validators/autocorr_upper.py

#!/usr/bin/env python3
"""
Validator for problem 039: Improve Upper Bound on Autocorrelation Constant C

The autocorrelation constant C is defined as:
    C = inf_f max_t (f*f)(t) / (∫f)^2
where f is non-negative and supported on [-1/4, 1/4].

Current best bounds: 1.2748 ≤ C ≤ 1.50286.

The model provides a step function as a list of non-negative values on
N equal-width subintervals of [-1/4, 1/4]. The validator computes the
autoconvolution ratio and checks if it improves the best known upper bound.

Expected input format:
    {"values": [v_0, v_1, ..., v_{N-1}]}
    or [v_0, v_1, ..., v_{N-1}]
"""

import argparse
from typing import Any

import numpy as np
from scipy.signal import fftconvolve

from . import ValidationResult, load_solution, output_result, success, failure


BEST_KNOWN_UPPER = 1.50286
LOWER_BOUND = 1.28
MIN_INTERVALS = 10
MAX_INTERVALS = 1_000_000


def compute_autoconvolution_ratio(values: np.ndarray) -> float:
    """
    Compute max_t (f*f)(t) / (∫f)^2 for a step function.

    The function f is defined on N equal-width subintervals of [-1/4, 1/4].
    Each subinterval has width h = (1/2) / N.

    The autoconvolution (f*f)(t) = ∫ f(t-x)f(x) dx is computed via
    discrete convolution of the step function values scaled by the
    subinterval width h.
    """
    n = len(values)
    h = 0.5 / n  # width of each subinterval

    # Discrete convolution: (f*f) sampled at points spaced by h
    # fftconvolve gives the convolution of the coefficient sequences;
    # multiply by h to account for the integral approximation
    conv = fftconvolve(values, values) * h

    max_conv = np.max(conv)
    integral_f = np.sum(values) * h

    if integral_f <= 0:
        return float('inf')

    return max_conv / (integral_f ** 2)


def validate(solution: Any) -> ValidationResult:
    """
    Validate an autocorrelation upper bound construction.

    Args:
        solution: Dict with 'values' key or list of non-negative values

    Returns:
        ValidationResult with autoconvolution ratio
    """
    try:
        if isinstance(solution, dict) and 'values' in solution:
            values_data = solution['values']
        elif isinstance(solution, list):
            values_data = solution
        else:
            return failure("Invalid format: expected dict with 'values' or list")

        values = np.array(values_data, dtype=np.float64)
    except (ValueError, TypeError) as e:
        return failure(f"Failed to parse values: {e}")

    if values.ndim != 1:
        return failure(f"Values must be a 1D array, got {values.ndim}D")

    n = len(values)
    if n < MIN_INTERVALS:
        return failure(f"Need at least {MIN_INTERVALS} intervals, got {n}")

    if n > MAX_INTERVALS:
        return failure(f"Too many intervals ({n}), maximum is {MAX_INTERVALS}")

    # Check all entries are finite reals (reject NaN/inf)
    if not np.all(np.isfinite(values)):
        return failure("All values must be finite real numbers (no NaN or inf)")

    # Check all values are non-negative
    if np.any(values < 0):
        neg_count = int(np.sum(values < 0))
        return failure(
            f"Function values must be non-negative ({neg_count} negative values found)"
        )

    # Check function is not identically zero
    if np.all(values == 0):
        return failure("Function is identically zero")

    ratio = compute_autoconvolution_ratio(values)

    if not np.isfinite(ratio):
        return failure(
            "Computed ratio is not finite, indicating a numerical issue",
            autoconvolution_ratio=float(ratio)
        )

    if ratio < LOWER_BOUND - 1e-6:
        return failure(
            f"Ratio {ratio:.6f} is below the proven lower bound {LOWER_BOUND}, "
            f"indicating a likely numerical error",
            autoconvolution_ratio=ratio
        )

    return success(
        f"Step function with {n} intervals achieves autoconvolution ratio {ratio:.6f} "
        f"(best known: {BEST_KNOWN_UPPER})",
        num_intervals=n,
        autoconvolution_ratio=ratio,
        best_known_upper=BEST_KNOWN_UPPER,
        improves_bound=ratio < BEST_KNOWN_UPPER
    )


def main():
    parser = argparse.ArgumentParser(
        description='Validate autocorrelation upper bound construction'
    )
    parser.add_argument('solution', help='Solution as JSON string or path to JSON file')
    parser.add_argument('--verbose', '-v', action='store_true', help='Verbose output')
    args = parser.parse_args()

    solution = load_solution(args.solution)
    result = validate(solution)
    output_result(result)


if __name__ == '__main__':
    main()