# From data Processing to Knowledge Processing: Working with Operational Schemas by Autopoietic Machines

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## Abstract

**:**

## 1. Introduction

## 2. Schemas and Elements of Their Mathematical Theory

- Operational schemas
- Descriptive or categorical schemas
- Representation schemas

- Object/node, port, and connection/edge constants.
- Object/node, port, and connection/edge variables.
- Objects/nodes, ports, and connections with variables.

**Definition**

**2.1.**

_{B}, V

_{NB}, P

_{B}, V

_{PB}, C

_{B}, V

_{CB}, p

_{IB}, c

_{B}, p

_{EB})

_{B}is the set of all objects (e.g., automata) from the schema B; the multiset V

_{NB}consists of all object variables (e.g., automaton variables) from B; the set C

_{B}is the set of all connections/links between the objects and object variables in the schema B; the multiset V

_{CB}consists of all link variables, i.e., variables that take values in the links between the objects and object variables in the schema B; the set P

_{B}= P

_{IB}∪ P

_{EB}(with P

_{IB}∩P

_{EB}= ∅) is the set of all ports of the schema B, P

_{IB}is the set of all ports (called internal ports) of the automata from A

_{B}, and P

_{EB}is the set of external ports of B, which are used for interaction of B with different external systems and are divided into the input and output ports; the multiset V

_{PB}consists of all port variables from B and is divided into two disjunctive sub-multisets V

_{PBin}that consists of all variable inlets from B, and V

_{PBout}consists of all outlets from the schema B; p

_{IB}: P

_{IB}∪ V

_{PB}→ A

_{B}∪ V

_{NB}is a (variable) total function, called the internal port assignment function, that assigns ports to automata; c

_{B}: C

_{B}∪ V

_{CB}→ ((P

_{Ibout}∪ V

_{PBout}) × (P

_{Ibin}∪ V

_{PBin})) ∪ (P′

_{IBin}∪ V′

_{PBin}) ∪ (P′

_{IBout}∪ V′

_{PBout}) is a (variable) function, called the port-link adjacency function, that assigns connections to ports, where P′

_{IGin}, P″

_{Igout}, V′

_{PBin}, and V′

_{PBout}are disjunctive copies of P′

_{IGin}, P″

_{Igout}, V′

_{PBin}, and V′

_{PBout}, correspondingly; and p

_{EB}: P

_{EB}∪ V

_{PB}→ A

_{B}∪ P

_{IB}∪ C

_{B}∪ V

_{NB}∪ V

_{PB}∪ V

_{CB}is a (variable) function, called the external port assignment function, that assigns ports to different elements from the schema B.

**Example**

**2.1.**

_{NTr}, P

_{Tr}, V

_{PTr}, C

_{Tr}, V

_{CTr}, p

_{ITr}, c

_{Tr}, p

_{ETr})

_{NTr}= {ID, IP, OD}, where ID is a variable that takes input devices as values, IP is a variable that takes information processors as values, and OD is a variable that takes output devices as values.

_{Tr}consists of ports from the three device variables, each of which has one input port and one output port.

_{PTr}consists of port variables attached to the three device variables.

_{B}is the set of all connections/links between ID, IP, and OD.

_{CTr}is the set of all connection/link variables, which connect ID, IP, and OD.

_{Tr}is the adjacency function.

**Example**

**2.2.**

_{NPTr}, P

_{PTr}, V

_{PTr}, C

_{PTr}, V

_{CPTr}, p

_{IPTr}, c

_{PTr}, p

_{EPTr})

_{NPTr}= {ID1, IP1, OD1, ID2, IP2, OD2}, where ID1 and ID2 are variables that take input devices as values, IP1 and IP2 are variables that take information processors as values, and OD1 and OD2 are variables that takes output devices as values. These variables are similar to the variables from Example 2.1 and can take similar values.

_{PTr}consists of ports from the device variables, each of which has one input port and one output port.

