Applicative computing systems, or ACS are the systems of object calculi founded on combinatory logic

Combinatory logic

Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming...

 and lambda calculus

Lambda calculus

In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...

.
The only essential notion which is under consideration in these systems is the representation of object

Object (computer science)

In computer science, an object is any entity that can be manipulated by the commands of a programming language, such as a value, variable, function, or data structure...

. In combinatory logic

Combinatory logic

Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming...

 the only metaoperator is application

Function application

In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range.-Representation:...

 in a sense of applying one object to other. In lambda calculus

Lambda calculus

In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...

 two metaoperators are used: application

Function application

In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range.-Representation:...

 – the same as in combinatory logic, and functional abstraction which binds the only variable in one object.

Features of ACS

The objects generated in these systems are the functional entities with the following features:

1. the number of argument places, or object arity is not fixed but is enabling step by step in interoperations with other objects;
2. in a process of generating the compound object one of its counterparts—function, -- is applied to other one—argument, -- but in other contexts they can change their roles, i.e. functions and arguments are considered on the equal rights;
3. the self-applying of functions is allowed, i.e. any object can be applied to itself.

ACS give a sound ground for applicative approach to programming.

Research challenge

Applicative computing systems' lack of storage and history sensitivity is the basic reason they have not provided a foundation for computer design. Moreover, most applicative systems employ the substitution operation of the lambda calculus

Lambda calculus

In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...

 as their basic operation. This operation is one of virtually unlimited power, but its complete and efficient realization presents great difficulties to the machine designer.

See also

• Combinatory logic
  Combinatory logic
  Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming...
  
  
• Lambda calculus
  Lambda calculus
  In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
  
  
• Categorical abstract machine
  Categorical abstract machine
  The categorical abstract machine is a model of computation for programs that preserves the abilities of applicative, functional, or compositional style. It is based on the techniques of applicative computing.- Overview :...
  
  
• Functional programming
  Functional programming
  In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...
  
  
• Applicative programming language

Further reading

[This volume reflects the research program and philosophy of H. Curry

Haskell Curry

Haskell Brooks Curry was an American mathematician and logician. Curry is best known for his work in combinatory logic; while the initial concept of combinatory logic was based on a single paper by Moses Schönfinkel, much of the development was done by Curry. Curry is also known for Curry's...

, one of the founders of computational models and the deductive framework for reasoning in terms of objects.]

• Wolfengagen, V.E. Combinatory logic in programming. Computations with objects through examples and exercises. -- 2-nd ed. -- M.: "Center JurInfoR" Ltd., 2003. -- x+337 с. ISBN 5-89158-101-9.