Revised by Hankui Zhuo, March, 08 Logical agents Chapter 7 Chapter 7
Outline Wumpus world Logic in general models and entailment Propositional (oolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution Chapter 7
Wumpus World PES description Performance measure gold +000, death -000 - per step, -0 for using the arrow Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square ctuators Left turn, Right turn, Forward, Grab, Release, Shoot Sensors, Glitter, Smell 4 3 Stench Stench Gold Stench STRT 3 4 Chapter 7 3
Observable?? Wumpus world characterization Chapter 7 4
Wumpus world characterization Observable?? No only local perception Deterministic?? Chapter 7 5
Wumpus world characterization Observable?? No only local perception Deterministic?? Yes outcomes exactly specified Episodic?? Chapter 7 6
Wumpus world characterization Observable?? No only local perception Deterministic?? Yes outcomes exactly specified Episodic?? No sequential at the level of actions Static?? Chapter 7 7
Wumpus world characterization Observable?? No only local perception Deterministic?? Yes outcomes exactly specified Episodic?? No sequential at the level of actions Static?? Yes Wumpus and Pits do not move Discrete?? Chapter 7 8
Wumpus world characterization Observable?? No only local perception Deterministic?? Yes outcomes exactly specified Episodic?? No sequential at the level of actions Static?? Yes Wumpus and Pits do not move Discrete?? Yes Single-agent?? Chapter 7 9
Wumpus world characterization Observable?? No only local perception Deterministic?? Yes outcomes exactly specified Episodic?? No sequential at the level of actions Static?? Yes Wumpus and Pits do not move Discrete?? Yes Single-agent?? Yes Wumpus is essentially a natural feature Chapter 7 0
Exploring a wumpus world Chapter 7
Exploring a wumpus world Chapter 7
Exploring a wumpus world P? P? Chapter 7 3
Exploring a wumpus world P? P? S Chapter 7 4
Exploring a wumpus world P P? P? S W Chapter 7 5
Exploring a wumpus world P P? P? S W Chapter 7 6
Exploring a wumpus world P P? P? S W Chapter 7 7
Exploring a wumpus world P P? P? GS S W Chapter 7 8
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 + y is a sentence; x+y> is not a sentence x + y is true iff the number x +is no less than the number y x + y is true in a world where x =7,y= x + y is false in a world where x =0,y=6 Chapter 7 9
Entailment Entailment means that one thing follows from another: K = α Knowledge base K entails sentence α if and only if α is true in all worlds where K is true E.g., the K 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., syntax) that is based on semantics Note: brains process syntax (of some sort) Chapter 7 0
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 asentenceα if α is true in m M(α) is the set of all models of α Then K = α if and only if M(K) M(α) E.g. K =GiantswonandRedswon α =Giantswon M( ) x x x x x x x x x x x x x x x x x x x x x x x x x xx x xx x x x x x x x M(K) x x x x x x x x x Chapter 7
Entailment in the wumpus world Situation after detecting nothing in [,], moving right, breeze in [,]?? Consider possible models for?s assuming only pits? 3 oolean choices 8 possible models Chapter 7
Wumpus models 3 3 3 3 3 3 3 3 Chapter 7 3
Wumpus models K 3 3 3 3 3 3 3 3 K = wumpus-world rules + observations Chapter 7 4
Wumpus models K 3 3 3 3 3 3 3 3 K = wumpus-world rules + observations α = [,] is safe, K = α,provedbymodel checking Chapter 7 5
Wumpus models K 3 3 3 3 3 3 3 3 K = wumpus-world rules + observations Chapter 7 6
Wumpus models K 3 3 3 3 3 3 3 3 K = wumpus-world rules + observations α = [,] is safe, K = α Chapter 7 7
Inference K i α =sentenceα can be derived from K by procedure i Soundness: i is sound if whenever K i α,itisalsotruethatk = α Completeness: i is complete if whenever K = α, itisalsotruethatk i α Chapter 7 8
Propositional logic: Syntax Propositional logic is the simplest logic illustrates basic ideas The proposition symbols P, P etc are sentences If S is a sentence, S is a sentence (negation) If S and S are sentences, S S is a sentence (conjunction) If S and S are sentences, S S is a sentence (disjunction) If S and S are sentences, S If S and S are sentences, S S is a sentence (implication) S is a sentence (biconditional) Chapter 7 9
Propositional logic: Semantics Each model specifies true/false for each proposition symbol E.g. P, P, P 3, true true false (With these symbols, 8 possible models, can be enumerated automatically.) Rules for evaluating truth with respect to a model m: S is true iff S is false S S is true iff S is true and S is true S S is true iff S is true or S is true S S is true iff S is false or S is true i.e., is false iff S is true and S is false S S is true iff S S is true and S S is true Simple recursive process evaluates an arbitrary sentence, e.g., P, (P, P 3, ) = true (false true)=true true = true Chapter 7 30