_{PTr}consists of port variables attached to the three device variables.

_{PTr}is the set of all connections/links between ID1, IP1, OD1, ID2, IP2, and OD2.

_{CPTr}is the set of all connection/link variables, which connect ID1, IP1, OD1, ID2, IP2, and OD2.

_{PTr}is the adjacency function.

**Definition**

**2.2.**

_{R}, V

_{NR}, C

_{R}, V

_{CR}, c

_{R}).

**Example**

**2.3.**

_{NTm}, C

_{Tm}, c

_{Tm})

_{NTm}= {cd, h, m}, where cd is a variable that takes values in accepting finite automata, h is a variable that takes computing finite automata as values, and m is a variable that takes different types of tapes, e.g., one-dimensional, two-dimensional, or n-dimensional tapes, as values.

_{Tm}is the set of all connections/links between cd, h, and m. That is, cd is connected to h, while h is connected to one cell in m.

_{Tm}is the adjacency function between elements/variables and links.

**Example**

**2.4.**

_{NITm}, C

_{ITm}, c

_{ITm})

_{NTr}= {

**ir, or, cd, h, m**}, where

**cd**is a variable that takes values in accepting finite automata,

**h**is a variable that takes computing finite automata as values,

**ir**denotes the input register,

**or**denotes the output register, and

**m**is a variable that takes different types of tapes, e.g., one-dimensional, two-dimensional, or n-dimensional tapes, as values.

_{ITm}is the set of all connections/links between

**cd**,

**h**, and

**m**. That is, the element

**cd**is connected to the element

**h**, while

**h**is connected to one cell in m.

_{ITm}is the adjacency function between elements/variables and links.

**Definition**

**2.3.**

_{G(B)}equal to c

_{B}.

**Example**

**2.5.**

**Example**

**2.6.**

## 3. Operations with Schemas

- Schema processing
- Schema utilization
- Schema transmission

- Creation/elaboration of schemas
- Transformation of schemas
- Decomposition/termination of schemas

- Creation of a schema from the existing material
- Formation of a schema instance (instantiation)
- Application of the schema instance

- Composition/aggregation of several schemas
- Monotransformation when one schema is changed
- Coordinated transformation of several schemas—polytransformation

- Outside clutching (also called external composition) con(A, B) of schemas A and B is composed by correct attaching some external ports of the schemas A and B to one another.

**Example**

**3.1.**

**Theorem**

**1.**

- 2.
- Mixed clutching (also called incorporation) inc(A, B) of schemas A and B is composed by correct attaching the external ports of the schema B to the internal ports of the schema A.

- 3.
- Inside clutching icl(A, B) of schemas A and B is composed by correct attaching the internal ports of schemas A and B.

- 4.
- Substitution:
- Node substitution sub(A, a; B) of a node a in the schema A by the schema B is constructed by eliminating the node a from the schema A and connecting the external links of the node a to the appropriate external ports of the schema B.
- Link substitution sub(A, l; B) of a link l in the schema A by the schema B is constructed by eliminating l from the schema A and connecting the input ports of the schema B to the source of l and the output ports of B to the target of l.
- Component substitution sub(A, C; B) of a component C in the schema A by the schema B is constructed by connecting external links of the component C to appropriate external ports of the schema B.

**Example**

**3.2.**

**Lemma**

**1.**

**Theorem**

**2.**

- 5.
- Free composition A
_{*}B of schemas A and B is constructed by taking unions of their elements, i.e., A_{A}_{*B}= A_{A}∪ A_{B}, V_{N(A*B)}= V_{NA}∪ V_{NB}, P_{A}_{*B}= P_{A}∪ P_{B}, V_{P(A*B)}= V_{PA}∪ V_{PB}, C_{A}_{*B}= C_{A}∪ C_{B}, and V_{C(A*B)}= V_{CA}∪ V_{CB}.

