### Artificial IntelligenceAIMA Exercises

A propositional 2-CNF expression is a conjunction of clauses, each containing exactly 2 literals, e.g., $$(A\lor B) \land (\lnot A \lor C) \land (\lnot B \lor D) \land (\lnot C \lor G) \land (\lnot D \lor G)\ .$$
1. Prove using resolution that the above sentence entails $G$.
2. Two clauses are semantically distinct if they are not logically equivalent. How many semantically distinct 2-CNF clauses can be constructed from $n$ proposition symbols?
3. Using your answer to (b), prove that propositional resolution always terminates in time polynomial in $n$ given a 2-CNF sentence containing no more than $n$ distinct symbols.
4. Explain why your argument in (c) does not apply to 3-CNF.

A propositional 2-CNF expression is a conjunction of clauses, each containing exactly 2 literals, e.g., $$(A\lor B) \land (\lnot A \lor C) \land (\lnot B \lor D) \land (\lnot C \lor G) \land (\lnot D \lor G)\ .$$
1. Prove using resolution that the above sentence entails $G$.
2. Two clauses are semantically distinct if they are not logically equivalent. How many semantically distinct 2-CNF clauses can be constructed from $n$ proposition symbols?
3. Using your answer to (b), prove that propositional resolution always terminates in time polynomial in $n$ given a 2-CNF sentence containing no more than $n$ distinct symbols.
4. Explain why your argument in (c) does not apply to 3-CNF.

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