@sschepis/resolang/assembly/examples/self-referential-complexity.ts

ResoLang - Core quantum resonance computation library for browser and Node.js

/**
 * Self-Referential Complexity Implementation for P ≠ NP Proof
 *
 * This module implements the core concept from Javier Muñoz de la Cuesta's proof:
 * Self-referential complexity S(M,x) = |{t | δ(qt, wt|pt) is self-referential}|
 *
 * A computational step is self-referential if it accesses:
 * - Prior configuration Cs (s < t)
 * - Machine description ⟨M⟩
 * - Computable functions of these
 */

import {
  ResonantFragment,
  EntangledNode,
  Prime
} from '../resolang';
import {
  tensor,
  collapse,
  entropy
} from '../operators';
import { toFixed } from '../utils';

// ============================================================================
// CORE DATA STRUCTURES
// ============================================================================

/**
 * Represents a single computational step in a Turing machine
 */
export class ComputationStep {
  public stepNumber: i32;
  public state: string;
  public tapePosition: i32;
  public tapeContent: string;
  public transition: string;
  public timestamp: i64;
  public godelNumber: u64;
  
  constructor(
    stepNumber: i32,
    state: string, 
    tapePosition: i32,
    tapeContent: string,
    transition: string
  ) {
    this.stepNumber = stepNumber;
    this.state = state;
    this.tapePosition = tapePosition;
    this.tapeContent = tapeContent;
    this.transition = transition;
    this.timestamp = Date.now();
    this.godelNumber = this.calculateGodelNumber();
  }
  
  /**
   * Calculate Gödel number for this configuration
   * Uses prime encoding: state × tape × position
   */
  private calculateGodelNumber(): u64 {
    const stateHash = this.hashString(this.state);
    const tapeHash = this.hashString(this.tapeContent);
    const positionPrime = this.getPrime(this.tapePosition + 1);
    
    return stateHash * tapeHash * positionPrime;
  }
  
  private hashString(str: string): u64 {
    let hash: u64 = 5381;
    for (let i = 0; i < str.length; i++) {
      hash = ((hash << 5) + hash) + str.charCodeAt(i);
    }
    return hash;
  }
  
  private getPrime(n: i32): u64 {
    const primes: u64[] = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47];
    return primes[n % primes.length];
  }
  
  /**
   * Check if this step references a prior configuration
   */
  public referencesConfiguration(otherStep: ComputationStep): bool {
    // Check for state reference
    if (this.transition.includes(otherStep.state)) {
      return true;
    }
    
    // Check for tape content reference
    if (this.tapeContent.includes(otherStep.tapeContent.substr(0, 3))) {
      return true;
    }
    
    // Check for position reference
    if (this.tapePosition === otherStep.tapePosition) {
      return true;
    }
    
    return false;
  }
}

/**
 * Tracks self-referential complexity during algorithm execution
 */
export class SelfReferentialTracker {
  private steps: Array<ComputationStep>;
  private selfreferentialSteps: Array<i32>;
  private holographicMemory: HolographicMemory;
  private machineDescription: ResonantFragment;
  
  constructor(machineDescription: string) {
    this.steps = new Array<ComputationStep>();
    this.selfreferentialSteps = new Array<i32>();
    this.holographicMemory = new HolographicMemory();
    this.machineDescription = ResonantFragment.encode(machineDescription);
  }
  
  /**
   * Add a new computation step and check for self-referential properties
   */
  public addStep(step: ComputationStep): void {
    this.steps.push(step);
    
    // Store step in holographic memory for quantum access
    const stepFragment = this.encodeStep(step);
    this.holographicMemory.store(`step_${step.stepNumber}`, stepFragment);
    
    // Check if this step is self-referential
    if (this.isSelfreferentialStep(step)) {
      this.selfreferentialSteps.push(step.stepNumber);
    }
  }
  
  /**
   * Core implementation of self-referential detection
   * A step is self-referential if it accesses prior configurations or machine description
   */
  private isSelfreferentialStep(currentStep: ComputationStep): bool {
    const currentFragment = this.encodeStep(currentStep);
    
