Kilo represents the letter K in the NATO phonetic alphabet. In international Morse code the letter K is DahDitDah: - · - In Braille the letter K is represented as ⠅ (in Unicode), the dot pattern, X. .. X. K - Computing. In Unicode the capital K is codepoint U+004B and the lowercase k is U+006B. The ASCII code for capital K is 75 and for lowercase k is 107; or in binary 01001011 and 01101011, correspondingly. The EBCDIC cod ...

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K, K - Alternative representations, K - Computing, K - Meanings for K

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Raymond Smullyan - Popular. The Tao is Silent The Chess Mysteries of Sherlock Holmes The Chess Mysteries of the Arabian Knights The Lady or the Tiger? The Riddle of Scheherazade This Book Needs No Title Alice in Puzzle-land - Shopping ... What Is the Name of This Book? Forever Undecided To Mock a Mockingbird Satan, Cantor and Infinity Some Interesting Memories ...

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Raymond Smullyan, Raymond Smullyan - Selected publications, Raymond Smullyan - Popular, Raymond Smullyan - Academic, Raymond Smullyan - Quotations, Raymond Smullyan - Quotations About Smullyan, Raymond Smullyan - External references

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Following the lambda calculus, we will use λx.E to denote the function with formal parameter x and body E. When applied to an argument, say a, this function yields E, with every free appearance of x replaced with a. Valid λ-calculus expressions have one of these forms: v (where v is a variable) λv.E (where v is a variable and E is a λ-calculus expression) (E F) (where E and FSee also:

Curry-Howard, Curry-Howard - Types, Curry-Howard - The type inhabitation problem, Curry-Howard - Intuitionistic logic, Curry-Howard - Hilbert-style proofs, Curry-Howard - Programs are proofs, Curry-Howard - Proof of α → α, Curry-Howard - Proof of β → α → γ → β → γ → α, Curry-Howard - Sequent calculus, Curry-Howard - Point of view of category theory

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In his 1943 paper Recursive Predicates and Quantifiers (reprinted in The Undecidable, p. 255) Stephen Kleene first proposed his "THESIS I": "This heuristic fact [general recursive functions are effectively calculable]...led Church to state the following thesis (Kleene's footnote 22). The same thesis is implicit in Turing's description of computing machines (Kleene's footnote 23). "THESIS I. Every effectively calculable function (effectively decidable predicate) is general r ...

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Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

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It's clear that λ-expressions can have quite complicated types. One might ask whether there is a λ-expression with any given type. The problem of finding a λ-expression with a particular type is called the type inhabitation problem. The answer turns out to be remarkable: There is a closed λ-expression with a particular type only if the type corresponds to a theorem of logic, when the ...

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Curry-Howard, Curry-Howard - Types, Curry-Howard - The type inhabitation problem, Curry-Howard - Intuitionistic logic, Curry-Howard - Hilbert-style proofs, Curry-Howard - Programs are proofs, Curry-Howard - Proof of α → α, Curry-Howard - Proof of β → α → γ → β → γ → α, Curry-Howard - Sequent calculus, Curry-Howard - Point of view of category theory

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One simple way to formally characterize intuitionistic logic is as follows. It has two axiom schemas. All formulas of the form α → β → α are axioms, as are all formulas of the form (α → β → γ) → (α → β) → α → γ The only deduction rule is modus ponens, which states that if we have proved two theorems, one of the form α → β and one of the form α, we may conclude that β is also a theorem. The set of formulas that can be deduced in this fashion is precisely the ...

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Curry-Howard, Curry-Howard - Types, Curry-Howard - The type inhabitation problem, Curry-Howard - Intuitionistic logic, Curry-Howard - Hilbert-style proofs, Curry-Howard - Programs are proofs, Curry-Howard - Proof of α → α, Curry-Howard - Proof of β → α → γ → β → γ → α, Curry-Howard - Sequent calculus, Curry-Howard - Point of view of category theory

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Hilbert-style proofs can be difficult to construct. A more intuitive way to prove theorems of logic is Gentzen's 'sequent calculus'. Sequent calculus proofs correspond to λ-calculus programs in the same way that Hilbert-style proofs correspond to combinator expressions. The sequent calculus rules for the implicational fragment of intuitionistic logic are: Γ represents a context, which is a set of hypotheses. indicates that we can prove a assuming the conte ...

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Curry-Howard, Curry-Howard - Types, Curry-Howard - The type inhabitation problem, Curry-Howard - Intuitionistic logic, Curry-Howard - Hilbert-style proofs, Curry-Howard - Programs are proofs, Curry-Howard - Proof of α → α, Curry-Howard - Proof of β → α → γ → β → γ → α, Curry-Howard - Sequent calculus, Curry-Howard - Point of view of category theory

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Raymond Smullyan - Quotations About Smullyan. I now introduce Professor Smullyan, who will prove to you that either he doesn't exist or you don't exist, but you won't know which. Melvin Fitting ...

