[Rule] 3SAT to Consistency of Database Frequency Tables

[isPANN]

Source: 3SAT
Target: Consistency of Database Frequency Tables
Motivation: Establishes NP-completeness of Consistency of Database Frequency Tables via polynomial-time reduction from 3SAT. This result has practical implications for statistical database security: it shows that no polynomial-time algorithm can determine whether a set of published frequency tables can be used to "compromise" a database by deducing specific attribute values of individual records, unless P = NP. The reduction encodes Boolean variables as attribute values and clauses as frequency table constraints, so that satisfying all clauses corresponds to finding an assignment of attribute values consistent with the given frequency tables.

Reference: Garey & Johnson, Computers and Intractability, Appendix A4.3, p.235

GJ Source Entry

[SR35] CONSISTENCY OF DATABASE FREQUENCY TABLES
INSTANCE: Set A of attribute names, domain set D_a for each a E A, set V of objects, collection F of frequency tables for some pairs a,b E A (where a frequency table for a,b E A is a function f_{a,b}: D_a × D_b → Z+ with the sum, over all pairs x E D_a and y E D_b, of f_{a,b}(x,y) equal to |V|), and a set K of triples (v,a,x) with v E V, a E A, and x E D_a, representing the known attribute values.
QUESTION: Are the frequency tables in F consistent with the known attribute values in K, i.e., is there a collection of functions g_a: V → D_a, for each a E A, such that g_a(v) = x if (v,a,x) E K and such that, for each f_{a,b} E F, x E D_a, and y E D_b, the number of v E V for which g_a(v) = x and g_b(v) = y is exactly f_{a,b}(x,y)?
Reference: [Reiss, 1977b]. Transformation from 3SAT.
Comment: Above result implies that no polynomial time algorithm can be given for "compromising" a data base from its frequency tables by deducing prespecified attribute values, unless P = NP (see reference for details).

Reduction Algorithm

Summary:
Given a 3SAT instance with variables x_1, ..., x_n and clauses C_1, ..., C_m (each clause having exactly 3 literals), construct a Consistency of Database Frequency Tables instance as follows:

1. Object construction: Create one object v_i for each variable x_i in the 3SAT formula. Thus |V| = n (the number of variables).

2. Attribute construction for variables: Create one attribute a_i for each variable x_i, with domain D_{a_i} = {T, F} (representing True and False). The assignment g_{a_i}(v_i) encodes the truth value of variable x_i.

3. Attribute construction for clauses: For each clause C_j = (l_{j1} ∨ l_{j2} ∨ l_{j3}), create an attribute b_j with domain D_{b_j} = {1, 2, 3, ..., 7} representing which of the 7 satisfying truth assignments for the 3 literals in C_j is realized. (There are 2^3 - 1 = 7 ways to satisfy a 3-literal clause.)

4. Frequency table construction: For each clause C_j involving variables x_{p}, x_{q}, x_{r}:

   • Create frequency tables f_{a_p, b_j}, f_{a_q, b_j}, and f_{a_r, b_j} that encode the relationship between the truth value of each variable and the satisfying assignment chosen for clause C_j.
   • The frequency table f_{a_p, b_j}(T, k) = 1 if the k-th satisfying assignment of C_j has x_p = True, and 0 otherwise (similarly for F). These tables enforce that the attribute value of object v_p (the truth value of x_p) is consistent with the satisfying assignment chosen for clause C_j.

5. Known attribute values (K): The set K is initially empty (no attribute values are pre-specified), or may contain specific triples to encode unit propagation constraints.

6. Marginal consistency constraints: Additional frequency tables between variable-attributes a_p and a_q for variables appearing together in clauses enforce that each object v_i has a unique, globally consistent truth value.

7. Solution extraction: The frequency tables in F are consistent with K if and only if there exists an assignment of truth values to x_1, ..., x_n that satisfies all clauses. A consistent set of functions g_a corresponds directly to a satisfying assignment.

Key invariant: Each object represents a Boolean variable, each variable-attribute encodes {T, F}, and the frequency tables between variable-attributes and clause-attributes ensure that every clause has at least one true literal — which is exactly the 3SAT satisfiability condition.

