Seminar on Decision Procedures
By Ruzica Piskac
Seminar on Decision Procedures
Ruzica Piskac
Problem 1
Semantic Tableaux
Using the semantic argument, prove that the following formulas are either valid, or
unsatisfiable, or neither. In the last case use a tableaux to construct a falsifying
and a satisfying interpretation. Make sure to respect the correct precedence of the
connectives.
• P ∧Q→P →Q
• (P → Q → R) → ¬R → ¬Q → ¬P
• (P → Q ∨ R) → P → R
Problem 2
Normal Forms
Convert the following formulas to CNF using the linear time transformations introduced in the lectures.
• ¬(P → Q)
• ¬(¬(P ∧ Q) → ¬R)
• (Q ∧ R → (P ∨ ¬Q)) ∧ (P ∨ R)
Problem 3
DPLL
Describe the execution of the DPLL algorithm on the following formulas
• (P ∨ ¬Q ∨ ¬R) ∧ (Q ∨ ¬P ∨ R) ∧ (R ∨ ¬Q)
• (P ∨ Q ∨ R) ∧ (¬P ∨ ¬Q ∨ ¬R) ∧ (¬P ∨ Q ∨ R) ∧ (¬Q ∨ R) ∧ (Q ∨ ¬R)
1
Seminar on Decision Procedures
Problem 4
By Ruzica Piskac
FOL validity
Which of the following formulae is valid. If it is valid, give a proof using the definition
of the semantics of first-order logic; otherwise, give a falsifying interpretation.
1. ∃x.equals(x, y)
2. ∀z.equals(z, z) → ∃x.equals(x, y)
3. ∀x, y.(p(x, y) ∨ p(y, x)) → ∀z.p(z, z)
4. ∃x.p(x) → ∀y.p(y)
5. ∃x.(p(x) → ∀y.p(y))
Problem 5
Clausal Normal Form
1. Transform the following formula into prenex normal form:
∀z. (∀x.q(x, z)) → p(x, g(y), z)
∧ ¬ ∀z.¬(∀x.q(f (x, y), z))
2. Let F be a formula derived in the first step. Universally quantify all the free
variables of F and denote this new formula with G
3. Compute the clausal (normal) form of G
Problem 6
Herbrand interpretations
Let Σ = (Ω, Π) with Ω = {a/0 , b/0 } and Π = {p/0 , q/1 , r/2 }.
(1)
How many different Herbrand interpretations over Σ do exist? Explain.
(2)
How many different Herbrand models over Σ does the universally quantified clause
∀x.(¬p ∨ q(a) ∨ ¬r(x, b)) have? Explain.
2