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Topic: Intuitionistic type theory

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	Intuitionistic type theory - Wikipedia, the free encyclopedia
		Intuitionistic type theory is based on a certain analogy or isomorphism between propositions and types: a proposition is identified with the type of its proofs.
		, Γ is a well-formed context of typing assumptions.
		, σ is a well-formed type in context Γ.
	en.wikipedia.org /wiki/Intuitionistic_Type_Theory (1338 words)

	Type theory - Wikipedia, the free encyclopedia
		At the broadest level, type theory is the branch of mathematics and logic that first creates a hierarchy of types, then assigns each mathematical (and possibly other) entity to a type.
		Types in this sense are related to the metaphysical notion of 'type'.
		ST is "simple" (relative to the type theory of Principia Mathematica) primarily because all members of the domain and codomain of any relation must be of the same type.
	en.wikipedia.org /wiki/Type_theory (1138 words)

	Type theory (Site not responding. Last check: 2007-10-31)
		At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into sets called types.
		In this sense, it is related to the metaphysical notion of 'type'.
		Modern type theory was invented partly in response to Russell's paradox, and features prominently in Russell and Whitehead's Principia Mathematica.
	bopedia.com /en/wikipedia/t/ty/type_theory.html (474 words)

	Typed lambda calculus - Wikipedia, the free encyclopedia
		System F allows polymorphism by using universal quantification over all types; from a logical perspective it can describe all functions which are provable total in second-order logic.
		Lambda calculi with dependent types are the base of intuitionistic type theory, the calculus of constructions and the logical framework (LF), a pure lambda calculus with dependent types.
		For example the dependently typed lambda calculus with a type of all types (Type : Type) is not normalizing due to Girard's paradox.
	en.wikipedia.org /wiki/Typed_lambda_calculus (571 words)

	Constructivism (mathematics) - Wikipedia, the free encyclopedia
		In this sense, propositions restricted to the finite are still regarded as being either true or false, as they are in classical mathematics, but this bivalence is not assumed to extend to those which talk about infinite collections.
		In fact, L.E.J. Brouwer, founder of the intuitionist school, viewed the law of the excluded middle as something which was abstracted from finite experience, and which was then applied by mathematicians to the infinite, without justification.
		Constructions can be defined as broadly as free choice sequences, which is the intuitionistic view, or as narrowly as algorithms (or more technically, the computable functions), or even left unspecified.
	en.wikipedia.org /wiki/Mathematical_constructivism (1331 words)

	Encyclopedia :: encyclopedia : Philosophy of mathematics (Site not responding. Last check: 2007-10-31)
		To the logicist, all mathematical statements are precisely of the same type; they are analytic truths, or tautologies.
		The most important output of this was new theories of truth, notably those appropriate to activism and grounding empirical methods.
		Although the social theories and quasi-empiricism, and especially the embodied mind theory, have focused more attention on the epistemology implied by current mathematical practices, they fall far short of actually relating this to ordinary human perception and everyday understandings of knowledge.
	www.hallencyclopedia.com /Philosophy_of_mathematics (3626 words)

	Type - Biocrawler (Site not responding. Last check: 2007-10-31)
		In biology, a type is the specimen or specimens upon which an original species description is based.
		In the decades of the 1900s and 1910s, in philosophy, Bertrand Russell used his Theory of Types to further discuss the mapping of mathematics to logic.
		In intuitionistic type theory, a type can be a proposition or a set.
	www.biocrawler.com /encyclopedia/Type (279 words)

	SEP: Type Theory
		The theory of types was introduced by Russell in order to cope with some contradictions he found in his account of set theory (Russell, 1903).
		Type theory can be used as a foundation for mathematics, and indeed, it was presented as such by Russell in his 1908 paper, which appeared the same year as Zermelo's paper, presenting set theory as a foundation for mathematics.
		We limit ourselves to presenting two applications of type theory to category theory: the constructions of the free cartesian closed category and of the free topos (see the entry on category theory for an explanation of "cartesian closed" and "topos").
	plato.stanford.edu /entries/type-theory (6518 words)

	Intuitionistic Logic
		Intuitionistic logic encompasses the principles of logical reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [1907].
		While intuitionistic arithmetic is a proper part of classical arithmetic, the intuitionistic attitude toward mathematical objects results in a theory of real numbers diverging from the classical.
		Kleene's [1969] formalizes the theory of partial recursive functionals, enabling precise formalizations of the function-realizability interpretation used in [1965] and of a related q-realizability interpretation which establishes a recursive uniformization rule for intuitionistic analysis.
	www.seop.leeds.ac.uk /archives/fall2003/entries/logic-intuitionistic (2215 words)

