Last update: March 4, 00 Logical agents CMSC 4: Chapter 7 CMSC 4: Chapter 7
Outline Knowledge-based agents Wumpus world Logic in general models and entailment Propositional (oolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution CMSC 4: Chapter 7
Knowledge bases Inference engine domain independent algorithms 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 CMSC 4: Chapter 7 3
simple knowledge-based agent function K-gent( percept) returns an action static: K, a knowledge base t, a counter, initially 0, indicating time Tell(K, Make-Percept-Sentence( percept, t)) action sk(k, Make-ction-Query(t)) Tell(K, Make-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 CMSC 4: Chapter 7 4
Wumpus World PES description Environment: One wumpus, one heap of gold P (pit) = 0. for each square Squares net to wumpus are smelly Shooting into wumpus s square kills it Shooting uses up the only arrow Squares net to pit are breezy Glitter iff the gold is in your square Grabbing picks it up Releasing drops it 4 3 Stench Stench STRT reeze Stench Gold reeze reeze reeze reeze reeze 3 4 Performance measure: gold +000, death 000, per step, 0 for using the arrow ctuators: Left turn, Right turn, Forward, Grab, Release, Shoot Sensors: reeze, Glitter, Smell CMSC 4: Chapter 7 5
Fully observable? Wumpus world characterization CMSC 4: Chapter 7 6
Wumpus world characterization Fully observable? No only local perception Deterministic? CMSC 4: Chapter 7 7
Wumpus world characterization Fully observable? No only local perception Deterministic? Yes outcomes eactly specified Episodic? CMSC 4: Chapter 7 8
Wumpus world characterization Fully observable? No only local perception Deterministic? Yes outcomes eactly specified Episodic? No sequential at the level of actions Static? CMSC 4: Chapter 7 9
Wumpus world characterization Fully observable? No only local perception Deterministic? Yes outcomes eactly specified Episodic? No sequential at the level of actions Static? Yes Wumpus, pits, and gold do not move Discrete? CMSC 4: Chapter 7 0
Wumpus world characterization Fully observable? No only local perception Deterministic? Yes outcomes eactly specified Episodic? No sequential at the level of actions Static? Yes Wumpus, pits, and gold do not move Discrete? Yes Single-agent? CMSC 4: Chapter 7
Wumpus world characterization Fully observable? No only local perception Deterministic? Yes outcomes eactly specified Episodic? No sequential at the level of actions Static? Yes Wumpus, pits, and gold do not move Discrete? Yes Single-agent? Yes Wumpus is essentially a natural feature CMSC 4: Chapter 7
Eploring a wumpus world CMSC 4: Chapter 7 3
Eploring a wumpus world CMSC 4: Chapter 7 4
Eploring a wumpus world P? P? CMSC 4: Chapter 7 5
Eploring a wumpus world P? P? S CMSC 4: Chapter 7 6
Eploring a wumpus world P P? P? S W CMSC 4: Chapter 7 7
Eploring a wumpus world P P? P? S W CMSC 4: Chapter 7 8
Eploring a wumpus world P P? P? S W CMSC 4: Chapter 7 9
Eploring a wumpus world P P? P? GS S W CMSC 4: Chapter 7 0
Other tight spots P? reeze in (,) and (,) no safe actions P? P? P(pit in (,)) 0.86 P(pits in (,3) and (3,)) 0.3 P? In a later chapter we ll see how to compute this S Smell in (,) cannot move safely Can use a strategy of coercion: shoot straight ahead wumpus was there dead safe wumpus wasn t there safe CMSC 4: Chapter 7
Logic in general Logics 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 at least as big as the number y + y is true in a world where = 7, y = + y is false in a world where = 0, y = 6 CMSC 4: Chapter 7
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., if a K contains Maryland won and Duke won, the K entails Maryland won or* Duke won E.g., + y = 4 entails 4 = + y Entailment is a relationship between sentences (i.e., synta) that is based on semantics Note: brains process synta (of some sort) *The or is inclusive, not eclusive. CMSC 4: Chapter 7 3
Models model is a formally structured world in which truth can be evaluated We say m is a model of a sentence α 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 = Maryland won and Duke won α = Maryland won M( ) M(α) M(K) M(K) CMSC 4: Chapter 7 4
Entailment in the wumpus world Situation after detecting nothing in [,], moving right, breeze in [,] For now, ignore the wumpus and gold. Which of the?s are pits? For each possible combination of pit locations, check whether it s a model. 3 oolean choices 8 possible models??? CMSC 4: Chapter 7 5
