From 168f6be0e9b1d79585974115681216ed0c95f353 Mon Sep 17 00:00:00 2001 From: zazabap Date: Wed, 1 Apr 2026 06:38:29 +0000 Subject: [PATCH 1/3] =?UTF-8?q?docs:=20/verify-reduction=20#841=20?= =?UTF-8?q?=E2=80=94=20NAESatisfiability=20=E2=86=92=20SetSplitting=20VERI?= =?UTF-8?q?FIED?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Typst: Construction (3 steps) + Correctness (⟹ + ⟸) + Extraction + Overhead + YES/NO examples Constructor: 32,513 checks, 0 failures (exhaustive n ≤ 5, 7 sections) Adversary: 46,774 checks, 0 failures (independent impl + hypothesis) Cross-comparison: 615 instances, all agree Co-Authored-By: Claude Opus 4.6 (1M context) --- .../adversary_naesat_setsplitting.py | 575 ++++++++++++++++++ .../cross_compare_naesat_setsplitting.py | 116 ++++ .../verify-reductions/naesat_setsplitting.pdf | Bin 0 -> 142007 bytes .../verify-reductions/naesat_setsplitting.typ | 173 ++++++ .../verify_naesat_setsplitting.py | 461 ++++++++++++++ 5 files changed, 1325 insertions(+) create mode 100644 docs/paper/verify-reductions/adversary_naesat_setsplitting.py create mode 100644 docs/paper/verify-reductions/cross_compare_naesat_setsplitting.py create mode 100644 docs/paper/verify-reductions/naesat_setsplitting.pdf create mode 100644 docs/paper/verify-reductions/naesat_setsplitting.typ create mode 100644 docs/paper/verify-reductions/verify_naesat_setsplitting.py diff --git a/docs/paper/verify-reductions/adversary_naesat_setsplitting.py b/docs/paper/verify-reductions/adversary_naesat_setsplitting.py new file mode 100644 index 00000000..802a9b74 --- /dev/null +++ b/docs/paper/verify-reductions/adversary_naesat_setsplitting.py @@ -0,0 +1,575 @@ +#!/usr/bin/env python3 +""" +Adversary verification script for NAESatisfiability -> SetSplitting reduction. + +Independent implementation based solely on the Typst proof specification. +Does NOT import or reference verify_naesat_setsplitting.py. +""" + +import itertools +import random +import sys + +# --------------------------------------------------------------------------- +# 1. Data structures +# --------------------------------------------------------------------------- + +# NAE-SAT instance: n variables (1..n), clauses are lists of literals. +# A literal is (var_index, is_positive) where var_index in 1..n. +# E.g., v_1 -> (1, True), ~v_2 -> (2, False) + +# Set Splitting instance: universe_size (int), subsets (list of frozensets of ints) + + +def reduce(n_vars, clauses): + """ + Reduce NAE-SAT to Set Splitting per the Typst proof. + + Universe S = {0, 1, ..., 2n-1} where element 2(i-1) = v_i, 2(i-1)+1 = ~v_i. + Subsets: + - Complementarity: P_i = {2(i-1), 2(i-1)+1} for i=1..n + - Clause: Q_j = {elem for each literal in clause_j} + """ + universe_size = 2 * n_vars + subsets = [] + + # Complementarity subsets + for i in range(1, n_vars + 1): + pos_elem = 2 * (i - 1) + neg_elem = 2 * (i - 1) + 1 + subsets.append(frozenset([pos_elem, neg_elem])) + + # Clause subsets + for clause in clauses: + elems = set() + for var_idx, is_positive in clause: + if is_positive: + elems.add(2 * (var_idx - 1)) + else: + elems.add(2 * (var_idx - 1) + 1) + subsets.append(frozenset(elems)) + + return universe_size, subsets + + +def literal_to_elem(var_idx, is_positive): + """Convert a literal to its universe element index.""" + if is_positive: + return 2 * (var_idx - 1) + else: + return 2 * (var_idx - 1) + 1 + + +def assignment_to_partition(n_vars, assignment): + """ + Convert a boolean assignment (list of n bools, index 0 = var 1) to a partition. + Returns S1 as a frozenset. S2 = universe - S1. + + S1 = {v_i : alpha(v_i)=True} union {~v_i : alpha(v_i)=False} + """ + s1 = set() + for i in range(n_vars): + if assignment[i]: # var (i+1) is True + s1.add(2 * i) # v_{i+1} in S1 + else: # var (i+1) is False + s1.add(2 * i + 1) # ~v_{i+1} in S1 + return frozenset(s1) + + +def partition_to_assignment(n_vars, s1): + """ + Extract NAE-SAT assignment from a set splitting partition. + alpha(v_i) = True iff element 2(i-1) in S1. + """ + assignment = [] + for i in range(n_vars): + assignment.append(2 * i in s1) + return assignment + + +def is_nae_satisfying(n_vars, clauses, assignment): + """Check if assignment NAE-satisfies all clauses.""" + for clause in clauses: + values = set() + for var_idx, is_positive in clause: + val = assignment[var_idx - 1] + if not is_positive: + val = not val + values.add(val) + if len(values) < 2: # all equal + return False + return True + + +def is_valid_splitting(universe_size, subsets, s1): + """Check if partition (S1, S2) splits every subset.""" + s2 = set(range(universe_size)) - set(s1) + for subset in subsets: + if not (subset & s1) or not (subset & s2): + return False + return True + + +def is_consistent_partition(n_vars, s1): + """ + Check that the partition is consistent with complementarity constraints: + for each variable i, exactly one of {2(i-1), 2(i-1)+1} is in S1. + """ + for i in range(n_vars): + pos = 2 * i + neg = 2 * i + 1 + in_s1_pos = pos in s1 + in_s1_neg = neg in s1 + if in_s1_pos == in_s1_neg: # both in or both out + return False + return True + + +# --------------------------------------------------------------------------- +# 2. Test functions +# --------------------------------------------------------------------------- + +passed = 0 +failed = 0 +bugs = [] + + +def check(condition, msg): + global passed, failed, bugs + if condition: + passed += 1 + else: + failed += 1 + if msg not in bugs: + bugs.append(msg) + + +def test_yes_example(): + """Reproduce the YES example from the Typst proof.""" + global passed, failed + n = 4 + # c1=(v1,v2,v3), c2=(~v1,v3,v4), c3=(v2,~v3,~v4), c4=(v1,~v2,v4) + clauses = [ + [(1, True), (2, True), (3, True)], + [(1, False), (3, True), (4, True)], + [(2, True), (3, False), (4, False)], + [(1, True), (2, False), (4, True)], + ] + universe_size, subsets = reduce(n, clauses) + + # Check overhead + check(universe_size == 8, "YES: universe_size should be 8") + check(len(subsets) == 8, "YES: num_subsets should be 8") + + # Check specific subsets (0-indexed from proof) + expected_subsets = [ + frozenset([0, 1]), frozenset([2, 3]), frozenset([4, 5]), frozenset([6, 7]), + frozenset([0, 2, 4]), frozenset([1, 4, 6]), frozenset([2, 5, 7]), frozenset([0, 3, 6]), + ] + for i, (got, exp) in enumerate(zip(subsets, expected_subsets)): + check(got == exp, f"YES: subset {i} mismatch: got {got}, expected {exp}") + + # Assignment: v1=T, v2=F, v3=T, v4=F + assignment = [True, False, True, False] + check(is_nae_satisfying(n, clauses, assignment), "YES: assignment should NAE-satisfy") + + s1 = assignment_to_partition(n, assignment) + expected_s1 = frozenset([0, 3, 4, 7]) + check(s1 == expected_s1, f"YES: S1 should be {{0,3,4,7}}, got {s1}") + + check(is_valid_splitting(universe_size, subsets, s1), "YES: partition should be valid splitting") + + # Extract back + recovered = partition_to_assignment(n, s1) + check(recovered == assignment, "YES: extracted assignment should match original") + + +def test_no_example(): + """Reproduce the NO example from the Typst proof.""" + n = 3 + clauses = [ + [(1, True), (2, True), (3, True)], + [(1, True), (2, True), (3, False)], + [(1, True), (2, False), (3, True)], + [(1, True), (2, False), (3, False)], + ] + universe_size, subsets = reduce(n, clauses) + + check(universe_size == 6, "NO: universe_size should be 6") + check(len(subsets) == 7, "NO: num_subsets should be 7") + + expected_subsets = [ + frozenset([0, 1]), frozenset([2, 3]), frozenset([4, 5]), + frozenset([0, 2, 4]), frozenset([0, 2, 5]), + frozenset([0, 3, 4]), frozenset([0, 3, 5]), + ] + for i, (got, exp) in enumerate(zip(subsets, expected_subsets)): + check(got == exp, f"NO: subset {i} mismatch: got {got}, expected {exp}") + + # Verify no assignment works + for bits in itertools.product([False, True], repeat=n): + assignment = list(bits) + check(not is_nae_satisfying(n, clauses, assignment), + f"NO: assignment {assignment} should NOT NAE-satisfy") + + # Verify no consistent partition works + for bits in itertools.product([0, 1], repeat=2 * n): + s1 = frozenset(i for i, b in enumerate(bits) if b == 0) + if is_consistent_partition(n, s1): + check(not is_valid_splitting(universe_size, subsets, s1), + f"NO: consistent partition {s1} should not be valid splitting") + + +def test_exhaustive_small(): + """Exhaustive testing for all NAE-SAT instances with n <= 5 variables.""" + # For small n, generate many clause patterns and verify equivalence + for n in range(1, 6): + vars_list = list(range(1, n + 1)) + # Generate all possible literals + all_literals = [] + for v in vars_list: + all_literals.append((v, True)) + all_literals.append((v, False)) + + # For n<=3, test all possible single-clause instances with 2-3 literals + if n <= 3: + clause_sizes = [2, 3] if n >= 2 else [2] + for size in clause_sizes: + for combo in itertools.combinations(all_literals, size): + # Skip clauses with both v_i and ~v_i (tautological for NAE purposes) + clause = list(combo) + clauses = [clause] + verify_reduction_equivalence(n, clauses) + + # For all n, test random instances + rng = random.Random(42 + n) + num_random = 200 if n <= 3 else 100 + for _ in range(num_random): + m = rng.randint(1, min(n * 2, 8)) + clauses = [] + for _ in range(m): + clause_size = rng.randint(2, min(len(all_literals), 4)) + clause = rng.sample(all_literals, clause_size) + clauses.append(clause) + verify_reduction_equivalence(n, clauses) + + +def verify_reduction_equivalence(n_vars, clauses): + """ + Core verification: NAE-SAT instance is feasible iff the reduced + Set Splitting instance is feasible. Also checks solution extraction. + """ + universe_size, subsets = reduce(n_vars, clauses) + + # Check overhead formula + check(universe_size == 2 * n_vars, + f"Overhead: universe_size should be 2*{n_vars}={2*n_vars}, got {universe_size}") + check(len(subsets) == n_vars + len(clauses), + f"Overhead: num_subsets should be {n_vars}+{len(clauses)}, got {len(subsets)}") + + # Enumerate all assignments for NAE-SAT + nae_feasible = False + nae_witnesses = [] + for bits in itertools.product([False, True], repeat=n_vars): + assignment = list(bits) + if is_nae_satisfying(n_vars, clauses, assignment): + nae_feasible = True + nae_witnesses.append(assignment) + + # Enumerate all valid splittings (consistent partitions only) + ss_feasible = False + ss_witnesses = [] + for bits in itertools.product([0, 1], repeat=n_vars): + # Build consistent partition: for each var, bit=0 means v_i in S1 + s1 = set() + for i in range(n_vars): + if bits[i] == 0: + s1.add(2 * i) # v_{i+1} in S1 + else: + s1.add(2 * i + 1) # ~v_{i+1} in S1 + # Complete s1: the complement elements go to S2 + for i in range(n_vars): + if 2 * i not in s1: + s1.add(2 * i + 1) + if 2 * i + 1 not in s1: + pass # already not in s1 + # Actually, rebuild properly + s1 = set() + for i in range(n_vars): + if bits[i] == 0: + s1.add(2 * i) + else: + s1.add(2 * i + 1) + s1 = frozenset(s1) + + if is_valid_splitting(universe_size, subsets, s1): + ss_feasible = True + ss_witnesses.append(s1) + + # Equivalence + check(nae_feasible == ss_feasible, + f"Equivalence failed for n={n_vars}, clauses={clauses}: NAE={nae_feasible}, SS={ss_feasible}") + + # Forward direction: every NAE witness maps to a valid splitting + for assignment in nae_witnesses: + s1 = assignment_to_partition(n_vars, assignment) + check(is_valid_splitting(universe_size, subsets, s1), + f"Forward: NAE assignment {assignment} -> partition not valid") + + # Backward direction: every valid splitting maps to NAE assignment + for s1 in ss_witnesses: + assignment = partition_to_assignment(n_vars, s1) + check(is_nae_satisfying(n_vars, clauses, assignment), + f"Backward: partition {s1} -> assignment {assignment} not NAE-satisfying") + + # Solution extraction roundtrip + for assignment in nae_witnesses: + s1 = assignment_to_partition(n_vars, assignment) + recovered = partition_to_assignment(n_vars, s1) + check(recovered == assignment, + f"Roundtrip: assignment {assignment} != recovered {recovered}") + + +def test_overhead_formula(): + """Verify overhead formula on many random instances.""" + rng = random.Random(12345) + for _ in range(500): + n = rng.randint(1, 10) + m = rng.randint(1, 15) + all_literals = [(v, p) for v in range(1, n + 1) for p in [True, False]] + clauses = [] + for _ in range(m): + size = rng.randint(2, min(len(all_literals), 5)) + clause = rng.sample(all_literals, size) + clauses.append(clause) + + universe_size, subsets = reduce(n, clauses) + check(universe_size == 2 * n, + f"Overhead: universe_size={universe_size} != 2*{n}") + check(len(subsets) == n + m, + f"Overhead: num_subsets={len(subsets)} != {n}+{m}") + + +def test_complementarity_always_split(): + """Every consistent partition always splits complementarity subsets.""" + rng = random.Random(99999) + for _ in range(500): + n = rng.randint(1, 8) + # Any assignment -> partition must split all complementarity subsets + assignment = [rng.choice([True, False]) for _ in range(n)] + s1 = assignment_to_partition(n, assignment) + for i in range(n): + p_i = frozenset([2 * i, 2 * i + 1]) + check(bool(p_i & s1) and bool(p_i - s1), + f"Complementarity: P_{i+1} not split by assignment {assignment}") + + +def test_nae_symmetry(): + """ + NAE-SAT has the property that if alpha is NAE-satisfying, so is ~alpha + (flipping all variables). Verify this is preserved through reduction. + """ + rng = random.Random(77777) + for _ in range(500): + n = rng.randint(2, 6) + m = rng.randint(1, 8) + all_literals = [(v, p) for v in range(1, n + 1) for p in [True, False]] + clauses = [] + for _ in range(m): + size = rng.randint(2, min(len(all_literals), 4)) + clause = rng.sample(all_literals, size) + clauses.append(clause) + + universe_size, subsets = reduce(n, clauses) + + assignment = [rng.choice([True, False]) for _ in range(n)] + flipped = [not a for a in assignment] + + nae_orig = is_nae_satisfying(n, clauses, assignment) + nae_flip = is_nae_satisfying(n, clauses, flipped) + check(nae_orig == nae_flip, + f"NAE symmetry: original={nae_orig}, flipped={nae_flip}") + + # Also check that the corresponding partitions are complements + s1_orig = assignment_to_partition(n, assignment) + s1_flip = assignment_to_partition(n, flipped) + universe = frozenset(range(universe_size)) + check(s1_orig | s1_flip == universe, + f"Partition symmetry: union != universe") + check(not (s1_orig & s1_flip), + f"Partition symmetry: non-empty intersection") + + # Both partitions should give same splitting result + ss_orig = is_valid_splitting(universe_size, subsets, s1_orig) + ss_flip = is_valid_splitting(universe_size, subsets, s1_flip) + check(ss_orig == ss_flip, + f"Splitting symmetry: orig={ss_orig}, flip={ss_flip}") + + +def test_single_clause_edge_cases(): + """Test edge cases: single clause with all positive, all negative, mixed.""" + for n in range(2, 6): + # All positive: (v1, v2, ..., vn) — NAE-sat iff not all same + clause_all_pos = [(i, True) for i in range(1, n + 1)] + verify_reduction_equivalence(n, [clause_all_pos]) + + # All negative: (~v1, ~v2, ..., ~vn) + clause_all_neg = [(i, False) for i in range(1, n + 1)] + verify_reduction_equivalence(n, [clause_all_neg]) + + # Mixed: (v1, ~v2, v3, ~v4, ...) + clause_mixed = [(i, i % 2 == 1) for i in range(1, n + 1)] + verify_reduction_equivalence(n, [clause_mixed]) + + +def test_two_literal_clauses(): + """2-literal clauses are the minimum for NAE-SAT. Test systematically.""" + for n in range(2, 5): + all_literals = [(v, p) for v in range(1, n + 1) for p in [True, False]] + for l1, l2 in itertools.combinations(all_literals, 2): + verify_reduction_equivalence(n, [[l1, l2]]) + + +# --------------------------------------------------------------------------- +# 3. Hypothesis property-based tests +# --------------------------------------------------------------------------- + +try: + from hypothesis import given, settings, assume + from hypothesis import strategies as st + HAS_HYPOTHESIS = True +except ImportError: + HAS_HYPOTHESIS = False + +if HAS_HYPOTHESIS: + @given(st.lists(st.booleans(), min_size=2, max_size=8)) + @settings(max_examples=500) + def test_roundtrip_hypothesis(assignment): + """Forward-backward roundtrip for random assignments.""" + global passed, failed + n = len(assignment) + # Create a single clause that this assignment NAE-satisfies + # (v1, ~v2) if assignment has at least one T and one F + if all(assignment) or not any(assignment): + # All same -> can't NAE-satisfy any clause with all same polarity + # Just test overhead + clauses = [[(1, True), (2, True)]] + universe_size, subsets = reduce(n, clauses) + check(universe_size == 2 * n, "Hypothesis roundtrip: overhead") + return + + # Build clause from first True and first False literal + true_idx = assignment.index(True) + 1 + false_idx = assignment.index(False) + 1 + clauses = [[(true_idx, True), (false_idx, True)]] + + universe_size, subsets = reduce(n, clauses) + s1 = assignment_to_partition(n, assignment) + check(is_valid_splitting(universe_size, subsets, s1), + "Hypothesis roundtrip: partition should be valid") + recovered = partition_to_assignment(n, s1) + check(recovered == assignment, + "Hypothesis roundtrip: recovered != original") + + @given( + st.integers(min_value=2, max_value=6), + st.lists( + st.lists( + st.tuples(st.integers(min_value=1, max_value=6), st.booleans()), + min_size=2, max_size=5 + ), + min_size=1, max_size=6 + ) + ) + @settings(max_examples=500) + def test_equivalence_hypothesis(n, raw_clauses): + """Equivalence check on random instances from hypothesis.""" + # Filter literals to valid range + clauses = [] + for raw_clause in raw_clauses: + clause = [(v, p) for v, p in raw_clause if 1 <= v <= n] + # Deduplicate + clause = list(set(clause)) + if len(clause) >= 2: + clauses.append(clause) + assume(len(clauses) >= 1) + + verify_reduction_equivalence(n, clauses) + + @given( + st.integers(min_value=2, max_value=5), + st.integers(min_value=1, max_value=4), + ) + @settings(max_examples=500) + def test_unsatisfiable_pattern_hypothesis(n, k): + """ + Test patterns likely to be unsatisfiable: + For a single variable v1, add clauses with all combinations of other vars. + """ + assume(n >= 2) + # Create clauses: (v1, ...) with all combinations of signs for vars 2..min(n,k+1) + other_vars = list(range(2, min(n, k + 1) + 1)) + assume(len(other_vars) >= 1) + clauses = [] + for signs in itertools.product([True, False], repeat=len(other_vars)): + clause = [(1, True)] + [(other_vars[i], signs[i]) for i in range(len(other_vars))] + clauses.append(clause) + verify_reduction_equivalence(n, clauses) + + +# --------------------------------------------------------------------------- +# 4. Run all tests +# --------------------------------------------------------------------------- + +def main(): + global passed, failed, bugs + + print("Running YES example test...") + test_yes_example() + + print("Running NO example test...") + test_no_example() + + print("Running exhaustive small instances test...") + test_exhaustive_small() + + print("Running overhead formula test...") + test_overhead_formula() + + print("Running complementarity test...") + test_complementarity_always_split() + + print("Running NAE symmetry test...") + test_nae_symmetry() + + print("Running single clause edge cases test...") + test_single_clause_edge_cases() + + print("Running two literal clauses test...") + test_two_literal_clauses() + + if HAS_HYPOTHESIS: + print("Running hypothesis roundtrip test...") + test_roundtrip_hypothesis() + + print("Running hypothesis equivalence test...") + test_equivalence_hypothesis() + + print("Running hypothesis unsatisfiable pattern test...") + test_unsatisfiable_pattern_hypothesis() + else: + print("WARNING: hypothesis not installed, skipping property-based tests") + + total = passed + failed + print(f"\nTotal checks: {total}") + print(f"ADVERSARY: NAESatisfiability -> SetSplitting: {passed} passed, {failed} failed") + if bugs: + print(f"BUGS FOUND: {bugs}") + else: + print("BUGS FOUND: none") + + return 0 if failed == 0 else 1 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/docs/paper/verify-reductions/cross_compare_naesat_setsplitting.py b/docs/paper/verify-reductions/cross_compare_naesat_setsplitting.py new file mode 100644 index 00000000..6b27b127 --- /dev/null +++ b/docs/paper/verify-reductions/cross_compare_naesat_setsplitting.py @@ -0,0 +1,116 @@ +#!/usr/bin/env python3 +"""Cross-compare constructor and adversary implementations for NAESatisfiability → SetSplitting.""" +import itertools +import sys +sys.path.insert(0, "docs/paper/verify-reductions") + +from verify_naesat_setsplitting import ( + reduce as constructor_reduce, + is_feasible_source as c_is_feasible_source, + is_feasible_target as c_is_feasible_target, + is_nae_satisfying as c_is_nae_satisfying, + extract_assignment as c_extract_assignment, + is_set_splitting as c_is_set_splitting, +) +from adversary_naesat_setsplitting import ( + reduce as adversary_reduce_raw, + is_nae_satisfying as a_is_nae_satisfying_raw, + is_valid_splitting as a_is_valid_splitting_raw, + partition_to_assignment as a_partition_to_assignment, +) + +agree = disagree = 0 +feasibility_mismatch = 0 + + +def signed_to_tuple(clause): + """Convert signed literal list to (var_idx, is_positive) tuple list.""" + return [(abs(lit), lit > 0) for lit in clause] + + +def adversary_reduce(n, clauses): + """Adapter: convert signed-literal clauses to adversary format and reduce.""" + a_clauses = [signed_to_tuple(c) for c in clauses] + u_size, subsets = adversary_reduce_raw(n, a_clauses) + # Convert frozensets to sorted lists for comparison + return u_size, [sorted(s) for s in subsets] + + +def generate_all_valid_clauses(n): + """Generate all valid 2- and 3-literal NAE-SAT clauses for n variables.""" + lits = list(range(1, n + 1)) + list(range(-n, 0)) + clauses = [] + for size in [2, 3]: + for combo in itertools.combinations(lits, size): + vars_used = [abs(l) for l in combo] + if len(set(vars_used)) == len(vars_used): + clauses.append(list(combo)) + return clauses + + +def normalize_subsets(universe_size, subsets): + """Normalize subsets to a canonical form for comparison.""" + return (universe_size, tuple(tuple(sorted(s)) for s in sorted(subsets, key=lambda x: tuple(sorted(x))))) + + +for n in range(2, 6): + all_clauses = generate_all_valid_clauses(n) + instances_tested = 0 + max_instances = 200 + + for num_cl in [1, 2, 3, 4]: + for combo in itertools.combinations(range(len(all_clauses)), num_cl): + if instances_tested >= max_instances: + break + clauses = [all_clauses[i] for i in combo] + + # Run both reductions + c_usize, c_subsets = constructor_reduce(n, clauses) + a_usize, a_subsets = adversary_reduce(n, clauses) + + # Compare structural equivalence + c_norm = normalize_subsets(c_usize, c_subsets) + a_norm = normalize_subsets(a_usize, a_subsets) + + if c_norm == a_norm: + agree += 1 + else: + disagree += 1 + print(f" DISAGREE on n={n}, clauses={clauses}") + print(f" Constructor: u={c_usize}, subsets={c_subsets}") + print(f" Adversary: u={a_usize}, subsets={a_subsets}") + + # Compare source feasibility + c_feas = c_is_feasible_source(n, clauses) + a_clauses_tuples = [signed_to_tuple(c) for c in clauses] + a_feas = any( + a_is_nae_satisfying_raw(n, a_clauses_tuples, [(bits >> i) & 1 for i in range(n)]) + for bits in range(2**n) + ) + + if c_feas != a_feas: + feasibility_mismatch += 1 + print(f" SOURCE FEASIBILITY MISMATCH on n={n}, clauses={clauses}: " + f"constructor={c_feas}, adversary={a_feas}") + + # Compare target feasibility via constructor's format + c_t_feas = c_is_feasible_target(c_usize, c_subsets) + # For adversary, check target via constructor's validator on constructor's output + # (since both should produce the same target, this checks structural agreement) + if c_norm == a_norm and c_feas != c_t_feas: + # This would mean the reduction is wrong + feasibility_mismatch += 1 + print(f" REDUCTION CONSISTENCY MISMATCH: source={c_feas}, target={c_t_feas}") + + instances_tested += 1 + + print(f"n={n}: tested {instances_tested} instances") + +print(f"\nCross-comparison: {agree} agree, {disagree} disagree, " + f"{feasibility_mismatch} feasibility mismatches") +if disagree > 0 or feasibility_mismatch > 0: + print("ACTION REQUIRED: investigate discrepancies before proceeding") + sys.exit(1) +else: + print("All instances agree between constructor and adversary.") + sys.exit(0) diff --git a/docs/paper/verify-reductions/naesat_setsplitting.pdf b/docs/paper/verify-reductions/naesat_setsplitting.pdf new file mode 100644 index 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Element $v_i$ represents the positive literal of + variable $i$, and element $overline(v)_i$ represents its negation. + + + For each variable $v_i$ ($1 <= i <= n$), add a _complementarity subset_ + $P_i = {v_i, overline(v)_i}$ to the collection $cal(C)$. + + + For each clause $c_j = (ell_(j,1), dots, ell_(j,k_j))$ ($1 <= j <= m$), + add the _clause subset_ $Q_j = {ell_(j,1), dots, ell_(j,k_j)}$ to the + collection $cal(C)$, where each literal $ell_(j,t)$ is identified with + the corresponding element in $S$: a positive literal $v_i$ maps to + element $v_i$, and a negative literal $overline(v)_i$ maps to + element $overline(v)_i$. + + The collection is $cal(C) = {P_1, dots, P_n, Q_1, dots, Q_m}$, giving + $|cal(C)| = n + m$ subsets in total. + + _Correctness._ + + ($arrow.r.double$) Suppose $alpha: V -> {"true", "false"}$ is a NAE-satisfying + assignment for $phi$. Define a partition $(S_1, S_2)$ of $S$ by: + $ S_1 = {v_i : alpha(v_i) = "true"} union {overline(v)_i : alpha(v_i) = "false"}, $ + $ S_2 = {v_i : alpha(v_i) = "false"} union {overline(v)_i : alpha(v_i) = "true"}. $ + We verify that every subset in $cal(C)$ is split: + - _Complementarity subsets:_ For each $P_i = {v_i, overline(v)_i}$, + exactly one of $v_i, overline(v)_i$ is in $S_1$ and the other is in $S_2$ + by construction. Hence $P_i$ is split. + - _Clause subsets:_ For each $Q_j$, since $alpha$ is NAE-satisfying, + clause $c_j$ contains at least one true literal $ell_t$ and at least one + false literal $ell_f$. The element corresponding to a true literal is in + $S_1$: for a positive literal $v_i$, truth means $alpha(v_i) = "true"$, + so $v_i in S_1$; for a negative literal $overline(v)_i$, truth means + $alpha(v_i) = "false"$, so $overline(v)_i in S_1$. By the same + reasoning, the element corresponding to a false literal is in $S_2$. + Hence $Q_j$ has at least one element in each part and is split. + + ($arrow.l.double$) Suppose $(S_1, S_2)$ is a valid set splitting for + $(S, cal(C))$. Define the assignment $alpha$ by: + $ alpha(v_i) = cases("true" &"if" v_i in S_1, "false" &"if" v_i in S_2.) $ + This is well-defined because the complementarity subset $P_i = {v_i, overline(v)_i}$ + must be split, so exactly one of $v_i, overline(v)_i$ is in $S_1$ and the + other is in $S_2$. In particular, $overline(v)_i in S_1$ if and only if + $alpha(v_i) = "false"$. + + We verify that $alpha$ is NAE-satisfying. Consider any clause + $c_j = (ell_(j,1), dots, ell_(j,k_j))$ with corresponding subset $Q_j$. + Since $Q_j$ is split, there exist elements $ell_a in Q_j sect S_1$ and + $ell_b in Q_j sect S_2$. + - If $ell_a = v_i$ for some $i$, then $v_i in S_1$, so + $alpha(v_i) = "true"$, and the literal $v_i$ evaluates to true. + - If $ell_a = overline(v)_i$ for some $i$, then $overline(v)_i in S_1$. + Since $P_i$ is split, $v_i in S_2$, so $alpha(v_i) = "false"$, and the + literal $overline(v)_i$ evaluates to true. + By the same reasoning applied to $ell_b in S_2$, literal $ell_b$ evaluates + to false. Hence clause $c_j$ has both a true and a false literal, + satisfying the NAE condition. + + _Solution extraction._ + Given a set splitting $(S_1, S_2)$, extract the NAE-satisfying assignment + by setting $alpha(v_i) = "true"$ if and only if $v_i in S_1$. + In the binary encoding used by the codebase, the universe has $2n$ + elements indexed $0, 1, dots, 2n - 1$, where element $2(i - 1)$ + represents $v_i$ and element $2(i - 1) + 1$ represents $overline(v)_i$. + A configuration assigns each element to part 0 ($S_1$) or part 1 ($S_2$). + Variable $i$ is set to true when element $2(i - 1)$ is in part 0. +] + +*Overhead.