CS 561, Sessions 10-111Knowledge and reasoning – second part• Knowledge representation• Logic and representation• Propositional (Boolean) logic• Normal forms• Inference in propositional logic• Wumpus world exampleCS 561, Sessions 10-112Knowledge-Based Agent• Agent that uses prior or acquiredknowledge 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 561, 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 561, Sessions 10-114Wumpus world exampleCS 561, Sessions 10-115Wumpus world characterization• Deterministic?• Accessible?• Static?• Discrete?• Episodic?CS 561, 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 561, Sessions 10-117Exploring a Wumpus worldCS 561, Sessions 10-118Exploring a Wumpus worldCS 561, Sessions 10-119Exploring a Wumpus worldCS 561, Sessions 10-1110Exploring a Wumpus worldCS 561, Sessions 10-1111Exploring a Wumpus worldCS 561, Sessions 10-1112Exploring a Wumpus worldCS 561, Sessions 10-1113Exploring a Wumpus worldCS 561, Sessions 10-1114Exploring a Wumpus worldCS 561, Sessions 10-1115Other tight spotsCS 561, Sessions 10-1116Another example solutionNo perception à 1,2 and 2,1 OKMove to 2,1B in 2,1 à 2,2 or 3,1 P?1,1 V à no P in 1,1Move to 1,2 (only option)CS 561, Sessions 10-1117Example solutionS and No S when in 2,1 à 1,3 or 1,2 has W1,2 OK à 1,3 WNo B in 1,2 à 2,2 OK & 3,1 PCS 561, Sessions 10-1118Logic in generalCS 561, Sessions 10-1119Types of logicCS 561, Sessions 10-1120EntailmentCS 561, Sessions 10-1121ModelsCS 561, Sessions 10-1122InferenceCS 561, Sessions 10-1123Basic 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 ó Q “P and Q are either both true or both false” equivalenceCS 561, Sessions 10-1124Propositional logic: syntaxCS 561, Sessions 10-1125Propositional logic: semanticsCS 561, Sessions 10-1126Truth 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:PQP V QTTTTFTFTTFFFCS 561, Sessions 10-1127Truth tables for basic connectivesP Q ¬P ¬Q P V Q P ^ Q P=>Q PóQTTFFTTTTTFFTTFFFFTTFTFTFFFTTFFTTCS 561, Sessions 10-1128Propositional 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 ó B = (A => B) ^ (B => A) by definition•¬(A ó B) = (A ^ (¬B))V(B ^ (¬A)) using negated and & or•…CS 561, Sessions 10-1129Propositional inference: enumeration methodCS 561, Sessions 10-1130Enumeration: SolutionCS 561, Sessions 10-1131Propositional inference: normal forms“sum of products of simple variables ornegated simple variables”“product of sums of simple variables ornegated simple variables”CS 561, Sessions 10-1132Deriving 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 functionPQ RESULTTT FTF T P ^ (¬Q)F T T (¬P) ^ QFF FRESULT = (P ^ (¬Q)) V ((¬P) ^ Q)CS 561, Sessions 10-1133A 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, where:- For each variable whose value is T in that row, include the variable in the minterm- For each variable whose value is F in that row, include the negation of the variable in the minterm- Link variables in minterm by conjunctions- The expression consists of the disjunction of all minterms.CS 561, Sessions 10-1134Example: adder with carryTakes 3 variables in: x, y and ci (carry-in); yields 2 results: sum (s) and carry-out (co). To get you used to other notations, here we assume T = 1, F = 0, V = OR, ^ = AND, ¬ = NOT.co is:s is:CS 561, Sessions 10-1135Tautologies• Logical expressions that are always true. Can be simplified out.Examples:TT V AA V (¬A)¬(A ^ (¬A))A ó A((P V Q) ó P) V (¬P ^ Q)(P ó Q) => (P => Q)CS 561, Sessions 10-1136Validity and satisfiability TheoremCS 561, Sessions 10-1137Proof methodsCS 561, Sessions 10-1138Inference rulesCS 561, Sessions 10-1139Wumpus world: example• Facts: Percepts inject (TELL) facts into the KB• [stench at 1,1 and 2,1] à S1,1 ; S2,1• Rules: if square has no stench then neither the square or adjacent square contain the wumpus• R1: !S1,1 Þ!W1,1 ∧ !W1,2 ∧ !W2,1• R2: !S2,1 Þ!W1,1 ∧!W2,1 ∧ !W2,2 ∧ !W3,1•…• Inference: • KB contains !S1,1 then using Modus Ponens we infer!W1,1 ∧ !W1,2 ∧ !W2,1• Using And-Elimination we get: !W1,1 !W1,2 !W2,1•…CS 561, Sessions 10-1140Limitations of Propositional Logic1. It is too weak, i.e., has very limited expressiveness:• Each rule has to be represented for each situation:e.g., “don’t go forward if the wumpus is in front of you” takes 64 rules2. It cannot keep track of changes:• If one needs to track changes, e.g., where the agent has been before then we need a timed-version of each rule. To track 100 steps we’ll then need 6400 rules for the previous example.Its hard to write and maintain such a huge rule-baseInference becomes intractableCS 561, Sessions
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