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TypeTheoretic Methodology For Practical Programming Languages
 DEPARTMENT OF COMPUTER SCIENCE, CORNELL UNIVERSITY
, 1998
"... The significance of type theory to the theory of programming languages has long been recognized. Advances in programming languages have often derived from understanding that stems from type theory. However, these applications of type theory to practical programming languages have been indirect; the ..."
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Cited by 23 (3 self)
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The significance of type theory to the theory of programming languages has long been recognized. Advances in programming languages have often derived from understanding that stems from type theory. However, these applications of type theory to practical programming languages have been indirect; the differences between practical languages and type theory have prevented direct connections between the two. This dissertation presents systematic techniques directly relating practical programming languages to type theory. These techniques allow programming languages to be interpreted in the rich mathematical domain of type theory. Such interpretations lead to semantics that are at once denotational and operational, combining the advantages of each, and they also lay the foundation for formal verification of computer programs in type theory. Previous type theories either have not provided adequate expressiveness to interpret practical languages, or have provided such expressiveness at the expense of essential features of the type theory. In particular, no previous type theory has supported a notion of partial functions (needed to interpret recursion in practical languages), and a notion of total functions and objects (needed to reason about data values), and an intrinsic notion of equality (needed for most interesting results). This dissertation presents the first type theory incorporating all three, and discusses issues arising in the design of that type theory. This type theory is used as the target of a typetheoretic semantics for a expressive programming calculus. This calculus may serve as an internal language for a variety of functional programming languages. The semantics is stated as a syntaxdirected embedding of the programming calculus into type theory. A critical point arising in both the type theory and the typetheoretic semantics is the issue of admissibility. Admissibility governs what types it is legal to form recursive functions over. To build a useful type theory for partial functions it is necessary to have a wide class of admissible types. In particular, it is necessary for all the types arising in the typetheoretic semantics to be admissible. In this dissertation I present a class of admissible types that is considerably wider than any previously known class.
From Operational to Denotational Semantics
 In MFPS 1991
, 1989
"... In this paper it is shown how operational semantic methods may be naturally extended to encompass many of the concepts of denotational semantics. This work builds on the standard development of an operational semantics as an interpreter and operational equivalence. The key addition is an operational ..."
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Cited by 18 (6 self)
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In this paper it is shown how operational semantic methods may be naturally extended to encompass many of the concepts of denotational semantics. This work builds on the standard development of an operational semantics as an interpreter and operational equivalence. The key addition is an operational ordering on sets of terms. From properties of this ordering a closure construction directly yields a fully abstract continuous cpo model. Furthermore, it is not necessary to construct the cpo, for principles such as soundness of fixedpoint induction may be obtained by direct reasoning from this new ordering. The end result is that traditional denotational techniques may be applied in a purely operational setting in a natural fashion, a matter of practical importance for developing semantics of realistic programming languages. 1 Introduction This paper aims to accomplish a degree of unification between operational and denotational approaches to programming language semantics by recasting d...
Partial computations in constructive type theory
 JOURNAL OF LOGIC AND COMPUTATION
, 1991
"... Constructive type theory as conceived by Per MartinLöf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects ..."
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Cited by 7 (5 self)
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Constructive type theory as conceived by Per MartinLöf has a very rich type system, but partial functions cannot be typed. This also makes it impossible to directly write recursive programs. In this paper a constructive type theory Red is defined which includes a partial type constructor A; objects in the type A may diverge, but if they converge, they must be members of A. A fixed point typing principle is given to allow typing of recursive functions. The extraction paradigm of type theory, whereby programs are automatically extracted from constructive proofs, is extended to allow extraction of fixed points. There is a Scott fixed point induction principle for reasoning about these functions. Soundness of the theory is proven. Type theory becomes a more expressive programming logic as a result.
Hybrid PartialTotal Type Theory
, 1995
"... In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be memb ..."
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Cited by 5 (0 self)
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In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type with the logical notion of total type into a single theory. A new partial type constructor A is added to the type theory: objects in A may diverge, but if they converge, they must be members of A. A fixed point typing rule is given to allow for typing of fixed points. The underlying theory is based on ideas from Feferman's Class Theory and Martin Lof's Intuitionistic Type Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary fixed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial functions, a partial ordering ! ¸ on computations, and a fixed point induction principle. The resulting theory is thus intended as a generalpurpose programming logic. Rules are presented and soundness of the theory established. Keywords: Constructive Type Theory, Logics...