Jean-Yves Girard

The Calculus of Constructions can be considered an extension of the Curry-Howard isomorphism. The Curry-Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic. The Calculus of Constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will a ...

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Calculus of constructions, Calculus of constructions - The basics of the calculus of constructions, Calculus of constructions - Terms, Calculus of constructions - Judgements, Calculus of constructions - Inference rules for calculus of constructions, Calculus of constructions - Defining logical operators, Calculus of constructions - Defining data types

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The French Academy of Sciences (Académie des sciences) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the forefront of scientific developments in Europe in the 17th and 18th centuries. Académie des Sciences - History. The Academy of Sciences owes its origin to Colbert's plan to create a general academy. He chose a small group of scholars who met on December 22, 1666 in the Ki ...

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The cut-elimination theorem is the central result establishing the significance of the sequent calculus. It was originally proved by Gerhard Gentzen in his landmark paper "Investigations in Logical Deduction" for the systems LJ and LK formalising intuitionistic and classical logic respectively. The cut-elimination theorem states that any judgement that possesses a proof in the sequent calculus that makes use of the cut rule also possesses a cut-free proof ...

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The Boolean type is defined as: , where α is a type variable. This produces the following two definitions for the boolean values TRUE and FALSE: TRUE := Λα.λxαλyα.x FALSE := Λα.λxαλyα.y Then, with these two λ-terms, we can define some logic operators: AND := λxB ...

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System F, System F - Logic and predicates, System F - System F Structures, System F - Use in programming languages

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The logical connectives are re-examined in this resource-interpretation; each connective splits into multiplicative and additive versions, which correspond to simultaneous and alternative presence, respectively. To motivate the connectives, let us use the example of a vending machine. Multiplicative conjunction, also called tensor (written ⊗), denotes simultaneous occurrence of resources. For example, if I insert 50 cents into the vending machine, then the vending machine simultaneouslySee also:

Linear logic, Linear logic - Linear connectives, Linear logic - Flavours of linear logic

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Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Peano, Russell and Dedekind, conventionally the story of modern proof theory is seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem seemed to bring Hilbert's problem of reducing all mathematics to a finitist formal system, then his incompleteness theorems showed that was unattainable. All of this work was carried out with the pr ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Formally, we start with a countably infinite set of identifiers, say {a, b, c, ..., x, y, z, x1, x2, ...}. The set of all lambda expressions can then be described by the following context-free grammar in BNF: <expr> ::= <identifier> <expr> ::= (λ <identifier>. <expr>) <expr ...

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Lambda calculus, Lambda calculus - History, Lambda calculus - Informal description, Lambda calculus - Formal definition, Lambda calculus - α-conversion, Lambda calculus - β-reduction, Lambda calculus - η-conversion, Lambda calculus - Arithmetic in lambda calculus, Lambda calculus - Logic and predicates, Lambda calculus - Recursion, Lambda calculus - Computable functions and lambda calculus, Lambda calculus - Undecidability of equivalence, Lambda calculus - Lambda calculus and programming languages

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Académie des Sciences - A. Anatole Abragam - Claude Jean Allègre - Christian André Amatore - Jean-Claude Pierre André - Jacques François Olivier Angelier - Vladimir Igorevich Arnol'd - Jacques Jean Arsac - Philippe Ascher - Alain Aspect - Ivan André Albert Assenmacher - Sir Michael Francis Atiyah - Thierry Émilien Flavien Aubin - Jean Armand Aubouin - Pierre Auge ...

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Académie des Sciences, Académie des Sciences - History, Académie des Sciences - The Academy today, Académie des Sciences - Current Members, Académie des Sciences - A, Académie des Sciences - B, Académie des Sciences - C, Académie des Sciences - D, Académie des Sciences - E, Académie des Sciences - F, Académie des Sciences - G, Académie des Sciences - H, Académie des Sciences - I, Académie des Sciences - J, Académie des Sciences - K, Académie des Sciences - L, Académie des Sciences - M, Académie des Sciences - N, Académie des Sciences - O, Académie des Sciences - P, Académie des Sciences - Q, Académie des Sciences - R, Académie des Sciences - S, Académie des Sciences - T, Académie des Sciences - V, Académie des Sciences - W, Académie des Sciences - Y, Académie des Sciences - Z, Académie des Sciences - External link

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In lambda calculus, every expression stands for a function with a single argument; the argument of the function is in turn a function with a single argument, and the value of the function is another function with a single argument. A function is anonymously defined by a lambda expression which expresses the function's action on its argument. For instance, the "add-two" function f such that  f(x) = x + 2  would be expressed in lambda calculus as  λ x. x + 2  (or ...

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Lambda calculus, Lambda calculus - History, Lambda calculus - Informal description, Lambda calculus - Formal definition, Lambda calculus - α-conversion, Lambda calculus - β-reduction, Lambda calculus - η-conversion, Lambda calculus - Arithmetic in lambda calculus, Lambda calculus - Logic and predicates, Lambda calculus - Recursion, Lambda calculus - Computable functions and lambda calculus, Lambda calculus - Undecidability of equivalence, Lambda calculus - Lambda calculus and programming languages

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Main article: Ordinal analysis Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for theories formalising arithmetic and analysis. ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Linear logic has many restrictions and variants. The primary axis of variation is along the classical/intuitionistic divide. Classical linear logic (CLL) is the original linear logic as proposed by Girard. In CLL every connective has a dual. The following is a two-sided presentation of CLL as a sequent calculus: Linear implication is definable in terms of linear negation and multiplicative disjunction in CLL: A ⊸ B ≡ ¬ A ⅋ B. This is familiar from other classical logics: for example, th ...

