@sschepis/resolang
Version:
ResoLang - Core quantum resonance computation library
595 lines (504 loc) • 18.9 kB
text/typescript
/**
* Gödel Encoding System for P ≠ NP Proof Implementation
*
* This module implements Gödel numbering for encoding Turing machine configurations,
* states, and transitions as prime-based numbers. This is essential for the
* self-referential complexity analysis in the P ≠ NP proof.
*
* Based on the work of Javier Muñoz de la Cuesta's formal proof using
* self-referential complexity S(M,x).
*/
import {
ResonantFragment,
EntangledNode,
Prime
} from '../resolang';
import {
tensor,
entropy
} from '../operators';
import { toFixed } from '../utils';
// ============================================================================
// CORE GÖDEL ENCODING FUNCTIONS
// ============================================================================
/**
* Prime number generator for Gödel encoding
* Uses deterministic prime sequence for consistent encoding
*/
export class PrimeGenerator {
private static primes: Array<Prime> = [
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151,
157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233,
239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317
];
/**
* Get the nth prime number (0-indexed)
*/
public static getPrime(n: i32): Prime {
if (n >= 0 && n < PrimeGenerator.primes.length) {
return PrimeGenerator.primes[n];
}
// Generate more primes if needed (simplified approach)
return PrimeGenerator.primes[n % PrimeGenerator.primes.length] + (n / PrimeGenerator.primes.length) * 1000;
}
/**
* Get the index of a prime number
*/
public static getPrimeIndex(prime: Prime): i32 {
for (let i = 0; i < PrimeGenerator.primes.length; i++) {
if (PrimeGenerator.primes[i] === prime) {
return i;
}
}
return -1;
}
}
/**
* Turing machine configuration for Gödel encoding
*/
export class TuringConfiguration {
public state: string; // Current state (e.g., "q0", "q1", "qaccept")
public tapeContent: string; // Tape content as string
public headPosition: i32; // Read/write head position
public stepNumber: i32; // Step number in computation
public godelNumber: u64; // Gödel number encoding
constructor(state: string, tapeContent: string, headPosition: i32, stepNumber: i32) {
this.state = state;
this.tapeContent = tapeContent;
this.headPosition = headPosition;
this.stepNumber = stepNumber;
this.godelNumber = this.calculateGodelNumber();
}
/**
* Calculate Gödel number for this configuration
* Uses formula: ⟨q,w,i⟩ = 2^⟨q⟩ × 3^⟨w⟩ × 5^i
*/
private calculateGodelNumber(): u64 {
const stateGodel = this.encodeState(this.state);
const tapeGodel = this.encodeTapeContent(this.tapeContent);
const positionPrime = PrimeGenerator.getPrime(2); // Use 5 for position
// Simplified calculation to avoid overflow
// In practice, would use modular arithmetic
let result: u64 = 1;
// 2^⟨q⟩ component (limited to prevent overflow)
const statePower = Math.min(stateGodel, 20);
for (let i = 0; i < statePower; i++) {
result *= 2;
}
// 3^⟨w⟩ component (limited)
const tapePower = Math.min(tapeGodel % 10, 10);
for (let i = 0; i < tapePower; i++) {
result *= 3;
}
// 5^i component (limited)
const positionPower = Math.min(this.headPosition, 8);
for (let i = 0; i < positionPower; i++) {
result *= positionPrime;
}
return result;
}
/**
* Encode state string to number
*/
private encodeState(state: string): u32 {
let hash: u32 = 0;
for (let i = 0; i < state.length; i++) {
hash = (hash * 31 + state.charCodeAt(i)) % 1000;
}
return hash;
}
/**
* Encode tape content to number
*/
private encodeTapeContent(tape: string): u32 {
let hash: u32 = 0;
for (let i = 0; i < tape.length; i++) {
const charCode = tape.charCodeAt(i);
hash = (hash * 127 + charCode) % 10000;
}
return hash;
}
/**
* Get string representation for debugging
*/
public toString(): string {
return `Config(${this.state}, "${this.tapeContent}", ${this.headPosition}, step=${this.stepNumber}, godel=${this.godelNumber})`;
}
}
/**
* Turing machine description encoder
*/
export class TuringMachineEncoder {
public states: Array<string>;
public alphabet: Array<string>;
public transitions: Array<Transition>;
public startState: string;
public acceptStates: Array<string>;
public machineGodelNumber: u64;
constructor(
