Formalizing Adequacy: A Case Study for Higher-order Abstract Syntax

Cheney, James; Norrish, Michael; Vestergaard, René

doi:10.1007/s10817-011-9221-6

Cited by 4 publications

(3 citation statements)

References 23 publications

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“…In particular, in this journal paper we cover environment models and the soundness of β-reduction and take a principled approach to adequacy of encodings in λ-calculus with constants and background β-reduction. The only other formalization of HOAS adequacy we are aware of is that of Cheney et al [31] using Nominal Isabelle, which covers a more complex case than ours: that of encoding λ-calculus in HOL. Admittedly, Nominal Isabelle already delivers well for the task of defining HOAS encodings and proving their adequacy.…”

Section: Similar Case Studies In Other Frameworkmentioning

confidence: 99%

Case Studies in Formal Reasoning About Lambda-Calculus: Semantics, Church-Rosser, Standardization and HOAS

Gheri¹,

Popescu²

2021

Preprint

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We have previously published the Isabelle/HOL formalization of a general theory of syntax with bindings. In this companion paper, we instantiate the general theory to the syntax of lambda-calculus and formalize the development leading to several fundamental constructions and results: sound semantic interpretation, the Church-Rosser and standardization theorems, and higher-order abstract syntax encoding. For Church-Rosser and standardization, our work covers both the call-by-name and call-by-value versions of the calculus, following classic papers by Takahashi and Plotkin. During the formalization, we were able to stay focused on the high-level ideas of the development-thanks to the arsenal provided by our general theory: a wealth of basic facts about the substitution, swapping and freshness operators, as well as recursive-definition and reasoning principles, including a specialization to semantic interpretation of syntax.

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Section: Similar Case Studies In Other Frameworkmentioning

confidence: 99%

Case Studies in Formal Reasoning About Lambda-Calculus: Semantics, Church-Rosser, Standardization and HOAS

Gheri¹,

Popescu²

2021

Preprint

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“…Application in -calculus is defined by a simple desugaring to the builtin application, using a different constant app; that is, 1 2 is defined as app 1 2 (rather than 1 2 ). Thus, the definition needs to be justified by proving adequacy theorems that establish a bijection between the expressions and formal proofs of -calculus, and the HOAS terms and type derivations, which is a tedious and nontrivial task [Cheney et al 2012].…”

Section: Related Work: Existing Approaches To Defining Bindersmentioning

confidence: 99%

A general approach to define binders using matching logic

Chen

Roşu

2020

Proc. ACM Program. Lang.

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We propose a novel definition of binders using matching logic, where the binding behavior of object-level binders is directly inherited from the built-in ∃ binder of matching logic. We show that the behavior of binders in various logical systems such as-calculus, System F,-calculus, pure type systems, can be axiomatically defined in matching logic as notations and logical theories. We show the correctness of our definitions by proving conservative extension theorems, which state that a sequent/judgment is provable in the original system if and only if it is provable in matching logic, in the corresponding theory. Our matching logic definition of binders also yields models to all binders, which are deductively complete with respect to formal reasoning in the original systems. For-calculus, we further show that the yielded models are representationally complete, a desired property that is not enjoyed by many existing-calculus semantics. This work is part of a larger effort to develop a logical foundation for the programming language semantics framework K (http://kframework.org).

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“…It is a significant concern whether a given signature correctly represents an object language we have in mind. This property is often referred to as adequacy in an LF settings [15,6]. As in LF, adequacy in λ Π N relies upon the existence of (unique) canonical forms.…”

Section: Adequacymentioning

confidence: 99%

A dependent nominal type theory

Cheney¹

2012

Logical Methods in Computer Science

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Abstract. Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of βη-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles.

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Formalizing Adequacy: A Case Study for Higher-order Abstract Syntax

Cited by 4 publications

References 23 publications

Case Studies in Formal Reasoning About Lambda-Calculus: Semantics, Church-Rosser, Standardization and HOAS

Case Studies in Formal Reasoning About Lambda-Calculus: Semantics, Church-Rosser, Standardization and HOAS

A general approach to define binders using matching logic

A dependent nominal type theory

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