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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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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.
TYPE THEORY AND HOMOTOPY
"... The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy ..."
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Cited by 4 (0 self)
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The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Geometry, Algebra, and Logic, which has recently come to light in the form of an interpretation of the constructive type theory of Per MartinLöf into homotopy
Calculating the Fundamental Group of the Circle in Homotopy Type Theory
"... Abstract—Recent work on homotopy type theory exploits an exciting new correspondence between MartinLof’s dependent type theory and the mathematical disciplines of category theory and homotopy theory. The category theory and homotopy theory suggest new principles to add to type theory, and type theo ..."
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Abstract—Recent work on homotopy type theory exploits an exciting new correspondence between MartinLof’s dependent type theory and the mathematical disciplines of category theory and homotopy theory. The category theory and homotopy theory suggest new principles to add to type theory, and type theory can be used in novel ways to formalize these areas of mathematics. In this paper, we formalize a basic result in algebraic topology, that the fundamental group of the circle is the integers. Though simple, this example is interesting for several reasons: it illustrates the new principles in homotopy type theory; it mixes ideas from traditional homotopytheoretic proofs of the result with typetheoretic inductive reasoning; and it provides a context for understanding an existing puzzle in type theory—that a universe (type of types) is necessary to prove that the constructors of inductive types are disjoint and injective. I.
A Syntactical Approach to Weak ωGroupoids
"... Abstract—When moving to a Type Theory without proof irrelevance the notion of a setoid has to be generalized to the notion of a weak ωgroupoid. As a first step in this direction we study the formalisation of weak ωgroupoids in Type Theory. This is motivated by Voevodsky’s proposal of univalent typ ..."
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Abstract—When moving to a Type Theory without proof irrelevance the notion of a setoid has to be generalized to the notion of a weak ωgroupoid. As a first step in this direction we study the formalisation of weak ωgroupoids in Type Theory. This is motivated by Voevodsky’s proposal of univalent type theory which is incompatible with proofirrelevance and the results by Lumsdaine and Garner/van de Berg showing that the standard eliminator for equality gives rise to a weak ωgroupoid.
WRITTEN IN LJUBLJANA, SLOVENIA.
, 2012
"... 1 A short guide to constructive type theory 7 1.1 A dependent type over a type........................ 8 1.1.1 Dependent products......................... 9 ..."
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1 A short guide to constructive type theory 7 1.1 A dependent type over a type........................ 8 1.1.1 Dependent products......................... 9
The miniworkshop The Homotopy Interpretation of Constructive Type Theory,
, 2011
"... Abstract. Over the past few years it has become apparent that there is a surprising and deep connection between constructive logic and higherdimensional structures in algebraic topology and category theory, in the form of an interpretation of the dependent type theory of Per MartinLöf into classica ..."
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Abstract. Over the past few years it has become apparent that there is a surprising and deep connection between constructive logic and higherdimensional structures in algebraic topology and category theory, in the form of an interpretation of the dependent type theory of Per MartinLöf into classical homotopy theory. The interpretation results in a bridge between the worlds of constructive and classical mathematics which promises to shed new light on both. This miniworkshop brought together researchers in logic, topology, and cognate fields in order to explore both theoretical and practical ramifications of this discovery.
Higher Inductive Types as HomotopyInitial Algebras
, 2014
"... Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between MartinLofs constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines we can use geometric intuition to formulate new concepts in type th ..."
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Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between MartinLofs constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines we can use geometric intuition to formulate new concepts in type theory and, conversely, use typetheoretic machinery to verify and often simplify existing mathematical proofs. A crucial ingredient in this new system are higher inductive types, which allow us to represent objects such as spheres, tori, pushouts, and quotients. We investigate a variant of higher inductive types whose computational behavior is determined up to a higher path. We show that in this setting, higher inductive types are characterized by the universal property of being a homotopyinitial algebra.