	Persistent Applicative Heaps and Knowledge Bases
		If we chose a type system which provides a "universal type" as well as the sort of expressiveness found in "intuitionistic type theory" then the user might be given the liberty to specify his program as precisely or as loosely as he choses and accept a corresponding level of error checking by the system.
		A type system in which some names are exempt from constraint provides no protection from paradoxes, and is logically pretty much the same as a Theory of Types which includes a Universal Type.
		However, type systems are not the only way to arrive at consistency, I propose therefore that consistency be ensured by means other than the type system, Adding a type system to a logic which is already consistent should permit the universal type to be retained without giving rise to inconsistency.
	www.rbjones.com /rbjpub/rbjcv/papers/wp32.htm (7215 words)

	More on Proof Theory
		Together with model theory, axiomatic set theory, and recursion theory, proof theory is one of the so-called four pillars of the foundations of mathematics.
		Proof theory can also be considered a branch of philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.
		Structural proof theory is connected to type theory by means of the Curry-Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus.
	www.artilifes.com /proof-theory.htm (1247 words)

	type \| English \| Dictionary & Translation by Babylon
		Theory of Types, a means by Bertrand Russell to further discuss the mapping of mathematics to logic in philosophical logic
		A type is a classification of data that tells the compiler or interpreter how the programmer intends to use it.
		Types supported by most programming languages include integers (usually limited to some range so they will fit in one word of storage), Booleans, floating point numbers, and characters.
	www.babylon.com /definition/type (786 words)

	Type Theory
		Martin-Löf, Per, An Intuitionistic Theory of Types, in Twenty-Five Years of Constructive Type Theory, eds G. Sambin and J. Smith, Oxford University Press, 1998 (reprinted version of an unpublished report from 1972).
		On the meaning of the logical constants and the justification of the logical laws, short course given at the meeting Teoria della Dimostrazione e Filosofia della Logica, organized in Siena, 6-9 April 1983, by the Scuola di Specializzazione in Logica Matematica of the Università degli Studi di Siena.
		Thierry Coquand's exercises for another course on inductive definitions and type theory are also relevant for this course.
	www.cs.chalmers.se /~peterd/kurser/tt03/index.html (544 words)

	Thesis on semantics for type theory
		My thesis entiteled Domains-with-totality semantics for intuitionistic type theory is now available by anonymous ftp from ftp.math.uio.no in the directory /pub/geirwa/ Both a mini-version (11 pages) and the full version is available (125 pages).
		Furthermore, this model is (at least to some extent) faithful to the fact that this logic is intuitionistic; e.g.\ the principle of the excluded middle does not come out as true.
		Using the solutions to the lifting problem, the relation of this model to Beeson's realizability interpretation of ITT is investigated.
	www.cis.upenn.edu /~bcpierce/types/archives/1997-98/msg00094.html (555 words)

	A Taste of Intuitionistic Type Theory (Site not responding. Last check: 2007-10-31)
		The aim of this course is to give an overview over the concepts of Intuitionistic Type Theory which is based on the ideas of Per Martin-Löf.
		This shortcoming is remedied by Type Theory where a proposition is identified with the set of its proofs.
		Thus Type Theory is at the same time a logic and a programming language with a quite sophisticated type system.
	www.cs.nott.ac.uk /~txa/itt (189 words)

	Intuitionistic Logic (Site not responding. Last check: 2007-10-31)
		Intuitionists conclude that the meaning of a statement resides not in its truth conditions but in the means of proof or verification.
		However, proof search in intuitionistic logic is more difficult than in first-order classsical logic; there are no normal forms like conjunctive normal form or prenex form and Skolemization cannot, in general, be applied to intuitionistic formulas.
		Type theory was originally developed with the aim of being a clarification of constructive mathematics.
	cs.wwc.edu /~aabyan/Logic/Intuitionistic.html (502 words)

	Computer Science: Publication: Static Analysis of Martin-L"of's Intuitionistic Type Theory
		Martin-Lof's intuitionistic type theory has been under investigation in recent years as a potential source for future functional programming languages.
		We show how a series of static analyses may be used to improve the efficiency of type theory as a lazy functional programming language.
		After an informal treatment of the application of abstract interpretation to type theory (where we discuss the features of type theory which make it particularly amenable to such an approach), we give formal proofs of correctness of our abstract interpretation techniques, with regard to the semantics of type theory.
	www.cs.kent.ac.uk /pubs/1995/500 (314 words)

	chains, etc & announcement on intuitionistic Tarski theorem
		The result can be proved by induction on cardinals: assuming that the given directed set actually has bounds within itself not just of pairs but of sets smaller than the cardinal under consideration, we can choose a transfinite increasing cofinal sequence.
		Now, I assume that the reason why John Mitchell and others on types should be interested in this is with a view to understanding semantics of programming languages, a subject which lies within CONSTRUCTIVE mathematics.
		However the theory of ordinals as usually presented depends on excluded middle.
	www.cis.upenn.edu /~bcpierce/types/archives/1991/msg00111.html (725 words)

