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Differential Analytic Turing Automata • Overview

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Author: Jon Awbrey


OverviewPart 1Part 2Document 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.

PERS Figure 01.jpg (1)
PERS Figure 02.jpg (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


OverviewPart 1Part 2Document History