states: Array<string>,
alphabet: Array<string>,
transitions: Array<Transition>,
startState: string,
acceptStates: Array<string>
) {
this.states = states;
this.alphabet = alphabet;
this.transitions = transitions;
this.startState = startState;
this.acceptStates = acceptStates;
this.machineGodelNumber = this.calculateMachineGodel();
}
/**
* Calculate Gödel number for entire machine description
* ⟨M⟩ = 2^⟨Q⟩ × 3^⟨Σ⟩ × 5^⟨δ⟩ × 7^⟨q0⟩ × 11^⟨F⟩
*/
private calculateMachineGodel(): u64 {
let result: u64 = 1;
// Encode states
const statesHash = this.encodeStringArray(this.states);
for (let i = 0; i < Math.min(statesHash % 5, 5); i++) {
result *= 2;
}
// Encode alphabet
const alphabetHash = this.encodeStringArray(this.alphabet);
for (let i = 0; i < Math.min(alphabetHash % 3, 3); i++) {
result *= 3;
}
// Encode transitions (simplified)
const transitionsHash = this.transitions.length;
for (let i = 0; i < Math.min(transitionsHash % 4, 4); i++) {
result *= 5;
}
return result;
}
private encodeStringArray(arr: Array<string>): u32 {
let hash: u32 = 0;
for (let i = 0; i < arr.length; i++) {
for (let j = 0; j < arr[i].length; j++) {
hash = (hash * 31 + arr[i].charCodeAt(j)) % 1000;
}
}
return hash;
}
/**
* Get string representation of the Turing machine
*/
public toString(): string {
return `TuringMachine(states=${this.states.length}, alphabet=${this.alphabet.length}, transitions=${this.transitions.length}, start=${this.startState}, godel=${this.machineGodelNumber})`;
}
}
/**
* Transition function representation
*/
export class Transition {
public fromState: string;
public readSymbol: string;
public toState: string;
public writeSymbol: string;
public direction: string; // "L" or "R"
public godelNumber: u64;
constructor(
fromState: string,
readSymbol: string,
toState: string,
writeSymbol: string,
direction: string
) {
this.fromState = fromState;
this.readSymbol = readSymbol;
this.toState = toState;
this.writeSymbol = writeSymbol;
this.direction = direction;
this.godelNumber = this.calculateTransitionGodel();
}
private calculateTransitionGodel(): u64 {
let hash: u32 = 0;
// Combine all transition components
const combined = this.fromState + this.readSymbol + this.toState + this.writeSymbol + this.direction;
for (let i = 0; i < combined.length; i++) {
hash = (hash * 37 + combined.charCodeAt(i)) % 100000;
}
return hash;
}
public toString(): string {
return `δ(${this.fromState}, ${this.readSymbol}) = (${this.toState}, ${this.writeSymbol}, ${this.direction})`;
}
}
// ============================================================================
// QUANTUM GÖDEL ENCODING
// ============================================================================
/**
* Quantum-enhanced Gödel encoder using ResoLang's holographic memory
*/
export class QuantumGodelEncoder {
private configurationHistory: Array<TuringConfiguration>;
private machineDescription: TuringMachineEncoder;
private quantumStates: Map<string, ResonantFragment>;
constructor(machine: TuringMachineEncoder) {
this.configurationHistory = new Array<TuringConfiguration>();
this.machineDescription = machine;
this.quantumStates = new Map<string, ResonantFragment>();
}
/**
* Add configuration and create quantum encoding
*/
public addConfiguration(config: TuringConfiguration): void {
this.configurationHistory.push(config);
// Create quantum state representation
const quantumState = this.createQuantumState(config);
this.quantumStates.set(`config_${config.stepNumber}`, quantumState);
}
/**
* Create quantum state from Turing configuration
*/
private createQuantumState(config: TuringConfiguration): ResonantFragment {
const configString = `${config.state}_${config.tapeContent}_${config.headPosition}`;
return ResonantFragment.encode(configString);
}
/**
* Check if a configuration access is self-referential
* Returns true if current config accesses prior configurations or machine description
*/
public isSelfreferentialAccess(currentStep: i32, accessedStep: i32): bool {
if (currentStep <= accessedStep) {
return false; // Can't access future configurations
}
const currentConfig = this.configurationHistory[currentStep];
const accessedConfig = this.configurationHistory[accessedStep];
// Check for direct state reference
if (currentConfig.state === accessedConfig.state) {
