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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 •