    // Check access to machine description
    const machineAccess = tensor(currentFragment, this.machineDescription);
    if (entropy(machineAccess) > 0.1) {
      return true;
    }
    
    // Check access to prior configurations
    for (let i = 0; i < this.steps.length - 1; i++) {
      const priorStep = this.steps[i];
      
      // Direct reference check
      if (currentStep.referencesConfiguration(priorStep)) {
        return true;
      }
      
      // Quantum entanglement check
      const priorFragment = this.encodeStep(priorStep);
      const entanglement = tensor(currentFragment, priorFragment);
      
      if (entropy(entanglement) > 0.05) {
        return true;
      }
    }
    
    return false;
  }
  
  /**
   * Encode computation step as quantum fragment
   */
  private encodeStep(step: ComputationStep): ResonantFragment {
    const configString = `${step.state}_${step.tapeContent}_${step.tapePosition}`;
    return ResonantFragment.encode(configString);
  }
  
  /**
   * Calculate total self-referential complexity S(M,x)
   */
  public getSelfreferentialComplexity(): i32 {
    return this.selfreferentialSteps.length;
  }
  
  /**
   * Get ratio of self-referential steps to total steps
   */
  public getComplexityRatio(): f64 {
    if (this.steps.length === 0) return 0.0;
    return f64(this.selfreferentialSteps.length) / f64(this.steps.length);
  }
  
  /**
   * Verify the fundamental lemma: T(n) ≥ c · S(M,x)
   */
  public verifyFundamentalLemma(totalTime: i32): VerificationResult {
    const selfreferentialComplexity = this.getSelfreferentialComplexity();
    
    if (selfreferentialComplexity === 0) {
      return new VerificationResult(false, 0.0, "No self-referential steps detected");
    }
    
    const ratio = f64(totalTime) / f64(selfreferentialComplexity);
    const verified = totalTime >= selfreferentialComplexity;
    
    return new VerificationResult(
      verified,
      ratio,
      `T(n)=${totalTime}, S(M,x)=${selfreferentialComplexity}, ratio=${toFixed(ratio, 3)}`
    );
  }
  
  /**
   * Get detailed complexity analysis
   */
  public getComplexityAnalysis(): ComplexityAnalysis {
    const totalSteps = this.steps.length;
    const selfreferentialCount = this.selfreferentialSteps.length;
    const ratio = this.getComplexityRatio();
    
    // Calculate step-by-step growth
    const growthPattern = new Array<i32>();
    let cumulativeSelfref = 0;
    
    for (let i = 0; i < totalSteps; i++) {
      if (this.selfreferentialSteps.includes(i)) {
        cumulativeSelfref++;
      }
      growthPattern.push(cumulativeSelfref);
    }
    
    return new ComplexityAnalysis(
      totalSteps,
      selfreferentialCount,
      ratio,
      growthPattern,
      this.classifyComplexity(ratio)
    );
  }
  
  private classifyComplexity(ratio: f64): string {
    if (ratio < 0.1) return "LOW";
    if (ratio < 0.3) return "MODERATE"; 
    if (ratio < 0.6) return "HIGH";
    return "EXTREME";
  }
}

// ============================================================================
// RESULT CLASSES
// ============================================================================

export class VerificationResult {
  public verified: bool;
  public constant: f64;
  public details: string;
  
  constructor(verified: bool, constant: f64, details: string) {
    this.verified = verified;
    this.constant = constant;
    this.details = details;
  }
}

export class ComplexityAnalysis {
  public totalSteps: i32;
  public selfreferentialSteps: i32;
  public ratio: f64;
  public growthPattern: Array<i32>;
  public classification: string;
  
  constructor(
    totalSteps: i32,
    selfreferentialSteps: i32,
    ratio: f64,
    growthPattern: Array<i32>,
    classification: string
  ) {
    this.totalSteps = totalSteps;
    this.selfreferentialSteps = selfreferentialSteps;
    this.ratio = ratio;
    this.growthPattern = growthPattern;
    this.classification = classification;
  }
}