Wumpus models reeze reeze 3 3 reeze 3 reeze 3 reeze 3??? reeze 3 reeze 3 reeze 3 K = wumpus-world rules + observations Eight possible combinations of pit locations: which ones are models of K? CMSC 4: Chapter 7 6
Wumpus models reeze reeze K 3 3 reeze 3 reeze 3 reeze 3??? reeze 3 reeze 3 reeze 3 K = wumpus-world rules + observations Three models CMSC 4: Chapter 7 7
Wumpus models reeze reeze K 3 3 reeze 3 reeze 3 reeze 3??? reeze 3 reeze 3 reeze 3 K = wumpus-world rules + observations α = [,] is safe, K = α, proved by model checking CMSC 4: Chapter 7 8
Wumpus models reeze reeze K 3 3 reeze 3 reeze 3 reeze 3??? reeze 3 reeze 3 reeze 3 K = wumpus-world rules + observations CMSC 4: Chapter 7 9
Wumpus models K reeze 3 reeze 3 reeze 3 reeze 3 reeze 3??? reeze 3 reeze 3 reeze 3 K = wumpus-world rules + observations α = [,] is safe, K = α CMSC 4: Chapter 7 30
Inference K i α = sentence α 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, it is also true that K = α Completeness: i is complete if whenever K = α, it is also true that K i α Model checking (what we just did) is one kind of inference procedure, but not the only one Model checking is sound Model checking is complete if the set of all possible models is finite CMSC 4: Chapter 7 3
Preview of where we re going Later, we ll 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. That is, the procedure will answer any question whose answer follows from what is known by the K. ut first, let s look at propositional logic. CMSC 4: Chapter 7 3
Propositional logic: Synta 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) CMSC 4: Chapter 7 33
Propositional logic: Semantics Each model specifies a true/false value for every proposition symbol E.g. P, P, P 3, true true f alse (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 CMSC 4: Chapter 7 34
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 CMSC 4: Chapter 7 35
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 CMSC 4: Chapter 7 36
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 CMSC 4: Chapter 7 37
Truth tables for inference,, P, P, P, P, P 3, K α false false false false false false false false don t care false. false. false. false. false. false. true. false. don t care. false true false false false false false false don t care 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. don t care. true true true true true true true false don t care Model checking in propositional logic = inference using truth tables Each row is a potential model: an assignment of truth values to symbols if K is true in row, is α true too? CMSC 4: Chapter 7 38
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, Etend(P, true, model)) and TT-Check-ll(K, α, rest, Etend(P, false, model)) O( n ) for n symbols; problem is co-np-complete CMSC 4: Chapter 7 39
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 CMSC 4: Chapter 7 40
Validity and satisfiability sentence is valid if it is true in all models, e.g., T rue,,, ( ( )) 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 at least one model 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 CMSC 4: Chapter 7 4
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 eponential in n) improved backtracking, e.g., Davis Putnam Logemann Loveland heuristic search in model space (sound but incomplete) e.g., min-conflicts-like hill-climbing algorithms CMSC 4: Chapter 7 4
Forward and backward chaining Horn Form K = conjunction of Horn clauses Horn clause = proposition symbol; or (conjunction of symbols) symbol E.g., C ( ) (C D ) This is a restricted subset of propositional logic e.g., the following is not a Horn clause, and can t be translated into one: 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 CMSC 4: Chapter 7 43
Forward chaining Idea: fire any rule whose premises are satisfied in the K, add its conclusion to the K, until query is found P Q L M P L M P L L L Q P M CMSC 4: Chapter 7 44
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(c), number of c s premises not yet inferred inferred(c), whether or not c has been inferred agenda, {all clauses that are ready to be inferred} while agenda is not empty do p Pop(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] = 0 then do if Head[c] = q then return true Push(Head[c], agenda) return false CMSC 4: Chapter 7 45
