Combinatorial Problem Solving (CPS)
Propositional Satisfiability. Propositional Logic.
1. Let F be a formula. Show that F is a tautology if and only if ¬F is unsatisfiable.
2. Let F and G be two formulas. Show that F |= G if and only if F ∧ ¬G is unsatisfiable.
3. Let F and G be two formulas. Show that F ≡ G if and only if (F ∧¬G)∨(G∧¬F ) is unsatisfiable.
4. (Substitution Lemma) Show that, if in a formula F we replace an occurrence of a subformula
G by another formula G0 which is logically equivalent, then the formula F 0 that is obtained is
logically equivalent to F .
5. Let us define the ↓ connective (called nor ) with the following semantics:
evalI (F ↓ G) = 1 − max(evalI (F ), evalI (G)).
Show that for all formula there exists a logically equivalent one with only ↓ connectives.
6. Given two interpretations I and I 0 over a vocabulary P, we write I ≤ I 0 if for all propositional
symbol p ∈ P it holds that I(p) ≤ I 0 (p).
We say that a formula F is monotone when for all pair of interpretations I and I 0 , if I ≤ I 0 then
evalI (F ) ≤ evalI 0 (F ).
(a) Show that any formula without negations is monotone.
(b) Show that for any monotone formula there is a logically equivalent one without negations.
7. Show that Tseitin transformation does not preserve logical equivalence.
8. Let Res(S) be the closure of a set of clauses S under resolution. Show that resolution is correct:
if C ∈ Res(S) then S |= C.
9. Show that resolution is refutationally complete: if S is unsatisfiable then ∈ Res(S).
10. Show that if S is finite then Res(S) is finite.
11. We say a literal in a formula in CNF is pure if it appears always positively or always negatively.
We say a clause is redundant if it contains at least one pure literal. Show that, if C 0 is a
redundant clause in the CNF {C1 , . . . , Cn , C 0 }, then this CNF is satisfiable iff {C1 , . . . , Cn } is
satisfiable.