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Linear logic, Linear logic - Linear connectives, Linear logic - Flavours of linear logic

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System F allows recursive constructions to be embedded in a natural manner, related to that in Martin-Löf's type theory. Suppose you want to create an abstract structure (call it S). The first thing you'll need are constructors. These will be functions whose type will be . Recursivity is manifested when S itself appears within one of the types Ki. If you have m of these constructors, you can define the type of See also:

System F, System F - Logic and predicates, System F - System F Structures, System F - Use in programming languages

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The Academy of Sciences owes its origin to Colbert's plan to create a general academy. He chose a small group of scholars who met on December 22, 1666 in the King's library, and thereafter held twice-weekly working meetings there. The first 30 years of the Academy's existence were relatively informal, since no statutes had as yet been laid down for the institution. On January 20, 1699, Louis XIV gave the Company its first rules. The Academy received the title of Royal Academy of Sciences and was installed in the Louvre in Paris ...

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Académie des Sciences, Académie des Sciences - History, Académie des Sciences - The Academy today, Académie des Sciences - Current Members, Académie des Sciences - A, Académie des Sciences - B, Académie des Sciences - C, Académie des Sciences - D, Académie des Sciences - E, Académie des Sciences - F, Académie des Sciences - G, Académie des Sciences - H, Académie des Sciences - I, Académie des Sciences - J, Académie des Sciences - K, Académie des Sciences - L, Académie des Sciences - M, Académie des Sciences - N, Académie des Sciences - O, Académie des Sciences - P, Académie des Sciences - Q, Académie des Sciences - R, Académie des Sciences - S, Académie des Sciences - T, Académie des Sciences - V, Académie des Sciences - W, Académie des Sciences - Y, Académie des Sciences - Z, Académie des Sciences - External link

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Main article: Tableau systems Tableau systems apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics. ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Main article: Structural proof theory Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz; the definition is slightly more complex, we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Gira ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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Most programming languages are equivalent to the lambda calculus extended with some additional programming language constructs. The classical work where this viewpoint was put forward was Peter Landin's "A Correspondence between ALGOL 60 and Church's Lambda-notation", published in CACM in 1965. The key point is that the lambda calculus expresses the kind of procedural abstraction and application useful for any programming language. Prominently, functional programming languages are basically the lambda calculus with some constants and datatyp ...

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Lambda calculus, Lambda calculus - History, Lambda calculus - Informal description, Lambda calculus - Formal definition, Lambda calculus - α-conversion, Lambda calculus - β-reduction, Lambda calculus - η-conversion, Lambda calculus - Arithmetic in lambda calculus, Lambda calculus - Logic and predicates, Lambda calculus - Recursion, Lambda calculus - Computable functions and lambda calculus, Lambda calculus - Undecidability of equivalence, Lambda calculus - Lambda calculus and programming languages

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There is no algorithm which takes as input two lambda expressions and outputs TRUE or FALSE depending on whether or not the two expressions are equivalent. This was historically the first problem for which the unsolvability could be proven. Of course, in order to do so, the notion of algorithm has to be cleanly defined; Church used a definition via recursive functions, which is now known to ...

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Lambda calculus, Lambda calculus - History, Lambda calculus - Informal description, Lambda calculus - Formal definition, Lambda calculus - α-conversion, Lambda calculus - β-reduction, Lambda calculus - η-conversion, Lambda calculus - Arithmetic in lambda calculus, Lambda calculus - Logic and predicates, Lambda calculus - Recursion, Lambda calculus - Computable functions and lambda calculus, Lambda calculus - Undecidability of equivalence, Lambda calculus - Lambda calculus and programming languages

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Recursion is the definition of a function using the function itself; on the face of it, lambda calculus does not allow this. However, this impression is misleading. Consider for instance the factorial function f(n) recursively defined by f(n) = 1, if n = 0; and n·f(n-1), if n>0. In lambda calculus, one cannot define a function which includes itself. To get around this, one may start by defining a function, here called g< ...

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Lambda calculus, Lambda calculus - History, Lambda calculus - Informal description, Lambda calculus - Formal definition, Lambda calculus - α-conversion, Lambda calculus - β-reduction, Lambda calculus - η-conversion, Lambda calculus - Arithmetic in lambda calculus, Lambda calculus - Logic and predicates, Lambda calculus - Recursion, Lambda calculus - Computable functions and lambda calculus, Lambda calculus - Undecidability of equivalence, Lambda calculus - Lambda calculus and programming languages

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There are several possible ways to define the natural numbers in lambda calculus, but by far the most common are the Church numerals, which can be defined as follows: 0 := λ f x. x 1 := λ f x. f x 2 := λ f x. f (f x) 3 := λ f< ...

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Lambda calculus, Lambda calculus - History, Lambda calculus - Informal description, Lambda calculus - Formal definition, Lambda calculus - α-conversion, Lambda calculus - β-reduction, Lambda calculus - η-conversion, Lambda calculus - Arithmetic in lambda calculus, Lambda calculus - Logic and predicates, Lambda calculus - Recursion, Lambda calculus - Computable functions and lambda calculus, Lambda calculus - Undecidability of equivalence, Lambda calculus - Lambda calculus and programming languages

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However, the proofs used in everyday mathematical practice are almost never like the formal proofs in proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, and given enough time and patience. For most mathematicians, writing a fully formal proof would have all the drawbacks of programming in machine code. Formal proofs are constructed, with the help of computers, in automated theorem proving. Significantly, these proofs can be checked automatically by com ...

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Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography

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