A Wisdom Archive on lambda calculus

For complete details about the lambda calculus, see the article under that head. We will summarize here. The lambda calculus is concerned with objects called lambda-terms, which are strings of symbols of one of the following forms: v λv.E1 (E1 E2) where v is a variable name drawn from a predefined infinite set of variable names, and E1 and E2 are lambda-terms. Terms of the form λv.E1 are called abstractions. The variable v is called the formal parameter of t ...

Read more here: » Combinatory logic: Encyclopedia II - Combinatory logic - Summary of the lambda calculus

Since abstraction is the only way to manufacture functions in the lambda calculus, something must replace it in the combinatory calculus. Instead of abstraction, combinatory calculus provides a limited set of primitive functions out of which other functions may be built. Combinatory logic - Combinatory terms. A combinatory term has one of the following forms: v

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In computer science, combinatory logic is used as a simplified model of computation, used in computability theory (the study of what can be computed) and proof theory (the study of what can be mathematically proven.) The theory, despite its simplicity, captures many essential features of the nature of computation. Combinatory logic can be looked at as a variation of the lambda calculus, in which lambda expressions (used to allow for functional abstraction) are replaced by a limited set of combinators, primitive functions which ...

Read more here: » Combinatory logic: Encyclopedia II - Combinatory logic - Combinatory logic in computing

In computer science the boolean datatype, sometimes called the logical datatype, is a primitive datatype having two values: one and zero (sometimes called true and false). It is the special case of a binary numeric datatype of only one digit, or bit, and can also be represented in any other radix by restricting the range of allowed values for certain operations. This datatype is used in boolean and other operations such as and (AND, &, *), or (O ... Including:

• Boolean datatype - Ada
• Boolean datatype - Algol
• Boolean datatype - BrainFuck
• Boolean datatype - C
• Boolean datatype - C++
• Boolean datatype - C#
• Boolean datatype - Fortran
• Boolean datatype - Java
• Boolean datatype - Lambda Calculus
• Boolean datatype - ML
• Boolean datatype - Ocaml
• Boolean datatype - Perl
• Boolean datatype - Visual Basic

Read more here: » Boolean datatype: Encyclopedia - Boolean datatype

It is undecidable whether a general combinatory term has a normal form; whether two combinatory terms are equivalent, etc. This is equivalent to the undecidability of the corresponding problems for lambda terms. However, a direct proof is as follows: First, observe that the term Ω = (S I I (S I I)) has no normal form, because it reduces to itself after three steps, as follows: (S I I (S I I)) ...

Read more here: » Combinatory logic: Encyclopedia II - Combinatory logic - Undecidability of combinatorial calculus

Combinatory logic was intended as a simple 'pre-logic' which would clarify the meaning of variables in logical notation, and indeed eliminate the need for them. See Curry, 1958-72. ...

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The Actor model built on previous models of computation. Actor model - Lambda calculus. The lambda calculus of Alonzo Church can be viewed as the earliest message passing programming language. For example the lambda expression below implements a tree data structure when supplied with parameters for a leftSubTree and rightSubTree. When such a tree is given a parameter message "getLeft", it returns leftSubTree and likewise when given th ...

Read more here: » Actor model: Encyclopedia II - Actor model - Models prior to the Actor model

BrainFuck does not have built in boolean datatypes: Indeed, there is no simple if-then-else structure. However, the character [ means "loop while the value of the current byte pointed to is non-zero. Thus, if can be replicated by the following, if the current byte contains either 1 (true) or 0 (false). [(Some brainfuck code)-] ...

Read more here: » Boolean datatype: Encyclopedia II - Boolean datatype - BrainFuck

The C++ programming language introduced the bool, true and false keywords, adding a native datatype to support boolean data. Preprocessor macros are used to turn bool into _Bool, false into 0 and true into 1, allowing compatibility with the aforementioned C99 use of the stdbool.h header. ...

Read more here: » Boolean datatype: Encyclopedia II - Boolean datatype - C++