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Dependently Typed Programming with DomainSpecific Logics
 SUBMITTED TO POPL ’09
, 2008
"... We define a dependent programming language in which programmers can define and compute with domainspecific logics, such as an accesscontrol logic that statically prevents unauthorized access to controlled resources. Our language permits programmers to define logics using the LF logical framework, ..."
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Cited by 6 (3 self)
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We define a dependent programming language in which programmers can define and compute with domainspecific logics, such as an accesscontrol logic that statically prevents unauthorized access to controlled resources. Our language permits programmers to define logics using the LF logical framework, whose notion of binding and scope facilitates the representation of the consequence relation of a logic, and to compute with logics by writing functional programs over LF terms. These functional programs can be used to compute values at runtime, and also to compute types at compiletime. In previous work, we studied a simplytyped framework for representing and computing with variable binding [LICS 2008]. In this paper, we generalize our previous type theory to account for dependently typed inference rules, which are necessary to adequately represent domainspecific logics, and we present examples of using our type theory for certified software and mechanized metatheory.
Positively Dependent Types
 SUBMITTED TO PLPV ’09
, 2008
"... This paper is part of a line of work on using the logical techniques of polarity and focusing to design a dependent programming language, with particular emphasis on programming with deductive systems such as programming languages and proof theories. Polarity emphasizes the distinction between posit ..."
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Cited by 5 (0 self)
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This paper is part of a line of work on using the logical techniques of polarity and focusing to design a dependent programming language, with particular emphasis on programming with deductive systems such as programming languages and proof theories. Polarity emphasizes the distinction between positive types, which classify data, and negative types, which classify computation. In previous work, we showed how to use Zeilberger’s higherorder formulation of focusing to integrate a positive function space for representing variable binding, an essential tool for specifying logical systems, with a standard negative computational function space. However, our previous work considers only a simplytyped language. The central technical contribution of the present paper is to extend higherorder focusing with a form of dependency that we call positively dependent types: We allow dependency on positive data, but not negative computation, and we present the syntax of dependent pair and function types using an iterated inductive definition, mapping positive data to types, which gives an account of typelevel computation. We construct our language inside the dependently typed programming language Agda 2, making essential use of coinductive types and inductionrecursion.
An Extensible Theory of Indexed Types
 PUBLISHED IN SUBMITTED TO POPL ’08
, 2008
"... Indexed families of types are a way of associating runtime data with compiletime abstractions that can be used to reason about them. We propose an extensible theory of indexed types, in which programmers can define the index data appropriate to their programs and use them to track properties of r ..."
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Indexed families of types are a way of associating runtime data with compiletime abstractions that can be used to reason about them. We propose an extensible theory of indexed types, in which programmers can define the index data appropriate to their programs and use them to track properties of runtime code. The essential ingredients in our proposal are (1) a logical framework, which is used to define index data, constraints, and proofs, and (2) computation with indices, both at the static and dynamic levels of the programming language. Computation with indices supports a variety of mechanisms necessary for programming with extensible indexed types, including the definition and implementation of indexed types, metatheoretic reasoning about indices, proofproducing runtime checks, computed index expressions, and runtime actions of index constraints. Programmerdefined index propositions and their proofs can be represented naturally in a logical framework such as LF, where variable binding is used to model the consequence relation of the logic. Consequently, the central technical challenge in our design is that computing with indices requires computing with the higherorder terms of a logical framework. The technical contributions of this paper are a novel method for computing with the higherorder data of a simple logical framework and the integration of such computation into the static and dynamic levels of a programming language. Further, we present examples showing how this computation supports indexed programming. 2008
An Extensible Theory of Indexed Types
 SUBMITTED TO POPL ’08
, 2008
"... Indexed families of types are a way of associating runtime data with compiletime abstractions that can be used to reason about them. We propose an extensible theory of indexed types, in which programmers can define the index data appropriate to their programs and use them to track properties of ru ..."
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Indexed families of types are a way of associating runtime data with compiletime abstractions that can be used to reason about them. We propose an extensible theory of indexed types, in which programmers can define the index data appropriate to their programs and use them to track properties of runtime code. The essential ingredients in our proposal are (1) a logical framework, which is used to define index data, constraints, and proofs, and (2) computation with indices, both at the static and dynamic levels of the programming language. Computation with indices supports a variety of mechanisms necessary for programming with extensible indexed types, including the definition and implementation of indexed types, metatheoretic reasoning about indices, proofproducing runtime checks, computed index expressions, and runtime actions of index constraints. Programmerdefined index propositions and their proofs can be represented naturally in a logical framework such as LF, where variable binding is used to model the consequence relation of the logic. Consequently, the central technical challenge in our design is that computing with indices requires computing with the higherorder terms of a logical framework. The technical contributions of this paper are a novel method for computing with the higherorder data of a simple logical framework and the integration of such computation into the static and dynamic levels of a programming language. Further, we present examples showing how this computation supports indexed programming.
Submitted to POPL ’09 Dependently Typed Programming with DomainSpecific Logics
"... We define a dependent programming language in which programmers can define and compute with domainspecific logics, such as an accesscontrol logic that statically prevents unauthorized access to controlled resources. Our language permits programmers to define logics using the LF logical framework ..."
Abstract
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We define a dependent programming language in which programmers can define and compute with domainspecific logics, such as an accesscontrol logic that statically prevents unauthorized access to controlled resources. Our language permits programmers to define logics using the LF logical framework, whose notion of binding and scope facilitates the representation of the consequence relation of a logic, and to compute with logics by writing functional programs over LF terms. These functional programs can be used to compute values at runtime, and also to compute types at compiletime. In previous work, we studied a simplytyped framework for representing and computing with variable binding [LICS 2008]. In this paper, we generalize our previous type theory to account for dependently typed inference rules, which are necessary to adequately represent domainspecific logics, and we present examples of using our type theory for certified software and mechanized metatheory. 1.