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Harris Gold Book Question Set #1: Ertel, Chapter 2: Propositional Logic 1.) What is propositional logic? With respect to AI, what is it good for? Propositional logic entails connecting propositions with logical operators. In regards to AI, propositional logic can grant a machine an extremely basic and limited form of reasoning. 2.) Using BNF, define the language of all “fully parenthesized” propositional logic formulas, where a propositional logical formula is said to be fully parenthesized only if it contains one set of parenthesis for each logical operator. (Hint: look up BNF – Backus-Naur Form) ( formula ) ::= ( a truth value ) ( a propositional variable ) ( formula ) ( formula ˄ formula ) ( formula ˅ formula ) ( formula → formula ) ( formula ↔ formula ) 3.) With respect to propositional calculus, what is an interpretation? An interpretation is an assignment of truth values to the propositional values. 4.) How many interpretations are there for a propositional formula involving 3 variables? 4 variables? 5 variables? N variables? There are 8 interpretations for a propositional formula involving 3 variables, 16 involving 4, and 32 involving 5. 5.) In words, define… a. ( NOT A ), where A is a propositional variable A negation of A b. ( A AND B ), where A and B are a propositional variable A conjunction of A and B c. ( A OR B ), where A and B are a propositional variable A disjunction of A and B Harris Gold d. ( A IMPLIES B ), where A and B are a propositional variable An implication that, ‘if A then B’ (also known as material implication) e. ( A EQUIVALENT-TO B ), where A and B are a propositional value A statement of equivalence where, ‘A if and only if B’ 6.) What does it mean for two propositional formulas to be logically equivalent? If two propositional formulas are logically equivalent, they must evaluate to the same truth values for all interpretations. 7.) What does it mean for a logical formula to be satisfiable? Give an example of a logical formula having 3 variables that is satisfiable – and provide a simple argument that it is, indeed, satisfiable. For a formula to be called satisfiable, it must be true for at least one interpretation. ((P ^ Q) → ¬R), where P = t, Q = t, and R = f This will always be valid as long as one variable is false. 8.) What does it mean for a logical formula to be valid? Give an example of a logical formula having 3 variables that is satisfiable – and provide a simple argument that it is, indeed valid. For a formula to be called valid, it must be true for all of its interpretations. True formulas are also called tautologies. ((A -> B) ^ (B -> C)) -> (A -> C) 9.) What does it mean for a logical formula to be invalid? Give an example of a logical formula having 3 variables that are satisfiable – and provide a simple argument that is, indeed, invalid. For a formula to be called invalid, it evaluates to false for all of its interpretations. (A^B^C) If all variables are false, it will evaluate to false. 10.) What does it mean for a logical formula to be unsatisfiable? Give an example of a logical formula having 3 variables that are satisfiable – and provide a simple argument that is, indeed, unsatisfiable. Harris Gold 11.) With respect to a logical formula, what is a model? Give an example of a model for some logical formula having three variables. A model is every interpretation that satisfies a formula. An example could be: ( ( A ^ B ) → C ) 12.) What is a proof system in the context of propositional calculus? A proof system in the context of propositional calculus is the system that allows us to compute equivalence and put together tautologies. 13.) What does it mean for a logical formula to follow from, or be entailed by a set of logical formulas? In every interpretation where KB (a knowledge base) is true, Q (a query) is also true. More over, whenever KB is true, Q is also true. 14.) Convert each of the formulas in Exercise 2.2 to a fully parenthesized logical formula. a. ( A ˄ B ) ↔ A ˄ B (((A˄B))↔(( A)˄( B))) b. A → B ↔ B →A ((A→B)↔(( B)→( A))) c. ( ( A → B ) ˄ ( B → A ) ) ↔ ( A ↔ B ) (((A→B)˄(B→A))↔(A↔B)) d. ( A ˅ B ) ˄ ( B ˅ C ) → ( A ˅ C ) (((A˅B)˄(( B)˅C))→(A˅C)) Harris Gold 15.) Show that each formula of the previous problem is a tautology. a.) ¬ ( A ^ B ) ↔ ¬ A ^ B A B T T F F T F T F ¬( A ^ B ) T F F T ¬A^¬B ¬(A^B)↔¬A^B T F F T T T T T b.) A → B ↔ ¬ B → ¬ A A B A→B ¬ B → ¬ A A→B↔ ¬B → ¬ A T T F F T F T F T F F T T F F T T T T T c.) ( ( A → B ) ˄ ( B → A ) ) ↔ ( A ↔ B ) A B A→B B→A A↔B T T F F T F T F T F F T T F F T T F F T ((A→B )˄(B→ A))↔( A↔B) T T T T Harris Gold d.) ( A ˅ B ) ˄ ( ¬ B ˅ C ) → ( A ˅ C ) A B C A˅B ¬B˅C A˅C (A˅B) ˄(¬B˅ C)→( A˅C) T T T T F F F F T T F F F F T T T F T F F T F T T T T T F F T T T F T T F T F T T T T T F T F T T F T T T F T T