Forward chaining eample Q P Q L M P L M P L L P M L CMSC 4: Chapter 7 46
Forward chaining eample Q P Q L M P L M P L L P M L CMSC 4: Chapter 7 47
Forward chaining eample Q P Q L M P L M P L L P L M 0 CMSC 4: Chapter 7 48
Forward chaining eample Q P Q L M P L M P L L P L M 0 0 CMSC 4: Chapter 7 49
Forward chaining eample Q P Q L M P L M P L L P 0 L M 0 0 CMSC 4: Chapter 7 50
Forward chaining eample Q 0 P Q L M P L M P L L 0 P 0 L M 0 0 CMSC 4: Chapter 7 5
Forward chaining eample Q 0 P Q L M P L M P L L 0 P 0 L M 0 0 CMSC 4: Chapter 7 5
Forward chaining eample Q 0 P Q L M P L M P L L 0 P 0 L M 0 0 CMSC 4: Chapter 7 53
Proof of completeness FC derives every atomic sentence that is entailed by K. t i th iteration of the while loop, can create a world m i as follows: assign true to every atomic symbol that has been derived assign false to all other atomic symbols.. There are only finitely many atomic sentences, so FC must reach a fied point where no new atomic sentences are derived. Let n be the iteration where this happens. 3. Every clause in the original K is true in m n Proof: Suppose a clause a... a k b is false in m i. Then a... a k is true in m i and b is false in m i. Thus count(b) = 0, so b will be inferred in a future iteration, so iteration i isn t a fied point. Hence at the fied point, m n is a model of K. 5. If q is atomic and K = q, then q is true in every model of K, including m. Hence FC must have derived q. CMSC 4: Chapter 7 54
ackward chaining Idea: work backwards from the query q: to prove q by C, check if q is known already, or recursively call C to prove all premises of some rule concluding q void loops: check if new subgoal is already on the recursion stack void repeated work: check if new subgoal ) has already been proved true, or ) has already failed CMSC 4: Chapter 7 55
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 56
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 57
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 58
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 59
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 60
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 6
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 6
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 63
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 64
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 65
ackward chaining eample Q P Q L M P L M P L L L P M CMSC 4: Chapter 7 66
Forward vs. backward chaining FC is data-driven data-driven algorithms can be used for automatic, unconscious processing, e.g., object recognition, routine decisions May 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 PhD program? Compleity of C can be much less than linear in size of K CMSC 4: Chapter 7 67
Resolution Conjunctive Normal Form (CNF): conjunction of disjunctions of literals }{{} clauses E.g., ( ) ( C D) Resolution inference rule: If l i and m j are negations of each other, l l i l k, m m j m n l l i l i+ l k m m j m j+ m n i.e., the disjunct of everything other than l i and m j Eample: P,3 P,, P, P,3 P P? P? S W CMSC 4: Chapter 7 68
Resolution Resolution is equivalent to Modus Ponens: clauses from CNF: C D rewrite as implications: C D apply modus ponens: C D rewrite as clauses: C D Resolution is sound and complete for propositional logic ut to use it, you need to convert all your propositional statements to CNF CMSC 4: Chapter 7 69
Conversion to CNF There s a breeze in (,) iff there s a pit in (,) or (,):, (P, P, ). Eliminate, by replacing α β with (α β) (β α). (, (P, P, )) ((P, P, ), ). Eliminate, by 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,, ) CMSC 4: Chapter 7 70
Resolution algorithm Proof by contradiction: to prove K α, show K α is 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 ;; compute all possible resolvents from clauses, and add them to clauses 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 CMSC 4: Chapter 7 7
Resolution eample K: there s a breeze in (,) iff there s a pit in (,) or (,); and there s no breeze in (,) K = (, (P, P, )), = (, P, P, ) ( P,, ) ( P,, ), α = P, we want to show there s no pit in (,) α = P, suppose there is one (for proof by contradiction) CMSC 4: Chapter 7 7
Summary Logical 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 Propositional logic lacks epressive power CMSC 4: Chapter 7 73