Multi-level Meta-reasoning with Higher-Order Abstract Syntax

Momigliano, Alberto; Ambler, Simon

doi:10.1007/3-540-36576-1_24

Cited by 18 publications

(28 citation statements)

References 16 publications

(39 reference statements)

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“…As already discussed in the introduction, the HOAS-tailored framework's solution is axiomatic: (*) would be an axiom in a logic L (hosting the representation of the object system), with L itself is viewed as an object by the meta-logic; in the meta-logic then, one can perform proofs by induction on derivations in L. Thus, HOAS-tailored frameworks solve the problems with (*) by stepping one level up to a meta-logic. Previous work in general-purpose frameworks, after several experiments, eventually proposed similar solutions, either of directly interfering with the framework axiomatically [45] or of employing the mentioned intermediate logic L [44].…”

Section: Induction Principle For Type Inferencementioning

confidence: 99%

“…On the other hand, the Hybrid package [7], written in Isabelle/HOL, is a successful realization of the general-purpose framework approach. Later versions of this system [44], [46], [22] also import the three-level architecture idea from the HOAStailored framework approach. Our context-free induction principle from Prop.…”

Section: More Related Workmentioning

confidence: 99%

“…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. In weak HOAS, some of the convenience is lost, since substitution of terms for variables is not mere function application, as in strong HOAS.…”

Section: More Related Workmentioning

confidence: 99%

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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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Section: Induction Principle For Type Inferencementioning

confidence: 99%

Section: More Related Workmentioning

confidence: 99%

Section: More Related Workmentioning

confidence: 99%

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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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“…Multi-level approaches [14,23] in Coq and Isabelle, respectively, have been adopted to facilitate reasoning by induction on object-level judgments. We should be able to directly adopt these ideas to create a multi-level version of our system.…”

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confidence: 99%

Higher-order abstract syntax in type theory

Capretta¹,

Felty²

2009

Logic Colloquium 2006

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Abstract. We develop a general tool to formalize and reason about languages expressed using higher-order abstract syntax in a proof-tool based on type theory (Coq). A language is specified by its signature, which consists of sets of sort and operation names and typing rules. These rules prescribe the sorts and bindings of each operation. An algebra of terms is associated to a signature, using de Bruijn notation. Then a higher-order notation is built on top of the de Bruijn level, so that the user can work with meta-variables instead of de Bruijn indices. We also provide recursion and induction principles formulated directly on the higher-order syntax. This generalizes work on the Hybrid approach to higher-order syntax in Isabelle and our earlier work on a constructive extension to Hybrid formalized in Coq. In particular, a large class of theorems that must be repeated for each object language in Hybrid is done once in the present work and can be applied directly to each object language. §1. Introduction. We aim to use proof assistants (in our specific case Coq [9,6]) to formally represent and reason about languages using higher-order syntax, i.e. object languages where binding operators are expressed using binding at the meta-level. This is an active and fertile field of research. Several methods contend to become the most elegant, efficient, and easy to use. The differences stem from the approach of the researchers and the characteristics of the proof tool used.Our starting point was the work on the Hybrid tool in Isabelle/HOL by Ambler, Crole, and Momigliano [2]. We began by replicating their development step by step in Coq, but soon realized that the different underlying meta-theory (the Calculus of Inductive Constructions [10,38], as opposed to higher-order logic) provided us with different tools and led us to diverge from a simple translation of their work. The final result [8] exploits the computational content of the Coq logic: we prove a recursion principle that can be used to program functions on the object language. These functions can be directly computed inside Coq. In the present work, we extend it and provide a general tool in which the user can easily define a language by giving its signature, and a set of tools (higher-order notation, recursion and induction principles) are automatically available.Let us introduce the problem of representing languages with bindings in a logical framework. While a first-order language contains operation symbols that take just elements of the domain(s) as arguments, a second-order language may have operations whose input consists of functions of arbitrary complexity. From a syntactic point of view these operations bind some of the free variables in their arguments. The simplest example of this phenomenon is the abstraction operation in λ-calculus. We can see the operation λ, syntactically, as a binder; 1

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“…More recent work by Miller and Tiu [16] includes a new quantifier for this style of logic, which provides an elegant way to handle abstractions at the level of proofs. Another approach uses multi-level encodings [8,17]. This approach also aims to capture more than term-level abstraction, and is inspired by the work of McDowell and Miller but uses Coq and Isabelle, respectively.…”

Section: Related Workmentioning

confidence: 99%

Combining de Bruijn Indices and Higher-Order Abstract Syntax in Coq

Capretta

Felty

Lecture Notes in Computer Science

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Abstract. The use of higher-order abstract syntax is an important approach for the representation of binding constructs in encodings of languages and logics in a logical framework. Formal meta-reasoning about such object languages is a particular challenge. We present a mechanism for such reasoning, formalized in Coq, inspired by the Hybrid tool in Isabelle. At the base level, we define a de Bruijn representation of terms with basic operations and a reasoning framework. At a higher level, we can represent languages and reason about them using higher-order syntax. We take advantage of Coq's constructive logic by formulating many definitions as Coq programs. We illustrate the method on two examples: the untyped lambda calculus and quantified propositional logic. For each language, we can define recursion and induction principles that work directly on the higher-order syntax.

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Multi-level Meta-reasoning with Higher-Order Abstract Syntax

Cited by 18 publications

References 16 publications

Strong Normalization for System F by HOAS on Top of FOAS

Strong Normalization for System F by HOAS on Top of FOAS

Higher-order abstract syntax in type theory

Combining de Bruijn Indices and Higher-Order Abstract Syntax in Coq

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