Knowledge and reasoning – second partKnowledge-Based AgentGeneric knowledge-based agentWumpus world exampleWumpus world characterizationSlide 6Exploring a Wumpus worldSlide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Other tight spotsAnother example solutionExample solutionLogic in generalTypes of logicThe Semantic WallTruth depends on InterpretationEntailmentLogic as a representation of the WorldModelsInferenceBasic symbolsPropositional logic: syntaxPropositional logic: semanticsTruth tablesTruth tables for basic connectivesPropositional logic: basic manipulation rulesPropositional inference: enumeration methodEnumeration: SolutionPropositional inference: normal formsDeriving expressions from functionsA more formal approachExample: adder with carryTautologiesValidity and satisfiabilityProof methodsInference RulesSlide 42Wumpus world: exampleLimitations of Propositional LogicSummaryNext timeCS 460, Sessions 10-111Knowledge and reasoning – second part•Knowledge representation•Logic and representation•Propositional (Boolean) logic•Normal forms•Inference in propositional logic•Wumpus world exampleCS 460, Sessions 10-112Knowledge-Based Agent•Agent that uses prior or acquired knowledge to achieve its goals•Can make more efficient decisions•Can make informed decisions•Knowledge Base (KB): contains a set of representations of facts about the Agent’s environment•Each representation is called a sentence •Use some knowledge representation language, to TELL it what to know e.g., (temperature 72F)•ASK agent to query what to do•Agent can use inference to deduce new facts from TELLed factsKnowledge BaseInference engineDomain independent algorithmsDomain specific contentTELLASKCS 460, Sessions 10-113Generic knowledge-based agent1. TELL KB what was perceivedUses a KRL to insert new sentences, representations of facts, into KB2. ASK KB what to do.Uses logical reasoning to examine actions and select best.CS 460, Sessions 10-114Wumpus world exampleCS 460, Sessions 10-115Wumpus world characterization•Deterministic?•Accessible?•Static?•Discrete?•Episodic?CS 460, Sessions 10-116Wumpus world characterization•Deterministic? Yes – outcome exactly specified.•Accessible? No – only local perception.•Static? Yes – Wumpus and pits do not move.•Discrete? Yes•Episodic? (Yes) – because static.CS 460, Sessions 10-117Exploring a Wumpus worldA= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = GlitterCS 460, Sessions 10-118Exploring a Wumpus worldA= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = GlitterCS 460, Sessions 10-119Exploring a Wumpus worldA= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = GlitterCS 460, Sessions 10-1110Exploring a Wumpus worldA= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = GlitterCS 460, Sessions 10-1111Exploring a Wumpus worldA= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = GlitterCS 460, Sessions 10-1112Exploring a Wumpus worldA= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = GlitterCS 460, Sessions 10-1113Exploring a Wumpus worldA= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = GlitterCS 460, Sessions 10-1114Exploring a Wumpus worldA= AgentB= BreezeS= SmellP= PitW= WumpusOK = SafeV = VisitedG = GlitterCS 460, Sessions 10-1115Other tight spotsCS 460, Sessions 10-1116Another example solutionNo perception  1,2 and 2,1 OKMove to 2,1B in 2,1  2,2 or 3,1 P?1,1 V  no P in 1,1Move to 1,2 (only option)CS 460, Sessions 10-1117Example solutionS and No S when in 2,1  1,3 or 1,2 has W1,2 OK  1,3 WNo B in 1,2  2,2 OK & 3,1 PCS 460, Sessions 10-1118Logic in generalCS 460, Sessions 10-1119Types of logicCS 460, Sessions 10-1120The Semantic WallPhysical Symbol System World+BLOCKA++BLOCKB++BLOCKC+P1:(IS_ON +BLOCKA+ +BLOCKB+)P2:((IS_RED +BLOCKA+)CS 460, Sessions 10-1121Truth depends on InterpretationRepresentation 1 WorldABON(A,B) TON(A,B) FON(A,B) F AON(A,B) T BCS 460, Sessions 10-1122EntailmentEntailment is different than inferenceCS 460, Sessions 10-1123Logic as a representation of the WorldFactsWorldFactfollowsRefers to (Semantics)Representation: SentencesSentenceentailsCS 460, Sessions 10-1124ModelsCS 460, Sessions 10-1125InferenceCS 460, Sessions 10-1126Basic symbols•Expressions only evaluate to either “true” or “false.”•P “P is true”•¬P “P is false” negation•P V Q “either P is true or Q is true or both”disjunction•P ^ Q “both P and Q are true” conjunction•P => Q “if P is true, the Q is true” implication•P  Q “P and Q are either both true or both false” equivalenceCS 460, Sessions 10-1127Propositional logic: syntaxCS 460, Sessions 10-1128Propositional logic: semanticsCS 460, Sessions 10-1129Truth tables•Truth value: whether a statement is true or false.•Truth table: complete list of truth values for a statement given all possible values of the individual atomic expressions.Example:P Q P V QT T TT F TF T TF F FCS 460, Sessions 10-1130Truth tables for basic connectivesP Q ¬P ¬Q P V Q P ^ Q P=>Q PQT T F F T T T TT F F T T F F FF T T F T F T FF F T T F F T TCS 460, Sessions 10-1131Propositional logic: basic manipulation rules•¬(¬A) = A Double negation•¬(A ^ B) = (¬A) V (¬B) Negated “and”•¬(A V B) = (¬A) ^ (¬B) Negated “or”•A ^ (B V C) = (A ^ B) V (A ^ C) Distributivity of ^ on V•A => B = (¬A) V B by definition•¬(A => B) = A ^ (¬B) using negated or•A  B = (A => B) ^ (B => A) by definition•¬(A  B) = (A ^ (¬B))V(B ^ (¬A)) using negated and & or•…CS 460, Sessions 10-1132Propositional inference: enumeration methodCS 460, Sessions 10-1133Enumeration: SolutionCS 460, Sessions 10-1134Propositional inference: normal forms“sum of products of simple variables ornegated simple variables”“product of sums of simple variables ornegated simple variables”CS 460, Sessions 10-1135Deriving expressions from functions•Given a boolean function in truth table form, find a propositional logic expression for it that uses only V, ^ and ¬.•Idea: We can easily do it by disjoining the “T” rows of the truth table.Example: XOR functionP Q RESULTT T FT F T P ^ (¬Q)F T T (¬P) ^ QF F FRESULT = (P ^ (¬Q)) V ((¬P) ^ Q)CS 460, Sessions 10-1136A more formal approach•To construct a logical expression in disjunctive normal form from a truth table:-Build a “minterm” for each row of the table,