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hw2Sol - CS 440 Introduction to AI Homework 2 Part A...

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CS 440: Introduction to AI Homework 2 - Part A Solution Due: September 23 11:59PM, 2010 Your answers must be concise and clear. Explain sufficiently that we can easily determine what you understand. We will give more points for a brief interesting discussion with no answer than for a bluffing answer. 1 Validity and Satisfiability (16 points) Determine whether each formula is valid, satisfiable (but not valid), or unsatisfiable. 1. P ( x ) Satisfiable since It holds for some denotational correspondence and some world. 2. x P ( x ) Satisfiable. 3. x y P ( x, y ) Satisfiable. 4. x ( P ( x ) ∨ ¬ P ( x )) Valid. This is a tautology, which is always true. 5. x y ( P ( x, y ) ⇒ ∀ w z P ( w, z )) Satisfiable. 6. x ( P ( x ) ⇒ ¬ P ( x )) Satisfiable. 7. x ¬ ( ¬ P ( x ) P ( x )) Unsatisfiable. ¬ P ( x ) P ( x ) is a tautology, and it is negated, which makes it always false. 8. x y ( ¬ P ( x ) P ( y )) ( P ( x ) P ( y )) Valid. This is the same as Θ Θ, which becomes ¬ Θ Θ, a tautology. 1
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2 Sentence Translation (24 points) This problem has two parts. In the first part, you are asked to translate some first order predicate calculus (FOPC) sentences into English sentences. The meanings of the predicates and functions in the FOPC sentences are given. You should give as natural the English translations as possible. Few points will be given for an English sentence like “For all x there exists a y such that either P of x and y or Q of y .” In the second part, you are asked to translate English sentences into FOPC sentences. If the meaning of the English sentence is ambiguous, give all possible readings. Note that sometimes
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