# Prolegomena to an Operator Theory of Computation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{Q}, while each such process (transformation) is an application of operator A

_{Q}to the input objects. For instance, a Turing machine operator describes transformation of the input words into the output results by a Turing machine.

## 2. The Concept of Operator

**Definition**

**1.**

**Definition**

**2.**

_{∅}that works with words in the alphabet {0, 1} but never halts independently of its input, we see the domain D(T

_{∅}) is the set of all words in the alphabet {0, 1} while the definability domain DD(T

_{∅}) is the empty set ∅.

_{∅}) is the set of all words in the alphabet {0, 1} while the range R(T

_{∅}) is the empty set ∅.

## 3. The Concept of the Information Operator

- -
- Substantial information operators transform physical objects into structural objects, i.e., their domain consists of physical objects while structural objects shape their range.
- -
- Co-substantial information operators transform structural objects into physical objects, i.e., their domain consists of structural objects while physical objects compose their range.
- -
- Pure information operators transform structural objects into structural objects, i.e., their domain and range consist of structural objects.

#### 3.1. Syntactic Information Spaces

#### 3.2. Semantic Information Spaces

## 4. The Concept of Computational Information Operator

- On the top (most general) level, computation is perceived as any transformation of information and/or information representation.
- On the middle level, computation is distinguished as a discrete process of transformation of information and/or information representation.
- On the bottom level, computation is defined as a discrete process of symbolic transformation of information and/or symbolic information representation in case of classical computation models. Alternatively, in case of natural and unconventional computing, physical/chemical/biological/cognitive processes that are interpreted as computation or the basis for computational behaviors of physical systems under consideration are at the bottom level.

- Symbolic computation when information is represented by physically- or mentally-given symbols.
- Material computation when information is represented by material objects, such as atoms, or molecules of a biological cell, and can be continuous as there are continuous phenomena in many branches of physics.

- Discrete computation with digital operations performed in elementary separate steps.
- Continuous computation when operation goes without breaks in time.
- Piecewise continuous computation, combining discrete and continuous computation.

- Sequential computation, which is performed in linear time.
- Parallel or branching computation, in which separate steps are synchronized in time.
- Concurrent computation, which does not have synchronization in time.

- Input, or initial, operands
- Processed operands
- Output, or resulting, operands

- Explicit computational information operators represent algorithms or computing automata, which function so that the last output is final and/or it is identified by the algorithm or computing automaton.
- Implicit computational information operators represent algorithms or computing automata, which function so that the final output is not always identified by the algorithm or computing automaton.
- Analytic computational information operators produce analytic outputs.

**Proposition**

**1.**

- A single-valued computational information operator produces at most one result for any valid input.
- A finite-valued computational information operator can produce a finite number of results for any valid input.
- An infinite-valued computational information operator can produce an infinite number of results for some valid input.

**Proposition**

**2.**

**Proposition**

**3.**

## 5. Computational Information Operators and Natural Computation

- (1)
- inspired by nature for the development of novel problem-solving techniques (e.g., cellular automata, neural computation, evolutionary computation, swarm intelligence, artificial immune systems, membrane computing, amorphous computing, cellular computing, molecular computing)
- (2)
- based on the use of computers to synthesize/simulate natural phenomena (e.g., artificial life, artificial chemistry); and
- (3)
- using natural materials (e.g., molecules) to compute (e.g., molecular computing or quantum computing).

## 6. Computational Information Operators as an Efficient Tool in Computer Science

**First possibility**: Operator representation of computing devices allows formulating and solving many problems about these of computing devices in a more general context of operating spaces of operators.

**Definability Problem**. Given a class

**K**of operators, is there an operator B in

**K**such that for any element x from the domain of operators from

**K**and any operator A from

**K,**B determines whether x belongs to the definability domain of A or does not belong.

**Second possibility**: Operator representation of computing devices allows constructing a variety of operator compositions (operations) and developing new schemas of computation as well as new network and computer architectures using operations with (a composition of) operators.

**Example**

**1.**

- (1)
- A(x) is defined and belongs to the domain of B;
- (2)
- B(A(x)) is defined.

**Example**

**2.**

**Third possibility**: Operator representation of computing devices allows efficient application of the axiomatic technique for investigation of computing devices, algorithms and computations.

**K**of algorithms (computing devices) and the corresponding class

**O**of operators, which model algorithms (computing devices) from

_{K}**K**.Here are some examples of axioms, which characterize the class

**O**.

_{K}**Totality axiom**: For any operator A from**O**, D(A) = DD(A)._{K}**Domain stability axiom**: For any operators A and B from**O**, D(A) = D(B)._{K}**Domain loop axiom**: For any operator A from**O**, D(A) = C(A)._{K}

**K**.

**Totality axiom**: For any computing device (algorithm) R from**K**, D(A) = DD(A).**Domain stability axiom**: For any computing devices (algorithms) R and Q from**K**, D(A) = D(B).**Domain loop axiom**: For any computing device (algorithm) R from**K**, D(A) = C(A).

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Burgin, Mark, and Gordana Dodig-Crnkovic. 2020. "Prolegomena to an Operator Theory of Computation" *Information* 11, no. 7: 349.
https://doi.org/10.3390/info11070349