Exploring a wumpus world B。K BGS OK 2 [1,2]
11 Exploring a wumpus world 1 2 3 4 1 2 [1,2]
Logic in general ogics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language Semantics define the meaning"of sentences i. e. define truth of a sentence in a world E.g. the language of arithmetic X+2 2y is a sentence; X2+y >t is not a sentence 12
12 Logic in general • Logics are formal languages for representing information such that conclusions can be drawn • • Syntax defines the sentences in the language • • Semantics define the "meaning" of sentences; • – i.e., define truth of a sentence in a world – • E.g., the language of arithmetic • – x+2 ≥ y is a sentence; x2+y > {} is not a sentence – – x+2 ≥ y is true iff the number x+2 is no less than the number y
Entailment Entailment means that one thing follows from another KB a Knowledge base KB entails sentence a if and only if a is true in all worlds where KB is true E.g., the KB containing the Rockets won"and"the Lakers won entails "Either the rockets won or the Lakers won -E.g., X+y =4 entails 4=X+y 13
13 Entailment • Entailment means that one thing follows from another: • KB ╞ α • Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true – E.g., the KB containing “the Rockets won” and “the Lakers won” entails “Either the Rockets won or the Lakers won” – – E.g., x+y = 4 entails 4 = x+y – – Entailment is a relationship between sentences (i.e
Models Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated We say m is a model of a sentence a f w n t X M(a)is the set of all models of a X X x/x Then KB Fa iff M(KB)C x XX E.g. KB= Giants won and Reds won a= giants won M(KB) How to check? truth tablel 14
14 Models • Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated • • We say m is a model of a sentence α if α is true in m • M(α) is the set of all models of α • • Then KB ╞ α iff M(KB) M(α) • – E.g. KB = Giants won and Reds won α = Giants won • How to check? – truth table! –
E.g. Entailment in NBA the KB the rockets won”and the lakers won entails Either the rockets won or the lakers Won Check with the xxxx table 15
15 E.g. Entailment in NBA • the KB – “the Rockets won” and – “the Lakers won” – entails “Either the Rockets won or the Lakers won” – ? • Check with the xxxx table