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Transcript

Propositional Logic Rather than jumping right into FOL, we begin with propositional logic A logic involves: Language (with a syntax) Semantics Proof (Inference) System Example of k-rep in prop calc R : “It is raining” B : “Take the bus to class” W : “Walk to class” Some things to tell our agent R B (“If it is raining, (then) take the bus to class”) R W (“If it is not raining, (then) walk to class”) Ideally, we would like our agent to sense that it is raining & then decide to take the bus Alphabet Non-Logical Symbols (meaning given by interpretation) Propositions P, Q, R,… • atomic statements (facts) about the world • R : it’s-raining-now • needn’t be a single letter Logical Symbols (fixed meaning) Alphabet Logical Symbols Connectives: not () and () or () implies () equivalent () Punctuation Symbols: ( , ) Truth symbols: TRUE, FALSE Well-formed formulae (wffs) Sentences just like in a programming language, there are rules (syntax) for legally creating compound statements remember: we’re always stating a truth about the world, hence every wff is something that has a Boolean value (it is either a true or a false statement about the world) Syntax rules Propositions (P, Q, R, …) are wffs Truth symbols (TRUE, FALSE) are wffs If A is a wff, so is A If A and B are wffs, so are AB A B A B A B There are no other wffs. Language: set of all wffs Are these WFFs? PQR (P Q) (R S) P (Q R) Semantics KB |= Q KB - Set of wffs Q- a wff |= Entailment Compositional Two-Valued What is an interpretation? An interpretation gives meaning to the nonlogical symbols of the language. An assignment of facts to atomic wffs a fact is taken to be either true or false about the world thus, by providing an interpretation, we also provide the truth value of each of the atoms example • P : it-is-raining-here-now • since this is either a true or false statement about the world, the value of P is either true or false a function that maps atomic formulas to truth values Truth tables Connectives Semantics How to evaluate a wff ((P U) R) (S V) First, we need an interpretation P : T; U : F; R : T; S : F; V : T Then using this interpretation, evaluate formula according to the fixed meanings of the connectives PU:T (P U) R : T SV:F whole formula : F Satisfiability and Models An interpretation I satisfies a wff iff I assigns the wff the value T An interpretation I satisfies a set of S of wffs iff I satisfies every wff in S. An interpretation that satisfies a (set of) wff is said to be a model of it. A (set of) wff is satisfiable iff there exists some interpretation that satisfies it Examples: P is satisfiable • simply let P be true P P is unsatisfiable • if P is false, the formula is false • if P is true, P is false, the formula is false P Q is satisfiable • three ways: P is true, Q is true; etc. A wff that is unsatisfiable is called a contradiction for example, a model for {A B, B C} is • A : true, B : true, C : true • note: there may be more than one model for a (set of) wff Entailment (Logical Consequence) KB |= Q iff for every interpretation I, If I satisfies KB then I satisfies Q. That is, if every model of KB is also a model of Q. For example: A B, A |= B Validity A formula G is valid if it is true for every interpretation P P is valid • if P is true, then the formula is true • if P is false, then ~P is true and the formula is true (P Q) (P Q) isn’t valid • when P is true & Q is true, the formula isn’t true • in order to not be valid, there only need exist one counter-example also called a tautology Some important Theorems a) KB |= Q iff KB U { Q} is unsatisfiable b) KB , A |= B iff KB |= (A B) c) Monotonicity: if KB KB’ then {Q | KB |= Q} {Q | KB’ |= Q}