Models and Propositional Logic In propositional logic, a model in general simply fixes the truth value – true or false – for every proposition symbol.

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Presentation transcript:

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

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

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.

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.