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 ...

See also:

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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Church numerals are the representations of natural numbers under Church encoding. The higher-order function that represents natural number n is a function that maps any other function f to its n-fold composition . Church encoding - Definition. Church numerals 0, 1, 2, ..., are defined as follows in the lambda calculus: 0 ≡ λf.λx. x 1 ≡ λf.λx. f x ...

See also:

Church encoding, Church encoding - Church numerals, Church encoding - Definition, Church encoding - Computation with Church numerals, Church encoding - Translation with other representations, Church encoding - Church booleans

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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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An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system. Abstraction of computing processes is used in both the computer science and computer engineering disciplines and usually assumes discrete time paradigm. In the theory of computation, abstract machines are often used in thought experiments regarding computability or to analyze the complexity of algorithms (see computational complexity theory). A typical abstract machine consists of a definition i ...

Including:

• Abstract machine - List of abstract machines

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Church booleans are the Church encoding of the boolean values true and false. Some programming languages use these as implementation model for boolean arithmetic; examples are Smalltalk and Pico. The boolean values are represented as functions of two values that evaluate to one or the other of their arguments. Formal definition in lambda calculus: true ≡ λa.λb. a ...

See also:

Church encoding, Church encoding - Church numerals, Church encoding - Definition, Church encoding - Computation with Church numerals, Church encoding - Translation with other representations, Church encoding - Church booleans

Read more here: » Church encoding: Encyclopedia II - Church encoding - Church booleans

Suppose that plus is a function taking two arguments x and y and returning x + y. In the ML programming language we would define it as follows: plus = fn(x, y) => x + y and plus(1, 2) returns 3 as we expect. The curried version of plus takes a single argument x and returns a new function which takes a single argument y and returns x + y. In ML we would define it as follows: curried_plus = fn(x) => fn(y) => x + y and now when we call curried_plus(1) we get a new ...

See also:

Currying, Currying - Examples, Currying - Mathematical view

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Purely functional programs have no side-effects. Since functions do not modify state, no data may be changed by parallel function calls. For this reason, pure functions are always thread-safe, a fact which is exploited by languages that use call-by-future evaluation. Because ordering of side-effects does not have to be preserved in their absence, some languages (such as Haskell) use call-by-need evaluation for pure functions. "Pure" functional programming languages typically enforce referential transparency, which is the notion ...

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Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

Read more here: » Functional programming: Encyclopedia II - Functional programming - Pure functions

1.TR.6 -- 100BaseFX -- 100BaseTX -- 100BaseT -- 100BaseVG -- 100VG-AnyLAN -- 10base2 -- 10base5 -- 10baseT -- 120 reset -- 16-bit -- 16-bit application -- 16550 UART -- 1NF -- 1TBS -- 2.PAK -- 20-Gate programming language -- 20-GATE -- 28-bit -- 2B1D -- 2B1Q -- 2D -- 2NF -- 3-tier (computing) -- 32-bit application -- 32-bit -- 320xx microprocessor -- 320xx -- 386BSD -- 386SPART.PAR -- 3Com Corporation -- 3DO -- 3D computer graphics -- 3GL -- 3NF -- 3Station -- 4.2BSD -- 404 error -- 431A -- 473L Query programming language -- 486SX -- 4GL -- 4NF -- 51forth programming language -- 56 kbit/s ...

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List of computing topics, List of computing topics - 0–9, List of computing topics - A, List of computing topics - B, List of computing topics - C, List of computing topics - D, List of computing topics - E, List of computing topics - F, List of computing topics - G, List of computing topics - H, List of computing topics - I, List of computing topics - J, List of computing topics - K, List of computing topics - L, List of computing topics - M, List of computing topics - N, List of computing topics - O, List of computing topics - P, List of computing topics - Q, List of computing topics - R, List of computing topics - S, List of computing topics - T, List of computing topics - U, List of computing topics - V, List of computing topics - W, List of computing topics - X, List of computing topics - Y, List of computing topics - Z

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List of mathematics categories - 0–9. 3-manifolds -- 4-dimensional geometry -- 4-manifolds -- List of mathematics categories - A. List of mathematics categories - B. Banach algebras -- Banach spaces -- Bilinear forms -- Birational geometry -- Boolean algebra -- Bourbaki -- List of mathematics categories - C. List of mathematics categories - D. Definitions of mathematical integration ...

