How To Write A Program That Is A Knowledge Based Computer Science Program

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1 Knowledge bases Inference engine domain independent algorithms ogical agents Chapter 7 Knowledge base domain specific content Knowledge base = set of sentences in a formal language Declarative approach to building an agent (or other system): Tell 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 Chapter 7 Chapter 7 3 Knowledge-based agents Wumpus world Outline ogic in general models and entailment ropositional (oolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution simple knowledge-based agent function K-gent( percept) returns an action static: K, a knowledge base t, a counter, initially, indicating time Tell(K, ake-ercept-sentence( percept, t)) action sk(k, ake-ction-uery(t)) Tell(K, ake-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 Chapter 7 Chapter 7 4

2 Wumpus World ES description Wumpus world characterization 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 4 3 Stench Stench Gold Stench STRT 3 4 Observable?? artially only local perception Deterministic?? ctuators eft turn, Right turn, Forward, Grab, Release, Shoot Sensors, Glitter, Smell Chapter 7 5 Chapter 7 7 Observable?? Wumpus world characterization Wumpus world characterization Observable?? artially only local perception Deterministic?? Yes outcomes eactly specified Episodic?? Chapter 7 6 Chapter 7 8

3 Wumpus world characterization Observable?? artially local perception Deterministic?? Yes outcomes eactly specified Episodic?? No sequential at the level of actions Static?? Wumpus world characterization Observable?? artially local perception Deterministic?? Yes outcomes eactly specified Episodic?? No sequential at the level of actions Static?? Yes Wumpus and its do not move Discrete?? Yes Single-agent?? Chapter 7 9 Chapter 7 Wumpus world characterization Observable?? artially local perception Deterministic?? Yes outcomes eactly specified Episodic?? No sequential at the level of actions Static?? Yes Wumpus and its do not move Discrete?? Wumpus world characterization Observable?? artially local perception Deterministic?? Yes outcomes eactly 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 Chapter 7 Chapter 7

4 Eploring a wumpus world Eploring a wumpus world?? ercept variables: =breeze, S=stench, G=glitter State variables: =pit, W=wumpus, ercept variables: =breeze, S=stench, G=glitter State variables: =pit, W=wumpus, Chapter 7 3 Chapter 7 5 Eploring a wumpus world Eploring a wumpus world?? S ercept variables: =breeze, S=stench, G=glitter State variables: =pit, W=wumpus, ercept variables: =breeze, S=stench, G=glitter State variables: =pit, W=wumpus, Chapter 7 4 Chapter 7 6

5 Eploring a wumpus world Eploring a wumpus world???? S W S W ercept variables: =breeze, S=stench, G=glitter State variables: =pit, W=wumpus, ercept variables: =breeze, S=stench, G=glitter State variables: =pit, W=wumpus, Chapter 7 7 Chapter 7 9 Eploring a wumpus world Eploring a wumpus world???? GS S W S W ercept variables: =breeze, S=stench, G=glitter State variables: =pit, W=wumpus, ercept variables: =breeze, S=stench, G=glitter State variables: =pit, W=wumpus, Chapter 7 8 Chapter 7

6 Other tight spots Entailment Entailment means that one thing follows from another:? K = α S??? in (,) and (,) no safe actions Smell in (,) cannot move Can use a strategy of coercion: shoot straight ahead wumpus was there dead safe wumpus wasn t there safe 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, the Reds won, (...) entails the Giants won or the Reds won [logical or, not eclusive or] (also the Gaints won and the Reds did not lose,...) E.g., +y=4 entails 4=+y (also...) Entailment is a relationship between sentences (i.e., synta) that is based on semantics Note: brains process synta (of some sort) Chapter 7 Chapter 7 3 ogic in general odels ogics are formal languages for representing information such that conclusions can be drawn Synta 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 + y is a sentence; +y > is not a sentence + y is true iff the number + is no less than the number y + y is true in a world where =7, y= + y is false in a world where =, y=6 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 a sentence α if α is true in m (α) is the set of all models of α Then K = α iff (K) (α) (ie ((K) (α)) = (K) ) E.g. K: Giants won and Reds won α: Giants won [α Gaints and Yankees won ] ( ) (K) Chapter 7 Chapter 7 4

