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# Topic: Calculus of Constructions

 Calculus Ambient calculus Ambient calculus is a form of notation devised by Luca Cardelli and 1998 and used to describe and theor... Calculus of variations Calculus of variations is a field of brachistochrone, the path along which a particle would desce... Stochastic calculus Stochastic calculus is a branch of random (i.e. www.brainyencyclopedia.com /topics/calculus.html   (680 words)

 Read about Calculus of Constructions at WorldVillage Encyclopedia. Research Calculus of Constructions and learn about ...   (Site not responding. Last check: 2007-09-17) It is thus possible, within the CoC, to define functions from, say, integers to types, types to types as well as functions from integers to integers. In the original CoC, inductive datatypes had to be emulated as their polymorphic destructor function. The valid judgements for the calculus of constructions are derivable from a set of inference rules. encyclopedia.worldvillage.com /s/b/Calculus_of_constructions   (430 words)

 Calculus of constructions   (Site not responding. Last check: 2007-09-17) Connected Curriculum Project: Multivariable Calculus Covers topics in multivarible calculus such as transformation of vectors, volume and torque in 3 dimensions, and directional derivatives. Calculus Preparation Features general information of what calculus is and how it is applied. Calculus of Variations and Geometric Measure Theory at Pisa Preprints on various topics on the calculus of variations. www.serebella.com /encyclopedia/article-Calculus_of_constructions.html   (281 words)

 ECC, an Extended Calculus of Constructions - Luo (ResearchIndex)   (Site not responding. Last check: 2007-09-17) Explicit Universes for the Calculus of Constructions - Courant (2002) 7: The Calculus of Constructions (context) - Coquand, rard - 1986 45 A Calculus of Constructions (context) - Coquand - 1986 citeseer.ist.psu.edu /luo89ecc.html   (604 words)

 Coq - Wikipedia, the free encyclopedia Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq means "rooster" in French - and Thierry Coquand (along with Gérard Huet) developed the aforementioned calculus of constructions. Benjamin Werner (of INRIA) and Georges Gonthier (of Microsoft Research, in Cambridge, England) used Coq to create a surveyable proof of the four color theorem, which was completed in September 2004. en.wikipedia.org /wiki/Coq   (179 words)

 OCC - Open Calculus of Constructions Pure type systems, on the other hand, in particular the calculus of constructions, provide higher-order (dependent) types, but they are based on a fixed notion of computation, namely beta-reduction. To close the gap between these two different paradigms of equational logic and higher-order type theory we are currently investigating the open calculus of constructions (OCC) an equational variant of the calculus of constructions with an open computational system and a flexible universe hierarchy. Calculus of Indexed Names and Named Indices (CINNI) and Uniform Pure Type Systems (UPTS), we have developed an experimental proof assistant for OCC that has additional features such as definitions and meta-variables. formal.cs.uiuc.edu /stehr/occ_eng.html   (955 words)

 Project-Team-LogiCal Another direction is the design of extensions of the Calculus of Constructions with arbitrary computation rules, while the original calculus had a fixed set of rules. Another property of the Calculus of Inductive Constructions is important for its use as the langage of a proof assistant. By insisting on this idea that constructive proofs must be distinguished from classical proofs, the project-team LogiCal participates to rise of a new form a constructivism, not trying to restrict mathematics to constructive mathematics, but trying to identify the part of mathematics that can be done constructively and the part that cannot. www.inria.fr /rapportsactivite/RA2004/logical/uid7.html   (640 words)

 Abstracts for TYPES 98 There is an obvious interpretation of the calculus of constructions in classical set theory ZF, with types being modelled as sets and the type Prop being modelled as a two element type. Using the construction of my type theoretic interpretation of CZF in Martin-Lof Type Theory there is a converse interpretation that models CZF+(P-scheme) in a suitable extension of the Calculus of Constructions. One novelty is that we use short constructive proofs of Dickson's lemma and Hilbert's basis theorem which are extracted from classical proofs using the techniques of [Coquand 92] based on open induction [Raoult 88]. www.tcs.informatik.uni-muenchen.de /~types98/abstract.html   (3504 words)

 Frédéric Blanqui's publications First, we prove that, in the Calculus of Algebraic Constructions with size annotations, the problems of type inference and type-checking are decidable, provided that the sets of constraints generated by size annotations are satisfiable and admit most general solutions. This paper presents general syntactic conditions ensuring the strong normalization and the logical consistency of the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. This paper is concerned with the foundations of the Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions by inductive data types. webloria.loria.fr /~blanqui/papers.html   (1709 words)