	SEP: Category Theory
		Connections with intuitionistic mathematics were noted early on, and toposes are still used to investigate models of various aspects of intuitionism (Lambek and Scott 1986, Mac Lane and Moerdijk 1992, Van der Hoeven and Moerdijk 1984a, 1984b, 1984c, Moerdijk 1984, Moerdijk 1995a, Moerdijk 1998, Moerdijk and Palmgren 1997, Moerdijk and Palmgren 2002).
		Still, it remains to be seen whether category theory should be "on the same plane," so to speak, as set theory, whether it should be taken as a serious alternative to set theory as a foundation for mathematics, or whether it is foundational in a different sense altogether.
		From the foregoing disussion, it should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics.
	plato.stanford.edu /entries/category-theory (11786 words)

	FMCS'06: Abstracts
		This is an introduction to the theory and applications of operads, with the emphasis on the theory.
		Again, such generalized operads can be viewed in two ways: (i) as algebraic theories of a rather sophisticated kind (including, for instance, various theories of n-categories); (ii) as higher categorical structures in their own right, of independent interest.
		Rather than the usual domain theory, which provides models for computation languages by virtue of the way categories of domains are defined, SDT aims to construct a theory where properties of certain categories of sets (which then allow a similar modelling of computational phenomena) follow axiomatically.
	pages.cpsc.ucalgary.ca /~robin/FMCS/FMCS_06/FMCSAbstracts.html (1828 words)

	Programming Language Semantics in Foundational Type Theory - Crary (ResearchIndex) (Site not responding. Last check: 2007-10-31)
		The primary mechanisms of this semantics for the core calculus are partial types, for typing recursion, set types, for encoding...
		Theorem proving systems based on type theory have been used for the verification of both hardware and software, and have also...
		7 Type' is not a type (context) - Meyer, Reinhold - 1986
	sherry.ifi.unizh.ch /1918.html (913 words)

	Amazon.ca: The Structure of Typed Programming Languages: Books: David A. Schmidt (Site not responding. Last check: 2007-10-31)
		Using classical and recent research from lambda calculus and type theory, it presents a rational reconstruction of the Algol-like imperative languages such as Pascal, Ada, and Modula-3, and the higher-order functional languages such as Scheme and ML.
		The text is unique in its tutorial presentation of higher-order lambda calculus and intuitionistic type theory.
		The latter in particular reveals that a programming language is a logic in which its typing system defines the propositions of the logic and its well-typed programs constitute the proofs of the propositions.
	www.amazon.ca /Structure-Typed-Programming-Languages/dp/0262193493 (423 words)

	SCHMIDT/STOUGHTON/HOWARD N00014-94-1-0866
		Lemon is based upon the principles of constructive type theory and possesses a strict structure that is especially amenable to writing correct code.
		The term constructors of the language strictly follow the introduction and elimination rules for the corresponding types; in particular, the elimination for mu is iteration and the introduction for nu is coiteration (also called generation).
		Given an intuitionistic propositional logic sequent, Gamma - phi, porgi either finds a minimally sized, normal natural deduction of phi from the assumptions in Gamma, or it finds a finite, tree-based Kripke model whose root node forces all of the formulas in Gamma but does not force phi.
	www.cis.ksu.edu /~bhoward/ONR/eoyl95.html (1400 words)

	JOT: JOT: Journal of Object Technology - Theory of Classification, Part 1: Perspectives on Type Compatibility (Site not responding. Last check: 2007-10-31)
		In the case of the Mars Climate Orbiter, the failure was due to inadequate characterisation of syntactic type, resulting in a confusion of metric and imperial units.
		This is where one type is replaced by another, which also systematically replaces the original functions with new ones appropriate to the new type.
		We are motivated to study object-oriented type theory out of a concern to understand better the notion of syntactic and semantic type compatibility.
	www.jot.fm /issues/issue_2002_05/column5 (2419 words)

	Type Theory publications
		Abstract: In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function is equivalent, when working within the framework of Martin-Löf's Intuitionistic Type Theory, to require that there exists a decision function.
		Abstract: An intuitionistic version of Cantor's theorem, which shows that there is no surjective function from the set of the natural numbers N into the set of the functions from N into N, is proved within Martin-Löf's Intuitionistic Type Theory with the universe of the small sets.
		Abstract: We will analyze some extensions of Martin-Löf's constructive type theory by means of extensional set constructors and we will show that often the most natural requirements over them lead to classical logic or even to inconsistency.
	www.math.unipd.it /~silvio/PublicationsTT.html (702 words)

	Contract Number N00014-94-1-0866 Report 95:2 (Site not responding. Last check: 2007-10-31)
		Intuitionistic type theory leads to a mechanical generation of correct code by using specifications.
		The idea is that the specification of a program is its type, and the specification can be expressed by logical formulas and proved by using mathematical axioms and inference rules of logic.
		Then, using the correspondence ``propositions are types'' and ``proofs are programs are values,'' a proof can be translated into correct programming code.
	www.cis.ksu.edu /~bhoward/ONR/95q2rept.html (433 words)

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