return true;
}
// Check for tape content overlap
if (this.hasContentOverlap(currentConfig.tapeContent, accessedConfig.tapeContent)) {
return true;
}
// Check quantum entanglement
const currentQuantumState = this.quantumStates.get(`config_${currentStep}`);
const accessedQuantumState = this.quantumStates.get(`config_${accessedStep}`);
if (currentQuantumState && accessedQuantumState) {
const entanglement = tensor(currentQuantumState, accessedQuantumState);
const entanglementEntropy = entropy(entanglement);
// High entanglement indicates self-referential access
if (entanglementEntropy > 0.1) {
return true;
}
}
return false;
}
/**
* Check if accessing machine description (always self-referential)
*/
public isMachineDescriptionAccess(config: TuringConfiguration): bool {
// In a real implementation, would check if transition function
// or state set is being accessed
// Simplified: check if state name suggests introspection
if (config.state.includes("check") || config.state.includes("verify") || config.state.includes("introspect")) {
return true;
}
return false;
}
private hasContentOverlap(content1: string, content2: string): bool {
// Check for substring overlap
const minLength = Math.min(content1.length, content2.length);
const threshold = Math.max(1, minLength / 3); // 33% overlap threshold
for (let i = 0; i <= content1.length - threshold; i++) {
const substring = content1.substr(i, threshold);
if (content2.includes(substring)) {
return true;
}
}
return false;
}
/**
* Calculate total self-referential complexity for the computation
*/
public calculateSelfreferentialComplexity(): i32 {
let selfreferentialSteps = 0;
for (let i = 0; i < this.configurationHistory.length; i++) {
const currentConfig = this.configurationHistory[i];
let isSelfreferential = false;
// Check machine description access
if (this.isMachineDescriptionAccess(currentConfig)) {
isSelfreferential = true;
}
// Check prior configuration access
if (!isSelfreferential) {
for (let j = 0; j < i; j++) {
if (this.isSelfreferentialAccess(i, j)) {
isSelfreferential = true;
break;
}
}
}
if (isSelfreferential) {
selfreferentialSteps++;
}
}
return selfreferentialSteps;
}
/**
* Get complexity analysis report
*/
public getComplexityReport(): GodelComplexityReport {
const totalSteps = this.configurationHistory.length;
const selfreferentialSteps = this.calculateSelfreferentialComplexity();
const ratio = totalSteps > 0 ? f64(selfreferentialSteps) / f64(totalSteps) : 0.0;
return new GodelComplexityReport(
totalSteps,
selfreferentialSteps,
ratio,
this.machineDescription.machineGodelNumber,
this.configurationHistory
);
}
}
/**
* Complexity analysis report
*/
export class GodelComplexityReport {
public totalSteps: i32;
public selfreferentialSteps: i32;
public complexityRatio: f64;
public machineGodelNumber: u64;
public configurations: Array<TuringConfiguration>;
constructor(
totalSteps: i32,
selfreferentialSteps: i32,
complexityRatio: f64,
machineGodelNumber: u64,
configurations: Array<TuringConfiguration>
) {
this.totalSteps = totalSteps;
this.selfreferentialSteps = selfreferentialSteps;
this.complexityRatio = complexityRatio;
this.machineGodelNumber = machineGodelNumber;
this.configurations = configurations;
}
public toString(): string {
return `GodelReport(total=${this.totalSteps}, selfref=${this.selfreferentialSteps}, ratio=${toFixed(this.complexityRatio, 3)}, machine_godel=${this.machineGodelNumber})`;
}
}
// ============================================================================
// EXAMPLE DEMONSTRATIONS
// ============================================================================
/**
* Example 1: Basic Gödel Encoding
*/
export function demonstrateBasicGodelEncoding(): void {
console.log("=== Example 1: Basic Gödel Encoding ===");
// Create simple configurations
const config1 = new TuringConfiguration("q0", "101", 0, 0);
const config2 = new TuringConfiguration("q1", "111", 1, 1);
const config3 = new TuringConfiguration("q0", "101", 2, 2); // Same state as config1
console.log(`Config 1: ${config1.toString()}`);
console.log(`Config 2: ${config2.toString()}`);
console.log(`Config 3: ${config3.toString()}`);
console.log(`Gödel numbers: ${config1.godelNumber}, ${config2.godelNumber}, ${config3.godelNumber}`);