 Project-Team-LEMME The calculus of inductive constructions is powerful enough to formalize complex mathematics, based on algebraic structures together with their dependences and operations (as was also done in the Axiom computer algebra system). The calculus of inductive constructions also makes it possible to write algorithms as recursive functional programs, based on rich data structures. A third important characteristic of the calculus of inductive constructions is that it is also a language for manipulating proofs, thanks to the Curry-Howard isomorphism. www.inria.fr /rapportsactivite/RA2004/lemme/uid10.html   (204 words)

 An Extended Calculus of Constructions - Luo (ResearchIndex)   (Site not responding. Last check: 2007-09-17) ECC integrates Coquand-Huet's (impredicative) calculus of constructions and Martin-Lof's (predicative) type theory with universes, and turns out to be a strong and expressive calculus for formalization of mathematics, structured proof development and program specification. 0.3: Explicit Universes for the Calculus of Constructions - Courant (2002) 708 The Lambda Calculus: its Syntax and Semantics (context) - Barendregt citeseer.ist.psu.edu /luo90extended.html   (1478 words)

 An Extended Calculus of Constructions Abstract: This thesis presents and studies a unifying thoery of dependent types ECC - Extended Calculus of Constructions. ECC integrates Coquand-Huet's (impredicative) calculus of constructions and Martin-Löf's (predicative) type theory with universes, and turns out to be a strong and expressive calculus for formalization of mathematics, structured proof development and program specification. The strong normalization result shows the proof-theoretic consistency of the calculus; in particular, it implies the consistency of the embedded intuitionistic higher-order logic and the decidability of the theory. www.lfcs.inf.ed.ac.uk /reports/90/ECS-LFCS-90-118   (358 words)

 TLCA 2003 - abstracts of accepted papers This means that constructions on open terms are necessarily parameterized in two different ways for both variables and names. Semantically, such a construction must be modeled by a bi-parameterized family of operators. In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), which is the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. www.dsic.upv.es /~rdp03/tlca/abstracts.html   (2047 words)

 Harry Mairson Optimal evaluation: An evaluator for lambda calculus (or more broadly speaking, a functional programming language) is said to be correct and optimal if it returns a normal form whenever there is one (i.e., it never diverges by choosing to evaluate the wrong redex), and never does inessential work (i.e., copying redexes). Expressibility and typed lambda calculus: I recently developed a new proof of a famous theorem of Richard Statman, that deciding equivalence of the normal forms of two simply typed lambda terms is not Kalmar-elementary. I have studied the interpretation of types as programming specifications for records, objects, and other constructs, showing how their relationship can be formalized in the calculus of constructions, a higher order logic. www.cs.brandeis.edu /~mairson   (1073 words)

 Uni Bham SCS - Research in CS and AI - 1999   (Site not responding. Last check: 2007-09-17) We aim to construct abstract machines for functional languages which address the issues of garbage collection and sharing from a novel perspective. Using the semantics, we then derive categorical combinators, which are a machine-oriented version of the calculus with explicit substitutions. The main difficulty in developing this calculus was to resolve a complex interaction between the linearity and the explicit substitution. www.cs.bham.ac.uk /research/overview_99/projects/node18.html   (785 words)

 [No title]   (Site not responding. Last check: 2007-09-17) On the one hand, the Calculus of Constructions is a powerful type system in which one can formalize the propositions and natural deduction proofs of higher-order logic. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. A closely related calculus is implemented in the Saturate system and has shown useful on many examples, in particular in set theory. protheo.loria.fr /biblio/bib2003-bib.html   (3132 words)

 MainFrame: The Lambda-calculus, Combinatory Logic, and Type Systems The lambda calculus was first devised by Alonzo Church, first to provide a foundation for mathematics and then to show the existence of unsolvable problems. The connection between the lambda calculus and pure combinatory logic was exploited to yield efficient techniques for the evaluation of functional programs by the reduction of graphs of combinators. The constructive type theories devised by the philosopher per Martin Löf influenced work on the design of programming languages and on the verification of programs. www.rbjones.com /rbjpub/logic/cl/cl017.htm   (1273 words)