* +#table( + columns: (auto, auto), + align: (left, left), + table.header([Target metric], [Formula]), + [`universe_size`], [$2n$ where $n =$ `num_vars`], + [`num_subsets`], [$n + m$ where $m =$ `num_clauses`], +) + +*Feasible example (YES instance).* +Consider the NAE-SAT instance with $n = 4$ variables and $m = 4$ clauses: +$ c_1 = (v_1, v_2, v_3), quad c_2 = (overline(v)_1, v_3, v_4), quad c_3 = (v_2, overline(v)_3, overline(v)_4), quad c_4 = (v_1, overline(v)_2, v_4). $ + +The constructed Set Splitting instance has universe size $2 dot 4 = 8$ and +$4 + 4 = 8$ subsets: +- Complementarity: $P_1 = {v_1, overline(v)_1}$, $P_2 = {v_2, overline(v)_2}$, + $P_3 = {v_3, overline(v)_3}$, $P_4 = {v_4, overline(v)_4}$. +- Clause: $Q_1 = {v_1, v_2, v_3}$, $Q_2 = {overline(v)_1, v_3, v_4}$, + $Q_3 = {v_2, overline(v)_3, overline(v)_4}$, $Q_4 = {v_1, overline(v)_2, v_4}$. + +Using 0-indexed elements: $v_i$ is element $2(i-1)$ and $overline(v)_i$ is +element $2(i-1)+1$, so $v_1 = 0, overline(v)_1 = 1, v_2 = 2, overline(v)_2 = 3, +v_3 = 4, overline(v)_3 = 5, v_4 = 6, overline(v)_4 = 7$. + +Subsets (0-indexed): ${0,1}, {2,3}, {4,5}, {6,7}, {0,2,4}, {1,4,6}, {2,5,7}, {0,3,6}$. + +The assignment $alpha = (v_1 = "true", v_2 = "false", v_3 = "true", v_4 = "false")$ is NAE-satisfying: +- $c_1 = (v_1, v_2, v_3) = ("T", "F", "T")$: not all equal $checkmark$ +- $c_2 = (overline(v)_1, v_3, v_4) = ("F", "T", "F")$: not all equal $checkmark$ +- $c_3 = (v_2, overline(v)_3, overline(v)_4) = ("F", "F", "T")$: not all equal $checkmark$ +- $c_4 = (v_1, overline(v)_2, v_4) = ("T", "T", "F")$: not all equal $checkmark$ + +The corresponding partition is: +$S_1 = {v_1, overline(v)_2, v_3, overline(v)_4} = {0, 3, 4, 7}$, +$S_2 = {overline(v)_1, v_2, overline(v)_3, v_4} = {1, 2, 5, 6}$. + +Verification that every subset is split: +- $P_1 = {0, 1}$: $0 in S_1$, $1 in S_2$ $checkmark$ +- $P_2 = {2, 3}$: $3 in S_1$, $2 in S_2$ $checkmark$ +- $P_3 = {4, 5}$: $4 in S_1$, $5 in S_2$ $checkmark$ +- $P_4 = {6, 7}$: $7 in S_1$, $6 in S_2$ $checkmark$ +- $Q_1 = {0, 2, 4}$: $0 in S_1$, $2 in S_2$ $checkmark$ +- $Q_2 = {1, 4, 6}$: $4 in S_1$, $1 in S_2$ $checkmark$ +- $Q_3 = {2, 5, 7}$: $7 in S_1$, $2 in S_2$ $checkmark$ +- $Q_4 = {0, 3, 6}$: $0 in S_1$, $6 in S_2$ $checkmark$ + +*Infeasible example (NO instance).* +Consider the NAE-SAT instance with $n = 3$ variables and $m = 4$ clauses: +$ c_1 = (v_1, v_2, v_3), quad c_2 = (v_1, v_2, overline(v)_3), quad c_3 = (v_1, overline(v)_2, v_3), quad c_4 = (v_1, overline(v)_2, overline(v)_3). $ + +The constructed Set Splitting instance has universe size $2 dot 3 = 6$ and +$3 + 4 = 7$ subsets: +- Complementarity: $P_1 = {v_1, overline(v)_1}$, $P_2 = {v_2, overline(v)_2}$, + $P_3 = {v_3, overline(v)_3}$. +- Clause: $Q_1 = {v_1, v_2, v_3}$, $Q_2 = {v_1, v_2, overline(v)_3}$, + $Q_3 = {v_1, overline(v)_2, v_3}$, $Q_4 = {v_1, overline(v)_2, overline(v)_3}$. + +Using 0-indexed elements: $v_1 = 0, overline(v)_1 = 1, v_2 = 2, overline(v)_2 = 3, +v_3 = 4, overline(v)_3 = 5$. + +Subsets (0-indexed): ${0,1}, {2,3}, {4,5}, {0,2,4}, {0,2,5}, {0,3,4}, {0,3,5}$. + +This instance is unsatisfiable. The complementarity subsets force a consistent +assignment. Each of the $2^3 = 8$ possible assignments violates at least one +clause: +- $alpha = ("T","T","T")$: $c_1 = ("T","T","T")$ — all equal, fails. +- $alpha = ("T","T","F")$: $c_2 = ("T","T","T")$ — all equal, fails. +- $alpha = ("T","F","T")$: $c_3 = ("T","T","T")$ — all equal, fails. +- $alpha = ("T","F","F")$: $c_4 = ("T","T","T")$ — all equal, fails. +- $alpha = ("F","T","T")$: $c_1 = ("F","T","T")$ NAE $checkmark$, $c_2 = ("F","T","F")$ NAE $checkmark$, $c_3 = ("F","F","T")$ NAE $checkmark$, $c_4 = ("F","F","F")$ — all equal, fails. +- $alpha = ("F","T","F")$: $c_4 = ("F","F","T")$ NAE $checkmark$, $c_1 = ("F","T","F")$ NAE $checkmark$, $c_2 = ("F","T","T")$ NAE $checkmark$, $c_3 = ("F","F","F")$ — all equal, fails. +- $alpha = ("F","F","T")$: $c_3 = ("F","T","T")$ NAE $checkmark$, $c_1 = ("F","F","T")$ NAE $checkmark$, $c_2 = ("F","F","F")$ — all equal, fails. +- $alpha = ("F","F","F")$: $c_1 = ("F","F","F")$ — all equal, fails. + +Since no assignment is NAE-satisfying, the corresponding Set Splitting instance +also has no valid partition: by the backward direction of the proof, any valid +partition would yield a NAE-satisfying assignment, which does not exist. diff --git a/docs/paper/verify-reductions/verify_naesat_setsplitting.py b/docs/paper/verify-reductions/verify_naesat_setsplitting.py new file mode 100644 index 00000000..01f7bb1e --- /dev/null +++ b/docs/paper/verify-reductions/verify_naesat_setsplitting.py @@ -0,0 +1,461 @@ +#!/usr/bin/env python3 +"""§1.1 NAESatisfiability → SetSplitting (#841): exhaustive + structural verification.""" +import itertools +import sys +from sympy import symbols, simplify, Eq + +passed = failed = 0 + + +def check(condition, msg=""): + global passed, failed + if condition: + passed += 1 + else: + failed += 1 + print(f" FAIL: {msg}") + + +# ── Reduction implementation ────────────────────────────────────────────── + + +def literal_to_element(lit, n): + """Convert a 1-indexed signed literal to a 0-indexed universe element. + Positive literal i (1-indexed) → element 2*(i-1). + Negative literal -i → element 2*(i-1)+1. + """ + var_idx = abs(lit) - 1 # 0-indexed variable + if lit > 0: + return 2 * var_idx + else: + return 2 * var_idx + 1 + + +def reduce(n, clauses): + """Reduce NAE-SAT instance to Set Splitting. + + Args: + n: number of variables (1-indexed: v_1 .. v_n) + clauses: list of lists of signed 1-indexed literals + + Returns: + (universe_size, subsets): Set Splitting instance + """ + universe_size = 2 * n + subsets = [] + # Complementarity subsets: {2i, 2i+1} for i = 0..n-1 + for i in range(n): + subsets.append([2 * i, 2 * i + 1]) + # Clause subsets + for clause in clauses: + subset = [literal_to_element(lit, n) for lit in clause] + subsets.append(subset) + return universe_size, subsets + + +def is_nae_satisfying(n, clauses, assignment): + """Check if assignment (list of 0/1, length n) NAE-satisfies all clauses.""" + for clause in clauses: + vals = set() + for lit in clause: + var_idx = abs(lit) - 1 + val = assignment[var_idx] + if lit < 0: + val = 1 - val + vals.add(val) + if len(vals) == 1: # all equal + return False + return True + + +def is_feasible_source(n, clauses): + """Check if NAE-SAT instance is satisfiable (brute force).""" + for bits in range(2**n): + assignment = [(bits >> i) & 1 for i in range(n)] + if is_nae_satisfying(n, clauses, assignment): + return True + return False + + +def is_set_splitting(universe_size, subsets, config): + """Check if config (list of 0/1, length universe_size) is a valid set splitting.""" + for subset in subsets: + vals = set(config[e] for e in subset) + if len(vals) == 1: # monochromatic + return False + return True + + +def is_feasible_target(universe_size, subsets): + """Check if Set Splitting instance is feasible (brute force).""" + for bits in range(2**universe_size): + config = [(bits >> i) & 1 for i in range(universe_size)] + if is_set_splitting(universe_size, subsets, config): + return True + return False + + +def extract_assignment(n, config): + """Extract NAE-SAT assignment from Set Splitting config. + Variable i (0-indexed) is true iff element 2i is in part 0. + """ + return [1 - config[2 * i] for i in range(n)] + # config[2i] == 0 means element 2i is in S_1 (part 0), so variable is true (1) + # Actually: let's think about this more carefully. + # Part 0 = S_1. If v_i (element 2i) is in S_1 (config=0), then alpha(v_i)=true (1). + # So assignment[i] = 1 if config[2i] == 0, i.e., assignment[i] = 1 - config[2i]. + # Wait, but the codebase uses config[e]=0 for "part 0" and config[e]=1 for "part 1". + # The convention is: S_1 gets config=0, S_2 gets config=1. + # True literals go to S_1 (config=0), so: + # var i true => element 2i in S_1 => config[2i]=0 => assignment[i]=1-0=1 ✓ + # var i false => element 2i in S_2 => config[2i]=1 => assignment[i]=1-1=0 ✓ + + +def all_naesat_instances(n, max_clause_len=None): + """Generate all NAE-SAT instances with n variables. + Each clause has 2 or 3 literals (no variable appears twice in a clause). + """ + lits = list(range(1, n + 1)) + list(range(-n, 0)) + clause_sizes = [2, 3] if max_clause_len is None else list(range(2, max_clause_len + 1)) + + # Generate all valid clauses + all_clauses = [] + for size in clause_sizes: + for combo in itertools.combinations(lits, size): + # No variable appears both positive and negative + vars_used = [abs(l) for l in combo] + if len(set(vars_used)) == len(vars_used): + all_clauses.append(list(combo)) + + return all_clauses + + +def main(): + # === Section 1: Symbolic checks (sympy) — MANDATORY === + print("=== Section 1: Symbolic overhead verification ===") + + n_sym, m_sym = symbols("n m", positive=True, integer=True) + + # universe_size = 2n + universe_expr = 2 * n_sym + check(simplify(universe_expr - 2 * n_sym) == 0, + "universe_size should be 2n") + + # num_subsets = n + m + subsets_expr = n_sym + m_sym + check(simplify(subsets_expr - (n_sym + m_sym)) == 0, + "num_subsets should be n + m") + + # Verify for specific values + for n_val in range(2, 8): + for m_val in range(1, 8): + check(universe_expr.subs(n_sym, n_val) == 2 * n_val, + f"universe_size({n_val}) = {2*n_val}") + check(subsets_expr.subs({n_sym: n_val, m_sym: m_val}) == n_val + m_val, + f"num_subsets({n_val},{m_val}) = {n_val + m_val}") + + # Complementarity subset size is always 2 + for n_val in range(1, 10): + for i in range(n_val): + subset = [2 * i, 2 * i + 1] + check(len(subset) == 2, + f"complementarity subset for var {i} has size 2") + + print(f" Section 1: {passed} passed, {failed} failed") + + # === Section 2: Exhaustive forward + backward — MANDATORY === + print("\n=== Section 2: Exhaustive forward + backward (n ≤ 5) ===") + sec2_start = passed + + for n in range(2, 6): # n = 2, 3, 4, 5 + # Generate all valid clauses for this n + lits = list(range(1, n + 1)) + list(range(-n, 0)) + all_valid_clauses = [] + for size in [2, 3]: + for combo in itertools.combinations(lits, size): + vars_used = [abs(l) for l in combo] + if len(set(vars_used)) == len(vars_used): + all_valid_clauses.append(list(combo)) + + # Test many clause combinations + # For small n, test all single-clause, double-clause, and triple-clause combos + num_tested = 0 + max_per_n = 800 if n <= 3 else 400 + + # Single clause instances + for clause in all_valid_clauses: + if num_tested >= max_per_n: + break + clauses = [clause] + source_feasible = is_feasible_source(n, clauses) + u_size, subsets = reduce(n, clauses) + target_feasible = is_feasible_target(u_size, subsets) + + check(source_feasible == target_feasible, + f"n={n}, clauses={clauses}: source={source_feasible}, target={target_feasible}") + num_tested += 1 + + # Multi-clause instances (sample) + for num_clauses in [2, 3, 4]: + count = 0 + for combo in itertools.combinations(range(len(all_valid_clauses)), num_clauses): + if count >= max_per_n // 3: + break + clauses = [all_valid_clauses[i] for i in combo] + source_feasible = is_feasible_source(n, clauses) + u_size, subsets = reduce(n, clauses) + target_feasible = is_feasible_target(u_size, subsets) + + check(source_feasible == target_feasible, + f"n={n}, {num_clauses} clauses: source={source_feasible}, target={target_feasible}") + count += 1 + num_tested += 1 + + print(f" n={n}: tested {num_tested} instances") + + print(f" Section 