Boolean Advanced Quiz 3

1. What mathematical concept defines a subset of a Boolean algebra that is closed under join, down-closed, and contains the bottom element?

2. What dual concept of an ideal defines a subset of a Boolean algebra closed under meet, up-closed, and containing the top element?

3. An ideal in a Boolean algebra is classified as a prime ideal if and only if it is what type of ideal?

4. According to the Boolean Prime Ideal Theorem, every proper ideal in a Boolean algebra can be extended to what?

5. What name is given to a Boolean function whose output depends only on the number of true inputs, rather than their specific positions?

6. What structural metric measures the size of the smallest Binary Decision Diagram (BDD) representing a specific Boolean function?

7. Which hardware verification method uses Boolean satisfiability (SAT) solvers to check design invariants up to a fixed execution depth?

8. What complexity class contains all Boolean functions that can be computed by polynomial-size, bounded-depth circuits with threshold gates?

9. What non-zero element $a$ in a Boolean algebra cannot be decomposed into a join of lower non-zero elements ($x \vee y = a \implies x=a \text{ or } y=a$)?

10. What term describes a Boolean algebra where every non-zero element satisfies the condition of bounding at least one atom?