Truth tables for connectives P Q P P Q P Q P Q P Q false false true false false true true false true true false true true false true false false false true false false true true false true true true true Chapter 7 3
Wumpus world sentences Let P i,j be true if there is a pit in [i, j]. Let i,j be true if there is a breeze in [i, j]. P,,, Pits cause breezes in adjacent squares Chapter 7 3
Wumpus world sentences Let P i,j be true if there is a pit in [i, j]. Let i,j be true if there is a breeze in [i, j]. P,,, Pits cause breezes in adjacent squares, (P, P, ), (P, P, P 3, ) square is breezy if and only if there is an adjacent pit Chapter 7 33
Truth tables for inference,, P, P, P, P, P 3, R R R 3 R 4 R 5 K false false false false false false false true true true true false false false. false. false. false. false. false. true. true. true. false. true. false. false. false true false false false false false true true false true true false false true false false false false true true true true true true true false true false false false true false true true true true true true false true false false false true true true true true true true true false. true. false. false. true. false. false. true. false. false. true. true. false. true true true true true true true false true true false true false Enumerate rows (different assignments to symbols), if K is true in row, check that α is too Chapter 7 34
Inference by enumeration Depth-first enumeration of all models is sound and complete function TT-Entails?(K, α) returns true or false inputs: K, the knowledge base, a sentence in propositional logic α, the query, a sentence in propositional logic symbols a list of the proposition symbols in K and α return TT-Check-ll(K, α, symbols, []) function TT-Check-ll(K, α, symbols, model) returns true or false if Empty?(symbols) then if PL-True?(K, model) then return PL-True?(α, model) else return true else do P First(symbols); rest Rest(symbols) return TT-Check-ll(K, α, rest, Extend(P,true, model)) and TT-Check-ll(K, α, rest, Extend(P,false, model)) O( n ) for n symbols; Chapter 7 35
Logical equivalence Two sentences are logically equivalent iff true in same models: α β if and only if α = β and β = α (α β) (β α) commutativity of (α β) (β α) commutativity of ((α β) γ) (α (β γ)) associativity of ((α β) γ) (α (β γ)) associativity of ( α) α double-negation elimination (α β) ( β α) contraposition (α β) ( α β) implication elimination (α β) ((α β) (β α)) biconditional elimination (α β) ( α β) De Morgan (α β) ( α β) De Morgan (α (β γ)) ((α β) (α γ)) distributivity of over (α (β γ)) ((α β) (α γ)) distributivity of over Chapter 7 36
Validity and satisfiability sentenceisvalid if it is true in all models, e.g., True,,, ( ( )) Validity is connected to inference via the Deduction Theorem: K = α if and only if (K α) is valid sentenceissatisfiable if it is true in some model e.g.,, C sentenceisunsatisfiable if it is true in no models e.g., Satisfiability is connected to inference via the following: K = α if and only if (K α) is unsatisfiable i.e., prove α by reductio ad absurdum Chapter 7 37
Proof methods Proof methods divide into (roughly) two kinds: pplication of inference rules Legitimate (sound) generation of new sentences from old Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search alg. Typically require translation of sentences into a normal form Model checking truth table enumeration (always exponential in n) improved backtracking heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms Chapter 7 38
Forward and backward chaining Horn Form (restricted) K = conjunction of Horn clauses Horn clause = proposition symbol; or (conjunction of symbols) symbol E.g., C ( ) (C D ) Modus Ponens (for Horn Form): complete for Horn Ks α,...,α n, α α n β β Can be used with forward chaining or backward chaining. These algorithms are very natural and run in linear time Chapter 7 39
Forward chaining Idea: fire any rule whose premises are satisfied in the K, add its conclusion to the K, until query is found Q P Q L M P L M P L L L P M Chapter 7 40
