Two-Level Hybrid: A System for Reasoning Using Higher-Order Abstract Syntax

Momigliano, Alberto; Martin, Alan J.; Felty, Amy P.

doi:10.1016/j.entcs.2007.09.019

Cited by 24 publications

(31 citation statements)

References 4 publications

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“…The Hybrid approach [MMF08] exploits the advantages of HOAS within general theorem proving systems. We use a pretty-printed version of Coq concrete syntax in this paper.…”

Section: Mechanization In Two-level Hybridmentioning

confidence: 99%

“…Hypothetical and parametric judgments are encoded in the SL layer. In this paper, we use a simple SL, a sequent formulation of a fragment of second-order minimal logic with backchaining, adapted from [MM02] (and also used in [MMF08]). Its syntax and inference rules can be encoded directly as follows:…”

Section: Mechanization In Two-level Hybridmentioning

confidence: 99%

“…On the other side of the spectrum of systems supporting HOAS, we have, for instance, the two-level Hybrid system [MMF08,FM08] as implemented in Coq [BC04] and Isabelle/HOL [NPW02], Abella [Gac08], and the Tac prover [BSM10], where contexts are manually represented as lists. While the substitution lemma is still obtained for free because it is an application of the cut-rule, structural properties about contexts such as weakening must typically be proven separately as lemmas.…”

Section: Introductionmentioning

confidence: 99%

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Reasoning with Higher-Order Abstract Syntax and Contexts: A Comparison

Felty

Pientka

2010

Interactive Theorem Proving

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Abstract. A variety of logical frameworks support the use of higherorder abstract syntax (HOAS) in representing formal systems given via axioms and inference rules and reasoning about them. In such frameworks, object-level binding is encoded directly using meta-level binding. Although these systems seem superficially the same, they differ in a variety of ways; for example, in how they handle a context of assumptions and in what theorems about a given formal system can be expressed and proven. In this paper, we present several case studies which highlight a variety of different aspects of reasoning using HOAS, with the intention of providing a basis for qualitative comparison of different systems. We then carry out such a comparison among three systems: Twelf, Beluga, and Hybrid. We also develop a general set of criteria for comparing such systems. We hope that others will implement these challenge problems, apply these criteria, and further our understanding of the trade-offs involved in choosing one system over another for this kind of reasoning.

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“…The Hybrid approach [MMF08] exploits the advantages of HOAS within general theorem proving systems. We use a pretty-printed version of Coq concrete syntax in this paper.…”

Section: Mechanization In Two-level Hybridmentioning

confidence: 99%

Section: Mechanization In Two-level Hybridmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reasoning with Higher-Order Abstract Syntax and Contexts: A Comparison

Felty

Pientka

2010

Interactive Theorem Proving

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“…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%

“…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%

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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Beluga: A Framework for Programming and Reasoning with Deductive Systems (System Description)

Pientka

Dunfield

2010

Automated Reasoning

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Abstract. Beluga is an environment for programming and reasoning about formal systems given by axioms and inference rules. It implements the logical framework LF for specifying and prototyping formal systems via higher-order abstract syntax. It also supports reasoning: the user implements inductive proofs about formal systems as dependently typed recursive functions. A distinctive feature of Beluga is that it not only represents binders using higher-order abstract syntax, but directly supports reasoning with contexts. Contextual objects represent hypothetical and parametric derivations, leading to compact and elegant proofs. Our test suite includes standard examples such as the Church-Rosser theorem, type uniqueness, proofs about compiler transformations, and preservation and progress for various ML-like languages. We also implemented proofs of structural properties of expressions and paths in expressions. Stating these properties requires nesting of quantifiers and implications, demonstrating the expressive power of Beluga.

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Two-Level Hybrid: A System for Reasoning Using Higher-Order Abstract Syntax

Cited by 24 publications

References 4 publications

Reasoning with Higher-Order Abstract Syntax and Contexts: A Comparison

Reasoning with Higher-Order Abstract Syntax and Contexts: A Comparison

Strong Normalization for System F by HOAS on Top of FOAS

Beluga: A Framework for Programming and Reasoning with Deductive Systems (System Description)

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