From 67d5b3a27ba3f1212da5d075f894f0e98445d7a7 Mon Sep 17 00:00:00 2001 From: SystymaticDev Date: Mon, 28 Apr 2025 18:00:24 -0400 Subject: [PATCH] Add files via upload --- certificates.py | 74 +++++++++++++++++++++++++ core.py | 59 ++++++++++++++++++++ examples.py | 23 ++++++++ experiments.py | 29 ++++++++++ lower_bounds.py | 86 +++++++++++++++++++++++++++++ proof_systems.py | 43 +++++++++++++++ reductions.py | 137 ++++++++++++++++++++++++++++++++++++++++++++++ test_suite.py | 24 ++++++++ tests.py | 24 ++++++++ topology_tools.py | 31 +++++++++++ 10 files changed, 530 insertions(+) create mode 100644 certificates.py create mode 100644 core.py create mode 100644 examples.py create mode 100644 experiments.py create mode 100644 lower_bounds.py create mode 100644 proof_systems.py create mode 100644 reductions.py create mode 100644 test_suite.py create mode 100644 tests.py create mode 100644 topology_tools.py diff --git a/certificates.py b/certificates.py new file mode 100644 index 0000000..76e910c --- /dev/null +++ b/certificates.py @@ -0,0 +1,74 @@ +# certificates.py + +# Import any necessary standard libraries or utility functions +import random +import hashlib + +class CertificateHandler: + def __init__(self): + # Initialization logic if needed + pass + + def generate_certificate(self, problem_instance): + """ + Generate a certificate for a given NP problem instance. + The certificate format depends on the problem’s structure. + """ + # Placeholder: Replace with actual certificate generation logic + certificate = { + "witness": self._generate_witness(problem_instance), + "hash": self._hash_instance(problem_instance) + } + return certificate + + def verify_certificate(self, problem_instance, certificate): + """ + Verify a given certificate for the given problem instance. + Returns True if the certificate is valid, False otherwise. + """ + # Extract witness and expected hash from the certificate + witness = certificate.get("witness") + expected_hash = certificate.get("hash") + + # Validate the witness and the hash + if witness and self._hash_instance(problem_instance) == expected_hash: + # Placeholder logic: Assume the witness is a solution if it passes certain checks + return self._is_valid_witness(problem_instance, witness) + return False + + def _generate_witness(self, problem_instance): + """ + Private helper to generate a "witness" for a given problem instance. + In a real implementation, this might involve producing a string that + satisfies certain conditions (e.g., a satisfying assignment for a SAT formula). + """ + # Simplified example: Return a random string as a stand-in for a valid witness + return "".join(random.choices("01", k=16)) + + def _hash_instance(self, problem_instance): + """ + Compute a hash of the problem instance to ensure certificates match + the correct instance. + """ + # Convert the problem instance to a string (or bytes) and hash it + instance_string = str(problem_instance) # Replace with proper serialization + return hashlib.sha256(instance_string.encode()).hexdigest() + + def _is_valid_witness(self, problem_instance, witness): + """ + Check whether the given witness satisfies the conditions of the problem. + This would involve problem-specific logic, such as verifying that + a Boolean formula is satisfied by the witness, or that a graph property holds. + """ + # Placeholder: Always return True as a simplification + # Replace with real checks in a full implementation + return True + +# If this module is run directly, you might test the CertificateHandler +if __name__ == "__main__": + handler = CertificateHandler() + sample_instance = {"some": "problem"} + cert = handler.generate_certificate(sample_instance) + is_valid = handler.verify_certificate(sample_instance, cert) + print("Certificate:", cert) + print("Is valid:", is_valid) diff --git a/core.py b/core.py new file mode 100644 index 0000000..d0a8784 --- /dev/null +++ b/core.py @@ -0,0 +1,59 @@ +# core.py + +# Imports: Assume we’re using various utility modules and libraries +from certificates import CertificateHandler +from reductions import Reducer +from lower_bounds import CircuitLowerBoundAnalyzer +from proof_systems import ProofVerifier +from topology_tools import TopologyAnalyzer + +class Core: + def __init__(self): + # Initialize various components and handlers + self.certificate_handler = CertificateHandler() + self.reducer = Reducer() + self.lower_bound_analyzer = CircuitLowerBoundAnalyzer() + self.proof_verifier = ProofVerifier() + self.topology_analyzer = TopologyAnalyzer() + + def verify_proof(self, problem_instance, proof): + """ + Verify the provided proof for a given problem instance. + """ + return self.proof_verifier.verify(problem_instance, proof) + + def simulate_algorithm(self, problem_instance): + """ + Simulate a hypothetical algorithm on a given problem instance + and return the result. + """ + # Placeholder: Replace with actual simulation logic + simulated_result = self.reducer.reduce(problem_instance) + is_valid = self.certificate_handler.verify_certificate(simulated_result) + return simulated_result, is_valid + + def analyze_hardness(self, problem_instance): + """ + Perform complexity analysis, checking circuit lower bounds and + other measures of computational difficulty. + """ + circuit_bound = self.lower_bound_analyzer.estimate_bound(problem_instance) + topological_analysis = self.topology_analyzer.analyze(problem_instance) + return { + "circuit_bound": circuit_bound, + "topological_analysis": topological_analysis + } + + def reduce_problem(self, source_problem, target_problem): + """ + Reduce one problem to another using the reduction utilities. + """ + return self.reducer.perform_reduction(source_problem, target_problem) + +# If this module is run directly, perhaps execute some default behavior +if __name__ == "__main__": + core = Core() + # Here you might add test cases, or simple calls to the core methods + # to demonstrate functionality, e.g.,: + # result = core.simulate_algorithm(some_problem_instance) + # print(result) diff --git a/examples.py b/examples.py new file mode 100644 index 0000000..f14aedc --- /dev/null +++ b/examples.py @@ -0,0 +1,23 @@ +# examples.py + +from proof_systems import PropositionalProofSystem +from topology_tools import TopologicalAnalyzer +from experiments import run_sat_solver, analyze_instance_complexity + +def example_workflow(): + # Demonstrate usage of the proof system + proof_system = PropositionalProofSystem("PROOF_OF(P)") + print("Proof valid:", proof_system.is_valid_proof("P")) + print("Proof size:", proof_system.proof_size()) + + # Demonstrate topology-based analysis + analyzer = TopologicalAnalyzer("ComplexBooleanFunction") + print("Topological complexity analysis:", analyzer.analyze_complexity()) + + # Demonstrate SAT solver experiments + print("SAT solver result:", run_sat_solver(10, 20)) + solve_times = analyze_instance_complexity(5) + print("Instance solve times:", solve_times) + +if __name__ == "__main__": + example_workflow() diff --git a/experiments.py b/experiments.py new file mode 100644 index 0000000..929b4c8 --- /dev/null +++ b/experiments.py @@ -0,0 +1,29 @@ +# experiments.py + +import random + +def run_sat_solver(num_vars, num_clauses): + """ + Simulates running a SAT solver on random 3-CNF formulas. + For demonstration, this is a placeholder that just outputs random results. + + Parameters: + num_vars (int): Number of variables in the formula. + num_clauses (int): Number of clauses in the formula. + + Returns: + str: "SATISFIABLE" or "UNSATISFIABLE". + """ + return random.choice(["SATISFIABLE", "UNSATISFIABLE"]) + +def analyze_instance_complexity(num_instances): + """ + Runs multiple SAT solver experiments and collects data on solve times. + + Parameters: + num_instances (int): Number of instances to analyze. + + Returns: + list: Simulated solve times for each instance. + """ + return [random.uniform(0.1, 5.0) for _ in range(num_instances)] diff --git a/lower_bounds.py b/lower_bounds.py new file mode 100644 index 0000000..dbd46c7 --- /dev/null +++ b/lower_bounds.py @@ -0,0 +1,86 @@ +# lower_bounds.py + +class CircuitLowerBound: + """ + A class to explore and establish circuit complexity lower bounds + for Boolean functions. It includes methods to construct functions + that are hard to compute by small circuits and to formally prove + lower bounds for certain restricted circuit classes. + """ + @staticmethod + def parity_lower_bound(n): + """ + Proves that computing the parity function on n variables requires + circuits of size exponential in the depth, if the depth is constant. + + Parameters: + n (int): The number of input variables. + + Returns: + str: A statement of the lower bound for parity in the given setting. + """ + # Known result: Parity requires depth-d circuits of size at least 2^{n^(1/(d-1))} + return f"The parity function on {n} variables requires circuit size at least 2^(n^(1/(d-1))) for constant depth-d." + + @staticmethod + def monotone_clique_lower_bound(n, k): + """ + Proves that the k-clique function on graphs with n vertices + requires monotone circuits of exponential size. + + Parameters: + n (int): The number of vertices in the graph. + k (int): The size of the clique. + + Returns: + str: A statement of the lower bound for monotone circuits computing k-clique. + """ + # Razborov (1985) result: Monotone circuits for k-clique require exponential size + return f"Monotone circuits for the {k}-clique function on graphs with {n} vertices require exponential size." + + @staticmethod + def circuit_size_lower_bound(function_description): + """ + Given a description of a Boolean function, attempts to apply known + lower bound techniques to estimate the size of circuits needed to + compute it. + + Parameters: + function_description (str): A textual or symbolic description of the Boolean function. + + Returns: + str: A lower bound statement, or a message indicating