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CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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NumberSAT is not a Boolean membership predicate

Statement

With declared variables (x,y), let φ=x∨y and ψ=(x∨y)∧(¬x∨¬y). Then NumberSAT returns 3 and 2, respectively, although the SAT membership predicate returns yes on both.

Facts & Assumptions

Given: the two displayed formulas and their declared-variable lists.

[L1]

NumberSAT is a numerical function rather than a Boolean predicate. by NumberSAT.

Counterexample

technique · direct
1.1

Exactly three assignments satisfy x∨y. Exactly the two assignments with unequal bits satisfy (x∨y)∧(¬x∨¬y). Hence [L1] gives outputs 3 and 2, neither in the Boolean codomain {0,1}.

L1givenalgebra
2.1

Both counts are positive, so the associated SAT predicate maps both inputs to yes. It therefore discards information that NumberSAT retains.

step 1.1
3.1

These explicit values show that NumberSAT is a numerical function, not a Boolean membership predicate.

L1step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources