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UIUC, CS498, Section EA - Autumn 04 - Midterm Sample
Given by: Eyal Amir
October 29, 2004
1 Short Questions (25%)
- Determine whether each of the following are valid, satisfiable (but not valid), or unsatisfiable. No explanation necessary.
(a) (happy ⇒ productive) ∧ happy ∧ ¬productive (b) (happy ⇒ productive) ∧ happy ∧ unproductive (c) ¬p ∧ (¬(h ⇒ p) ∨ (¬p ∧ h))) ∨ (¬h ∧ ¬p)|¬(¬p ∧ ¬h) (d) ¬(¬r ∨ t ∨ ¬s) ∨ ¬(r ⇒ s) ∨ (r ⇒ t) (e) (¬r ∧ (s ⇐⇒ ¬(q ∨ r))) ⇐⇒ (¬s ∨ ¬(t ⇐ (r ∧ q)))
- Consider the following set of sentences we will refer to by ∆.
logician(post) logician(godel) logician(turing) logician(church)
intelligent(post) intelligent(godel) intelligent(turing) intelligent(church) intelligent(f rodo)
(a) Does ∆ |= ∀x.(logician(x) ∧ intelligent(x))? Answer Yes or No. If no give a model that demon- strates this fact. (b) Does ∆ |= ∀x.(logician(x) ⇒ intelligent(x))? Answer Yes or No. If no give a model that demonstrates this fact.
2 Inference (15%)
- Convert this sentence to clausal form
¬p ∨ (q ⇒ ¬(r ⇐⇒ q))
- Using propositional linear resolution, show the following propositional sentence is unsatisfiable.
(p ∨ q ∨ ¬r) ∧ ((¬r ∨ q ∨ p) ⇒ ((r ∨ q) ∧ ¬q ∧ ¬p))
To do this, convert this sentence to clausal form and derive the empty clause using resolution.
3 Relational Statements (15%)
- Translate the following english sentences into relational logic. No explanation necessary.
Object constant: 0 Relations: W(x): x is a whole number, i.e. 1, 2, 3, ... N(x): x is a natural number, i.e. 0, 1, 2, 3, ... Q(x): x is a rational number R(x): x is a real number x > y: x is greater than y
(a) ”Every natural number greater than 0 is a whole number.” (b) ”Some real numbers are rational numbers, but some aren’t.” (c) ”For every pair of natural numbers there is a rational number between them.”
4 Propositional Logic (25%)
Lyndon’s Interpolation Theorem for propositional logic is the following: If α |= β, and α, β propositional sentences, then there is a sentence γ that includes only literals that appear in both α, β such that α |= γ and γ |= β. Prove Lyndon’s Interpolation Theorem for propositional logic from Craig’s interpolation theorem for propositional logic (hint: create new symbols).
5 First-Order Logic (20%)
Let ∆ be an arbitrarily-sized but finite set of first-order sentences. Let φ be a first-order sentence. Prove ∆ |= φ ⇐⇒ ∆ ∪ {¬φ} |= {}.