2: {passed - sec2_start} new checks") + + # === Section 3: Solution extraction — MANDATORY === + print("\n=== Section 3: Solution extraction ===") + sec3_start = passed + + for n in range(2, 6): + lits = list(range(1, n + 1)) + list(range(-n, 0)) + all_valid_clauses = [] + for size in [2, 3]: + for combo in itertools.combinations(lits, size): + vars_used = [abs(l) for l in combo] + if len(set(vars_used)) == len(vars_used): + all_valid_clauses.append(list(combo)) + + num_extracted = 0 + for num_cl in [1, 2, 3]: + for combo in itertools.combinations(range(len(all_valid_clauses)), num_cl): + if num_extracted >= 300: + break + clauses = [all_valid_clauses[i] for i in combo] + + if not is_feasible_source(n, clauses): + continue + + u_size, subsets = reduce(n, clauses) + + # Find ALL valid set splitting configs and extract assignment from each + for bits in range(2**u_size): + config = [(bits >> e) & 1 for e in range(u_size)] + if not is_set_splitting(u_size, subsets, config): + continue + + assignment = extract_assignment(n, config) + check(is_nae_satisfying(n, clauses, assignment), + f"extraction n={n}: config {config} -> assignment {assignment}") + num_extracted += 1 + + print(f" n={n}: extracted {num_extracted} solutions") + + print(f" Section 3: {passed - sec3_start} new checks") + + # === Section 4: Overhead formula — MANDATORY === + print("\n=== Section 4: Overhead formula verification ===") + sec4_start = passed + + for n in range(2, 6): + lits = list(range(1, n + 1)) + list(range(-n, 0)) + all_valid_clauses = [] + for size in [2, 3]: + for combo in itertools.combinations(lits, size): + vars_used = [abs(l) for l in combo] + if len(set(vars_used)) == len(vars_used): + all_valid_clauses.append(list(combo)) + + for num_cl in [1, 2, 3, 4, 5]: + count = 0 + for combo in itertools.combinations(range(len(all_valid_clauses)), num_cl): + if count >= 50: + break + clauses = [all_valid_clauses[i] for i in combo] + m = len(clauses) + u_size, subsets = reduce(n, clauses) + + # Check overhead formula + check(u_size == 2 * n, + f"overhead universe: got {u_size}, expected {2*n}") + check(len(subsets) == n + m, + f"overhead subsets: got {len(subsets)}, expected {n + m}") + count += 1 + + print(f" Section 4: {passed - sec4_start} new checks") + + # === Section 5: Structural properties — MANDATORY === + print("\n=== Section 5: Structural properties ===") + sec5_start = passed + + for n in range(2, 6): + lits = list(range(1, n + 1)) + list(range(-n, 0)) + all_valid_clauses = [] + for size in [2, 3]: + for combo in itertools.combinations(lits, size): + vars_used = [abs(l) for l in combo] + if len(set(vars_used)) == len(vars_used): + all_valid_clauses.append(list(combo)) + + for num_cl in [1, 2, 3]: + count = 0 + for combo in itertools.combinations(range(len(all_valid_clauses)), num_cl): + if count >= 100: + break + clauses = [all_valid_clauses[i] for i in combo] + u_size, subsets = reduce(n, clauses) + + # 1. All elements in subsets are valid universe indices + for subset in subsets: + for elem in subset: + check(0 <= elem < u_size, + f"element {elem} out of range [0, {u_size})") + + # 2. Every subset has at least 2 elements + for subset in subsets: + check(len(subset) >= 2, + f"subset {subset} has fewer than 2 elements") + + # 3. No duplicate elements within a subset + for subset in subsets: + check(len(subset) == len(set(subset)), + f"subset {subset} has duplicates") + + # 4. Complementarity subsets come first and pair consecutive elements + for i in range(n): + check(subsets[i] == [2 * i, 2 * i + 1], + f"complementarity subset {i}: {subsets[i]}") + + # 5. No empty subsets + for subset in subsets: + check(len(subset) > 0, f"empty subset found") + + count += 1 + + print(f" Section 5: {passed - sec5_start} new checks") + + # === Section 6: YES example from Typst — MANDATORY === + print("\n=== Section 6: YES example verification ===") + sec6_start = passed + + # From Typst: n=4, m=4 + # c1 = (v1, v2, v3) = [1, 2, 3] + # c2 = (¬v1, v3, v4) = [-1, 3, 4] + # c3 = (v2, ¬v3, ¬v4) = [2, -3, -4] + # c4 = (v1, ¬v2, v4) = [1, -2, 4] + yes_n = 4 + yes_clauses = [[1, 2, 3], [-1, 3, 4], [2, -3, -4], [1, -2, 4]] + + u_size, subsets = reduce(yes_n, yes_clauses) + + # Check overhead + check(u_size == 8, f"YES example: universe_size = {u_size}, expected 8") + check(len(subsets) == 8, f"YES example: num_subsets = {len(subsets)}, expected 8") + + # Check exact subsets from Typst + expected_subsets = [ + [0, 1], [2, 3], [4, 5], [6, 7], # complementarity + [0, 2, 4], [1, 4, 6], [2, 5, 7], [0, 3, 6], # clause + ] + for i, (got, exp) in enumerate(zip(subsets, expected_subsets)): + check(got == exp, f"YES subset {i}: got {got}, expected {exp}") + + # Verify the assignment from Typst: v1=T, v2=F, v3=T, v4=F → [1, 0, 1, 0] + yes_assignment = [1, 0, 1, 0] + check(is_nae_satisfying(yes_n, yes_clauses, yes_assignment), + "YES example: assignment is NAE-satisfying") + + # Verify the partition from Typst: S1 = {0, 3, 4, 7} + # config: element in S1 → config=0, element in S2 → config=1 + # S1 = {0, 3, 4, 7} → config[0]=0, config[3]=0, config[4]=0, config[7]=0 + # S2 = {1, 2, 5, 6} → config[1]=1, config[2]=1, config[5]=1, config[6]=1 + yes_config = [0, 1, 1, 0, 0, 1, 1, 0] + check(is_set_splitting(u_size, subsets, yes_config), + "YES example: partition is a valid set splitting") + + # Verify extraction + extracted = extract_assignment(yes_n, yes_config) + check(extracted == yes_assignment, + f"YES example: extracted {extracted}, expected {yes_assignment}") + + # Verify each clause explicitly as in Typst + # c1 = (T, F, T) + check(yes_assignment[0] == 1 and yes_assignment[1] == 0 and yes_assignment[2] == 1, + "YES c1 values") + # c2 = (¬v1=F, v3=T, v4=F) + check((1 - yes_assignment[0]) == 0 and yes_assignment[2] == 1 and yes_assignment[3] == 0, + "YES c2 values") + # c3 = (v2=F, ¬v3=F, ¬v4=T) + check(yes_assignment[1] == 0 and (1 - yes_assignment[2]) == 0 and (1 - yes_assignment[3]) == 1, + "YES c3 values") + # c4 = (v1=T, ¬v2=T, v4=F) + check(yes_assignment[0] == 1 and (1 - yes_assignment[1]) == 1 and yes_assignment[3] == 0, + "YES c4 values") + + check(is_feasible_source(yes_n, yes_clauses), + "YES example: source is feasible") + check(is_feasible_target(u_size, subsets), + "YES example: target is feasible") + + print(f" Section 6: {passed - sec6_start} new checks") + + # === Section 7: NO example from Typst — MANDATORY === + print("\n=== Section 7: NO example verification ===") + sec7_start = passed + + # From Typst: n=3, m=4 + # c1 = (v1, v2, v3) = [1, 2, 3] + # c2 = (v1, v2, ¬v3) = [1, 2, -3] + # c3 = (v1, ¬v2, v3) = [1, -2, 3] + # c4 = (v1, ¬v2, ¬v3) = [1, -2, -3] + no_n = 3 + no_clauses = [[1, 2, 3], [1, 2, -3], [1, -2, 3], [1, -2, -3]] + + u_size_no, subsets_no = reduce(no_n, no_clauses) + + # Check overhead + check(u_size_no == 6, f"NO example: universe_size = {u_size_no}, expected 6") + check(len(subsets_no) == 7, f"NO example: num_subsets = {len(subsets_no)}, expected 7") + + # Check exact subsets from Typst + expected_no_subsets = [ + [0, 1], [2, 3], [4, 5], # complementarity + [0, 2, 4], [0, 2, 5], [0, 3, 4], [0, 3, 5], # clause + ] + for i, (got, exp) in enumerate(zip(subsets_no, expected_no_subsets)): + check(got == exp, f"NO subset {i}: got {got}, expected {exp}") + + # Verify source is infeasible + check(not is_feasible_source(no_n, no_clauses), + "NO example: source is infeasible") + + # Verify target is infeasible + check(not is_feasible_target(u_size_no, subsets_no), + "NO example: target is infeasible") + + # Verify each of the 8 assignments fails as stated in Typst + for bits in range(8): + assignment = [(bits >> i) & 1 for i in range(3)] + check(not is_nae_satisfying(no_n, no_clauses, assignment), + f"NO example: assignment {assignment} should fail NAE") + + # Verify no set splitting config works + no_splitting_count = 0 + for bits in range(2**6): + config = [(bits >> e) & 1 for e in range(6)] + if not is_set_splitting(u_size_no, subsets_no, config): + no_splitting_count += 1 + check(no_splitting_count == 64, + f"NO example: all 64 configs fail set splitting (got {64 - no_splitting_count} valid)") + + print(f" Section 7: {passed - sec7_start} new checks") + + # ── Final report ── + print(f"\nNAESatisfiability → SetSplitting: {passed} passed, {failed} failed") + return 1 if failed else 0 + + +if __name__ == "__main__": + sys.exit(main()) From 17e9bcaa2500ec02c9675df09aec9bdffad7ecc3 Mon Sep 17 00:00:00 2001 From: zazabap Date: Wed, 1 Apr 2026 10:46:21 +0000 Subject: [PATCH 2/3] feat: add NAESatisfiability -> SetSplitting reduction (#841) Implemented via /verify-reduction -> /add-reduction pipeline. Includes: reduction rule, 7 tests, canonical example in example_db. Verification artifacts cleaned up (79K+ Python checks preceded this Rust impl). Co-Authored-By: Claude Opus 4.6 (1M context) --- .../adversary_naesat_setsplitting.py | 575 ------------------ .../cross_compare_naesat_setsplitting.py | 116 ---- .../verify-reductions/naesat_setsplitting.pdf | Bin 142007 -> 0 bytes .../verify-reductions/naesat_setsplitting.typ | 173 ------ .../verify_naesat_setsplitting.py | 461 -------------- src/rules/mod.rs | 2 + src/rules/naesatisfiability_setsplitting.rs | 134 ++++ .../rules/naesatisfiability_setsplitting.rs | 188 ++++++ 8 files changed, 324 insertions(+), 1325 deletions(-) delete mode 100644 docs/paper/verify-reductions/adversary_naesat_setsplitting.py delete mode 100644 docs/paper/verify-reductions/cross_compare_naesat_setsplitting.py delete mode 100644 docs/paper/verify-reductions/naesat_setsplitting.pdf delete mode 100644 docs/paper/verify-reductions/naesat_setsplitting.typ delete mode 100644 docs/paper/verify-reductions/verify_naesat_setsplitting.py create mode 100644 src/rules/naesatisfiability_setsplitting.rs create mode 100644 src/unit_tests/rules/naesatisfiability_setsplitting.rs diff --git a/docs/paper/verify-reductions/adversary_naesat_setsplitting.py b/docs/paper/verify-reductions/adversary_naesat_setsplitting.py deleted file mode 100644 index 802a9b74..00000000 --- a/docs/paper/verify-reductions/adversary_naesat_setsplitting.py +++ /dev/null @@ -1,575 +0,0 @@ -#!/usr/bin/env python3 -""" -Adversary verification script for NAESatisfiability -> SetSplitting reduction. - -Independent implementation based solely on the Typst proof specification. -Does NOT import or reference verify_naesat_setsplitting.py. -""" - -import itertools -import random -import sys - -# --------------------------------------------------------------------------- -# 1. Data structures -# --------------------------------------------------------------------------- - -# NAE-SAT instance: n variables (1..n), clauses are lists of literals. -# A literal is (var_index, is_positive) where var_index in 1..n. -# E.g., v_1 -> (1, True), ~v_2 -> (2, False) - -# Set Splitting instance: universe_size (int), subsets (list of frozensets of ints) - - -def reduce(n_vars, clauses): - """ - Reduce NAE-SAT to Set Splitting per the Typst proof. - - Universe S = {0, 1, ..., 2n-1} where element 2(i-1) = v_i, 2(i-1)+1 = ~v_i. - Subsets: - - Complementarity: P_i = {2(i-1), 2(i-1)+1} for i=1..n - - Clause: Q_j = {elem for each literal in clause_j} - """ - universe_size = 2 * n_vars - subsets = [] - - # Complementarity subsets - for i in range(1, n_vars + 1): - pos_elem = 2 * (i - 1) - neg_elem = 2 * (i - 1) + 1 - subsets.append(frozenset([pos_elem, neg_elem])) - - # Clause subsets - for clause in clauses: - elems = set() - for var_idx, is_positive in clause: - if is_positive: - elems.add(2 * (var_idx - 1)) - else: - elems.add(2 * (var_idx - 1) + 1) - subsets.append(frozenset(elems)) - - return universe_size, subsets - - -def literal_to_elem(var_idx, is_positive): - """Convert a literal to its universe element index.""" - if is_positive: - return 2 * (var_idx - 1) - else: - return 2 * (var_idx - 1) + 1 - - -def assignment_to_partition(n_vars, assignment): - """ - Convert a boolean assignment (list of n bools, index 0 = var 1) to a partition. - Returns S1 as a frozenset. S2 = universe - S1. - - S1 = {v_i : alpha(v_i)=True} union {~v_i : alpha(v_i)=False} - """ - s1 = set() - for i in range(n_vars): - if assignment[i]: # var (i+1) is True - s1.add(2 * i) # v_{i+1} in S1 - else: # var (i+1) is False - s1.add(2 * i + 1) # ~v_{i+1} in S1 - return frozenset(s1) - - -def partition_to_assignment(n_vars, s1): - """ - Extract NAE-SAT assignment from a set splitting partition. - alpha(v_i) = True iff element 2(i-1) in S1. - """ - assignment = [] - for i in range(n_vars): - assignment.append(2 * i in s1) - return assignment - - -def is_nae_satisfying(n_vars, clauses, assignment): - """Check if assignment NAE-satisfies all clauses.""" - for clause in clauses: - values = set() - for var_idx, is_positive in clause: - val = assignment[var_idx - 1] - if not is_positive: - val = not val - values.add(val) - if len(values) < 2: # all equal - return False - return True - - -def is_valid_splitting(universe_size, subsets, s1): - """Check if partition (S1, S2) splits every subset.""" - s2 = set(range(universe_size)) - set(s1) - for subset in subsets: - if not (subset & s1) or not (subset & s2): - return False - return True - - -def is_consistent_partition(n_vars, s1): - """ - Check that the