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25
A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF c ..."
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Cited by 217 (44 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
Eliminating Array Bound Checking Through Dependent Types
 In Proceedings of ACM SIGPLAN Conference on Programming Language Design and Implementation
, 1998
"... We present a typebased approach to eliminating array bound checking and list tag checking by conservatively extending Standard ML with a restricted form of dependent types. This enables the programmer to capture more invariants through types while typechecking remains decidable in theory and can s ..."
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Cited by 169 (24 self)
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We present a typebased approach to eliminating array bound checking and list tag checking by conservatively extending Standard ML with a restricted form of dependent types. This enables the programmer to capture more invariants through types while typechecking remains decidable in theory and can still be performed efficiently in practice. We illustrate our approach through concrete examples and present the result of our preliminary experiments which support support the feasibility and effectiveness of our approach. 1 Introduction The absence of runtime array bound checks is an infamous source of fatal errors for programs in languages such as C. Nonetheless, compilers offer the option to omit array bound checks, since they can turn out to be expensive in practice (Chow 1983; Gupta 1994). In statically typed languages such as ML, one would like to provide strong guarantees about the safety of all operations, so array bound checks cannot be omitted in general. The same is true for Ja...
Combining programming with theorem proving
 In ICFP ’05: Proceedings of the tenth ACM SIGPLAN international conference on Functional programming
, 2005
"... 1. Introduction The notion of type equality plays a pivotal r^ole in type systemdesign. However, the importance of this role is often less evident in commonly studied type systems. For instance, in the simplytyped ..."
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Cited by 85 (7 self)
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1. Introduction The notion of type equality plays a pivotal r^ole in type systemdesign. However, the importance of this role is often less evident in commonly studied type systems. For instance, in the simplytyped
Structural Cut Elimination  I. Intuitionistic and Classical Logic
 Information and Computation
, 2000
"... this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced b ..."
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Cited by 53 (17 self)
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this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as variables bound in derivations, thus naturally capturing occurrence conditions. A starting point for the proofs is Kleene's sequent system G 3 [Kle52], which we derive systematically from the point of view that a sequent calculus should be a calculus of proof search for natural deductions. It can easily be related to Gentzen's original and other sequent calculi. We augment G 3 with proof terms that are stable under weakening. These proof terms enable the structural induction and furthermore form the basis of the representation of the proof in LF. The most closely related work on cut elimination is MartinLo# f 's proof of admissibility [ML68]. In MartinLo# f 's system the cut rule incorporates aspects of both weakening and contraction which enables a structural induction argument closely related to ours. However, without the introduction of proof terms, the implicit weakening in the cut rule makes it difficult to implement this proof directly. Herbelin [Her95] restates this proof and proceeds by assigning proof terms only to restricted sequent calculi LJT and LKT which correspond more immediately to
Moving proofsasprograms into practice
 In: Proceedings of the 12 th IEEE International Conference on Automated Software Engineering, IEEE Computer Society
, 1997
"... Proofs in the Nuprl system, an implementation of a constructive type theory, yield “correctbyconstruction ” programs. In this paper a new methodology is presented for extracting efficient and readable programs from inductive proofs. The resulting extracted programs are in a form suitable for use i ..."
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Cited by 18 (5 self)
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Proofs in the Nuprl system, an implementation of a constructive type theory, yield “correctbyconstruction ” programs. In this paper a new methodology is presented for extracting efficient and readable programs from inductive proofs. The resulting extracted programs are in a form suitable for use in hierarchical verifications in that they are amenable to clean partial evaluation via extensions to the Nuprl rewrite system. The method is based on two elements: specifications written with careful use of the Nuprl settype to restrict the extracts to strictly computational content; and on proofs that use induction tactics that generate extracts using familiar fixedpoint combinators of the untyped lambda calculus. In this paper the methodology is described and its application is illustrated by example. 1.
A Structural Proof of Cut Elimination and Its Representation in a Logical Framework
, 1994
"... We present new proofs of cut elimination for intuitionistic and classical sequent calculi. In both cases the proofs proceed by three nested structural inductions, avoiding the explicit use of multisets and termination measures on sequent derivations. This makes them amenable to elegant and concise r ..."
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Cited by 17 (4 self)
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We present new proofs of cut elimination for intuitionistic and classical sequent calculi. In both cases the proofs proceed by three nested structural inductions, avoiding the explicit use of multisets and termination measures on sequent derivations. This makes them amenable to elegant and concise representations in LF, which are given in full detail. This work was supported by NSF Grant CCR9303383 The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of NSF or the U.S. government. Keywords: Logic, Cut Elimination, Logical Framework Contents 1 Introduction 1 2 Intuitionistic Sequent Calculus 2 3 Proof Terms for the Sequent Calculus 8 4 Representing Sequent Derivations in LF 10 5 Admissibility of Cut 13 6 Extension to Classical Logic 18 7 Conclusion 24 A Detailed Admissibility Proofs for Cut 26 A.1 Intuitionistic Calculus : : : : : : : : : : : : : : : : : : :...
Dependently Typed Data Structures
, 1999
"... The mechanism for declaring datatypes in functional programming languages such as ML and Haskell is of great use in practice. This mechanism, however, often suffers from its imprecision in capturing the invariants inherent in data structures. We remedy the situation with the introduction of dependen ..."
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Cited by 14 (3 self)
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The mechanism for declaring datatypes in functional programming languages such as ML and Haskell is of great use in practice. This mechanism, however, often suffers from its imprecision in capturing the invariants inherent in data structures. We remedy the situation with the introduction of dependent datatypes so that we can model data structures with significantly more accuracy. We present a few interesting examples such as implementations of redblack trees and binomial heaps to illustrate the use of dependent datatypes in capturing some sophisticated invariants in data structures. We claim that dependent datatypes can enable the programmer to implement algorithms in a way that is more robust and easier to understand.
Elimination of Negation in a Logical Framework
, 2000
"... Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with ..."
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Cited by 10 (3 self)
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Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgements, we adapt the idea of elimination of negation introduced in [21] for Horn logic to a fragment of higherorder HHF. This entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks; the main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.
Tradeoffs in the Intensional Representation of Lambda Terms
 Rewriting Techniques and Applications (RTA 2002), volume 2378 of LNCS
, 2002
"... Higherorder representations of objects such as programs, specifications and proofs are important to many metaprogramming and symbolic computation tasks. Systems that support such representations often depend on the implementation of an intensional view of the terms of suitable typed lambda calculi. ..."
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Cited by 10 (3 self)
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Higherorder representations of objects such as programs, specifications and proofs are important to many metaprogramming and symbolic computation tasks. Systems that support such representations often depend on the implementation of an intensional view of the terms of suitable typed lambda calculi. Refined lambda calculus notations have been proposed that can be used in realizing such implementations. There are, however, choices in the actual deployment of such notations whose practical consequences are not well understood. Towards addressing this lacuna, the impact of three specific ideas is examined: the de Bruijn representation of bound variables, the explicit encoding of substitutions in terms and the annotation of terms to indicate their independence on external abstractions. Qualitative assessments are complemented by experiments over actual computations using the lambdaProlog language.