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 Giants won"”and"“the Reds won"entails "Either the Giants won or the Reds won -E.g.,x+y =4 entails 4=x+y
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 Giants won” and “the Reds won” entails “Either the Giants won or the Reds 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 i~ M(a)is the set of all models of a M() 。 Then KB卡a iff M(KB)sMa) E.g.KB=Giants won and Reds won a Giants won M(KB)
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 –
Entailment in the wumpus world Situation after detecting nothing in [1,1],moving right,breeze in [2,1] Consider possible models for KB assuming only pits 3 Boolean choices→8 possible models
Entailment in the wumpus world Situation after detecting nothing in [1,1], moving right, breeze in [2,1] Consider possible models for KB assuming only pits 3 Boolean choices 8 possible models
Wumpus models
Wumpus models
Wumpus models KB KB wumpus-world rules observations ●
Wumpus models • KB = wumpus-world rules + observations •