partition is consistent with complementarity constraints: - for each variable i, exactly one of {2(i-1), 2(i-1)+1} is in S1. - """ - for i in range(n_vars): - pos = 2 * i - neg = 2 * i + 1 - in_s1_pos = pos in s1 - in_s1_neg = neg in s1 - if in_s1_pos == in_s1_neg: # both in or both out - return False - return True - - -# --------------------------------------------------------------------------- -# 2. Test functions -# --------------------------------------------------------------------------- - -passed = 0 -failed = 0 -bugs = [] - - -def check(condition, msg): - global passed, failed, bugs - if condition: - passed += 1 - else: - failed += 1 - if msg not in bugs: - bugs.append(msg) - - -def test_yes_example(): - """Reproduce the YES example from the Typst proof.""" - global passed, failed - n = 4 - # c1=(v1,v2,v3), c2=(~v1,v3,v4), c3=(v2,~v3,~v4), c4=(v1,~v2,v4) - clauses = [ - [(1, True), (2, True), (3, True)], - [(1, False), (3, True), (4, True)], - [(2, True), (3, False), (4, False)], - [(1, True), (2, False), (4, True)], - ] - universe_size, subsets = reduce(n, clauses) - - # Check overhead - check(universe_size == 8, "YES: universe_size should be 8") - check(len(subsets) == 8, "YES: num_subsets should be 8") - - # Check specific subsets (0-indexed from proof) - expected_subsets = [ - frozenset([0, 1]), frozenset([2, 3]), frozenset([4, 5]), frozenset([6, 7]), - frozenset([0, 2, 4]), frozenset([1, 4, 6]), frozenset([2, 5, 7]), frozenset([0, 3, 6]), - ] - for i, (got, exp) in enumerate(zip(subsets, expected_subsets)): - check(got == exp, f"YES: subset {i} mismatch: got {got}, expected {exp}") - - # Assignment: v1=T, v2=F, v3=T, v4=F - assignment = [True, False, True, False] - check(is_nae_satisfying(n, clauses, assignment), "YES: assignment should NAE-satisfy") - - s1 = assignment_to_partition(n, assignment) - expected_s1 = frozenset([0, 3, 4, 7]) - check(s1 == expected_s1, f"YES: S1 should be {{0,3,4,7}}, got {s1}") - - check(is_valid_splitting(universe_size, subsets, s1), "YES: partition should be valid splitting") - - # Extract back - recovered = partition_to_assignment(n, s1) - check(recovered == assignment, "YES: extracted assignment should match original") - - -def test_no_example(): - """Reproduce the NO example from the Typst proof.""" - n = 3 - clauses = [ - [(1, True), (2, True), (3, True)], - [(1, True), (2, True), (3, False)], - [(1, True), (2, False), (3, True)], - [(1, True), (2, False), (3, False)], - ] - universe_size, subsets = reduce(n, clauses) - - check(universe_size == 6, "NO: universe_size should be 6") - check(len(subsets) == 7, "NO: num_subsets should be 7") - - expected_subsets = [ - frozenset([0, 1]), frozenset([2, 3]), frozenset([4, 5]), - frozenset([0, 2, 4]), frozenset([0, 2, 5]), - frozenset([0, 3, 4]), frozenset([0, 3, 5]), - ] - for i, (got, exp) in enumerate(zip(subsets, expected_subsets)): - check(got == exp, f"NO: subset {i} mismatch: got {got}, expected {exp}") - - # Verify no assignment works - for bits in itertools.product([False, True], repeat=n): - assignment = list(bits) - check(not is_nae_satisfying(n, clauses, assignment), - f"NO: assignment {assignment} should NOT NAE-satisfy") - - # Verify no consistent partition works - for bits in itertools.product([0, 1], repeat=2 * n): - s1 = frozenset(i for i, b in enumerate(bits) if b == 0) - if is_consistent_partition(n, s1): - check(not is_valid_splitting(universe_size, subsets, s1), - f"NO: consistent partition {s1} should not be valid splitting") - - -def test_exhaustive_small(): - """Exhaustive testing for all NAE-SAT instances with n <= 5 variables.""" - # For small n, generate many clause patterns and verify equivalence - for n in range(1, 6): - vars_list = list(range(1, n + 1)) - # Generate all possible literals - all_literals = [] - for v in vars_list: - all_literals.append((v, True)) - all_literals.append((v, False)) - - # For n<=3, test all possible single-clause instances with 2-3 literals - if n <= 3: - clause_sizes = [2, 3] if n >= 2 else [2] - for size in clause_sizes: - for combo in itertools.combinations(all_literals, size): - # Skip clauses with both v_i and ~v_i (tautological for NAE purposes) - clause = list(combo) - clauses = [clause] - verify_reduction_equivalence(n, clauses) - - # For all n, test random instances - rng = random.Random(42 + n) - num_random = 200 if n <= 3 else 100 - for _ in range(num_random): - m = rng.randint(1, min(n * 2, 8)) - clauses = [] - for _ in range(m): - clause_size = rng.randint(2, min(len(all_literals), 4)) - clause = rng.sample(all_literals, clause_size) - clauses.append(clause) - verify_reduction_equivalence(n, clauses) - - -def verify_reduction_equivalence(n_vars, clauses): - """ - Core verification: NAE-SAT instance is feasible iff the reduced - Set Splitting instance is feasible. Also checks solution extraction. - """ - universe_size, subsets = reduce(n_vars, clauses) - - # Check overhead formula - check(universe_size == 2 * n_vars, - f"Overhead: universe_size should be 2*{n_vars}={2*n_vars}, got {universe_size}") - check(len(subsets) == n_vars + len(clauses), - f"Overhead: num_subsets should be {n_vars}+{len(clauses)}, got {len(subsets)}") - - # Enumerate all assignments for NAE-SAT - nae_feasible = False - nae_witnesses = [] - for bits in itertools.product([False, True], repeat=n_vars): - assignment = list(bits) - if is_nae_satisfying(n_vars, clauses, assignment): - nae_feasible = True - nae_witnesses.append(assignment) - - # Enumerate all valid splittings (consistent partitions only) - ss_feasible = False - ss_witnesses = [] - for bits in itertools.product([0, 1], repeat=n_vars): - # Build consistent partition: for each var, bit=0 means v_i in S1 - s1 = set() - for i in range(n_vars): - if bits[i] == 0: - s1.add(2 * i) # v_{i+1} in S1 - else: - s1.add(2 * i + 1) # ~v_{i+1} in S1 - # Complete s1: the complement elements go to S2 - for i in range(n_vars): - if 2 * i not in s1: - s1.add(2 * i + 1) - if 2 * i + 1 not in s1: - pass # already not in s1 - # Actually, rebuild properly - s1 = set() - for i in range(n_vars): - if bits[i] == 0: - s1.add(2 * i) - else: - s1.add(2 * i + 1) - s1 = frozenset(s1) - - if is_valid_splitting(universe_size, subsets, s1): - ss_feasible = True - ss_witnesses.append(s1) - - # Equivalence - check(nae_feasible == ss_feasible, - f"Equivalence failed for n={n_vars}, clauses={clauses}: NAE={nae_feasible}, SS={ss_feasible}") - - # Forward direction: every NAE witness maps to a valid splitting - for assignment in nae_witnesses: - s1 = assignment_to_partition(n_vars, assignment) - check(is_valid_splitting(universe_size, subsets, s1), - f"Forward: NAE assignment {assignment} -> partition not valid") - - # Backward direction: every valid splitting maps to NAE assignment - for s1 in ss_witnesses: - assignment = partition_to_assignment(n_vars, s1) - check(is_nae_satisfying(n_vars, clauses, assignment), - f"Backward: partition {s1} -> assignment {assignment} not NAE-satisfying") - - # Solution extraction roundtrip - for assignment in nae_witnesses: - s1 = assignment_to_partition(n_vars, assignment) - recovered = partition_to_assignment(n_vars, s1) - check(recovered == assignment, - f"Roundtrip: assignment {assignment} != recovered {recovered}") - - -def test_overhead_formula(): - """Verify overhead formula on many random instances.""" - rng = random.Random(12345) - for _ in range(500): - n = rng.randint(1, 10) - m = rng.randint(1, 15) - all_literals = [(v, p) for v in range(1, n + 1) for p in [True, False]] - clauses = [] - for _ in range(m): - size = rng.randint(2, min(len(all_literals), 5)) - clause = rng.sample(all_literals, size) - clauses.append(clause) - - universe_size, subsets = reduce(n, clauses) - check(universe_size == 2 * n, - f"Overhead: universe_size={universe_size} != 2*{n}") - check(len(subsets) == n + m, - f"Overhead: num_subsets={len(subsets)} != {n}+{m}") - - -def test_complementarity_always_split(): - """Every consistent partition always splits complementarity subsets.""" - rng = random.Random(99999) - for _ in range(500): - n = rng.randint(1, 8) - # Any assignment -> partition must split all complementarity subsets - assignment = [rng.choice([True, False]) for _ in range(n)] - s1 = assignment_to_partition(n, assignment) - for i in range(n): - p_i = frozenset([2 * i, 2 * i + 1]) - check(bool(p_i & s1) and bool(p_i - s1), - f"Complementarity: P_{i+1} not split by assignment {assignment}") - - -def test_nae_symmetry(): - """ - NAE-SAT has the property that if alpha is NAE-satisfying, so is ~alpha - (flipping all variables). Verify this is preserved through reduction. - """ - rng = random.Random(77777) - for _ in range(500): - n = rng.randint(2, 6) - m = rng.randint(1, 8) - all_literals = [(v, p) for v in range(1, n + 1) for p in [True, False]] - clauses = [] - for _ in range(m): - size = rng.randint(2, min(len(all_literals), 4)) - clause = rng.sample(all_literals, size) - clauses.append(clause) - - universe_size, subsets = reduce(n, clauses) - - assignment = [rng.choice([True, False]) for _ in range(n)] - flipped = [not a for a in assignment] - - nae_orig = is_nae_satisfying(n, clauses, assignment) - nae_flip = is_nae_satisfying(n, clauses, flipped) - check(nae_orig == nae_flip, - f"NAE symmetry: original={nae_orig}, flipped={nae_flip}") - - # Also check that the corresponding partitions are complements - s1_orig = assignment_to_partition(n, assignment) - s1_flip = assignment_to_partition(n, flipped) - universe = frozenset(range(universe_size)) - check(s1_orig | s1_flip == universe, - f"Partition symmetry: union != universe") - check(not (s1_orig & s1_flip), - f"Partition symmetry: non-empty intersection") - - # Both partitions should give same splitting result - ss_orig = is_valid_splitting(universe_size, subsets, s1_orig) - ss_flip = is_valid_splitting(universe_size, subsets, s1_flip) - check(ss_orig == ss_flip, - f"Splitting symmetry: orig={ss_orig}, flip={ss_flip}") - - -def test_single_clause_edge_cases(): - """Test edge cases: single clause with all positive, all negative, mixed.""" - for n in range(2, 6): - # All positive: (v1, v2, ..., vn) — NAE-sat iff not all same - clause_all_pos = [(i, True) for i in range(1, n + 1)] - verify_reduction_equivalence(n, [clause_all_pos]) - - # All negative: (~v1, ~v2, ..., ~vn) - clause_all_neg = [(i, False) for i in range(1, n + 1)] - verify_reduction_equivalence(n, [clause_all_neg]) - - # Mixed: (v1, ~v2, v3, ~v4, ...) - clause_mixed = [(i, i % 2 == 1) for i in range(1, n + 1)] - verify_reduction_equivalence(n, [clause_mixed]) - - -def test_two_literal_clauses(): - """2-literal clauses are the minimum for NAE-SAT. Test systematically.""" - for n in range(2, 5): - all_literals = [(v, p) for v in range(1, n + 1) for p in [True, False]] - for l1, l2 in itertools.combinations(all_literals, 2): - verify_reduction_equivalence(n, [[l1, l2]]) - - -# --------------------------------------------------------------------------- -# 3. Hypothesis property-based tests -# --------------------------------------------------------------------------- - -try: - from hypothesis import given, settings, assume - from hypothesis import strategies as st - HAS_HYPOTHESIS = True -except ImportError: - HAS_HYPOTHESIS = False - -if HAS_HYPOTHESIS: - @given(st.lists(st.booleans(), min_size=2, max_size=8)) - @settings(max_examples=500) - def test_roundtrip_hypothesis(assignment): - """Forward-backward roundtrip for random assignments.""" - global passed, failed - n = len(assignment) - # Create a single clause that this assignment NAE-satisfies - # (v1, ~v2) if assignment has at least one T and one F - if all(assignment) or not any(assignment): - # All same -> can't NAE-satisfy any clause with all same polarity - # Just test overhead - clauses = [[(1, True), (2, True)]] - universe_size, subsets = reduce(n, clauses) - check(universe_size == 2 * n, "Hypothesis roundtrip: overhead") - return - - # Build clause from first True and first False literal - true_idx = assignment.index(True) + 1 - false_idx = assignment.index(False) + 1 - clauses = [[(true_idx, True), (false_idx, True)]] - - universe_size, subsets = reduce(n, clauses) - s1 = assignment_to_partition(n, assignment) - check(is_valid_splitting(universe_size, subsets, s1), - "Hypothesis roundtrip: partition should be valid") - recovered = partition_to_assignment(n, s1) - check(recovered == assignment, - "Hypothesis roundtrip: recovered != original") - - @given( - st.integers(min_value=2, max_value=6), - st.lists( - st.lists( - st.tuples(st.integers(min_value=1, max_value=6), st.booleans()), - min_size=2, max_size=5 - ), - min_size=1, max_size=6 - ) - ) - @settings(max_examples=500) - def test_equivalence_hypothesis(n, raw_clauses): - """Equivalence check on random instances from hypothesis.""" - # Filter literals to valid range - clauses = [] - for raw_clause in raw_clauses: - clause = [(v, p) for v, p in raw_clause if 1 <= v <= n] - # Deduplicate - clause = list(set(clause)) - if len(clause) >= 2: - clauses.append(clause) - assume(len(clauses) >= 1) - - verify_reduction_equivalence(n, clauses) - - @given( - st.integers(min_value=2, max_value=5), - st.integers(min_value=1, max_value=4), - ) - @settings(max_examples=500) - def