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Models and Propositional Logic In propositional logic, a model in general simply fixes the truth value – true or false – for every proposition symbol. A model for a formula (set) is an assignment that makes the formula(s) true. One convention is to make a list of the symbols that are true in the model. Example: { P22, B21} is a model for the formula B21 P1 (B11 (P12 P21) ) (B21 (P11 P22 P31) ) B11

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Implicative Normal Form Clauses on Conjunctive Normal Form, which is the basis for Resolution, can be presented in a more readable form. This means that all clauses, which originally are in conjunctive normal form (CNF) like A1 B1 A2 ... B2 B3... An (order of literals is immaterial) is transformed to the equivalent formula (B1 B2 B3 … Bm ) (A1 A2 … An) If m=0, we write True (A1 A2 … An) If n=0, we write (B1 B2 B3 … Bm ) False

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Implicative Normal Form (LOUIS Style) For some implementations, the clauses in INF are represented using some conventions inherited from Prolog: - identifiers starting with small letters denote constants - identifiers starting with capital letters denote variables Operators ( ) are printed as (and,or,not,=>) and each clause is ended with a ‘.’ b1 and b2 and b3...bn => a1 or a2... an. If m = 0 (no negative literal), we write t => a1 or a2... an. where t denotes the logical constant TRUE. If n = 0 (no positive literals), we write b1 and b2 and b3...bn => f. where f denotes is the logical constant FALSE.

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LOUIS Theorem Prover (A Model Builder for Logical Expressions) LOUIS Therorem Prover is a model builder for clauses in Implicative Normal Form. It will try to find a model for the formulas, represented as a list of symbols that are true. If so, the formulas are satisfiable.If the formulas are unsatisfiable, a message is given. The system can be used to prove theorems of the kind KB |= if the negation of the formula is added to KB, and showing that the combined set is unsatisfiable.

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Propositional and First Order Reasoning. Terminology Propositional variable: boolean variable (p) Literal: propositional variable or its negation p

Propositional and First Order Reasoning. Terminology Propositional variable: boolean variable (p) Literal: propositional variable or its negation p

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