File size: 3,172 Bytes
848d4b7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | #!/usr/bin/env python3
"""
Validator for problem 058: Universal Formula Improving Keich's Construction
Keich's construction gives a Kakeya set with area tending to 0 as n → ∞.
This problem asks for a universal formula that improves upon Keich's approach.
Expected input format:
{
"formula": "description of the construction",
"test_cases": [
{"n": 64, "area": computed_area},
{"n": 128, "area": computed_area},
...
]
}
"""
import argparse
from typing import Any
from . import ValidationResult, load_solution, output_result, success, failure
def keich_bound(n: int) -> float:
"""Compute Keich's construction area bound for n directions."""
# Keich's construction: area ≈ π/8 * (1/log n)
# This is a simplified bound
import math
if n <= 1:
return float('inf')
return math.pi / 8 / math.log(n)
def validate(solution: Any) -> ValidationResult:
"""
Validate a universal Kakeya construction formula.
Args:
solution: Dict with formula description and test cases
Returns:
ValidationResult with comparison to Keich bound
"""
try:
if not isinstance(solution, dict):
return failure("Invalid format: expected dict")
formula = solution.get('formula', 'not provided')
test_cases = solution.get('test_cases', [])
if not test_cases:
return failure("Need at least one test case to validate")
except (ValueError, TypeError) as e:
return failure(f"Failed to parse solution: {e}")
# Verify each test case improves on Keich
improvements = []
for tc in test_cases:
n = int(tc['n'])
area = float(tc['area'])
keich_area = keich_bound(n)
improvement = (keich_area - area) / keich_area * 100
improvements.append({
'n': n,
'area': area,
'keich_bound': keich_area,
'improvement_percent': improvement
})
# Check if all cases improve on Keich
all_improve = all(imp['improvement_percent'] > 0 for imp in improvements)
if all_improve:
avg_improvement = sum(imp['improvement_percent'] for imp in improvements) / len(improvements)
return success(
f"Universal formula improves on Keich by avg {avg_improvement:.2f}%",
formula=formula,
test_cases=improvements,
average_improvement=avg_improvement
)
else:
failing = [imp for imp in improvements if imp['improvement_percent'] <= 0]
return failure(
f"Does not improve on Keich for n={failing[0]['n']}",
test_cases=improvements
)
def main():
parser = argparse.ArgumentParser(description='Validate universal Kakeya 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()
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