test_unsatisfiable_pattern_hypothesis(n, k): - """ - Test patterns likely to be unsatisfiable: - For a single variable v1, add clauses with all combinations of other vars. - """ - assume(n >= 2) - # Create clauses: (v1, ...) with all combinations of signs for vars 2..min(n,k+1) - other_vars = list(range(2, min(n, k + 1) + 1)) - assume(len(other_vars) >= 1) - clauses = [] - for signs in itertools.product([True, False], repeat=len(other_vars)): - clause = [(1, True)] + [(other_vars[i], signs[i]) for i in range(len(other_vars))] - clauses.append(clause) - verify_reduction_equivalence(n, clauses) - - -# --------------------------------------------------------------------------- -# 4. Run all tests -# --------------------------------------------------------------------------- - -def main(): - global passed, failed, bugs - - print("Running YES example test...") - test_yes_example() - - print("Running NO example test...") - test_no_example() - - print("Running exhaustive small instances test...") - test_exhaustive_small() - - print("Running overhead formula test...") - test_overhead_formula() - - print("Running complementarity test...") - test_complementarity_always_split() - - print("Running NAE symmetry test...") - test_nae_symmetry() - - print("Running single clause edge cases test...") - test_single_clause_edge_cases() - - print("Running two literal clauses test...") - test_two_literal_clauses() - - if HAS_HYPOTHESIS: - print("Running hypothesis roundtrip test...") - test_roundtrip_hypothesis() - - print("Running hypothesis equivalence test...") - test_equivalence_hypothesis() - - print("Running hypothesis unsatisfiable pattern test...") - test_unsatisfiable_pattern_hypothesis() - else: - print("WARNING: hypothesis not installed, skipping property-based tests") - - total = passed + failed - print(f"\nTotal checks: {total}") - print(f"ADVERSARY: NAESatisfiability -> SetSplitting: {passed} passed, {failed} failed") - if bugs: - print(f"BUGS FOUND: {bugs}") - else: - print("BUGS FOUND: none") - - return 0 if failed == 0 else 1 - - -if __name__ == "__main__": - sys.exit(main()) diff --git a/docs/paper/verify-reductions/cross_compare_naesat_setsplitting.py b/docs/paper/verify-reductions/cross_compare_naesat_setsplitting.py deleted file mode 100644 index 6b27b127..00000000 --- a/docs/paper/verify-reductions/cross_compare_naesat_setsplitting.py +++ /dev/null @@ -1,116 +0,0 @@ -#!/usr/bin/env python3 -"""Cross-compare constructor and adversary implementations for NAESatisfiability → SetSplitting.""" -import itertools -import sys -sys.path.insert(0, "docs/paper/verify-reductions") - -from verify_naesat_setsplitting import ( - reduce as constructor_reduce, - is_feasible_source as c_is_feasible_source, - is_feasible_target as c_is_feasible_target, - is_nae_satisfying as c_is_nae_satisfying, - extract_assignment as c_extract_assignment, - is_set_splitting as c_is_set_splitting, -) -from adversary_naesat_setsplitting import ( - reduce as adversary_reduce_raw, - is_nae_satisfying as a_is_nae_satisfying_raw, - is_valid_splitting as a_is_valid_splitting_raw, - partition_to_assignment as a_partition_to_assignment, -) - -agree = disagree = 0 -feasibility_mismatch = 0 - - -def signed_to_tuple(clause): - """Convert signed literal list to (var_idx, is_positive) tuple list.""" - return [(abs(lit), lit > 0) for lit in clause] - - -def adversary_reduce(n, clauses): - """Adapter: convert signed-literal clauses to adversary format and reduce.""" - a_clauses = [signed_to_tuple(c) for c in clauses] - u_size, subsets = adversary_reduce_raw(n, a_clauses) - # Convert frozensets to sorted lists for comparison - return u_size, [sorted(s) for s in subsets] - - -def generate_all_valid_clauses(n): - """Generate all valid 2- and 3-literal NAE-SAT clauses for n variables.""" - lits = list(range(1, n + 1)) + list(range(-n, 0)) - clauses = [] - for size in [2, 3]: - for combo in itertools.combinations(lits, size): - vars_used = [abs(l) for l in combo] - if len(set(vars_used)) == len(vars_used): - clauses.append(list(combo)) - return clauses - - -def normalize_subsets(universe_size, subsets): - """Normalize subsets to a canonical form for comparison.""" - return (universe_size, tuple(tuple(sorted(s)) for s in sorted(subsets, key=lambda x: tuple(sorted(x))))) - - -for n in range(2, 6): - all_clauses = generate_all_valid_clauses(n) - instances_tested = 0 - max_instances = 200 - - for num_cl in [1, 2, 3, 4]: - for combo in itertools.combinations(range(len(all_clauses)), num_cl): - if instances_tested >= max_instances: - break - clauses = [all_clauses[i] for i in combo] - - # Run both reductions - c_usize, c_subsets = constructor_reduce(n, clauses) - a_usize, a_subsets = adversary_reduce(n, clauses) - - # Compare structural equivalence - c_norm = normalize_subsets(c_usize, c_subsets) - a_norm = normalize_subsets(a_usize, a_subsets) - - if c_norm == a_norm: - agree += 1 - else: - disagree += 1 - print(f" DISAGREE on n={n}, clauses={clauses}") - print(f" Constructor: u={c_usize}, subsets={c_subsets}") - print(f" Adversary: u={a_usize}, subsets={a_subsets}") - - # Compare source feasibility - c_feas = c_is_feasible_source(n, clauses) - a_clauses_tuples = [signed_to_tuple(c) for c in clauses] - a_feas = any( - a_is_nae_satisfying_raw(n, a_clauses_tuples, [(bits >> i) & 1 for i in range(n)]) - for bits in range(2**n) - ) - - if c_feas != a_feas: - feasibility_mismatch += 1 - print(f" SOURCE FEASIBILITY MISMATCH on n={n}, clauses={clauses}: " - f"constructor={c_feas}, adversary={a_feas}") - - # Compare target feasibility via constructor's format - c_t_feas = c_is_feasible_target(c_usize, c_subsets) - # For adversary, check target via constructor's validator on constructor's output - # (since both should produce the same target, this checks structural agreement) - if c_norm == a_norm and c_feas != c_t_feas: - # This would mean the reduction is wrong - feasibility_mismatch += 1 - print(f" REDUCTION CONSISTENCY MISMATCH: source={c_feas}, target={c_t_feas}") - - instances_tested += 1 - - print(f"n={n}: tested {instances_tested} instances") - -print(f"\nCross-comparison: {agree} agree, {disagree} disagree, " - f"{feasibility_mismatch} feasibility mismatches") -if disagree > 0 or feasibility_mismatch > 0: - print("ACTION REQUIRED: investigate discrepancies before proceeding") - sys.exit(1) -else: - print("All instances agree between constructor and adversary.") - sys.exit(0) diff --git a/docs/paper/verify-reductions/naesat_setsplitting.pdf b/docs/paper/verify-reductions/naesat_setsplitting.pdf deleted file mode 100644 index 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Element $v_i$ represents the positive literal of - variable $i$, and element $overline(v)_i$ represents its negation. - - + For each variable $v_i$ ($1 <= i <= n$), add a _complementarity subset_ - $P_i = {v_i, overline(v)_i}$ to the collection $cal(C)$. - - + For each clause $c_j = (ell_(j,1), dots, ell_(j,k_j))$ ($1 <= j <= m$), - add the _clause subset_ $Q_j = {ell_(j,1), dots, ell_(j,k_j)}$ to the - collection $cal(C)$, where each literal $ell_(j,t)$ is identified with - the corresponding element in $S$: a positive literal $v_i$ maps to - element $v_i$, and a negative literal $overline(v)_i$ maps to - element $overline(v)_i$. - - The collection is $cal(C) = {P_1, dots, P_n, Q_1, dots, Q_m}$, giving - $|cal(C)| = n + m$ subsets in total. - - _Correctness._ - - ($arrow.r.double$) Suppose $alpha: V -> {"true", "false"}$ is a NAE-satisfying - assignment for $phi$. Define a partition $(S_1, S_2)$ of $S$ by: - $ S_1 = {v_i : alpha(v_i) = "true"} union {overline(v)_i : alpha(v_i) = "false"}, $ - $ S_2 = {v_i : alpha(v_i) = "false"} union {overline(v)_i : alpha(v_i) = "true"}. $ - We verify that every subset in $cal(C)$ is split: - - _Complementarity subsets:_ For each $P_i = {v_i, overline(v)_i}$, - exactly one of $v_i, overline(v)_i$ is in $S_1$ and the other is in $S_2$ - by construction. Hence $P_i$ is split. - - _Clause subsets:_ For each $Q_j$, since $alpha$ is NAE-satisfying, - clause $c_j$ contains at least one true literal $ell_t$ and at least one - false literal $ell_f$. The element corresponding to a true literal is in - $S_1$: for a positive literal $v_i$, truth means $alpha(v_i) = "true"$, - so $v_i in S_1$; for a negative literal $overline(v)_i$, truth means - $alpha(v_i) = "false"$, so $overline(v)_i in S_1$. By the same - reasoning, the element corresponding to a false literal is in $S_2$. - Hence $Q_j$ has at least one element in each part and is split. - - ($arrow.l.double$) Suppose $(S_1, S_2)$ is a valid set splitting for - $(S, cal(C))$. Define the assignment $alpha$ by: - $ alpha(v_i) = cases("true" &"if" v_i in S_1, "false" &"if" v_i in S_2.) $ - This is well-defined because the complementarity subset $P_i = {v_i, overline(v)_i}$ - must be split, so exactly one of $v_i, overline(v)_i$ is in $S_1$ and the - other is in $S_2$. In particular, $overline(v)_i in S_1$ if and only if - $alpha(v_i) = "false"$. - - We verify that $alpha$ is NAE-satisfying. Consider any clause - $c_j = (ell_(j,1), dots, ell_(j,k_j))$ with corresponding subset $Q_j$. - Since $Q_j$ is split, there exist elements $ell_a in Q_j sect S_1$ and - $ell_b in Q_j sect S_2$. - - If $ell_a = v_i$ for some $i$, then $v_i in S_1$, so - $alpha(v_i) = "true"$, and the literal $v_i$ evaluates to true. - - If $ell_a = overline(v)_i$ for some $i$, then $overline(v)_i in S_1$. - Since $P_i$ is split, $v_i in S_2$, so $alpha(v_i) = "false"$, and the - literal $overline(v)_i$ evaluates to true. - By the same reasoning applied to $ell_b in S_2$, literal $ell_b$ evaluates - to false. Hence clause $c_j$ has both a true and a false literal, - satisfying the NAE condition. - - _Solution extraction._ - Given a set splitting $(S_1, S_2)$, extract the NAE-satisfying assignment - by setting $alpha(v_i) = "true"$ if and only if $v_i in S_1$. - In the binary encoding used by the codebase, the universe has $2n$ - elements indexed $0, 1, dots, 2n - 1$, where element $2(i - 1)$ - represents $v_i$ and element $2(i - 1) + 1$ represents $overline(v)_i$. - A configuration assigns each element to part 0 ($S_1$) or part 1 ($S_2$). - Variable $i$ is set to true when element $2(i - 1)$ is in part 0. -] - -*Overhead.* -#table( - columns: (auto, auto), - align: (left, left), - table.header([Target metric], [Formula]), - [`universe_size`], [$2n$ where $n =$ `num_vars`], - [`num_subsets`], [$n + m$ where $m =$ `num_clauses`], -) - -*Feasible example (YES instance).* -Consider the NAE-SAT instance with $n = 4$ variables and $m = 4$ clauses: -$ c_1 = (v_1, v_2, v_3), quad c_2 = (overline(v)_1, v_3, v_4), quad c_3 = (v_2, overline(v)_3, overline(v)_4), quad c_4 = (v_1, overline(v)_2, v_4). $ - -The constructed Set Splitting instance has universe size $2 dot 4 = 8$ and -$4 + 4 = 8$ subsets: -- Complementarity: $P_1 = {v_1, overline(v)_1}$, $P_2 = {v_2, overline(v)_2}$, - $P_3 = {v_3, overline(v)_3}$, $P_4 = {v_4, overline(v)_4}$. -- Clause: $Q_1 = {v_1, v_2, v_3}$, $Q_2 = {overline(v)_1, v_3, v_4}$, - $Q_3 = {v_2, overline(v)_3, overline(v)_4}$, $Q_4 = {v_1, overline(v)_2, v_4}$. - -Using 0-indexed elements: $v_i$ is element $2(i-1)$ and $overline(v)_i$ is -element $2(i-1)+1$, so $v_1 = 0, overline(v)_1 = 1, v_2 = 2, overline(v)_2 = 3, -v_3 = 4, overline(v)_3 = 5, v_4 = 6, overline(v)_4 = 7$. - -Subsets (0-indexed): ${0,1}, {2,3}, {4,5}, {6,7}, {0,2,4}, {1,4,6}, {2,5,7}, {0,3,6}$. - -The assignment $alpha = (v_1 = "true", v_2 = "false", v_3 = "true", v_4 = "false")$ is NAE-satisfying: -- $c_1 = (v_1, v_2, v_3) = ("T", "F", "T")$: not all equal $checkmark$ -- $c_2 = (overline(v)_1, v_3, v_4) = ("F", "T", "F")$: not all equal $checkmark$ -- $c_3 = (v_2, overline(v)_3, overline(v)_4) = ("F", "F", "T")$: not all equal $checkmark$ -- $c_4 = (v_1, overline(v)_2, v_4) = ("T", "T", "F")$: not all equal $checkmark$ - -The corresponding partition is: -$S_1 = {v_1, overline(v)_2, v_3, overline(v)_4} = {0, 3, 4, 7}$, -$S_2 = {overline(v)_1, v_2, overline(v)_3, v_4} = {1, 2, 5, 6}$. - -Verification that every subset is split: -- $P_1 = {0, 1}$: $0 in S_1$, $1 in S_2$ $checkmark$ -- $P_2 = {2, 3}$: $3 in S_1$, $2 in S_2$ $checkmark$ -- $P_3 = {4, 5}$: $4 in S_1$, $5 in S_2$ $checkmark$ -- $P_4 = {6, 7}$: $7 in S_1$, $6 in S_2$ $checkmark$ -- $Q_1 = {0, 2, 4}$: $0 in S_1$, $2 in S_2$ $checkmark$ -- $Q_2 = {1, 4, 6}$: $4 in S_1$, $1 in S_2$ $checkmark$ -- $Q_3 = {2, 5, 7}$: $7 in S_1$, $2 in S_2$ $checkmark$ -- $Q_4 = {0, 3, 6}$: $0 in S_1$, $6 in S_2$ $checkmark$ - -*Infeasible example (NO instance).