 ICALP W3: Accepted Papers   (Site not responding. Last check: 2007-09-17) However, the (deterministic) construction presented (which is the best construction to date) yields only a bound of O(exp(sqrt(log n))) on the average stretch. We obtain immediate proofs of strong normalisation for (1) the Calculus of Constructions with Fixpoints (2) the Algebraic Calculus of Constructions with first-order rewriting (3) a fragment of the Inductive Calculus of Constructions. Regarding the one-to-all communication pattern, we show that constructing competitive IRS for a given network is an intractable problem, both for the static and the dynamic case, that is respectively when the root node is fixed and when it can change along the time. www.cs.auc.dk /icalp98/AccptAbstr.html   (9811 words)

 Math Forum - Ask Dr. Math Archives: High School Calculus I'm soon to be a Calculus student in high school. Here is a 'proof' we ran up against during high school in the Netherlands that makes use of integral calculus. I am a grade twelve student taking calculus and was wondering if you could help me with this problem: y=2x(6x+5)exp4 - solve to the second derivative. mathforum.org /library/drmath/sets/high_calculus.html   (843 words)

 1992 research reports / rapport de recherche 1992 The first one is constructive and leads to a construction method in linear time with respect to the surface. In the pure Calculus of Constructions, it is possible to represent data structures and predicates using higher-order quantification. For these reasons, the calculus was extended with a primitive notion of inductive definitions. www.ens-lyon.fr /LIP/Pub/Sauve/rr1992.html   (796 words)

 LRI, DÉMONS: Logic and Types The Calculus of Inductive Constructions does not allow for satisfying representation of certain common notions like mathematical theories or quotient classes. It is based on the Calculus of Inductive Constructions which integrates functional features and structured data types. Coq is an interactive environment which allows for the specification of mathematical and computational problems, the construction of functional programs and the constructive demonstration of proofs by arbitrary composition of elementary proofs. www.lri.fr /demons/theme2.en.html   (799 words)

 FI Abstracts vol. 65 Semantically, such a construction must be modeled by a biparameterized family of operators. We show that the construction of traces for both kinds of relations follows the same principles of construction. Second, for each n > 0, we construct a countable RCC model that is a sub-model of the standard model over the Euclidean unit n-cube; and show that all these countable models are non-isomorphic. www.mimuw.edu.pl /~szczuka/FI/abs65.html   (2111 words)

 Formal Methods and Safety Critical Systems With the open calculus of constructions we explore the interaction between two key features: conditional rewriting modulo equational theories as in rewriting logic (with its membership equational sublogic) and dependent types with universes as they are known from Martin-Loef's type theory and the calculus of constructions. Since the formalism comes with both a classical set-theoretic semantics and an operational semantics, it is especially suitable to formalize mathematical theories in a computational fashion, category theory being one example as indicated for instance by a formalization of the categorical semantics of rewriting logic itself. A prototype of the open calculus of constructions has been developed on the top of the Maude rewriting engine and several examples will be presented in this talk. fmc.cs.uiuc.edu   (681 words)

 CSL01: Abstract for Paper 9   (Site not responding. Last check: 2007-09-17) We present a new method to specify a certain class of quotient in intentional type theory, and in the calculus of inductive constructions in particular. This method is described as an extension of the calculus of constructions allowing normalized types. We prove that this calculus has the properties of strong normalization, subject reduction, and confluence modulo conversion. www.lsv.ens-cachan.fr /csl01/Abstracts/9.html   (157 words)

 WRLA2002: Abstracts The representability extends from higher-order procedures to imperative constructs (such as references and control) to objects to generic computation to concurrency to elements of distributed computing. We show how techniques for specifying the operational semantics of imperative functional programs (syntax-based semantics) and for formalizing variable binding constructs and mobile environments (CINNI calculus) are used in combination with the natural representation of concurrency and distribution provided by rewriting logic to develop a faithful description of the informal PLAN semantics. Based on these ideas, the second part of the tutorial motivates and develops the open calculus of constructions (OCC), a type theory with dependent types that has membership equational logic and rewriting logic as computational sublanguages. www.di.unipi.it /wrla2002/abstracts.html   (2243 words)

 INRIA Futurs   (Site not responding. Last check: 2007-09-17) In particular, the document should unify the existing consistency proofs, on one side of the calculus of constructions with strong and dependent elimination of inductive types, and on the other side with (floating) type universes, taking care of the guard conditions for inductive and co-inductive recursion and of Prop to Set singleton elimination. Coquand, Metamathematical Investigations of a Calculus of Constructions, Logic and Computer Science, 1990. Luo, An Extended Calculus of Constructions, PhD thesis, 1990. www-futurs.inria.fr /emplois/logical.en.html   (334 words)

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