}
/**
* Example 2: Turing Machine Encoding
*/
export function demonstrateTuringMachineEncoding(): void {
console.log("\n=== Example 2: Turing Machine Encoding ===");
// Create a simple Turing machine
const states = ["q0", "q1", "qaccept", "qreject"];
const alphabet = ["0", "1", "B"]; // B = blank
const transitions = [
new Transition("q0", "0", "q1", "1", "R"),
new Transition("q0", "1", "q0", "0", "R"),
new Transition("q1", "0", "qaccept", "0", "R"),
new Transition("q1", "1", "qreject", "1", "R")
];
const machine = new TuringMachineEncoder(
states,
alphabet,
transitions,
"q0",
["qaccept"]
);
console.log(`Machine Gödel number: ${machine.machineGodelNumber}`);
console.log(`Number of states: ${machine.states.length}`);
console.log(`Number of transitions: ${machine.transitions.length}`);
for (let i = 0; i < transitions.length; i++) {
console.log(` ${transitions[i].toString()} [Gödel: ${transitions[i].godelNumber}]`);
}
}
/**
* Example 3: Self-Referential Complexity Detection
*/
export function demonstrateSelfreferentialDetection(): void {
console.log("\n=== Example 3: Self-Referential Complexity Detection ===");
// Create machine
const states = ["q0", "q1", "q_check", "qaccept"];
const alphabet = ["0", "1"];
const transitions = [
new Transition("q0", "0", "q1", "1", "R"),
new Transition("q1", "1", "q_check", "0", "L"), // Check state - potentially self-referential
new Transition("q_check", "0", "q0", "0", "R"), // References q0 - self-referential!
new Transition("q_check", "1", "qaccept", "1", "R")
];
const machine = new TuringMachineEncoder(states, alphabet, transitions, "q0", ["qaccept"]);
const encoder = new QuantumGodelEncoder(machine);
// Simulate computation steps
const steps = [
new TuringConfiguration("q0", "01", 0, 0),
new TuringConfiguration("q1", "11", 1, 1),
new TuringConfiguration("q_check", "10", 0, 2), // Self-referential state
new TuringConfiguration("q0", "10", 1, 3) // Back to q0 - self-referential!
];
for (let i = 0; i < steps.length; i++) {
encoder.addConfiguration(steps[i]);
}
const report = encoder.getComplexityReport();
console.log(`Complexity Report: ${report.toString()}`);
// Check specific self-referential accesses
console.log(`Step 2->0 self-referential: ${encoder.isSelfreferentialAccess(2, 0)}`);
console.log(`Step 3->0 self-referential: ${encoder.isSelfreferentialAccess(3, 0)}`);
console.log(`Machine description access in step 2: ${encoder.isMachineDescriptionAccess(steps[2])}`);
}
/**
* Example 4: Quantum Entanglement in Gödel Encoding
*/
export function demonstrateQuantumGodelEntanglement(): void {
console.log("\n=== Example 4: Quantum Gödel Entanglement ===");
const machine = new TuringMachineEncoder(
["q0", "q1"],
["0", "1"],
[new Transition("q0", "0", "q1", "1", "R")],
"q0",
["q1"]
);
const encoder = new QuantumGodelEncoder(machine);
// Create configurations with quantum entanglement
const config1 = new TuringConfiguration("q0", "quantum_state_1", 0, 0);
const config2 = new TuringConfiguration("q1", "quantum_state_2", 1, 1);
const config3 = new TuringConfiguration("q0", "quantum_state_1", 2, 2); // Similar to config1
encoder.addConfiguration(config1);
encoder.addConfiguration(config2);
encoder.addConfiguration(config3);
// Check quantum entanglement
const state1 = ResonantFragment.encode("quantum_state_1");
const state2 = ResonantFragment.encode("quantum_state_2");
const entanglement = tensor(state1, state2);
console.log(`Quantum state 1 entropy: ${toFixed(entropy(state1), 4)}`);
console.log(`Quantum state 2 entropy: ${toFixed(entropy(state2), 4)}`);
console.log(`Entanglement entropy: ${toFixed(entropy(entanglement), 4)}`);
const report = encoder.getComplexityReport();
console.log(`Final complexity: ${report.toString()}`);
}
/**
* Run all Gödel encoding examples
*/
export function runAllGodelEncodingExamples(): void {
console.log("🔢 Gödel Encoding System for P ≠ NP Proof");
console.log("Implementing prime-based configuration encoding\n");
demonstrateBasicGodelEncoding();
demonstrateTuringMachineEncoding();
demonstrateSelfreferentialDetection();
demonstrateQuantumGodelEntanglement();
console.log("\n✅ Gödel encoding system operational!");
console.log("Ready for Turing machine simulation in Phase 1.");
}