Using typed lambda calculus to implement formal systems on a machine

Avron, Arnon; Honsell, Furio; Mason, Ian; Pollack, Robert

doi:10.1007/bf00245294

Cited by 78 publications

(72 citation statements)

References 20 publications

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“…We showed that our general-purpose framework offers access to some of the HOAS advanced conveniences, such as impredicative and context-free representations of (originally context-based) type systems. Another purpose was to bring, via an extensive HOAS exercise, more evidence to a belief seemingly shared by the whole HOAS community (beyond the large variety of proposed technical solutions), but not yet sustained by many examples in the literature (apart from those from [9]): that a HOAS representation of a system is in principle able not only to allow hassle-free manipulation and study of a system, but also to actually shed more light on the deep properties of a system. We believe that our general-purpose HOAS machinery does simplify and clarify the setting and justification of a notoriously hard result in type theory.…”

Section: Conclusion Related Work and Future 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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Section: Conclusion Related Work and Future 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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“…A number of logics, particularly higher-order logics based on typed lambda calculi, have been proposed as logical frameworks, including the Edinburgh logical framework LF [35,2,27], generic theorem provers such as Isabelle [56], λProlog [54,25], and Elf [57], and the work of Basin and Constable [4] on metalogical frameworks. Other approaches, such as Feferman's logical framework FS 0 [24]-that has been used in the work of Matthews, Smaill, and Basin [46]earlier work by Smullyan [59], and the 2OBJ generic theorem prover of Goguen, Stevens, Hobley, and Hilberdink [33] are instead first-order.…”

Section: A Reflective Logical Frameworkmentioning

confidence: 99%

“…Given a rewrite theory T = (Ω, E, R), a system module has essentially the form mod T endm, that is, it is expressed with a syntax quite close to the corresponding mathematical notation for its corresponding rewrite theory. 2 The equations E in the equational theory (Ω, E) underlying the rewrite theory T = (Ω, E, R) are presented as a union E = A ∪ E , with A a set of equational axioms introduced as attributes of certain operators in the signature Ω-for example, a conjunction operator ∧ can be declared associative and commutative by keywords assoc and comm-and where E is a set of equations that are assumed to be ChurchRosser and terminating modulo the axioms A. Maude supports rewriting modulo different combinations of such equational attributes: operators can be declared associative, commutative, with identity, and idempotent [13]. Maude contains a sublanguage of functional modules of the form fmod (Ω, E) endfm, with the equational theory (Ω, E) satisfying the conditions already mentioned.…”

Section: Maude's Metalanguage Featuresmentioning

confidence: 99%

Maude as a formal meta-tool

Clavel¹,

Durán

Eker

et al. 1999

FM’99 — Formal Methods

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Abstract. Given the different perspectives from which a complex software system has to be analyzed, the multiplicity of formalisms is unavoidable. This poses two important technical challenges: how to rigorously meet the need to interrelate formalisms, and how to reduce the duplication of effort in tool and specification building across formalisms. These challenges could be answered by adequate formal meta-tools that, when given the specification of a formal inference system, generate an efficient inference engine, and when given a specification of two formalisms and a translation, generate an actual translator between them. Similarly, module composition operations that are logic-independent, but that at present require costly implementation efforts for each formalism, could be provided for logics in general by module algebra generator meta-tools. The foundations of meta-tools of this kind can be based on a metatheory of general logics. Their actual design and implementation can be based on appropriate logical frameworks having efficient implementations. This paper explains how the reflective logical framework of rewriting logic can be used, in conjunction with an efficient reflective implementation such as the Maude language, to design formal meta-tools such as those described above. The feasibility of these ideas and techniques has been demonstrated by a number of substantial experiments in which new formal tools and new translations between formalisms, efficient enough to be used in practice, have been generated.

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“…Figure 3 contains a speci cation of Gentzen's natural deduction system NJ. This speci cation is similar to those given using intuitionistic meta-logics 6, 1 9 ] a n d dependent t yped calculi 11,3]. Let NJ be the set of clauses displayed in A proof of this Proposition can be done similar to the proof of Proposition 2.…”

Section: ?(Left B) This Equivalence Implies the Equivalence (Right Bmentioning

confidence: 93%

A multiple-conclusion meta-logic

Miller

Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science

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The theory of cut-free sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two such languages, λProlog and its linear logic refinement, Lolli [12], provide for various forms of abstraction (modules, abstract data types, higher-order programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) [2] provides for concurrency but lacks abstraction mechanisms. In this paper we present Forum, a logic programming presentation of all of linear logic that modularly extends the languages λProlog, Lolli, and LO. Forum, therefore, allows specifications to incorporate both abstractions and concurrency. As a meta-language, Forum greatly extends the expressiveness of these other logic programming languages. To illustrate its expressive strength, we specify in Forum a sequent calculus proof system and the operational semantics of a functional programming language that incorporates such nonfunctional features as counters and references. AbstractThe theory of cut-free sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two s u c h languages, Prolog and its linear logic re nement, Lolli 12], provide for various forms of abstraction (modules, abstract data types, higher-order programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) 2] provides for concurrency but lacks abstraction mechanisms. In this paper we present F orum, a logic programming presentation of all of linear logic that modularly extends the languages Prolog, Lolli, and LO. Forum, therefore, allows speci cations to incorporate both abstractions and concurrency. As a meta-language, Forum greatly extends the expressiveness of these other logic programming languages. To illustrate its expressive strength, we specify in Forum a sequent calculus proof system and the operational semantics of a functional programming language that incorporates such n o nfunctional features as counters and references.

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Using typed lambda calculus to implement formal systems on a machine

Cited by 78 publications

References 20 publications

Strong Normalization for System F by HOAS on Top of FOAS

Strong Normalization for System F by HOAS on Top of FOAS

Maude as a formal meta-tool

A multiple-conclusion meta-logic

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