Size Overhead

Symbols:

• n = number of variables in the 3SAT instance
• m = number of clauses in the 3SAT instance

Target metric (code name)	Polynomial (using symbols above)
num_objects	num_variables
num_attributes	num_variables + num_clauses
num_frequency_tables	3 * num_clauses

Derivation:

• Objects: one per Boolean variable -> |V| = n
• Attributes: one per variable (domain {T, F}) plus one per clause (domain {1,...,7}) -> |A| = n + m
• Frequency tables: 3 tables per clause (one for each literal's variable paired with the clause attribute) -> |F| = 3m
• Domain sizes: variable attributes have |D| = 2; clause attributes have |D| <= 7
• Known values: |K| = O(n) at most (possibly empty)

Validation Method

• Closed-loop test: reduce a 3SAT instance to a Consistency of Database Frequency Tables instance, solve the consistency problem by brute-force enumeration of all possible attribute-value assignments, extract the truth assignment, and verify it satisfies all original clauses
• Check that the number of objects, attributes, and frequency tables matches the overhead formula
• Test with a 3SAT instance that is satisfiable and verify that at least one consistent assignment exists
• Test with an unsatisfiable 3SAT instance and verify that no consistent assignment exists
• Verify that frequency table marginals sum to |V| as required by the problem definition

Example

Source instance (3SAT):
Variables: x_1, x_2, x_3, x_4, x_5, x_6
Clauses (7 clauses):

• C_1 = (x_1 ∨ x_2 ∨ x_3)
• C_2 = (¬x_1 ∨ x_4 ∨ x_5)
• C_3 = (¬x_2 ∨ ¬x_3 ∨ x_6)
• C_4 = (x_1 ∨ ¬x_4 ∨ ¬x_6)
• C_5 = (¬x_1 ∨ x_3 ∨ ¬x_5)
• C_6 = (x_2 ∨ ¬x_5 ∨ x_6)
• C_7 = (¬x_3 ∨ x_4 ∨ ¬x_6)

Satisfying assignment: x_1=T, x_2=T, x_3=F, x_4=T, x_5=F, x_6=T

• C_1: x_1=T ✓
• C_2: ¬x_1=F, x_4=T ✓
• C_3: ¬x_2=F, ¬x_3=T ✓
• C_4: x_1=T ✓
• C_5: ¬x_1=F, x_3=F, ¬x_5=T ✓
• C_6: x_2=T ✓
• C_7: ¬x_3=T ✓

Constructed target instance (Consistency of Database Frequency Tables):
Objects V = {v_1, v_2, v_3, v_4, v_5, v_6} (6 objects, one per variable)
Attributes A:

• Variable attributes: a_1, a_2, a_3, a_4, a_5, a_6 (domain {T, F} each)
• Clause attributes: b_1, b_2, b_3, b_4, b_5, b_6, b_7 (domain {1,...,7} each)

Total: 13 attributes

Frequency tables F (21 tables, 3 per clause):

• For C_1 = (x_1 ∨ x_2 ∨ x_3): tables f_{a_1,b_1}, f_{a_2,b_1}, f_{a_3,b_1}
• For C_2 = (¬x_1 ∨ x_4 ∨ x_5): tables f_{a_1,b_2}, f_{a_4,b_2}, f_{a_5,b_2}
• (... similarly for C_3 through C_7 ...)

Example frequency table f_{a_1, b_1} (for variable x_1 in clause C_1 = (x_1 ∨ x_2 ∨ x_3)):
The 7 satisfying assignments of (x_1 ∨ x_2 ∨ x_3) are:
1: (T,T,T), 2: (T,T,F), 3: (T,F,T), 4: (T,F,F), 5: (F,T,T), 6: (F,T,F), 7: (F,F,T)

a_1 \ b_1	1	2	3	4	5	6	7
T	*	*	*	*	0	0	0
F	0	0	0	0	*	*	*

(Entries marked * are determined by the assignment; each column sums to the number of objects that realize that satisfying pattern.)

Known values K = {} (empty)

Solution mapping:

• The satisfying assignment x_1=T, x_2=T, x_3=F, x_4=T, x_5=F, x_6=T corresponds to:
  • g_{a_1}(v_1) = T, g_{a_2}(v_2) = T, g_{a_3}(v_3) = F, g_{a_4}(v_4) = T, g_{a_5}(v_5) = F, g_{a_6}(v_6) = T
• For clause C_1 = (x_1 ∨ x_2 ∨ x_3) with assignment (T, T, F): this matches satisfying pattern Implement remaining reduction rules #2 (T,T,F)
• The frequency tables are consistent with these attribute functions ✓
• All frequency table marginals sum to |V| = 6 ✓

References

• [Reiss, 1977b]: [Reiss1977b] S. P. Reiss (1977). "Statistical database confidentiality". Dept. of Statistics, University of Stockholm.

[zazabap]

Issue Quality Check — Rule

Check	Result	Details
Usefulness	✅ Pass	No existing reduction path from KSatisfiability to ConsistencyOfDatabaseFrequencyTables
Non-trivial	✅ Pass	Encodes Boolean variables as attribute values and clauses as frequency table constraints
Correctness	⚠️ Warn	GJ reference is reliable but the cited Reiss 1977b is a department technical report that could not be independently verified online
Well-written	❌ Fail	Multiple issues: overhead metric names mismatch target getters, algorithm has significant gaps, example is incomplete

Overall: 2 passed, 1 warning, 1 failed

---

Usefulness

pred path KSatisfiability ConsistencyOfDatabaseFrequencyTables returns "No reduction path." This is a novel reduction that would connect KSatisfiability(K3) to a currently isolated problem (ConsistencyOfDatabaseFrequencyTables currently only reduces to ILP, with nothing reducing to it). Adding this rule would establish the NP-hardness provenance of the problem in the reduction graph.