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List of mathematics categories, List of mathematics categories - Mathematics categories, List of mathematics categories - 0–9, List of mathematics categories - A, List of mathematics categories - B, List of mathematics categories - C, List of mathematics categories - D, List of mathematics categories - E, List of mathematics categories - F, List of mathematics categories - G, List of mathematics categories - H, List of mathematics categories - I, List of mathematics categories - J, List of mathematics categories - K, List of mathematics categories - L, List of mathematics categories - M, List of mathematics categories - N, List of mathematics categories - O, List of mathematics categories - P, List of mathematics categories - Q, List of mathematics categories - R, List of mathematics categories - S, List of mathematics categories - T, List of mathematics categories - U, List of mathematics categories - V, List of mathematics categories - W, List of mathematics categories - Z, List of mathematics categories - Mathematicians categories, List of mathematics categories - 0–9, List of mathematics categories - A, List of mathematics categories - B, List of mathematics categories - C, List of mathematics categories - D, List of mathematics categories - E, List of mathematics categories - F, List of mathematics categories - G, List of mathematics categories - H, List of mathematics categories - I, List of mathematics categories - J, List of mathematics categories - L, List of mathematics categories - M, List of mathematics categories - N, List of mathematics categories - P, List of mathematics categories - R, List of mathematics categories - S, List of mathematics categories - T, List of mathematics categories - U, List of mathematics categories - V, List of mathematics categories - W, List of mathematics categories - Mathematics-related categories, List of mathematics categories - A, List of mathematics categories - B, List of mathematics categories - C, List of mathematics categories - D, List of mathematics categories - E, List of mathematics categories - F, List of mathematics categories - G, List of mathematics categories - H, List of mathematics categories - I, List of mathematics categories - K, List of mathematics categories - L, List of mathematics categories - M, List of mathematics categories - N, List of mathematics categories - O, List of mathematics categories - P, List of mathematics categories - Q, List of mathematics categories - R, List of mathematics categories - S, List of mathematics categories - T, List of mathematics categories - V, List of mathematics categories - W

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The computational systems (algebras, calculi) that are discussed as Turing complete systems are those intended for studying theoretical computer science. They are intended to be as simple as posible, so that it would be easier to understand the limits of computation. Here are a few: Automata theory the standard for teaching universal Turing machine the classic Lambda calculus the original (Alonzo Church's paper predated Turing's, but Turing is credited for fuller explanation of the implications) formal grammar (language generators) formal language (language recognizers) ...

See also:

Turing completeness, Turing completeness - Related Work, Turing completeness - Examples, Turing completeness - Bibliography

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Lambda calculus-based languages (such as Lisp, ISWIM, and Scheme) are in actual practice value-level languages, although they are not thus restricted by design. To see why typical lambda style programs are primarily value-level, consider the usual definition of a value-to-value function, say f = λx.E here, x must be a value variable (since the argument of f is a value by definition) and E must denote a value too (since f's result is a value by definiti ...

See also:

Value-level programming, Value-level programming - Connection with Data Types, Value-level programming - Connection with Lambda Calculus languages

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In the following: H refers to the source "Hodges" U refers to the source "Undecidable" W refers to definitions from Websters Ninth New Collegiate Dictionary Marriam-Webster Inc., Springfield Mass, 1990 PM refers to the source Principia Mathematica circa B.C.-- Pythagoras shows the existence of numbers that are not rational, i.e. numbers exist that are not the natural numbers or ratios of the counting numbers. Numbers that are either natural numbers or ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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SKI combinator calculus - Self-application and recursion. is an expression that takes an argument and applies that argument to itself: One interesting property of this is that it makes the expression irreducible: Another thing that results from this is that it allows you to write a function that applies something to t ...

See also:

SKI combinator calculus, SKI combinator calculus - Informal description, SKI combinator calculus - Formal definition, SKI combinator calculus - SKI expressions, SKI combinator calculus - Self-application and recursion, SKI combinator calculus - The reversal expression, SKI combinator calculus - Boolean logic

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Programming languages can achieve some of the same algorithmic results as are obtained through higher-order functions by dynamically executing code (sometimes called "Eval" or "Execute" operations) in the scope of evaluation. Unfortunately, there are significant drawbacks to this approach: The argument code to be executed is usually not statically typed; these languages generally rely on dynamic typing to determine the well-formedness and safety of the code to be executed. The argument is usually provided as a string, t ...

