that the arity of employed nonterminals and the size of the automaton are bounded by a constant; this results in an FPT algorithm for the model-checking problem.Our transformation is a generalization of a previous transformation of the author (2020), working for reachability automata in place of parity automata. The step-by-step approach can be opposed to previous algorithms solving the considered problem "in one step", being compulsorily more complicated.

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“…Proof. This follows by a proof completely analogous to [11,Theorem 6.4], [10,Theorem 48] or [20,Theorem 3]. The technique originates from [20].…”

Section: Infinitary Rewritingmentioning

confidence: 99%

“…Formally, they could be justified as in e.g. [35,39,26,11]. Our use of coinduction in this paper is not very involved, and there are no implicit corecursive function definitions like in [11].…”

Section: Introductionmentioning

confidence: 99%

An operational interpretation of coinductive types

Czajka

2018

Preprint

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We introduce an infinitary rewriting semantics for strictly positive nested higher-order (co)inductive types. This may be seen as a refinement and generalization of the notion of productivity in term rewriting to a setting with higher-order functions and with data specifed by nested higher-order inductive and coinductive definitions. We prove an approximation theorem which essentially states that if a term reduces to an arbitrarily large finite approximation of an infinite object in the interpretation of a coinductive type, then it infinitarily reduces to an infinite object in the interpretation of this type. We introduce a sufficient syntactic correctness criterion, in the form of a type system, for finite terms decorated with type information. Using the approximation theorem, we show that each well-typed term has a well-defined interpretation in our infinitary rewriting semantics. This gives an operational interpretation of typable terms which takes into account the "limits" of infinite reduction sequences.

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A Coinductive Confluence Proof for Infinitary Lambda-Calculus

Cited by 10 publications

References 45 publications

Every λ-Term is Meaningful for the Infinitary Relational Model

Every λ-Term is Meaningful for the Infinitary Relational Model

Higher-Order Model Checking Step by Step

An operational interpretation of coinductive types

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