that no known lower bound applies. + """ + # This is just a placeholder for now. In a real implementation, + # you might attempt to match the function to known results or + # apply structural arguments. + return f"Lower bounds for {function_description} are currently not established." + + @staticmethod + def topological_obstruction_bounds(n): + """ + Uses topological methods to argue that certain Boolean functions require + super-polynomial size circuits. + + Parameters: + n (int): The number of input variables. + + Returns: + str: A statement about lower bounds derived from topological obstructions. + """ + # Hypothetical: Using topological invariants of the Boolean cube to argue lower bounds + return f"Topological arguments suggest that functions with certain invariants require circuits of size at least 2^{sqrt(n)}." + +# Example usage +if __name__ == "__main__": + # Example: Parity lower bound + print(CircuitLowerBound.parity_lower_bound(10)) + + # Example: Monotone clique lower bound + print(CircuitLowerBound.monotone_clique_lower_bound(50, 5)) + + # Example: Generic lower bound attempt + print(CircuitLowerBound.circuit_size_lower_bound("Majority function")) + + # Example: Topological lower bound argument + print(CircuitLowerBound.topological_obstruction_bounds(100)) diff --git a/proof_systems.py b/proof_systems.py new file mode 100644 index 0000000..c832158 --- /dev/null +++ b/proof_systems.py @@ -0,0 +1,43 @@ +# proof_systems.py + +class PropositionalProofSystem: + """ + A class representing a generic propositional proof system. + Includes methods to check proof validity, estimate proof size, and + explore the relationship between proof length and complexity. + """ + def __init__(self, proof_structure): + self.proof_structure = proof_structure + + def is_valid_proof(self, formula): + """ + Verifies whether the given proof proves the formula. + + Parameters: + formula (str): A propositional formula in CNF. + + Returns: + bool: True if the proof is valid, False otherwise. + """ + # Placeholder logic + return "PROOF_OF(" + formula + ")" in self.proof_structure + + def proof_size(self): + """ + Calculates the size of the proof in some normalized units. + + Returns: + int: The size of the proof. + """ + return len(self.proof_structure) + + def check_efficiency(self): + """ + Determines if the proof is considered efficient by comparing its size + against known lower bounds for propositional tautologies. + + Returns: + bool: True if the proof is efficient, False otherwise. + """ + # For demonstration purposes, assume 1000 is the threshold + return self.proof_size() < 1000 diff --git a/reductions.py b/reductions.py new file mode 100644 index 0000000..adf0918 --- /dev/null +++ b/reductions.py @@ -0,0 +1,137 @@ +# reductions.py + +class ReductionError(Exception): + """Custom exception for errors encountered during reductions.""" + pass + +class ProblemReducer: + """ + This class provides methods for reducing instances of one NP-complete + problem into another. Each reduction function implements a known, + standard transformation between problems. + """ + @staticmethod + def sat_to_3sat(cnf_formula): + """ + Reduce a general SAT instance (in CNF form) into a 3-SAT instance. + + Parameters: + cnf_formula (list of lists): Each sublist represents a clause, and each + element in the sublist is a literal. + + Returns: + list of lists: A new CNF formula in 3-SAT form (each clause has at most 3 literals). + """ + three_sat_formula = [] + + for clause in cnf_formula: + if len(clause) <= 3: + # Clause is already in 3-SAT form, just add it + three_sat_formula.append(clause) + else: + # Break down larger clauses into multiple 3-literal clauses + current_clause = clause[:3] + remaining_literals = clause[3:] + + while remaining_literals: + # Create a new auxiliary variable + aux_var = f"aux_{len(three_sat_formula)}" + current_clause[-1] = aux_var + three_sat_formula.append(current_clause) + + # Prepare the next clause segment + current_clause = [f"~{aux_var}"] + remaining_literals[:2] + remaining_literals = remaining_literals[2:] + + # Add the final clause with 3 literals + three_sat_formula.append(current_clause) + + return three_sat_formula + + @staticmethod + def vertex_cover_to_clique(graph, k): + """ + Reduce the Vertex Cover problem on a graph to the Clique problem on its complement. + + Parameters: + graph (dict): An adjacency list representation of the graph. + k (int): Size of the vertex cover to find. + + Returns: + tuple: (complement_graph, k) where complement_graph is the complement graph, + and k is the size of the clique to find. + """ + # Create the complement graph + nodes = list(graph.keys()) + complement_graph = {node: set() for node in nodes} + + for u in nodes: + for v in nodes: + if u != v and v not in graph[u]: + complement_graph[u].add(v) + + # The complement graph and the desired clique size + return complement_graph, len(nodes) - k + + @staticmethod + def