// ============================================================================
// HOLOGRAPHIC MEMORY IMPLEMENTATION
// ============================================================================

/**
 * Simplified holographic memory for storing quantum computation states
 */
class HolographicMemory {
  private storage: Map<string, ResonantFragment>;
  private entanglements: Map<string, Array<string>>;
  
  constructor() {
    this.storage = new Map<string, ResonantFragment>();
    this.entanglements = new Map<string, Array<string>>();
  }
  
  /**
   * Store a quantum fragment with given key
   */
  public store(key: string, fragment: ResonantFragment): void {
    this.storage.set(key, fragment);
    
    // Create entanglements with existing fragments
    const keys = this.storage.keys();
    const entangled = new Array<string>();
    
    for (let i = 0; i < keys.length; i++) {
      const otherKey = keys[i];
      if (otherKey !== key) {
        const otherFragment = this.storage.get(otherKey)!;
        const entanglement = tensor(fragment, otherFragment);
        
        if (entropy(entanglement) > 0.01) {
          entangled.push(otherKey);
        }
      }
    }
    
    this.entanglements.set(key, entangled);
  }
  
  /**
   * Retrieve fragment by key
   */
  public retrieve(key: string): ResonantFragment | null {
    return this.storage.get(key) || null;
  }
  
  /**
   * Get all fragments entangled with given key
   */
  public getEntangled(key: string): Array<ResonantFragment> {
    const entangledKeys = this.entanglements.get(key);
    const fragments = new Array<ResonantFragment>();
    
    if (entangledKeys) {
      for (let i = 0; i < entangledKeys.length; i++) {
        const fragment = this.storage.get(entangledKeys[i]);
        if (fragment) {
          fragments.push(fragment);
        }
      }
    }
    
    return fragments;
  }
  
  /**
   * Check if accessing key would be self-referential
   */
  public isSelfreferentialAccess(key: string, accessorKey: string): bool {
    const entangledKeys = this.entanglements.get(accessorKey);
    
    if (entangledKeys) {
      return entangledKeys.includes(key);
    }
    
    return false;
  }
}

// ============================================================================
// EXAMPLE DEMONSTRATIONS
// ============================================================================

/**
 * Example 1: Basic Self-Referential Complexity Tracking
 */
export function demonstrateBasicTracking(): void {
  console.log("=== Example 1: Basic Self-Referential Complexity ===");
  
  const tracker = new SelfReferentialTracker("SIMPLE_MACHINE");
  
  // Simulate a simple computation
  const step1 = new ComputationStep(0, "q0", 0, "101", "read_0");
  const step2 = new ComputationStep(1, "q1", 1, "101", "read_1");
  const step3 = new ComputationStep(2, "q0", 2, "101", "check_q0"); // Self-referential!
  const step4 = new ComputationStep(3, "q2", 0, "101", "check_tape_101"); // Self-referential!
  
  tracker.addStep(step1);
  tracker.addStep(step2);
  tracker.addStep(step3);
  tracker.addStep(step4);
  
  const analysis = tracker.getComplexityAnalysis();
  
  console.log(`Total steps: ${analysis.totalSteps}`);
  console.log(`Self-referential steps: ${analysis.selfreferentialSteps}`);
  console.log(`Complexity ratio: ${toFixed(analysis.ratio, 3)}`);
  console.log(`Classification: ${analysis.classification}`);
  
  const verification = tracker.verifyFundamentalLemma(100);
  console.log(`Fundamental lemma verified: ${verification.verified}`);
  console.log(`Details: ${verification.details}`);
}

/**
 * Example 2: Holographic Memory Self-Reference Detection
 */
export function demonstrateHolographicMemory(): void {
  console.log("\n=== Example 2: Holographic Memory Self-Reference ===");
  
  const memory = new HolographicMemory();
  
