ogical gents Chapter 6, Ie Chapter 7 Outline Knowledge-based agents Wumpus world ogic in general models and entailment ropositional (oolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution
Knowledge bases Inference engine Knowledge base domain independent algorithms domain specific content Knowledge base = set of sentences in a formal language Declarative approach to building an agent (or other system): TE it what it needs to know Then it can SK itself what to do answers should follow from the K gents can be viewed at the knowledge level i.e., what they know, regardless of how implemented Or at the implementation level i.e., data structures in K and algorithms that manipulate them 3 simple knowledge-based agent function K-GENT( percept) returns an action static: K, a knowledge base t, a counter, initially, indicating time TE(K, KE-ERCET-SENTENCE( percept, t)) action SK(K,KE-CTION-UERY(t)) TE(K, KE-CTION-SENTENCE(action, t)) t t + return action The agent must be able to: Represent states, actions, etc. Incorporate new percepts Update internal representations of the world Deduce hidden properties of the world Deduce appropriate actions 4
Wumpus World ES description erformance measure gold +, death - - per step, - 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 Sensors, Glitter, Smell ctuators eft turn, Right turn, Forward, Grab, Release, Shoot 5 4 3 Stench Stench Gold Stench STRT 3 4 Wumpus world characterization Observable?? 6
Wumpus world characterization Observable?? No only local perception Deterministic?? 7 Wumpus world characterization Observable?? No only local perception Deterministic?? Yes outcomes exactly specified Episodic?? 8
Wumpus world characterization Observable?? No only local perception Deterministic?? Yes outcomes exactly specified Episodic?? No sequential at the level of actions Static?? 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 its do not move Discrete??
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 its do not move Discrete?? Yes Single-agent?? 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 its do not move Discrete?? Yes Single-agent?? Yes Wumpus is essentially a natural feature
Exploring a wumpus world 3 Exploring a wumpus world 4
Exploring a wumpus world?? 5 Exploring a wumpus world?? S 6
Exploring a wumpus world?? S W 7 Exploring a wumpus world?? S W 8
Exploring a wumpus world?? S W 9 Exploring a wumpus world?? GS S W
Other tight spots??? in (,) and (,) no safe actions? ssuming pits uniformly distributed, (,) has pit w/ prob.86, vs..3 S Smell in (,) cannot move Canuseastrategyofcoercion: shoot straight ahead wumpus was there dead safe wumpus wasn t there safe ogic 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 + 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 =,y=6
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 =4entails 4=x + y Entailment is a relationship between sentences (i.e., syntax) that is based on semantics Note: brains process syntax (of some sort) 3 odels ogicians 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 (α) is the set of all models of α Then K = α if and only if (K) (α) E.g. K = Giants won and Reds won α = Giants won ( ) 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 (K) x x x x x x x x x 4
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 5 Wumpus models 3 3 3 3 3 3 3 3 6
Wumpus models K 3 3 3 3 3 3 3 3 K = wumpus-world rules + observations 7 Wumpus models K 3 3 3 3 3 3 3 3 K = wumpus-world rules + observations α = [,] is safe, K = α, proved by model checking 8
Wumpus models K 3 3 3 3 3 3 3 3 K = wumpus-world rules + observations 9 Wumpus models K 3 3 3 3 3 3 3 3 K = wumpus-world rules + observations α = [,] is safe, K = α 3
Inference K i α =sentenceα can be derived from K by procedure i Consequences of K are a haystack; α is a needle. Entailment = needle in haystack; inference = finding it Soundness: i is sound if whenever K i α,itisalsotruethatk = α Completeness: i is complete if whenever K = α, itisalsotruethatk i α review: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure. That is, the procedure will answer any question whose answer follows from what is known by the K. 3 ropositional logic: Syntax ropositional logic is the simplest logic illustrates basic ideas The proposition symbols, 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 S is a sentence (implication) If S and S are sentences, S S is a sentence (biconditional) 3
ropositional logic: Semantics Each model specifies true/false for each proposition symbol E.g.,, 3, true true f alse (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.,, (, 3, ) = true (false true)=true true = true 33 Truth tables for connectives 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 34