clique_to_independent_set(graph, k): + """ + Reduce the Clique problem to the Independent Set problem. + + Parameters: + graph (dict): An adjacency list representation of the graph. + k (int): Size of the clique to find. + + Returns: + tuple: (complement_graph, k) where complement_graph is the complement graph, + and k is the size of the independent set to find. + """ + # The Independent Set problem on a graph is equivalent to + # the Clique problem on its complement + return ProblemReducer.vertex_cover_to_clique(graph, k) + + @staticmethod + def subset_sum_to_knapsack(weights, target): + """ + Convert a Subset Sum instance into a Knapsack problem instance. + + Parameters: + weights (list of int): The weights (and values) of items. + target (int): The target sum. + + Returns: + tuple: (weights, weights, target) where the first weights list represents + values, the second weights list represents weights, and target is the knapsack capacity. + """ + # For Subset Sum, treat weights as both the values and the weights + # of items in the Knapsack problem, and the target sum as the capacity. + return weights, weights, target + +# Example usage of reduction methods +if __name__ == "__main__": + # Example SAT instance + sat_formula = [[1, 2, 3], [4, 5, 6, 7], [8, 9]] + reduced_3sat = ProblemReducer.sat_to_3sat(sat_formula) + print("Reduced 3-SAT Formula:") + print(reduced_3sat) + + # Example graph (as adjacency list) for Vertex Cover -> Clique + graph = { + "A": {"B", "C"}, + "B": {"A", "C"}, + "C": {"A", "B", "D"}, + "D": {"C"} + } + clique_graph, clique_size = ProblemReducer.vertex_cover_to_clique(graph, 2) + print("\nComplement Graph (for Clique problem):") + print(clique_graph) + print("Clique Size to find:", clique_size) + + # Example Subset Sum instance + weights = [1, 3, 5, 7] + target = 8 + knapsack_instance = ProblemReducer.subset_sum_to_knapsack(weights, target) + print("\nKnapsack Problem instance:") + print("Values:", knapsack_instance[0]) + print("Weights:", knapsack_instance[1]) + print("Capacity:", knapsack_instance[2]) diff --git a/test_suite.py b/test_suite.py new file mode 100644 index 0000000..e4b9df4 --- /dev/null +++ b/test_suite.py @@ -0,0 +1,24 @@ +# test_suite.py + +import unittest + +class TestReductionFunctions(unittest.TestCase): + def test_reduction_validity(self): + self.assertTrue(True, "Reduction should be valid.") # Placeholder test + +class TestProofSystems(unittest.TestCase): + def test_proof_validity(self): + proof_system = PropositionalProofSystem("PROOF_OF(P)") + self.assertTrue(proof_system.is_valid_proof("P")) + + def test_proof_size(self): + proof_system = PropositionalProofSystem("PROOF_OF(P)") + self.assertGreater(proof_system.proof_size(), 0) + +class TestTopologyTools(unittest.TestCase): + def test_topological_analysis(self): + analyzer = TopologicalAnalyzer("BooleanFunction") + self.assertGreater(analyzer.calculate_topological_invariant(), 0) + +if __name__ == "__main__": + unittest.main() diff --git a/tests.py b/tests.py new file mode 100644 index 0000000..e4b9df4 --- /dev/null +++ b/tests.py @@ -0,0 +1,24 @@ +# test_suite.py + +import unittest + +class TestReductionFunctions(unittest.TestCase): + def test_reduction_validity(self): + self.assertTrue(True, "Reduction should be valid.") # Placeholder test + +class TestProofSystems(unittest.TestCase): + def test_proof_validity(self): + proof_system = PropositionalProofSystem("PROOF_OF(P)") + self.assertTrue(proof_system.is_valid_proof("P")) + + def test_proof_size(self): + proof_system = PropositionalProofSystem("PROOF_OF(P)") + self.assertGreater(proof_system.proof_size(), 0) + +class TestTopologyTools(unittest.TestCase): + def test_topological_analysis(self): + analyzer = TopologicalAnalyzer("BooleanFunction") + self.assertGreater(analyzer.calculate_topological_invariant(), 0) + +if __name__ == "__main__": + unittest.main() diff --git a/topology_tools.py b/topology_tools.py new file mode 100644 index 0000000..df3f3bc --- /dev/null +++ b/topology_tools.py @@ -0,0 +1,31 @@ +# topology_tools.py + +class TopologicalAnalyzer: + """ + A set of tools that leverage topological and algebraic constructs + to analyze computational complexity and circuit lower bounds. + """ + def __init__(self, function_representation): + self.function_representation = function_representation + + def calculate_topological_invariant(self): + """ + Computes a topological invariant that may relate to circuit complexity. + + Returns: + float: A numerical invariant. + """ + # Placeholder for an actual topological calculation + return len(self.function_representation) ** 0.5 + + def analyze_complexity(self): + """ + Uses topological methods to infer properties of the given function's + complexity. + + Returns: + str: A description of complexity properties. + """ + invariant = self.calculate_topological_invariant() + return f"Function exhibits a topological invariant of {invariant:.2f}, suggesting certain complexity thresholds." +