meta-log-db/dist/extensions/homology/validator.d.ts

Native database package for Meta-Log (ProLog, DataLog, R5RS)

/**
 * Homology Validator
 *
 * Validates chain complexes using algebraic topology principles.
 * Ensures ∂² = 0 property holds for all boundary operators.
 */
import { ChainComplex, HomologyResult } from './types.js';
/**
 * Homology Validator
 *
 * Validates chain complexes and computes homology groups
 */
export declare class HomologyValidator {
    private complex;
    constructor(complex: ChainComplex);
    /**
     * Validate that ∂_{n-1} ∘ ∂_n = 0 for all dimensions
     *
     * @param n - Dimension to validate (1, 2, 3, or 4)
     * @returns true if ∂² = 0 holds, false otherwise
     */
    validateComposition(n: 1 | 2 | 3 | 4): boolean;
    /**
     * Compute Betti number for dimension n
     *
     * Betti number β_n = dim(ker(∂_n)) - dim(im(∂_{n+1}))
     *
     * @param n - Dimension (0-4)
     * @returns Betti number β_n
     */
    computeBetti(n: number): number;
    /**
     * Compute Euler characteristic
     *
     * χ = Σ(-1)ⁿ|Cₙ| = |C₀| - |C₁| + |C₂| - |C₃| + |C₄|
     *
     * @returns Euler characteristic
     */
    computeEulerCharacteristic(): number;
    /**
     * Full validation of chain complex
     *
     * @returns HomologyResult with validation status, Betti numbers, and violations
     */
    validate(): HomologyResult;
    /**
     * Get cells for dimension n
     */
    private getCells;
    /**
     * Check if boundary forms a closed cycle
     *
     * For edges: each vertex should appear exactly twice (or zero times for isolated)
     * For faces: each edge should appear exactly twice with opposite orientations
     *
     * @param boundary - Array of boundary cell IDs
     * @returns true if boundary forms a closed cycle
     */
    private isCycle;
    /**
     * Compute kernel of boundary operator ∂_n
     *
     * Kernel = {cells where ∂_n(cell) = 0}
     *
     * @param n - Dimension
     * @returns Array of cell IDs in the kernel
     */
    private computeKernel;
    /**
     * Compute image of boundary operator ∂_n
     *
     * Image = {cells that are boundaries of (n+1)-cells}
     *
     * @param n - Dimension
     * @returns Array of cell IDs in the image
     */
    private computeImage;
}
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