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Programming Language Semantics in Foundational Type Theory
 In Proc. the IFIP TC2/WG2.2,2.3 International Conference on Programming Concepts and Methods (PROCOMET’98
, 1996
"... There are compelling benefits to using foundational type theory as a framework for programming language semantics. I give a semantics of an expressive programming calculus in the foundational type theory of Nuprl. Previous typetheoretic semantics have used less expressive type theories, or have sacr ..."
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There are compelling benefits to using foundational type theory as a framework for programming language semantics. I give a semantics of an expressive programming calculus in the foundational type theory of Nuprl. Previous typetheoretic semantics have used less expressive type theories, or have sacrificed important programming constructs such as recursion and modules. The primary mechanisms of this semantics are partial types, for typing recursion, set types, for encoding power and singleton kinds, which are used for subtyping and module programming, and very dependent function types, for encoding signatures. Keywords Semantics, program verification, type theory, functional programming 1 INTRODUCTION Type theory has become a popular framework for formal reasoning in computer science and has formed the basis for a number of automated deduction systems, including Automath, Nuprl, HOL and Coq, among others. In addition to formalizing mathematics, these systems are widely used for the a...
Parametricity in an Impredicative Sort
"... Reynold’s abstraction theorem is now a wellestablished result for a large class of type systems. We propose here a definition of relational parametricity and a proof of the abstraction theorem in the Calculus of Inductive Constructions (CIC), the underlying formal language of Coq, in which parametr ..."
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Reynold’s abstraction theorem is now a wellestablished result for a large class of type systems. We propose here a definition of relational parametricity and a proof of the abstraction theorem in the Calculus of Inductive Constructions (CIC), the underlying formal language of Coq, in which parametricity relations ’ codomain is the impredicative sort of propositions. To proceed, we need to refine this calculus by splitting the sort hierarchy to separate informative terms from noninformative terms. This refinement is very close to CIC, but with the property that typing judgments can distinguish informative terms. Among many applications, this natural encoding of parametricity inside CIC serves both theoretical purposes (proving the independence of propositions with respect to the logical system) as well as practical aspirations (proving properties of finite algebraic structures). We finally discuss how we can simply build, on top of our calculus, a new reflexive Coq tactic that constructs proof terms by parametricity.
Abstract Syntactic Finitism in the Metatheory of Programming Languages
, 2010
"... One of the central goals of programminglanguage research is to develop mathematically sound formal methods for precisely specifying and reasoning about the behavior of programs. However, just as software developers sometimes make mistakes when programming, researchers sometimes make mistakes when p ..."
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One of the central goals of programminglanguage research is to develop mathematically sound formal methods for precisely specifying and reasoning about the behavior of programs. However, just as software developers sometimes make mistakes when programming, researchers sometimes make mistakes when proving that a formal method is mathematically sound. As the field of programminglanguage research has grown, these proofs have become larger and more complex, and thus harder to verify on paper. This phenomenon has motivated a great deal of research into the development of logical systems that provide an automated means to apply— and verify the application of—trusted reasoning principles to concrete proofs. The boundary between trusted and untrusted reasoning principles is inherently blurry, and different researchers draw the line in different places. However, just as certain principles are widely recognized to allow the proofs of contradictory statements, others are so uncontroversially ubiquitous in practice that they can be considered beyond reproach. We posit the following questions: (1) what are these principles and (2) how much can we do with them?