Higher-order abstract syntax in type theory

Capretta, Venanzio; Felty, Amy P.

doi:10.1017/cbo9780511605321.005

Cited by 8 publications

(10 citation statements)

References 28 publications

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“…On one hand, these well-formedness predicates can provide a convenient form of induction within the context of the two-level approach; on the other hand this is a potential area for improvement. Some work in this direction has been done in the Coq version of Hybrid [4].…”

Section: Resultsmentioning

confidence: 99%

An Improved Implementation and Abstract Interface for Hybrid

Martin

Felty

2011

Electron. Proc. Theor. Comput. Sci.

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Hybrid is a formal theory implemented in Isabelle/HOL that provides an interface for representing and reasoning about object languages using higher-order abstract syntax (HOAS). This interface is built around an HOAS variable-binding operator that is constructed definitionally from a de Bruijn index representation. In this paper we make a variety of improvements to Hybrid, culminating in an abstract interface that on one hand makes Hybrid a more mathematically satisfactory theory, and on the other hand has important practical benefits. We start with a modification of Hybrid's type of terms that better hides its implementation in terms of de Bruijn indices, by excluding at the type level terms with dangling indices. We present an improved set of definitions, and a series of new lemmas that provide a complete characterization of Hybrid's primitives in terms of properties stated at the HOAS level. Benefits of this new package include a new proof of adequacy and improvements to reasoning about object logics. Such proofs are carried out at the higher level with no involvement of the lower level de Bruijn syntax.

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Section: Resultsmentioning

confidence: 99%

An Improved Implementation and Abstract Interface for Hybrid

Martin

Felty

2011

Electron. Proc. Theor. Comput. Sci.

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“…We have already mentioned that Venanzio Capretta and Amy Felty have implemented a Hybrid system in Coq (Capretta and Felty 2007). Further material on HOAS in type theory can be found in Capretta and Felty (2009), which contains generalisations both of our own work on Hybrid and of that in Capretta and Felty (2007). Roughly speaking, Capretta and Felty (2009) provides a language of universal algebra with bindings (and higher order signatures) that has an underlying de Bruijn syntax.…”

Section: Functional Abstraction Bindersmentioning

confidence: 99%

The representational adequacy of Hybrid

Crole

2011

Math. Struct. Comp. Sci.

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The Hybrid system (Ambler et al. 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general, and principles of (co)induction in particular. The form of HOAS provided by Hybrid is essentially a lambda calculus with constants.Of fundamental interest is the form of the lambda abstractions provided by Hybrid. The user has the convenience of writing lambda abstractions using names for the binding variables. However, each abstraction is actually a definition of a de Bruijn expression, andHybrid can unwind the user's abstractions (written with names) to machine friendly de Bruijn expressions (without names). In this sense the formal system contains a hybrid of named and nameless bound variable notation.In this paper, we present a formal theory in a logical framework, which can be viewed as a model of core Hybrid, and state and prove that the model is representationally adequate for HOAS. In particular, it is the canonical translation function from λ-expressions to Hybrid that witnesses adequacy. We also prove two results that characterise how Hybrid represents certain classes of λ-expression.We provide the first detailed proof to be published that proper locally nameless de Bruijn expressions and α-equivalence classes of λ-expressions are in bijective correspondence. This result is presented as a form of de Bruijn representational adequacy, and is a key component of the proof of Hybrid adequacy.The Hybrid system contains a number of different syntactic classes of expression, and associated abstraction mechanisms. Hence, this paper also aims to provide a self-contained theoretical introduction to both the syntax and key ideas of the system. Although this paper will be of considerable interest to those who wish to work with Hybrid in Isabelle/HOL, a background in automated theorem proving is not essential.

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“…For example, the aforementioned paper (painfully) succeeded in checking non-trivial results such as a Howe-style proof of congruence of applicative (bi)simulation [54]. 16 Another example [2] is the quite intricate verification of subject reduction of MIL-LITE [9], the intermediate language of the MLj compiler [10].…”

Section: Mc-theorem 13mentioning

confidence: 99%

“…15 A compromise is the "weak" HOAS view mentioned earlier and discussed in Section 6.4. 16 However, it would take a significant investment in man-months to extend the result from the lazy λ-calculus to more interesting calculi such as [58].…”

Section: Mc-theorem 13mentioning

confidence: 99%

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Hybrid

Felty

Momigliano

2010

J Autom Reasoning

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Combining higher-order abstract syntax and (co)-induction in a logical framework is well known to be problematic. We describe the theory and the practice of a tool called Hybrid, within Isabelle/HOL and Coq, which aims to address many of these difficulties. It allows object logics to be represented using higher-order abstract syntax, and reasoned about using tactical theorem proving and principles of (co)induction. Moreover, it is definitional, which guarantees consistency within a classical type theory. The idea is to have a de Bruijn representation of λ-terms providing a definitional layer that allows the user to represent object languages using higher-order abstract syntax, while offering tools for reasoning about them at the higher level. In this paper we describe how to use Hybrid in a multi-level reasoning fashion, similar in spirit to other systems such as Twelf and Abella. By explicitly referencing provability in a middle layer called a specification logic, we solve the problem of reasoning by (co)induction in the presence of non-stratifiable hypothetical judgments, which allow very elegant and succinct specifications of object logic inference rules. We first demonstrate the method on a simple example, formally proving type soundness (subject reduction) for a fragment of a pure functional language, using a minimal intuitionistic logic as the specification logic. We then prove an analogous result for a continuation-machine presentation of the operational semantics of the same language, encoded this time in an ordered linear logic that serves as the specification layer. This example demonstrates the ease with which we can incorporate new specification logics, and also illustrates a significantly more complex object logic whose encoding is elegantly expressed using features of the new specification logic.

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Higher-order abstract syntax in type theory

Cited by 8 publications

References 28 publications

An Improved Implementation and Abstract Interface for Hybrid

An Improved Implementation and Abstract Interface for Hybrid

The representational adequacy of Hybrid

Hybrid

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