See also:

Higher-order function, Higher-order function - Alternatives

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In the lambda calculus, a beta redex is a term of the form where A(x) is a term (possibly) involving variable x. A beta reduction in head position is an application of the following rewrite rule to a beta redex where A(t) is the result of substituting the term t for the variable x in the te ...

See also:

Beta normal form, Beta normal form - Beta reduction, Beta normal form - Reduction strategies, Beta normal form - Eta reduction

Read more here: » Beta normal form: Encyclopedia II - Beta normal form - Beta reduction

Mathematical functions have great strengths in terms of flexibility and analysis. For example, if a function is known to be idempotent, then a call to a function which has its own output as its argument, and which is known to have no side-effects, may be efficiently computed without multiple calls. A function in this sense has zero or more parameters and a single return value. The parameters—or arguments, as they are sometimes called—are the inputs to the function, and the return value is the function's output. The definition of a ...

See also:

Functional programming, Functional programming - Introduction, Functional programming - History, Functional programming - Comparison with imperative programming, Functional programming - Functional programming languages, Functional programming - Higher-order functions, Functional programming - Speed and space considerations, Functional programming - Functional languages

Read more here: » Functional programming: Encyclopedia II - Functional programming - Introduction

Functional languages have long been criticised as resource-hungry, both in terms of CPU resources and memory. This was mainly due to two things: some early functional languages were implemented with no concern for efficiency non-functional languages achieved speed at least in part by leaving out features such as bounds checking or garbage collection which are viewed as essential parts of modern computing frameworks, the ...

See also:

Functional programming, Functional programming - Introduction, Functional programming - History, Functional programming - Comparison with imperative programming, Functional programming - Functional programming languages, Functional programming - Higher-order functions, Functional programming - Speed and space considerations, Functional programming - Functional languages

Read more here: » Functional programming: Encyclopedia II - Functional programming - Speed and space considerations

The first computer-based functional programming language was Information Processing Language (IPL) from the RAND corporation. Another very old functional language is Lisp, though neither the original LISP nor modern Lisps such as Common Lisp are pure-functional. Some Lisp variants include Scheme, Dylan, and Logo (though Logo is an imperitive language). The modern canonical examples are Haskell and members of the ML family including SML and OCaml. Others include Erlang, Clean, and Miranda. A third type of a commonly used functional language is Xslt. Ano ...

See also:

Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

Read more here: » Functional programming: Encyclopedia II - Functional programming - Functional languages

The oldest example of a functional language is Lisp, though neither the original LISP nor modern Lisps such as Common Lisp are pure-functional. Lisp variants include Logo, Scheme, Dylan. The modern canonical examples are Haskell and members of the ML family including SML and OCaml. Others include Erlang, Clean, and Miranda. A third type of a commonly used functional language is Xslt. Another subset is the mathematics languages Maple and Mathematica. Some computer languages, for example Tcl, Perl, Python & Ruby, can also be used in a functional style, ...

See also:

Functional programming, Functional programming - Introduction, Functional programming - History, Functional programming - Comparison with imperative programming, Functional programming - Functional programming languages, Functional programming - Higher-order functions, Functional programming - Speed and space considerations, Functional programming - Functional languages

Read more here: » Functional programming: Encyclopedia II - Functional programming - Functional languages

Informally, the combinators in this system are defined as follows (x, y, and z represent expressions made from the functions , , and , and set values): takes one argument and returns it: takes two arguments, throws away the second, and returns the first: is a substitution operator. It takes three arguments and then returns the first argument applied to the third, which is then applied to the result of the second argument app ...

See also:

SKI combinator calculus, SKI combinator calculus - Informal description, SKI combinator calculus - Formal definition, SKI combinator calculus - SKI expressions, SKI combinator calculus - Self-application and recursion, SKI combinator calculus - The reversal expression, SKI combinator calculus - Boolean logic

Read more here: » SKI combinator calculus: Encyclopedia II - SKI combinator calculus - Informal description