**Example**

**3.3.**

**Theorem**

**3.**

## 4. Structural Machines as Schema Processors

**Definition**

**4.1.**

**A**= (A, r, R)

- Set A, which is called the substance of the structure
**A**and consists of elements of the structure**A**, which are called structure elements of the structure**A** - Set R, which is called the arrangement of the structure
**A**and consists of relations between elements from A in the structure**A**, which have the first order and are called structure relations of the structure**A** - The incidence relation r, which connects groups of elements from A with the names of relations from R

**Definition**

**4.2.**

- The unified control device C
_{M}regulates the state of the machine M. - The unified processor P
_{M}performs transformation of the processed structures and its actions (operations) depend on the state of the machine M and the state of the processed structures. - The functional space Sp
_{M}is the space where operated structures are situated.

_{M}, in turn, consists of three components:

- The input space In
_{M}, which contains the input structure. - The output space Out
_{M}, which contains the output structure. - The processing space PS
_{M}, in which the input structure(s) is transformed into the output structure(s).

_{M}and USp

_{M}:

- Sp
_{M}is the set of all structures that can be processed by the structural machine M and is called a categorical functional space. - USp
_{M}is a structure for which all structures that can be processed by the structural machine M are substructures and is called a universal functional space.

- It can be one central control device, which controls all processors of the structural machine.
- It can consist of cluster control devices, each of which controls a cluster of processors in the structural machine.
- It can consist of individual control devices, each of which controls a single processor in the structural machine.

- A localized processor is a single abstract device, which consists of one or several processor units or unit processors functioning as a unified whole.
- A distributed cluster processor consists of unit processors (processor units) from a cluster, which performs definite functions in a structural machine.
- A distributed total processor consists of a system of all unit processors (processor units) from a structural machine.

- Recursive structural machines, in which all processors work in the recursive mode.
- Inductive structural machines, in which all processors work in the inductive mode.
- Combined structural machines, in which some processors work in the recursive mode, while other processors work in the inductive mode.

**Theorem**

**4.**

**Theorem**

**5.**

## 5. Computing Structures for Operation with Schemas

- Reacting to large fluctuations in the demand for resources or the availability of resources in a widely distributed computing structure executing on different provider hardware and software increases complexity and cost of end-to-end operational visibility and control while increasing reaction latency.
- When the distributed components are communicating asynchronously, data consistency, availability, and partitioning cause problems for executing non-stop high-reliability applications at scale without service disruption.
- Insufficient scalability, especially in processing so-called big data, and the widely distributed nature of the access to both the sources and consumers of data necessitate pushing information processing closer to the edge.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Data and the program stored in the computer memory are processed by the CPU in the information processor.

**Figure 7.**The schema with a triadic automaton represents a knowledge structure containing various object, inter-object, and intra-object relationships and behaviors, which emerge when an event occurs, changing the objects or their relationships.

**Figure 9.**Schema managing infware, hardware, and software for the deployment, configuring, monitoring, and managing distributed application workloads on cloud resources.

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**MDPI and ACS Style**

Burgin, M.; Mikkilineni, R. From data Processing to Knowledge Processing: Working with Operational Schemas by Autopoietic Machines. *Big Data Cogn. Comput.* **2021**, *5*, 13.
https://doi.org/10.3390/bdcc5010013

**AMA Style**

Burgin M, Mikkilineni R. From data Processing to Knowledge Processing: Working with Operational Schemas by Autopoietic Machines. *Big Data and Cognitive Computing*. 2021; 5(1):13.
https://doi.org/10.3390/bdcc5010013

**Chicago/Turabian Style**

Burgin, Mark, and Rao Mikkilineni. 2021. "From data Processing to Knowledge Processing: Working with Operational Schemas by Autopoietic Machines" *Big Data and Cognitive Computing* 5, no. 1: 13.
https://doi.org/10.3390/bdcc5010013