The motivation is substantive and well-articulated: it explains the practical implications for statistical database security and the connection between satisfying clauses and frequency table consistency.

Non-trivial

The reduction involves genuine structural transformation: Boolean variables are encoded as database objects with binary-domain attributes, clauses are encoded as attributes with 7-element domains (representing satisfying assignments), and frequency tables enforce clause satisfaction. This is not a variable relabeling or subtype coercion.

Correctness

Garey & Johnson (1979), Appendix A4.3, p. 235: This is a highly reliable, well-known source. The GJ entry for SR35 (Consistency of Database Frequency Tables) is quoted verbatim and appears accurate — the problem definition matches what is implemented in the codebase.

[Reiss, 1977b]: Cited as "S. P. Reiss (1977). 'Statistical database confidentiality'. Dept. of Statistics, University of Stockholm." This is a department technical report from 1977 and could not be independently located via web search (not indexed in standard digital libraries). However, since Garey & Johnson cite it as the source of the 3SAT transformation, and GJ is a thoroughly peer-reviewed and canonical reference, the claim is credible. Not marking as a failure, but flagging as a warning since the original construction details cannot be independently verified.

Algorithm correctness concern: The described reduction has a fundamental conceptual gap (see Well-written section below) — it is unclear whether the algorithm as written actually constitutes a valid reduction. The GJ entry confirms the existence of such a reduction, but the issue's reconstruction of the algorithm may not be faithful to Reiss's original construction.

Well-written

Failures:

1. Size overhead metric names do not match target getter methods. The overhead table uses num_objects, num_attributes, num_frequency_tables, but pred show ConsistencyOfDatabaseFrequencyTables --json reports the actual size_fields as: num_assignment_indicators, num_assignment_variables, num_auxiliary_frequency_indicators, num_frequency_cells, num_known_values. The overhead table must use these actual field names.

2. Algorithm Step 4 is incomplete. The frequency table f_{a_1, b_1} shows entries marked * described as "determined by the assignment" without giving concrete values. A frequency table must contain non-negative integers that sum to |V|, but the algorithm does not specify what these integers should be. Since |V| = n and each column of the frequency table represents a specific satisfying pattern of the clause, the entries need to count how many objects realize that pattern — but the algorithm never explains this counting.

3. Algorithm Step 6 is a hand-wave. "Additional frequency tables between variable-attributes... enforce that each object has a unique, globally consistent truth value" is not a step-by-step procedure. It is critical to specify exactly which frequency tables are created, what their entries are, and why they enforce global consistency. Without this, the algorithm is not implementable.

4. Fundamental gap in the object-attribute mapping. The construction creates one object per variable (|V| = n) and one attribute per variable. But in the CDFT problem, every object must have a value for every attribute. The algorithm defines g_{a_i}(v_i) as the truth value of x_i, but never defines g_{a_i}(v_j) for i != j. What is object v_2's value for attribute a_1? This is a critical gap — the construction needs to specify these cross-assignments or restructure the object model entirely.

5. Example is incomplete. The example says "(... similarly for C_3 through C_7 ...)" instead of showing all frequency tables. The frequency table shown uses * placeholders instead of concrete values. A fully worked example must show the complete construction with actual numbers so it can be verified by hand.

6. Example round-trip quality is uncertain. While the source instance has 6 variables and 7 clauses (which is a reasonable size), the example does not demonstrate a complete round-trip: it does not show the full target instance with all 21 frequency tables and their concrete entries, making it impossible to verify the reduction is correct.

7. Symbol n and m are defined in the overhead section but the algorithm uses them inconsistently. The algorithm uses both n (number of variables) and symbolic names like x_1, ..., x_n but the overhead table switches to num_variables and num_clauses without bridging the notation.

Recommendations

• The algorithm appears to be an AI-generated reconstruction rather than a faithful reproduction of Reiss's original construction. Consider locating the original paper (or a textbook that reproduces the construction in detail) to verify the reduction algorithm.
• The object-attribute mapping issue (point 4 above) may indicate that the construction needs a fundamentally different approach — perhaps one object per (variable, clause) pair, or a different encoding altogether.
• The overhead table should be rewritten using the actual target size_fields: num_assignment_indicators, num_assignment_variables, num_auxiliary_frequency_indicators, num_frequency_cells, num_known_values.