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Dependent Intersection: A New Way of Defining Records in Type Theory
"... Record types are an important tool for programming and are essential in objectoriented calculi. Dependent record types are proven to be very useful for program specification and verification. Unfortunately, all known embedding of the dependent record type in the type theory had some imperfections. I ..."
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Cited by 22 (2 self)
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Record types are an important tool for programming and are essential in objectoriented calculi. Dependent record types are proven to be very useful for program specification and verification. Unfortunately, all known embedding of the dependent record type in the type theory had some imperfections. In this paper we present a new type constructor, dependent intersection, i.e., the intersection of two types, where the second type may depend on elements of the first one (do not confuse it with the intersection of a family of types). This new type constructor allows us to define dependent records in a very simple way.
MetaPRL  A Modular Logical Environment
, 2003
"... MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive ..."
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Cited by 10 (2 self)
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MetaPRL is the latest system to come out of over twenty five years of research by the Cornell PRL group. While initially created at Cornell, MetaPRL is currently a collaborative project involving several universities in several countries. The MetaPRL system combines the properties of an interactive LCFstyle tacticbased proof assistant, a logical framework, a logical programming environment, and a formal methods programming toolkit. MetaPRL is distributed under an opensource license and can be downloaded from http://metaprl.org/. This paper provides an overview of the system focusing on the features that did not exist in the previous generations of PRL systems.
Quotient Types: A Modular Approach
 ITUT Recommendation H.324
, 2002
"... In this paper we introduce a new approach to axiomatizing quotient types in type theory. We suggest replacing the existing monolithic rule set by a modular set of rules for a specially chosen set of primitive operations. This modular formalization of quotient types turns out to be much easier to use ..."
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Cited by 7 (3 self)
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In this paper we introduce a new approach to axiomatizing quotient types in type theory. We suggest replacing the existing monolithic rule set by a modular set of rules for a specially chosen set of primitive operations. This modular formalization of quotient types turns out to be much easier to use and free of many limitations of the traditional monolithic formalization. To illustrate the advantages of the new approach, we show how the type of collections (that is known to be very hard to formalize using traditional quotient types) can be naturally formalized using the new primitives. We also show how modularity allows us to reuse one of the new primitives to simplify and enhance the rules for the set types.
Type Theoretical Foundations for Data Structures, Classes, and Objects
, 2004
"... In this thesis we explore the question of how to represent programming data structures in a constructive type theory. The basic data structures in programing languages are records and objects. Most known papers treat such data structure as primitive. That is, they add new primitive type constructors ..."
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Cited by 6 (0 self)
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In this thesis we explore the question of how to represent programming data structures in a constructive type theory. The basic data structures in programing languages are records and objects. Most known papers treat such data structure as primitive. That is, they add new primitive type constructors and supporting axioms for records and objects. This approach is not satisfactory. First of all it complicates a type theory a lot. Second, the validity of the new axioms is not easily established. As we will see the naive choice of axioms can lead to contradiction even in the simplest cases. We will show that records and objects can be defined in a powerful enough type theory. We will also show how to use these type constructors to define abstract data structure.
Logic of subtyping
 Theoretical Computer Science
, 2005
"... We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutuallyrecursive types defined using those type constructors. Basic Logic of Subtyping S extends classical propositional logic by two new binary modalities ..."
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Cited by 3 (2 self)
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We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutuallyrecursive types defined using those type constructors. Basic Logic of Subtyping S extends classical propositional logic by two new binary modalities ⊗ and ⊕. An interpretation of S is a function that maps standard connectives into settheoretical operations (intersection, union, and complement) and modalities into Cartesian product and disjoint union type constructors. This allows S to capture many subtyping properties of the above type constructors. We also consider logics Sρ and S ω ρ that incorporate into S mutuallyrecursive types over arbitrary and wellfounded universes correspondingly. The main results are completeness of the above three logics with respect to appropriate type universes. In addition, we prove Cut elimination theorem for S and establish decidability of S and S ω ρ.
Pavel Naumov On Modal Logics of Partial Recursive Functions
"... Abstract. The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the a ..."
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Abstract. The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and nondeterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established. Keywords: modal logic, recursive function, CurryHoward isomorphism
Helsinki 2009
"... Constructive (intuitionist, antirealist) semantics has thus far been lacking an adequate concept of truth in innity concerning factual (i.e., empirical, nonmathematical) sentences. One consequence of this problem is the difculty of incorporating inductive reasoning in constructive semantics. It is ..."
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Constructive (intuitionist, antirealist) semantics has thus far been lacking an adequate concept of truth in innity concerning factual (i.e., empirical, nonmathematical) sentences. One consequence of this problem is the difculty of incorporating inductive reasoning in constructive semantics. It is not possible to formulate a notion for probable truth in innity if there is no adequate notion of what truth in innity is. One needs a notion of a constructive possible world based on sensory experience. Moreover, a constructive probability measure must be dened over these constructively possible empirical worlds. This study denes a particular kind of approach to the concept of truth in innity for Rudolf Carnap's inductive logic. The new approach is based on truth in the consecutive nite domains of individuals. This concept will be given a constructive interpretation. What can be veriably said about an empirical statement with respect to this concept of truth, will be explained, for which purpose a constructive