Forward chaining algorithm function PL-FC-Entails?(K, q) returns true or false inputs: K, the knowledge base, a set of propositional Horn clauses q, the query, a proposition symbol local variables: count, a table, indexed by clause, initially the number of premises inferred, a table, indexed by symbol, each entry initially false agenda, a list of symbols, initially the symbols known in K while agenda is not empty do p Pop(agenda) unless inferred[p] do inferred[p] true for each Horn clause c in which premise p appears do decrement count[c] if count[c] =0then do if Head[c] =q then return true Push(Head[c], agenda) return false Chapter 7 4
Forward chaining example P Q L M P L M P L L Q P M L Chapter 7 4
Forward chaining example P Q L M P L M P L L Q P M L Chapter 7 43
Forward chaining example P Q L M P L M P L L Q P L M 0 Chapter 7 44
Forward chaining example P Q L M P L M P L L Q P L M 0 0 Chapter 7 45
Forward chaining example P Q L M P L M P L L Q P 0 L M 0 0 Chapter 7 46
Forward chaining example P Q L M P L M P L L 0 Q 0 P 0 L M 0 0 Chapter 7 47
Forward chaining example P Q L M P L M P L L 0 Q 0 P 0 L M 0 0 Chapter 7 48
Forward chaining example P Q L M P L M P L L 0 Q 0 P 0 L M 0 0 Chapter 7 49
Proof of completeness FC derives every atomic sentence that is entailed by K. FC reaches a fixed point where no new atomic sentences are derived. Consider the final state as a model m, assigning true/false to symbols 3. Every clause in the original K is true in m Proof: Suppose a clause a... a k b is false in m Then a... a k is true in m and b is false in m Therefore the algorithm has not reached a fixed point! 4. Hence m is a model of K 5. If K = q, q is true in every model of K, including m General idea: construct any model of K by sound inference, check α Chapter 7 50
ackward chaining Idea: work backwards from the query q: to prove q by C, check if q is known already, or prove by C all premises of some rule concluding q void loops: check if new subgoal is already on the goal stack void repeated work: check if new subgoal ) has already been proved true, or ) has already failed Chapter 7 5
ackward chaining example Q P M L Chapter 7 5
ackward chaining example Q P M L Chapter 7 53
ackward chaining example Q P M L Chapter 7 54
ackward chaining example Q P M L Chapter 7 55
ackward chaining example Q P M L Chapter 7 56
ackward chaining example Q P M L Chapter 7 57
ackward chaining example Q P M L Chapter 7 58
ackward chaining example Q P M L Chapter 7 59
ackward chaining example Q P M L Chapter 7 60
ackward chaining example Q P M L Chapter 7 6
ackward chaining example Q P M L Chapter 7 6
Forward vs. backward chaining FC is data-driven May do lots of work that is irrelevant to the goal C is goal-driven, appropriate for problem-solving Complexity of C can be much less than linear in size of K Chapter 7 63
Resolution Conjunctive Normal Form (CNF universal) conjunction of disjunctions of literals }{{} clauses E.g., ( ) ( C D) Resolution inference rule (for CNF): complete for propositional logic l l k, m m n l l i l i+ l k m m j m j+ m n where l i and m j are complementary literals. E.g., P,3 P,, P, P,3 Resolution is sound and complete for propositional logic P P? S P? W Chapter 7 64
Conversion to CNF, (P, P, ). Eliminate, replacing α β with (α β) (β α). (, (P, P, )) ((P, P, ), ). Eliminate, replacing α β with α β. (, P, P, ) ( (P, P, ), ) 3. Move inwards using de Morgan s rules and double-negation: (, P, P, ) (( P, P, ), ) 4. pply distributivity law ( over ) and flatten: (, P, P, ) ( P,, ) ( P,, ) Chapter 7 65
Resolution algorithm Proof by contradiction, i.e., show K α unsatisfiable function PL-Resolution(K, α) returns true or false inputs: K, the knowledge base, a sentence in propositional logic α, the query, a sentence in propositional logic clauses the set of clauses in the CNF representation of K α new {} loop do for each C i, C j in clauses do resolvents PL-Resolve(C i, C j ) if resolvents contains the empty clause then return true new new resolvents if new clauses then return false clauses clauses new Chapter 7 66
Resolution example K =(, (P, P, )), α = P, P,,, P, P, P,,, P,, P,, P, P, P,, P,, P, P, P, P, P, Chapter 7 67
Summary Logical agents apply inference to a knowledge base to derive new information and make decisions asic concepts of logic: syntax: formal structure of sentences semantics: truth of sentences wrt models entailment: necessary truth of one sentence given another inference: deriving sentences from other sentences soundess: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences Wumpus world requires the ability to represent partial and negated information, reason by cases, etc. Forward, backward chaining are linear-time, complete for Horn clauses Resolution is complete for propositional logic Propositional logic lacks expressive power Chapter 7 68
End of Chapter 7 Thanks & Questions! Chapter 7 69