# Turing Diagrams: Systems of Calculus Based on Ordinal Logics Open Access

## Haig, Asher Dreskin (2014)

Permanent URL: https://etd.library.emory.edu/concern/etds/gh93gz96w?locale=en
Published

## Abstract

Alan Turing effectively created the foundation for the modern computing age. The imitation game, more commonly known and frequently dismissed in a severely reduced form as the "Turing Test", is the heart of Turing's endeavors. The imitation game consolidates Turing's previous work with computability, universal machines, ordinal logic, cryptography, and intelligent machines. It also functions as the predecessor to Turing's subsequent work in applied biochemistry known as "morphogenesis". This text examines Turing's imitation game in detail, following its cumulative development through eight distinct iterations. The iterations are shown to be developmentally related, and the imitation game is shown to be a problem of determination that constitutes the core of Turing's work with artificial intelligence by extending his earlier work with the universal machine to a context that concerns the space of interaction between multiple machines.

Introduction

Turing Machines 1

The Universal Machine

λ: A Definition for "Algorithm" 12

A: Problem 15

B: Structure 28

C: Condition 45

The Imitation Game

Mλ0: A Mathematical Meta-Model 54

Diagram: Turing's Imitation Game 57

A: Dynamic 58

Diagram: Figure A 60

B: Distribution 66

Diagram: Figure B 66

C: Context 70

Diagram: Figure C 72

Chapter 1

Actuality 76

Diagram: An n-Bodies Problem 79

Diagram: Incommensurate Modes 83

System

Mλ1: Computability 85

Diagram: Iteration A 87

Diagram: Substitution A 89

A: Consistency 91

Diagram: Machine A 94

B: Dimension 98

Diagram:Figure B 100

C: Direction 112

Diagram: Figure C 114

Syntax

Mλ2: Computation 116

Diagram: Iteration B 118

Diagram: Substitution B 119

A: Reference 120

Diagram: Machine A 121

B: Action 126

Diagram: Machine B 129

C: Intelligibility 131

Diagram: Figure C 131

Statement

Mλ3: Computable Number 135

Diagram: Iteration C 138

Diagram: Substitution C 142

A: Composition 143

Diagram: Figure A 143

Diagram: Machine A 144

Diagram: Machine C 145

Diagram: Machine C--A 146

B: Contingency 148

Diagram: Figure B 152

Diagram: Figure C 153

C: Expression 159

Diagram: Machine C 159

Diagram: Machine A 163

Chapter 2

Potentiality 165

The Image of Thought

M: The Pivot 167

A: Indeterminacy 170 B: Model 174 C: Ordinal Topology 178

Selection

Mλ4: Ramified Time 185

Diagram: Iteration C` 188

Diagram: Substitution C` 190

A: Incompleteness 191

Diagram: Machine A--B 192

Diagram: Machine A--C 193

B: Intuition 198

Diagram: Machine B--A 200

Diagram: Machine B--C 205

C: Reversibility 210

Diagram: Machine C--A 211

Diagram: Machine C--B 213

Chapter 3

Potential 228

Intrinsic Consistency

Mλ5: Terms of Dialogue 240

Diagram: Iteration C--C` 243

A: Artifice 247

Diagram: Machine A--B--Å 249

Diagram: Machine A--C--Å 250

B: Edifice 254

Diagram: Machine B--A--B 254

Diagram: Machine B--C--B 255

C: Intellection 262

Diagram: Machine C--B--C 264

Diagram: Machine C--A--C 265

Extrinsic Consistency

Mλ6: Encapsulation 270

Diagram: Iteration M--M` 274

A: Domain 275

Diagram: Machine A--B--C--B 276

Diagram: Machine A--C--B--C 278

Diagram: A-Type Machines 288

Diagram: B-Type Machines 289

B: Institution 290

Diagram: B-Type Machines Linearized 290

Diagram: B-Type Machine Example 294

Diagram: Machine B--A--C--A 295

Diagram: Machine B--C--A--C 298

C: Encounter 301

Diagram: Machine C--A--B--A 303 Diagram: Machine C--B--A--B 305

Conclusion

Turing Circuits 313

Halting Conditions

Mλ7: Iteration 316

Diagram: Iteration M`--M`` 318

A: Zero 333

Diagram: Machine A Extrinsic Articulation 334

Diagram: Machine A Intrinsic Determination 336

B: Non-Zero 340

Diagram: Machine B Extrinsic Articulation 342

Diagram: Machine B Intrinsic Determination 344

C: Continuity 349

Diagram: Machine C Extrinsic Articulation 352

Diagram: Machine C Intrinsic Determination 359

Synchronization

Mλ8: Iterability 360

Diagram: M--M`--M``--M 362

A: Catalysis 373

Diagram: Machine A Intrinsic Consistency 380

Diagram: Machine A Extrinsic Consistency 383

B: Reaction Potential 385

Diagram: Machine B Intrinsic Consistency 397

Diagram: Machine B Extrinsic Consistency 399

C: Opinion 412

Diagram: Machine C Intrinsic Consistency 417

Diagram: Machine C Extrinsic Consistency 422

Works Cited 432

Other Relevant Works 436

Endnotes 441