This site is supported by donations to The OEIS Foundation.

# Differential Analytic Turing Automata • Overview

**Author: Jon Awbrey**

• Overview • Part 1 • Part 2 • Document History •

The task ahead is to chart a course from general ideas about *transformational equivalence classes of graphs* to a comprehensive concept of *differential analytic turing automata*. Getting within sight of that goal will take some time but I thought it made for a better measure of motivation to name the thread after its envisioned end instead of its more prosaic starting place.

The basic idea is as follows. One has a set of graphs and a set of transformation rules, and each rule has the effect of transforming graphs into graphs, In the cases we shall be studying this set of transformation rules partitions the set of graphs into *transformational equivalence classes* (TECs).

There are many interesting excursions to be had from this point but I will be focusing on logical applications, so the transformational equivalence classes of interest here will almost always have the character of *logical equivalence classes* (LECs).

An example figuring heavily in the sequel is given by rooted trees as the species of graphs and a pair of equational transformation rules deriving from the graphical calculi of C.S. Peirce, as revived and extended by George Spencer Brown.

Here are the fundamental transformation rules, commonly known as the *arithmetic axioms* or more precisely as the *arithmetic initials*.

(1) | |

(2) |

That should be enough to get started.

## Differential Logic

### Cactus Language

### Example 1

### Example 2

### Example 3

### Example 4

### Differential Analysis

### Symbolic Method

#### Computation Summary for Logical Disjunction

#### Computation Summary for Logical Equality

### Differential as Locally Linear Approximation

### Notions of Approximation

### Analytic Series

#### Computation Summary for Logical Disjunction

#### Computation Summary for Logical Equality

### Visualization

## Turing Machine Examples

### Finite Approximations

### Basic Propositions

### Initial Conditions

#### Initial Conditions for Tape Input "0"

#### Initial Conditions for Tape Input "1"

### Propositional Program

#### Mediate Conditions

#### Terminal Conditions

#### State Partition

#### Register Partition

#### Symbol Partition

#### Interaction Conditions

#### Transition Relations

### Interpretation of the Propositional Program

#### Mediate Conditions

#### Terminal Conditions

#### State Partition

#### Register Partition

#### Symbol Partition

#### Interaction Conditions

#### Transition Relations

### Computation

### Output

#### Output Conditions for Tape Input "0"

#### Output Conditions for Tape Input "1"

## Document History

### Jan 2001 • Arisbe List

### Jan 2001 • Ontology List

### Feb–Mar 2004 • Ontology List

### Feb–Jun 2004 • NKS Forum

### Feb–Jun 2004 • Inquiry List

• Overview • Part 1 • Part 2 • Document History •