Wumpus world sentences et i,j be true if there is a pit in [i, j]. et i,j be true if there is a breeze in [i, j].,,, its cause breezes in adjacent squares 35 Wumpus world sentences et i,j be true if there is a pit in [i, j]. et i,j be true if there is a breeze in [i, j].,,, its cause breezes in adjacent squares, (,, ), (,, 3, ) square is breezy if and only if there is an adjacent pit 36
Truth tables for inference,,,,,, 3, K α false false false false false false false false true false false false false false false true false true......... false true false false false false false false true false true false false false false true true true false true false false false true false true true false true false false false true true true true false true false false true false false false true......... true true true true true true true false false 37 Inference by enumeration Depth-first enumeration of all models is sound and complete function TT-ENTIS?(K, α) returns true or false symbols a list of the proposition symbols in K and α return TT-CHECK-(K, α, symbols, []) function TT-CHECK-(K, α, symbols, model) returns true or false if ETY?(symbols) then if -TRUE?(K, model) then return -TRUE?(α, model) else return true else do FIRST(symbols); rest REST(symbols) return TT-CHECK-(K, α, rest, EXTEND(,true, model) and TT-CHECK-(K, α, rest, EXTEND(,false, model) O( n ) for n symbols; problem is co-n-complete 38
ogical 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 organ (α β) ( α β) de organ (α (β γ)) ((α β) (α γ)) distributivity of over (α (β γ)) ((α β) (α γ)) distributivity of over 39 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 4
roof methods roof methods divide into (roughly) two kinds: pplication of inference rules egitimate (sound) generation of new sentences from old roof = 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 odel checking truth table enumeration (always exponential in n) improved backtracking, e.g., Davis utnam ogemann oveland heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms 4 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 ) odus onens (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 4
Forward chaining Idea: fire any rule whose premises are satisfied in the K, add its conclusion to the K, until query is found 43 Forward chaining algorithm function -FC-ENTIS?(K, q) returns true or false 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 to be true while agenda is not empty do p O(agenda) unless inferred[p] do inferred[p] true for each Horn clause c in whose premise p appears do decrement count[c] if count[c] =then do if HED[c] =q then return true USH(HED[c], agenda) return false 44
Forward chaining example 45 Forward chaining example 46
Forward chaining example 47 Forward chaining example 48
Forward chaining example 49 Forward chaining example 5
Forward chaining example 5 Forward chaining example 5
roof 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 roof: 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 53 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 54
ackward chaining example 55 ackward chaining example 56
ackward chaining example 57 ackward chaining example 58
ackward chaining example 59 ackward chaining example 6
ackward chaining example 6 ackward chaining example 6
ackward chaining example 63 ackward chaining example 64
ackward chaining example 65 Forward vs. backward chaining FC is data-driven, cf. automatic, unconscious processing, e.g., object recognition, routine decisions ay do lots of work that is irrelevant to the goal C is goal-driven, appropriate for problem-solving, e.g., Where are my keys? How do I get into a hd program? Complexity of C can be much less than linear in size of K 66
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.,,3,,,,3?? Resolution is sound and complete for propositional logic S W 67 Conversion to CNF, (,, ). Eliminate, replacing α β with (α β) (β α). (, (,, )) ((,, ), ). Eliminate, replacing α β with α β. (,,, ) ( (,, ), ) 3. ove inwards using de organ s rules and double-negation: (,,, ) ((,, ), ) 4. pply distributivity law ( over ) and flatten: (,,, ) (,, ) (,, ) 68
Resolution algorithm roof by contradiction, i.e., show K α unsatisfiable function -RESOUTION(K, α) returns true or false 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 -RESOVE(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 69 Resolution example K =(, (,, )), α =,,,,,,,,,,,,,,,,,,,,,,,, 7
Summary ogical 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 ropositional logic lacks expressive power 7