  // Store some quantum states
  const state1 = ResonantFragment.encode("initial_config");
  const state2 = ResonantFragment.encode("intermediate_config");
  const state3 = ResonantFragment.encode("initial_config_modified"); // Similar to state1
  
  memory.store("config_0", state1);
  memory.store("config_1", state2);
  memory.store("config_2", state3);
  
  // Check for self-referential access patterns
  const isSelfref_2_0 = memory.isSelfreferentialAccess("config_0", "config_2");
  const isSelfref_1_0 = memory.isSelfreferentialAccess("config_0", "config_1");
  
  console.log(`config_2 accessing config_0 is self-referential: ${isSelfref_2_0}`);
  console.log(`config_1 accessing config_0 is self-referential: ${isSelfref_1_0}`);
  
  // Show entanglement patterns
  const entangled = memory.getEntangled("config_0");
  console.log(`config_0 is entangled with ${entangled.length} other states`);
}

/**
 * Example 3: Complexity Growth Analysis
 */
export function demonstrateComplexityGrowth(): void {
  console.log("\n=== Example 3: Complexity Growth Pattern ===");
  
  const tracker = new SelfReferentialTracker("GROWTH_ANALYSIS_MACHINE");
  
  // Simulate increasing self-referential behavior
  for (let i = 0; i < 10; i++) {
    const state = i < 3 ? "q0" : "q1";
    const transition = i > 2 ? `check_step_${i-3}` : `normal_${i}`;
    
    const step = new ComputationStep(
      i,
      state,
      i % 5,
      `tape_${i}`,
      transition
    );
    
    tracker.addStep(step);
  }
  
  const analysis = tracker.getComplexityAnalysis();
  
  console.log("Growth pattern (cumulative self-referential steps):");
  for (let i = 0; i < analysis.growthPattern.length; i++) {
    console.log(`Step ${i}: ${analysis.growthPattern[i]} self-referential steps`);
  }
  
  console.log(`\nFinal complexity ratio: ${toFixed(analysis.ratio, 3)}`);
  console.log(`Classification: ${analysis.classification}`);
}

/**
 * Example 4: Machine Description Access Detection
 */
export function demonstrateMachineDescriptionAccess(): void {
  console.log("\n=== Example 4: Machine Description Access ===");
  
  const machineDesc = "TM{states:[q0,q1,q2], alphabet:[0,1], transitions:...}";
  const tracker = new SelfReferentialTracker(machineDesc);
  
  // Steps that access machine description
  const step1 = new ComputationStep(0, "q0", 0, "01", "normal");
  const step2 = new ComputationStep(1, "q1", 1, "01", "check_states"); // References machine desc
  const step3 = new ComputationStep(2, "q2", 2, "01", "verify_alphabet"); // References machine desc
  
  tracker.addStep(step1);
  tracker.addStep(step2);
  tracker.addStep(step3);
  
  const analysis = tracker.getComplexityAnalysis();
  
  console.log("Self-referential steps detected:");
  console.log(`Total: ${analysis.selfreferentialSteps} out of ${analysis.totalSteps}`);
  console.log(`Ratio: ${toFixed(analysis.ratio, 3)}`);
  
  // This should show high self-referential complexity due to machine description access
  console.log(`Classification: ${analysis.classification}`);
}

// ============================================================================
// MAIN EXECUTION
// ============================================================================

export function runSelfReferentialComplexityExamples(): void {
  console.log("🧠 Self-Referential Complexity Implementation");
  console.log("Implementing core concepts from P ≠ NP proof by Muñoz de la Cuesta\n");
  
  demonstrateBasicTracking();
  demonstrateHolographicMemory();
  demonstrateComplexityGrowth();
  demonstrateMachineDescriptionAccess();
  
  console.log("\n✅ Self-referential complexity tracking system operational!");
  console.log("Ready for SAT solver implementation in Phase 2.");
}

// Note: Auto-execution removed for AssemblyScript compatibility
// Call runSelfReferentialComplexityExamples() manually to execute examples