7 Entailment in the wumpus world Wumpus models Situation after detecting nothing in [,], moving right, breeze in [,] Consider possible models for?s assuming only pits 3 oolean choices 8 possible models??? K K = wumpus-world rules + observations Chapter 7 5 Chapter 7 7 Wumpus models Wumpus models 3 3 K K = wumpus-world rules + observations α = [,] is safe, K = α, proved by model checking Chapter 7 6 Chapter 7 8

8 Wumpus models Inference K i α = sentence α can be derived from K by procedure i K 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, it is also true that K = α Completeness: i is complete if whenever K = α, it is also true that K i α 3 3 review: we will define a logic (first-order logic) which is epressive enough to say almost anything of interest, and for which there eists a sound and complete inference procedure. K = wumpus-world rules + observations 3 That is, the procedure will answer any question whose answer follows from what is known by the K. Chapter 7 9 Chapter 7 3 Wumpus models ropositional logic: Synta ropositional logic is the simplest logic illustrates basic ideas K 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 K = wumpus-world rules + observations α = [,] is safe, K = α Chapter 7 3 Chapter 7 3

9 ropositional logic: Semantics Each model specifies true/false for each proposition symbol E.g.,, 3, true true false (With these symbols, 8 possible models, can be enumerated automatically.) 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]. 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 Chapter 7 33 Chapter 7 35 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 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]., (,, ), (,, 3, ) square is breezy if and only if there is an adjacent pit Chapter 7 34 Chapter 7 36

10 Truth tables for inference,,,,,, 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/models (different assignments to symbols,... 3, ). R,R,... are (true) sentences in K (what we know). When all R,R,... are true, K is true. There are only three rows,, (α) is all false (ie, infer no pit in [,]). 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 Chapter 7 37 Chapter 7 39 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 -True?(K, model) then return -True?(α, model) else return true else do First(symbols); rest Rest(symbols) return TT-Check-ll(K, α, rest, Etend(, true, model)) and TT-Check-ll(K, α, rest, Etend(, false, model)) Validity and satisfiability sentence is valid 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 sentence is satisfiable if it is true in some models e.g.,, C sentence is unsatisfiable 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 (proof by contradiction) O( n ) for n symbols Chapter 7 38 Chapter 7 4

11 roof methods Forward chaining roof methods divide into (roughly) two kinds: Idea: fire any rule whose premises are satisfied in the K, add its conclusion to the K, until query is found 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 eponential 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 Chapter 7 4 Chapter 7 43 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 Forward chaining algorithm function -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, indeed by clause, initially the number of premises inferred, a table, indeed by symbol, each entry initially false agenda, a list of symbols, initially the symbols known in K while agenda is not empty do p op(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 Head[c] = q then return true ush(head[c], agenda) return false Chapter 7 4 Chapter 7 44

12 Forward chaining eample Chapter 7 45 Forward chaining eample Chapter 7 46 Forward chaining eample Chapter 7 47 Forward chaining eample Chapter 7 48

13 Forward chaining eample Chapter 7 49 Forward chaining eample Chapter 7 5 Forward chaining eample Chapter 7 5 Forward chaining eample Chapter 7 5

14 roof of completeness (FC with Horn clauses) FC derives every atomic sentence that is entailed by K. FC reaches a fied point where no new atomic sentences are derived ackward chaining eample. 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 fied 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 53 Chapter 7 55 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 ackward chaining eample void repeated work: check if new subgoal ) has already been proved true, or ) has already failed Chapter 7 54 Chapter 7 56

15 ackward chaining eample ackward chaining eample Chapter 7 57 Chapter 7 59 ackward chaining eample ackward chaining eample Chapter 7 58 Chapter 7 6

16 ackward chaining eample ackward chaining eample Chapter 7 6 Chapter 7 63 ackward chaining eample ackward chaining eample Chapter 7 6 Chapter 7 64

17 ackward chaining eample 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 Chapter 7 65 Chapter 7 67 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? Compleity of C can be much less than linear in size of K 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: (,,, ) (,, ) (,, ) Chapter 7 66 Chapter 7 68

18 Resolution algorithm roof by contradiction, i.e., show K α unsatisfiable function -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 loop do new {} // updated from book for each C i, C j in clauses do resolvents -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 Summary ogical agents apply inference to a knowledge base to derive new information and make decisions asic concepts of logic: synta: 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 epressive power Chapter 7 69 Chapter 7 7 Resolution eample K = (, (,, )), α =,,,,,,,,,,,,,,,,,,,,,,,, Chapter 7 7