* -Consider the NAE-SAT instance with $n = 3$ variables and $m = 4$ clauses: -$ c_1 = (v_1, v_2, v_3), quad c_2 = (v_1, v_2, overline(v)_3), quad c_3 = (v_1, overline(v)_2, v_3), quad c_4 = (v_1, overline(v)_2, overline(v)_3). $ - -The constructed Set Splitting instance has universe size $2 dot 3 = 6$ and -$3 + 4 = 7$ subsets: -- Complementarity: $P_1 = {v_1, overline(v)_1}$, $P_2 = {v_2, overline(v)_2}$, - $P_3 = {v_3, overline(v)_3}$. -- Clause: $Q_1 = {v_1, v_2, v_3}$, $Q_2 = {v_1, v_2, overline(v)_3}$, - $Q_3 = {v_1, overline(v)_2, v_3}$, $Q_4 = {v_1, overline(v)_2, overline(v)_3}$. - -Using 0-indexed elements: $v_1 = 0, overline(v)_1 = 1, v_2 = 2, overline(v)_2 = 3, -v_3 = 4, overline(v)_3 = 5$. - -Subsets (0-indexed): ${0,1}, {2,3}, {4,5}, {0,2,4}, {0,2,5}, {0,3,4}, {0,3,5}$. - -This instance is unsatisfiable. The complementarity subsets force a consistent -assignment. Each of the $2^3 = 8$ possible assignments violates at least one -clause: -- $alpha = ("T","T","T")$: $c_1 = ("T","T","T")$ — all equal, fails. -- $alpha = ("T","T","F")$: $c_2 = ("T","T","T")$ — all equal, fails. -- $alpha = ("T","F","T")$: $c_3 = ("T","T","T")$ — all equal, fails. -- $alpha = ("T","F","F")$: $c_4 = ("T","T","T")$ — all equal, fails. -- $alpha = ("F","T","T")$: $c_1 = ("F","T","T")$ NAE $checkmark$, $c_2 = ("F","T","F")$ NAE $checkmark$, $c_3 = ("F","F","T")$ NAE $checkmark$, $c_4 = ("F","F","F")$ — all equal, fails. -- $alpha = ("F","T","F")$: $c_4 = ("F","F","T")$ NAE $checkmark$, $c_1 = ("F","T","F")$ NAE $checkmark$, $c_2 = ("F","T","T")$ NAE $checkmark$, $c_3 = ("F","F","F")$ — all equal, fails. -- $alpha = ("F","F","T")$: $c_3 = ("F","T","T")$ NAE $checkmark$, $c_1 = ("F","F","T")$ NAE $checkmark$, $c_2 = ("F","F","F")$ — all equal, fails. -- $alpha = ("F","F","F")$: $c_1 = ("F","F","F")$ — all equal, fails. - -Since no assignment is NAE-satisfying, the corresponding Set Splitting instance -also has no valid partition: by the backward direction of the proof, any valid -partition would yield a NAE-satisfying assignment, which does not exist. diff --git a/docs/paper/verify-reductions/verify_naesat_setsplitting.py b/docs/paper/verify-reductions/verify_naesat_setsplitting.py deleted file mode 100644 index 01f7bb1e..00000000 --- a/docs/paper/verify-reductions/verify_naesat_setsplitting.py +++ /dev/null @@ -1,461 +0,0 @@ -#!/usr/bin/env python3 -"""§1.1 NAESatisfiability → SetSplitting (#841): exhaustive + structural verification.""" -import itertools -import sys -from sympy import symbols, simplify, Eq - -passed = failed = 0 - - -def check(condition, msg=""): - global passed, failed - if condition: - passed += 1 - else: - failed += 1 - print(f" FAIL: {msg}") - - -# ── Reduction implementation ────────────────────────────────────────────── - - -def literal_to_element(lit, n): - """Convert a 1-indexed signed literal to a 0-indexed universe element. - Positive literal i (1-indexed) → element 2*(i-1). - Negative literal -i → element 2*(i-1)+1. - """ - var_idx = abs(lit) - 1 # 0-indexed variable - if lit > 0: - return 2 * var_idx - else: - return 2 * var_idx + 1 - - -def reduce(n, clauses): - """Reduce NAE-SAT instance to Set Splitting. - - Args: - n: number of variables (1-indexed: v_1 .. v_n) - clauses: list of lists of signed 1-indexed literals - - Returns: - (universe_size, subsets): Set Splitting instance - """ - universe_size = 2 * n - subsets = [] - # Complementarity subsets: {2i, 2i+1} for i = 0..n-1 - for i in range(n): - subsets.append([2 * i, 2 * i + 1]) - # Clause subsets - for clause in clauses: - subset = [literal_to_element(lit, n) for lit in clause] - subsets.append(subset) - return universe_size, subsets - - -def is_nae_satisfying(n, clauses, assignment): - """Check if assignment (list of 0/1, length n) NAE-satisfies all clauses.""" - for clause in clauses: - vals = set() - for lit in clause: - var_idx = abs(lit) - 1 - val = assignment[var_idx] - if lit < 0: - val = 1 - val - vals.add(val) - if len(vals) == 1: # all equal - return False - return True - - -def is_feasible_source(n, clauses): - """Check if NAE-SAT instance is satisfiable (brute force).""" - for bits in range(2**n): - assignment = [(bits >> i) & 1 for i in range(n)] - if is_nae_satisfying(n, clauses, assignment): - return True - return False - - -def is_set_splitting(universe_size, subsets, config): - """Check if config (list of 0/1, length universe_size) is a valid set splitting.""" - for subset in subsets: - vals = set(config[e] for e in subset) - if len(vals) == 1: # monochromatic - return False - return True - - -def is_feasible_target(universe_size, subsets): - """Check if Set Splitting instance is feasible (brute force).""" - for bits in range(2**universe_size): - config = [(bits >> i) & 1 for i in range(universe_size)] - if is_set_splitting(universe_size, subsets, config): - return True - return False - - -def extract_assignment(n, config): - """Extract NAE-SAT assignment from Set Splitting config. - Variable i (0-indexed) is true iff element 2i is in part 0. - """ - return [1 - config[2 * i] for i in range(n)] - # config[2i] == 0 means element 2i is in S_1 (part 0), so variable is true (1) - # Actually: let's think about this more carefully. - # Part 0 = S_1. If v_i (element 2i) is in S_1 (config=0), then alpha(v_i)=true (1). - # So assignment[i] = 1 if config[2i] == 0, i.e., assignment[i] = 1 - config[2i]. - # Wait, but the codebase uses config[e]=0 for "part 0" and config[e]=1 for "part 1". - # The convention is: S_1 gets config=0, S_2 gets config=1. - # True literals go to S_1 (config=0), so: - # var i true => element 2i in S_1 => config[2i]=0 => assignment[i]=1-0=1 ✓ - # var i false => element 2i in S_2 => config[2i]=1 => assignment[i]=1-1=0 ✓ - - -def all_naesat_instances(n, max_clause_len=None): - """Generate all NAE-SAT instances with n variables. - Each clause has 2 or 3 literals (no variable appears twice in a clause). - """ - lits = list(range(1, n + 1)) + list(range(-n, 0)) - clause_sizes = [2, 3] if max_clause_len is None else list(range(2, max_clause_len + 1)) - - # Generate all valid clauses - all_clauses = [] - for size in clause_sizes: - for combo in itertools.combinations(lits, size): - # No variable appears both positive and negative - vars_used = [abs(l) for l in combo] - if len(set(vars_used)) == len(vars_used): - all_clauses.append(list(combo)) - - return all_clauses - - -def main(): - # === Section 1: Symbolic checks (sympy) — MANDATORY === - print("=== Section 1: Symbolic overhead verification ===") - - n_sym, m_sym = symbols("n m", positive=True, integer=True) - - # universe_size = 2n - universe_expr = 2 * n_sym - check(simplify(universe_expr - 2 * n_sym) == 0, - "universe_size should be 2n") - - # num_subsets = n + m - subsets_expr = n_sym + m_sym - check(simplify(subsets_expr - (n_sym + m_sym)) == 0, - "num_subsets should be n + m") - - # Verify for specific values - for n_val in range(2, 8): - for m_val in range(1, 8): - check(universe_expr.subs(n_sym, n_val) == 2 * n_val, - f"universe_size({n_val}) = {2*n_val}") - check(subsets_expr.subs({n_sym: n_val, m_sym: m_val}) == n_val + m_val, - f"num_subsets({n_val},{m_val}) = {n_val + m_val}") - - # Complementarity subset size is always 2 - for n_val in range(1, 10): - for i in range(n_val): - subset = [2 * i, 2 * i + 1] - check(len(subset) == 2, - f"complementarity subset for var {i} has size 2") - - print(f" Section 1: {passed} passed, {failed} failed") - - # === Section 2: Exhaustive forward + backward — MANDATORY === - print("\n=== Section 2: Exhaustive forward + backward (n ≤ 5) ===") - sec2_start = passed - - for n in range(2, 6): # n = 2, 3, 4, 5 - # Generate all valid clauses for this n - lits = list(range(1, n + 1)) + list(range(-n, 0)) - all_valid_clauses = [] - for size in [2, 3]: - for combo in itertools.combinations(lits, size): - vars_used = [abs(l) for l in combo] - if len(set(vars_used)) == len(vars_used): - all_valid_clauses.append(list(combo)) - - # Test many clause combinations - # For small n, test all single-clause, double-clause, and triple-clause combos - num_tested = 0 - max_per_n = 800 if n <= 3 else 400 - - # Single clause instances - for clause in all_valid_clauses: - if num_tested >= max_per_n: - break - clauses = [clause] - source_feasible = is_feasible_source(n, clauses) - u_size, subsets = reduce(n, clauses) - target_feasible = is_feasible_target(u_size, subsets) - - check(source_feasible == target_feasible, - f"n={n}, clauses={clauses}: source={source_feasible}, target={target_feasible}") - num_tested += 1 - - # Multi-clause instances (sample) - for num_clauses in [2, 3, 4]: - count = 0 - for combo in itertools.combinations(range(len(all_valid_clauses)), num_clauses): - if count >= max_per_n // 3: - break - clauses = [all_valid_clauses[i] for i in combo] - source_feasible = is_feasible_source(n, clauses) - u_size, subsets = reduce(n, clauses) - target_feasible = is_feasible_target(u_size, subsets) - - check(source_feasible == target_feasible, - f"n={n}, {num_clauses} clauses: source={source_feasible}, target={target_feasible}") - count += 1 - num_tested += 1 - - print(f" n={n}: tested {num_tested} instances") - - print(f" Section 2: {passed - sec2_start} new checks") - - # === Section 3: Solution extraction — MANDATORY === - print("\n=== Section 3: Solution extraction ===") - sec3_start = passed - - for n in range(2, 6): - lits = list(range(1, n + 1)) + list(range(-n, 0)) - all_valid_clauses = [] - for size in [2, 3]: - for combo in itertools.combinations(lits, size): - vars_used = [abs(l) for l in combo] - if len(set(vars_used)) == len(vars_used): - all_valid_clauses.append(list(combo)) - - num_extracted = 0 - for num_cl in [1, 2, 3]: - for combo in itertools.combinations(range(len(all_valid_clauses)), num_cl): - if num_extracted >= 300: - break - clauses = [all_valid_clauses[i] for i in combo] - - if not is_feasible_source(n, clauses): - continue - - u_size, subsets = reduce(n, clauses) - - # Find ALL valid set splitting configs and extract assignment from each - for bits in range(2**u_size): - config = [(bits >> e) & 1 for e in range(u_size)] - if not is_set_splitting(u_size, subsets, config): - continue - - assignment = extract_assignment(n, config) - check(is_nae_satisfying(n, clauses, assignment), - f"extraction n={n}: config {config} -> assignment {assignment}") - num_extracted += 1 - - print(f" n={n}: extracted {num_extracted} solutions") - - print(f" Section 3: {passed - sec3_start} new checks") - - # === Section 4: Overhead formula — MANDATORY === - print("\n=== Section 4: Overhead formula verification ===") - sec4_start = passed - - for n in range(2, 6): - lits = list(range(1, n + 1)) + list(range(-n, 0)) - all_valid_clauses = [] - for size in [2, 3]: - for combo in itertools.combinations(lits, size): - vars_used = [abs(l) for l in combo] - if len(set(vars_used)) == len(vars_used): - all_valid_clauses.append(list(combo)) - - for num_cl in [1, 2, 3, 4, 5]: - count = 0 - for combo in itertools.combinations(range(len(all_valid_clauses)), num_cl): - if count >= 50: - break - clauses = [all_valid_clauses[i] for i in combo] - m = len(clauses) - u_size, subsets = reduce(n, clauses) - - # Check overhead formula - check(u_size == 2 * n, - f"overhead universe: got {u_size}, expected {2*n}") - check(len(subsets) == n + m, - f"overhead subsets: got {len(subsets)}, expected {n + m}") - count += 1 - - print(f" Section 4: {passed - sec4_start} new checks") - - # === Section 5: Structural properties — MANDATORY === - print("\n=== Section 5: Structural properties ===") - sec5_start = passed - - for n in range(2, 6): - lits = list(range(1, n + 1)) + list(range(-n, 0)) - all_valid_clauses = [] - for size in [2, 3]: - for combo in itertools.combinations(lits, size): - vars_used = [abs(l) for l in combo] - if len(set(vars_used)) == len(vars_used): - all_valid_clauses.append(list(combo)) - - for num_cl in [1, 2, 3]: - count = 0 - for combo in itertools.combinations(range(len(all_valid_clauses)), num_cl): - if count >= 100: - break - clauses = [all_valid_clauses[i] for i in combo] - u_size, subsets = reduce(n, clauses) - - # 1. All elements in subsets are valid universe indices - for subset in subsets: - for elem in subset: - check(0 <= elem < u_size, - f"element {elem} out of range [0, {u_size})") - - # 2. Every subset has at least 2 elements - for subset in subsets: - check(len(subset) >= 2, - f"subset {subset} has fewer than 2 elements") - - # 3. No duplicate elements within a subset - for subset in subsets: - check(len(subset) == len(set(subset)), - f"subset {subset} has duplicates") - - # 4. Complementarity subsets come first and pair consecutive elements - for i in range(n): - check(subsets[i] == [2 * i, 2 * i + 1], - f"complementarity subset {i}: {subsets[i]}") - - # 5. No empty subsets - for subset in subsets: - check(len(subset) > 0, f"empty subset found") - - count += 1 - - print(f" Section 5: {passed - sec5_start} new checks") - - # === Section 6: YES example from Typst — MANDATORY === - print("\n=== Section 6: YES example verification ===") - sec6_start = passed - - # From Typst: n=4, m=4 - # c1 = (v1, v2, v3) = [1, 2, 3] - # c2 = (¬v1, v3, v4) = [-1, 3, 4] - # c3 = (v2, ¬v3, ¬v4) = [2, -3, -4] - # c4 = (v1, ¬v2, v4) = [1, -2, 4] - yes_n = 4 - yes_clauses = [[1, 2, 3], [-1, 3, 4], [2, -3, -4], [1, -2, 4]] - - u_size, subsets = reduce(yes_n, yes_clauses) - - # Check overhead - check(u_size == 8, f"YES example: universe_size = {u_size}, expected 8") - check(len(subsets) == 8, f"YES example: num_subsets = {len(subsets)}, expected 8") - - # Check exact subsets from Typst - expected_subsets = [ - [0, 1], [2, 3], [4, 5], [6, 7], # complementarity - [0, 2, 4], [1, 4, 6], [2, 5, 7], [0, 3, 6], # clause - ] - for i, (got, exp) in enumerate(zip(subsets, expected_subsets)): - check(got == exp, f"YES subset {i}: got {got}, expected {exp}") - - # Verify the assignment from Typst: v1=T, v2=F, v3=T, v4=F → [1, 0, 1, 0] - yes_assignment = [1, 0, 1, 0] - check(is_nae_satisfying(yes_n, yes_clauses, yes_assignment), - "YES example: assignment is NAE-satisfying") - - # Verify the partition from Typst: S1 = {0, 3, 4, 7} - # config: element in S1 → config=0, element in S2 → config=1 - # S1 = {0, 3, 4, 7} → config[0]=0, config[3]=0, config[4]=0, config[7]=0 - # S2 = {1, 2, 5, 6} → config[1]=1, config[2]=1, config[5]=1, config[6]=1 - yes_config = [0, 1, 1, 0, 0, 1, 1, 0] - check(is_set_splitting(u_size, subsets, yes_config), - "YES example: partition is a valid set splitting") - - # Verify extraction - extracted = extract_assignment(yes_n, yes_config) - check(extracted == yes_assignment, - f"YES example: extracted {extracted}, expected {yes_assignment}") - - # Verify each clause explicitly as in Typst - # c1 = (T, F, T) - check(yes_assignment[0] == 1 and yes_assignment[1] == 0 and yes_assignment[2] == 1, - "YES c1 values") - # c2 = (¬v1=F, v3=T, v4=F) - check((1 - yes_assignment[0]) == 0 and yes_assignment[2] == 1 and yes_assignment[3] == 0, - "YES c2 values") - # c3 = (v2=F, ¬v3=F, ¬v4=T) - check(yes_assignment[1] == 0 and (1 - yes_assignment[2]) == 0 and (1 - yes_assignment[3]) == 1, - "YES c3 values") - # c4 = (v1=T, ¬v2=T, v4=F) - check(yes_assignment[0] == 1 and (1 - yes_assignment[1]) == 1 and yes_assignment[3] == 0, - "YES c4 values") - - check(is_feasible_source(yes_n, yes_clauses), - "YES example: source is feasible") - check(is_feasible_target(u_size, subsets), - "YES example: target is feasible") - - print(f" Section 6: {passed - sec6_start} new checks") - - # === Section 7: NO example from Typst — MANDATORY === - print("\n=== Section 7: NO example verification ===") - sec7_start = passed - - # From Typst: n=3, m=4 - # c1 = (v1, v2, v3) = [1, 2, 3] - # c2 = (v1, v2, ¬v3) = [1, 2, -3] - # c3 = (v1, ¬v2, v3) = [1, -2, 3] - # c4 = (v1, ¬v2, ¬v3) = [1, -2, -3] - no_n = 3 - no_clauses = [[1, 2, 3], [1, 2, -3], [1, -2, 3], [1, -2, -3]] - - u_size_no, subsets_no = reduce(no_n, no_clauses) - - # Check overhead - check(u_size_no == 6, f"NO example: universe_size = {u_size_no}, expected 6") - check(len(subsets_no) == 7, f"NO example: num_subsets = {len(subsets_no)}, expected 7") - - # Check exact subsets from Typst - expected_no_subsets = [ - [0, 1], [2, 3], [4, 5], # complementarity - [0, 2, 4], [0, 2, 5], [0, 3, 4], [0, 3, 5], # clause - ] - for i, (got, exp) in enumerate(zip(subsets_no, expected_no_subsets)): - check(got == exp, f"NO subset {i}: got {got}, expected {exp}") - - # Verify source is infeasible - check(not is_feasible_source(no_n, no_clauses), - "NO example: source is infeasible") - - # Verify target is infeasible - check(not is_feasible_target(u_size_no, subsets_no), - "NO example: target is infeasible") - - # Verify each of the 8 assignments fails as stated in Typst - for bits in range(8): - assignment = [(bits >> i) & 1 for i in range(3)] - check(not is_nae_satisfying(no_n, no_clauses, assignment), - f"NO example: assignment {assignment} should fail NAE") - - # Verify no set splitting config works - no_splitting_count = 0 - for bits in range(2**6): - config = [(bits >> e) & 1 for e in range(6)] - if not is_set_splitting(u_size_no, subsets_no, config): - no_splitting_count += 1 - check(no_splitting_count == 64, - f"NO example: all 64 configs fail set splitting (got {64 - no_splitting_count} valid)") - - print(f" Section 7: {passed - sec7_start} new checks") - - # ── Final report ── - print(f"\nNAESatisfiability → SetSplitting: {passed} passed, {failed} failed") - return 1 if failed else 0 - - -if __name__ == "__main__": - sys.exit(main()) diff --git a/src/rules/mod.rs b/src/rules/mod.rs index fabfa8c2..f87cf124 100644 --- a/src/rules/mod.rs +++ b/src/rules/mod.rs @@ -59,6 +59,7 @@ pub(crate) mod minimumvertexcover_minimumfeedbackvertexset; pub(crate) mod minimumvertexcover_minimumhittingset; pub(crate) mod minimumvertexcover_minimumsetcovering; pub(crate) mod naesatisfiability_maxcut; +pub(crate) mod naesatisfiability_setsplitting; pub(crate) mod paintshop_qubo; pub(crate) mod partition_binpacking; pub(crate) mod partition_cosineproductintegration; @@ -341,6 +342,7 @@ pub(crate) fn canonical_rule_example_specs() -> Vec &Self::Target { + &self.target + } + + fn extract_solution(&self, target_solution: &[usize]) -> Vec { + // Variable i is true iff element 2i is in part 0 (color 0). + // So assignment[i] = 1 - target_solution[2*i]. + (0..self.num_vars) + .map(|i| 1 - target_solution[2 * i]) + .collect() + } +} + +/// Convert a 1-indexed signed literal to a 0-indexed universe element. +/// +/// Positive literal i -> element 2*(i-1). +/// Negative literal -i -> element 2*(i-1)+1. +fn literal_to_element(lit: i32) -> usize { + let var_idx = (lit.unsigned_abs() as usize) - 1; + if lit > 0 { + 2 * var_idx + } else { + 2 * var_idx + 1 + } +} + +#[reduction(overhead = { + universe_size = "2 * num_vars", + num_subsets = "num_clauses + num_vars", +})] +impl ReduceTo for NAESatisfiability { + type Result = ReductionNAESATToSetSplitting; + + fn reduce_to(&self) -> Self::Result { + let n = self.num_vars(); + let universe_size = 2 * n; + + let mut subsets = Vec::with_capacity(n + self.num_clauses()); + + // Complementarity subsets: {2i, 2i+1} for each variable i + for i in 0..n { + subsets.push(vec![2 * i, 2 * i + 1]); + } + + // Clause subsets: map each literal to its universe element + for clause in self.clauses() { + let subset: Vec = clause + .literals + .iter() + .map(|&lit| literal_to_element(lit)) + .collect(); + subsets.push(subset); + } + + let target = SetSplitting::new(universe_size, subsets); + + ReductionNAESATToSetSplitting { + num_vars: n, + target, + } + } +} + +#[cfg(feature = "example-db")] +pub(crate) fn canonical_rule_example_specs() -> Vec { + use crate::export::SolutionPair; + use crate::models::formula::CNFClause; + + vec![crate::example_db::specs::RuleExampleSpec { + id: "naesatisfiability_to_setsplitting", + build: || { + // YES instance from test vectors: n=4, + // clauses=[[1,2,3],[-1,3,4],[2,-3,-4],[1,-2,4]] + let source = NAESatisfiability::new( + 4, + vec![ + CNFClause::new(vec![1, 2, 3]), + CNFClause::new(vec![-1, 3, 4]), + CNFClause::new(vec![2, -3, -4]), + CNFClause::new(vec![1, -2, 4]), + ], + ); + // source_solution: [1,0,1,0] means x1=T, x2=F, x3=T, x4=F + // target_config: element 2i = 1 - assignment[i] + // x1=T -> config[0]=0, config[1]=1 + // x2=F -> config[2]=1, config[3]=0 + // x3=T -> config[4]=0, config[5]=1 + // x4=F -> config[6]=1, config[7]=0 + crate::example_db::specs::rule_example_with_witness::<_, SetSplitting>( + source, + SolutionPair { + source_config: vec![1, 0, 1, 0], + target_config: vec![0, 1, 1, 0, 0, 1, 1, 0], + }, + ) + }, + }] +} + +#[cfg(test)] +#[path = "../unit_tests/rules/naesatisfiability_setsplitting.rs"] +mod tests; diff --git a/src/unit_tests/rules/naesatisfiability_setsplitting.rs b/src/unit_tests/rules/naesatisfiability_setsplitting.rs new file mode 100644 index 00000000..e0522a12 --- /dev/null +++ b/src/unit_tests/rules/naesatisfiability_setsplitting.rs @@ -0,0 +1,188 @@ +use super::*; +use crate::models::formula::CNFClause; +use crate::rules::test_helpers::assert_satisfaction_round_trip_from_satisfaction_target; +use crate::solvers::BruteForce; +use crate::traits::Problem; + +#[test] +fn test_naesatisfiability_to_setsplitting_closed_loop() { + // YES instance from test vectors: n=4 + // clauses: (x1,x2,x3), (-x1,x3,x4), (x2,-x3,-x4), (x1,-x2,x4) + let naesat = NAESatisfiability::new( + 4, + vec![ + CNFClause::new(vec![1, 2, 3]), + CNFClause::new(vec![-1, 3, 4]), + CNFClause::new(vec![2, -3, -4]), + CNFClause::new(vec![1, -2, 4]), + ], + ); + + let reduction = ReduceTo::::reduce_to(&naesat); + + assert_satisfaction_round_trip_from_satisfaction_target( + &naesat, + &reduction, + "NAESAT->SetSplitting closed loop", + ); +} + +#[test] +fn test_naesatisfiability_to_setsplitting_infeasible() { + // NO instance from test vectors: n=3 + // clauses: (x1,x2,x3), (x1,x2,-x3), (x1,-x2,x3), (x1,-x2,-x3) + // This is infeasible because x1 must be both true and false. + let naesat = NAESatisfiability::new( + 3, + vec![ + CNFClause::new(vec![1, 2, 3]), + CNFClause::new(vec![1, 2, -3]), + CNFClause::new(vec![1, -2, 3]), + CNFClause::new(vec![1, -2, -3]), + ], + ); + + let reduction = ReduceTo::::reduce_to(&naesat); + let target = reduction.target_problem(); + + let solver = BruteForce::new(); + let naesat_solutions = solver.find_all_witnesses(&naesat); + let splitting_solutions = solver.find_all_witnesses(target); + + assert!(naesat_solutions.is_empty(), "NAESAT should be infeasible"); + assert!( + splitting_solutions.is_empty(), + "SetSplitting should also be infeasible" + ); +} + +#[test] +fn test_reduction_structure() { + let naesat = NAESatisfiability::new( + 4, + vec![ + CNFClause::new(vec![1, 2, 3]), + CNFClause::new(vec![-1, 3, 4]), + CNFClause::new(vec![2, -3, -4]), + CNFClause::new(vec![1, -2, 4]), + ], + ); + + let reduction = ReduceTo::::reduce_to(&naesat); + let target = reduction.target_problem(); + + // universe_size = 2 * num_vars = 8 + assert_eq!(target.universe_size(), 8); + // num_subsets = num_vars + num_clauses = 4 + 4 = 8 + assert_eq!(target.num_subsets(), 8); + + // First 4 subsets are complementarity: {0,1}, {2,3}, {4,5}, {6,7} + let subsets = target.subsets(); + for i in 0..4 { + assert_eq!(subsets[i], vec![2 * i, 2 * i + 1]); + } + + // Clause subsets follow the literal mapping + // Clause [1,2,3] -> [0,2,4] + assert_eq!(subsets[4], vec![0, 2, 4]); + // Clause [-1,3,4] -> [1,4,6] + assert_eq!(subsets[5], vec![1, 4, 6]); + // Clause [2,-3,-4] -> [2,5,7] + assert_eq!(subsets[6], vec![2, 5, 7]); + // Clause [1,-2,4] -> [0,3,6] + assert_eq!(subsets[7], vec![0, 3, 6]); +} + +#[test] +fn test_solution_extraction() { + let naesat = NAESatisfiability::new( + 4, + vec![ + CNFClause::new(vec![1, 2, 3]), + CNFClause::new(vec![-1, 3, 4]), + ], + ); + + let reduction = ReduceTo::::reduce_to(&naesat); + + // target_solution: [0, 1, 1, 0, 0, 1, 1, 0] + // assignment[i] = 1 - target_solution[2*i] + // x1 = 1-0 = 1 (true), x2 = 1-1 = 0 (false), x3 = 1-0 = 1 (true), x4 = 1-1 = 0 (false) + let extracted = reduction.extract_solution(&[0, 1, 1, 0, 0, 1, 1, 0]); + assert_eq!(extracted, vec![1, 0, 1, 0]); +} + +#[test] +fn test_all_satisfying_assignments_map_back() { + // Small instance: verify every SetSplitting solution maps to a valid NAESAT solution + let naesat = NAESatisfiability::new( + 3, + vec![ + CNFClause::new(vec![1, 2, -3]), + CNFClause::new(vec![-1, 3]), + ], + ); + + let reduction = ReduceTo::::reduce_to(&naesat); + let target = reduction.target_problem(); + + let solver = BruteForce::new(); + let splitting_solutions = solver.find_all_witnesses(target); + + assert!( + !splitting_solutions.is_empty(), + "SetSplitting should have solutions" + ); + + for sol in &splitting_solutions { + let naesat_sol = reduction.extract_solution(sol); + assert_eq!(naesat_sol.len(), 3); + assert!( + naesat.evaluate(&naesat_sol).0, + "Extracted solution {:?} from splitting solution {:?} does not satisfy NAESAT", + naesat_sol, + sol + ); + } +} + +#[test] +fn test_larger_instance_closed_loop() { + let naesat = NAESatisfiability::new( + 5, + vec![ + CNFClause::new(vec![1, 2, -3]), + CNFClause::new(vec![-1, 3, 4]), + CNFClause::new(vec![2, -4, 5]), + CNFClause::new(vec![-2, 3, -5]), + CNFClause::new(vec![1, -3, 5]), + ], + ); + + let reduction = ReduceTo::::reduce_to(&naesat); + + assert_satisfaction_round_trip_from_satisfaction_target( + &naesat, + &reduction, + "NAESAT->SetSplitting larger instance", + ); +} + +#[test] +fn test_two_variable_instance() { + // Minimal instance with 2 variables + let naesat = NAESatisfiability::new(2, vec![CNFClause::new(vec![1, -2])]); + + let reduction = ReduceTo::::reduce_to(&naesat); + let target = reduction.target_problem(); + + assert_eq!(target.universe_size(), 4); + // 2 complementarity + 1 clause = 3 subsets + assert_eq!(target.num_subsets(), 3); + + assert_satisfaction_round_trip_from_satisfaction_target( + &naesat, + &reduction, + "NAESAT->SetSplitting two-variable", + ); +} From bd516ef626b4d8c46cf7ed94f2863cb853b9bce2 Mon Sep 17 00:00:00 2001 From: zazabap Date: Wed, 1 Apr 2026 14:02:55 +0000 Subject: [PATCH 3/3] =?UTF-8?q?fix:=20replace=20needless=5Frange=5Floop=20?= =?UTF-8?q?with=20enumerate().take(4)=20in=20NAE=E2=86=92SS=20test?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Resolves clippy -D warnings failure on Rust 1.94 CI. Co-Authored-By: Claude Sonnet 4.6 --- src/unit_tests/rules/naesatisfiability_setsplitting.rs | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/unit_tests/rules/naesatisfiability_setsplitting.rs b/src/unit_tests/rules/naesatisfiability_setsplitting.rs index e0522a12..5c2282b3 100644 --- a/src/unit_tests/rules/naesatisfiability_setsplitting.rs +++ b/src/unit_tests/rules/naesatisfiability_setsplitting.rs @@ -78,8 +78,8 @@ fn test_reduction_structure() { // First 4 subsets are complementarity: {0,1}, {2,3}, {4,5}, {6,7} let subsets = target.subsets(); - for i in 0..4 { - assert_eq!(subsets[i], vec![2 * i, 2 * i + 1]); + for (i, subset) in subsets.iter().enumerate().take(4) { + assert_eq!(*subset, vec![2 * i, 2 * i + 1]); } // Clause subsets follow the literal mapping