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Raymond Smullyan, Raymond Smullyan - Selected publications, Raymond Smullyan - Popular, Raymond Smullyan - Academic, Raymond Smullyan - Quotations, Raymond Smullyan - Quotations About Smullyan, Raymond Smullyan - External references

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Although it is true that all inhabited types correspond to theorems of logic, the converse is not true. Even if we restrict our attention to logical formulas that contain only the → operator, the so-called implicational fragment of logic, not every classically valid logical formula is an inhabited type. For example, the type ((α → β) → α) → α is uninhabited, even though the corresponding lo ...

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Curry-Howard, Curry-Howard - Types, Curry-Howard - The type inhabitation problem, Curry-Howard - Intuitionistic logic, Curry-Howard - Hilbert-style proofs, Curry-Howard - Programs are proofs, Curry-Howard - Proof of α → α, Curry-Howard - Proof of β → α → γ → β → γ → α, Curry-Howard - Sequent calculus, Curry-Howard - Point of view of category theory

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Starting from the point of view that functional programming supports programming languages that are typed and have higher-order functions, the primary content of the "isomorphism" is to identify that amount of structure with that occurring in type theories of intuitionistic logic. Under the influence of category theory there are a number of further heuristic ways of looking at the overall position. These are syntax-lite, one could say; if Curry-Howard were some sort of compiler, they would try to explain what structure is preserved in ...

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Curry-Howard, Curry-Howard - Types, Curry-Howard - The type inhabitation problem, Curry-Howard - Intuitionistic logic, Curry-Howard - Hilbert-style proofs, Curry-Howard - Programs are proofs, Curry-Howard - Proof of α → α, Curry-Howard - Proof of β → α → γ → β → γ → α, Curry-Howard - Sequent calculus, Curry-Howard - Point of view of category theory

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The types of the simply typed lambda calculus are constructed from base types (or type variables) and given types σ,τ we can construct . Church used only two base types o for the type of propositions and ι for the type of individuals. Frequently the calculus with only one base type, usually o, is considered. associates to the right: we read as . To each type σ we assign a ...

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Simply typed lambda calculus, Simply typed lambda calculus - Types, Simply typed lambda calculus - Terms, Simply typed lambda calculus - Important results

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Mathematical logic was the name given by Giuseppe Peano to what is also known as symbolic logic. In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra. Attempts to treat the operations of formal logic in a symbolic or algebraic way were made by some of the more philosophical mathematicians, such as Leibniz and Lambert; but their labors remained little known and isolated. It was George Boole and then Augustus De Morgan, in the middle of the ninetee ...

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Mathematical logic, Mathematical logic - History, Mathematical logic - Topics in mathematical logic, Mathematical logic - Some fundamental results, Mathematical logic - Technical reference, Mathematical logic - First-order languages and structures, Mathematical logic - Terms formulas and sentences, Mathematical logic - Assignment functions, Mathematical logic - Logical satisfaction, Mathematical logic - Logical implication and truth, Mathematical logic - Variable substitution, Mathematical logic - Substitutability

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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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By convention, the following two definitions (known as Church booleans) are used for the boolean values TRUE and FALSE: TRUE := λ x y. x FALSE := λ x y. y (Note that FALSE is equivalent to the Church numeral zero defined above) Then, with these two λ-terms, we can define some logic operators: AND := λ p q. p q FALSE 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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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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Variable-binding mechanisms occur in different contexts in mathematics, logic and computer science but in all cases they are purely syntactic properties of expressions and variables in them. For this section we can summarize syntax by identifying expressions with trees whose leaf nodes are variables, function constants or predicate constants and whose nodes are logical operators. Variable-binding operators are logical operators that occur in almost every formal language. Indeed languages which do not have them are either extremely inexpressi ...

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Free variables and bound variables, Free variables and bound variables - Examples, Free variables and bound variables - Variable-binding operators, Free variables and bound variables - Formal explanation

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Since that time, many other formalisms for describing effective computability have been proposed, including recursive functions, the lambda calculus, register machines, Post systems, combinatory logic, and Markov algorithms. All these systems have been shown to compute the same functions as Turing machines; systems like this are called Turing-complete. Because all these different attempts of formalizing the concept of algorithm have yielded equivalent results, it is now generally assumed that the Church–Turing thesis is correct. However, t ...

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Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

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The Church–Turing thesis has some profound implications for the philosophy of mind. There are also some important open questions which cover the relationship between the Church–Turing thesis and physics, and the possibility of hypercomputation. When applied to physics, the thesis has several possible meanings: The universe is equivalent to a Turing machine (and thus, computing non-recursive functions is physically impossible). This has been termed the strong Church–Turing thesis (not to be confused with the previous ...

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Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

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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 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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In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form In this formula, t is the argument of the function F on the left-hand side, and the parameter that the integral depends on, on the right-hand side. The quantity x is a dummy variable or variable (or parameter) of integration. Now, if we performed the substitution x=g( ...

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Parameter, Parameter - Types of parameter, Parameter - Mathematical, Parameter - Computer science, Parameter - Logic, Parameter - Engineering, Parameter - Analytic geometry, Parameter - Mathematical analysis, Parameter - Probability theory, Parameter - Statistics

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