A definitional approach to primitivexs recursion over higher order abstract syntax

Ambler, Simon; Crole, Roy L.; Momigliano, Alberto

doi:10.1145/976571.976572

Cited by 18 publications

(24 citation statements)

References 30 publications

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“…Inducting HOAS-style over open terms is a major challenge [20]; in this setting generic judgments are particularly problematic, but can be dealt with by switching to a more expressive SL, based on a eigenvariable encoding [11]. The new theory of termsin-infinite-context underlying the new version of Hybrid [3] directly supports this syntax. With that in place, we will be able, for example, to replay in a full HOAS style a notion of program equivalence based on bisimilarity [13] and finally approach at the right level of abstraction the verification of the compiler optimizations of MIL-lite [4].…”

Section: Discussionmentioning

confidence: 99%

A formalization of an Ordered Logical Framework in Hybrid with applications to continuation machines

Momigliano

Polakow²

2003

Proceedings of the 2003 ACM SIGPLAN Workshop on Mechanized Reasoning About Languages With Variable Binding

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We report on work in progress devoted to the formalization of an Ordered Logical Framework (OLF) [16] based on a twolevels architecture [10] in the Hybrid system [2]. OLF here is a second-order version of ordered linear logic to be used as a meta-language for the verification of the (meta)theory of deductive systems. It is implemented roughly as a metainterpreter on top of the Hybrid system, which provides the full HOAS language. We apply the framework to the formal verification of type preservation of a simple continuation machine for Mini-ML.

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Section: Discussionmentioning

confidence: 99%

A formalization of an Ordered Logical Framework in Hybrid with applications to continuation machines

Momigliano

Polakow²

2003

Proceedings of the 2003 ACM SIGPLAN Workshop on Mechanized Reasoning About Languages With Variable Binding

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“…Both capture object-level bindings by meta-level functional bindings; "weak" refers to the considered functions mapping variables to terms, while "strong" refers to these functions mapping terms to terms. Weak HOAS approaches are taken in [18], [34], [57], [30], including in category-theoretic form (with a denotational-semantics flavor) in [23], [33], [8], [24]. Our work in this paper, the above HOAS-tailored approaches, as well as [19], the work on Hybrid [7], [44], [46], [22], as well as parametric HOAS [17], parametricity-based HOAS [35], 6 and de-Bruijn-mixed-HOAS [32], fall within strong HOAS.…”

Section: More Related Workmentioning

confidence: 99%

Strong Normalization for System F by HOAS on Top of FOAS

Popescu

Gunter

Osborn

2010

2010 25th Annual IEEE Symposium on Logic in Computer Science

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Abstract-We present a point of view concerning HOAS (Higher-Order Abstract Syntax) and an extensive exercise in HOAS along this point of view. The point of view is that HOAS can be soundly and fruitfully regarded as a definitional extension on top of FOAS (First-Order Abstract Syntax). As such, HOAS is not only an encoding technique, but also a higher-order view of a first-order reality. A rich collection of concepts and proof principles is developed inside the standard mathematical universe to give technical life to this point of view. The exercise consists of a new proof of Strong Normalization for System F. HOAS makes our proof considerably more direct than previous proofs. The concepts and results presented here have been formalized in the theorem prover Isabelle/HOL. [49] and has ever since been extensively developed in frameworks with a wide variety of features and flavors. We distinguish two main (overlapping) directions in these developments. Keywords--(I) First, the employment of a chosen meta logic as a pure logical framework, used for defining object systems for the purpose of reasoning inside those systems. A standard example is higher-order logic (HOL) as the meta logic and firstorder logic (FOL) as the object system. Thanks to affinities between the mechanisms of these two logics, one obtains an adequate encoding of FOL in HOL by merely declaring in HOL types and constants and stating the FOL axioms and rules as HOL axioms -then the mechanisms for building FOL deductions (including substitution, instantiation, etc.) are already present in the meta logic, HOL.-(II) Second, the employment of the meta-logic to reason about the represented object systems, i.e., to represent not only the object systems, but also (some of) their metatheory. (E.g., cut elimination is a property about Gentzenstyle FOL, not expressible in a standard HOAS-encoding of FOL into HOL.) While direction (I) has been quasi-saturated by the achievement of quasi-maximally convenient logical frameworks (such Edinburgh LF [31] and generic Isabelle [49]), this second direction undergoes these days a period of active research. We distinguish two main approaches here:

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“…Another approach is to leave the type var abstract, though programming with such a representation requires some extra semantically justified axioms about variables [Bucalo et al, 2006]. Ambler et al [2003] present a variation on weak HOAS in which free variables are represented as projections from an infinite context (a stream of variables); this approach mitigates Isabelle's lack of dependent types, which could be used to characterize terms' contexts more precisely.…”

Section: Related Workmentioning

confidence: 99%

Focusing on Binding and Computation

Licata

Zeilberger

Harper

2008

2008 23rd Annual IEEE Symposium on Logic in Computer Science

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Variable binding is a prevalent feature of the syntax and proof theory of many logical systems. In this paper, we define a programming language that provides intrinsic support for both representing and computing with binding. This language is extracted as the Curry-Howard interpretation of a focused sequent calculus with two kinds of implication, of opposite polarity. The representational arrow extends systems of definitional reflection with the notion of a scoped inference rule, which permits the adequate representation of binding via higher-order abstract syntax. On the other hand, the usual computational arrow classifies recursive functions over such higher-order data. Unlike many previous approaches, both binding and computation can mix freely. Following Zeilberger [POPL 2008], the computational function space admits a form of openendedness, in that it is represented by an abstract map from patterns to expressions. As we demonstrate with Coq and Agda implementations, this has the important practical benefit that we can reuse the pattern coverage checking of these tools.

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A definitional approach to primitivexs recursion over higher order abstract syntax

Cited by 18 publications

References 30 publications

A formalization of an Ordered Logical Framework in Hybrid with applications to continuation machines

A formalization of an Ordered Logical Framework in Hybrid with applications to continuation machines

Strong Normalization for System F by HOAS on Top of FOAS

Focusing on Binding and Computation

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