7. Logical Agents. COMP9414/ 9814/ 3411: Artificial Intelligence. Outline. Knowledge base. Models and Planning. Russell & Norvig, Chapter 7.
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1 COMP944/984/34 6s Logic COMP944/ 984/ 34: rtificial Intelligence 7. Logical gents Outline Knowledge-based agents Wumpus world Russell & Norvig, Chapter 7. Logic in general models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution COMP944/984/34 6s Logic Models and Planning COMP944/984/34 6s Logic 3 Knowledge bases World Model transition table dynamical system parametric model knowledge base Perception Planning state based seach simulation goals / utility logical inference ction 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 Environment Then it can sk itself what to do answers should follow from the KB
2 COMP944/984/34 6s Logic 4 simple knowledge-based agent COMP944/984/34 6s Logic 5 Wumpus World PES description Environment The agent must be able to: represent states, actions, etc. Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy incorporate new percepts update internal representations of the world deduce hidden properties of the world Glitter iff gold is in the same square Shooting kills wumpus if you are facing it uses up the only arrow 4 3 Stench Stench Gold determine appropriate actions Grabbing picks up gold if in same square Releasing drops the gold in same square Stench STRT 3 4 COMP944/984/34 6s Logic 6 Wumpus World PES description COMP944/984/34 6s Logic 7 Eploring a wumpus world Performance measure gold +000, death per step, -0 for using the arrow ctuators Left turn, Right turn, Forward, Grab, Release, Shoot Sensors, Glitter, Smell P B S W
3 COMP944/984/34 6s Logic 8 Eploring a wumpus world COMP944/984/34 6s Logic 9 Other tight spots in (,) and (,) no safe actions P B S BGS W B S B ssuming pits uniformly distributed, pit more likely in (,) than in (3,) Eercise: How much more likely? Smell in (,) cannot move Can use a strategy of coercion: shoot straight ahead wumpus was there dead safe wumpus wasn t there safe COMP944/984/34 6s Logic 0 Logic in general COMP944/984/34 6s Logic Entailment 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 no less than the number y + y is true in a world where =7, y= + y is false in a world where =0, y=6 Entailment means that one thing follows from another: KB = α Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true e.g. the KB containing the Giants won and the Reds won entails Either the Giants won or the Reds won e.g. +y=4 entails 4=+y Entailment is a relationship between sentences (i.e. synta) that is based on semantics.
4 COMP944/984/34 6s Logic Models COMP944/984/34 6s Logic 3 Entailment in the wumpus world 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 a sentence α if α is true in m M(α) is the set of all models of α Then KB = α if and only if M(KB) M(α) M( ) M(KB) Situation after detecting nothing in [,], moving right, breeze in [,] Consider possible models for?s assuming only pits 3 Boolean choices 8 possible models? B?? COMP944/984/34 6s Logic 4 Wumpus models COMP944/984/34 6s Logic 5 Wumpus models 3 3 KB KB = wumpus-world rules + observations
5 COMP944/984/34 6s Logic 6 Wumpus models COMP944/984/34 6s Logic 7 Wumpus models KB 3 3 KB KB = wumpus-world rules + observations α = [,] is safe, KB = α, proved by model checking KB = wumpus-world rules + observations α = [,] is safe, KB = α COMP944/984/34 6s Logic 8 Inference COMP944/984/34 6s Logic 9 Propositional logic: Synta KB i α = sentence α can be derived from KB by procedure i Consequences of KB are a haystack; α is a needle. Entailment = needle in haystack; inference = finding it Soundness: i is sound if whenever KB i α, it is also true that KB = α Completeness: i is complete if whenever KB = α, it is also true that KB i α 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 S is a sentence (implication) If S and S are sentences, S S is a sentence (biconditional)
6 COMP944/984/34 6s Logic 0 Propositional logic: Semantics COMP944/984/34 6s Logic Propositional logic: Semantics Each model specifies true/false for each proposition symbol E.g. P, P, P 3, TRUE TRUE FLSE (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 FLSE 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 FLSE or S is TRUE i.e. is FLSE iff S is TRUE and S is FLSE 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 (FLSE TRUE)= TRUE TRUE = TRUE COMP944/984/34 6s Logic Truth tables for connectives COMP944/984/34 6s Logic 3 Wumpus world sentences P Q P P Q P Q P Q F F T F F T F T T F T T T F F F T F T T F T T T Let P i, j be true if there is a pit in[i, j]. Let B i, j be true if there is a breeze in[i, j]. P, B, B, Pits cause breezes in adjacent squares
7 COMP944/984/34 6s Logic 4 Wumpus world sentences COMP944/984/34 6s Logic 5 Logical equivalence Let P i, j be true if there is a pit in[i, j]. Let B i, j be true if there is a breeze in[i, j]. P, Two sentences are logically equivalent iff true in same models: α β if and only if α = β and β = α B, B, Pits cause breezes in adjacent squares B, (P, P, ) B, (P, P, P 3, ) square is breezy if and only if there is an adjacent pit COMP944/984/34 6s Logic 6 Validity and satisfiability COMP944/984/34 6s Logic 7 Proof methods sentence is valid if it is true in all models, e.g. TRUE,,, ( ( B)) B Validity is connected to inference via the Deduction Theorem: KB = α if and only if(kb α) is valid sentence is satisfiable if it is true in some model e.g. B, C sentence is unsatisfiable if it is true in no models e.g. Satisfiability is connected to inference via the following: KB = α if and only if(kb α) is unsatisfiable i.e. prove α by reductio ad absurdum 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 algebra 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)
8 COMP944/984/34 6s Logic 8 Forward and backward chaining COMP944/984/34 6s Logic 9 Forward chaining KB = conjunction of Horn clauses Horn clause = proposition symbol; or Idea: fire any rule whose premises are satisfied in the KB, add its conclusion to the KB, until query is found. (conjunction of symbols) symbol e.g. C (B ) (C D B) Modus Ponens (for Horn Form): complete for Horn KBs α,...,α n, α α n β β Can be used with forward chaining or backward chaining. COMP944/984/34 6s Logic 30 Backward chaining COMP944/984/34 6s Logic 3 Forward vs. backward chaining Idea: work backwards from the query q: check if q is known already, or prove by BC 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 FC is data-driven automatic, unconscious processing e.g. object recognition, routine decisions May do lots of work that is irrelevant to the goal BC is goal-driven, appropriate for problem-solving, e.g. Where are my keys? How do I get into a PhD program? Compleity of BC can be much less than linear in size of KB
9 COMP944/984/34 6s Logic 3 Resolution COMP944/984/34 6s Logic 33 Conversion to CNF Conjunctive Normal Form (CNF universal) conjunction of disjunctions of literals }{{} clauses e.g. ( B) (B 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. B, (P, P, ). Eliminate,replacing α βwith(α β) (β α). (B, (P, P, )) ((P, P, ) B, ). Eliminate,replacing α βwith α β. ( B, P, P, ) ( (P, P, ) B, ) 3. Move inwards using de Morgan s rules and double-negation: ( B, P, P, ) (( P, P, ) B, ) 4. pply distributivity law ( over ) and flatten: ( B, P, P, ) ( P, B, ) ( P, B, ) COMP944/984/34 6s Logic 34 Limitations of Propositional Logic COMP944/984/34 6s Logic 35 Summary square is breezy if and only if there is an adjacent pit. This statement must be converted into a separate sentence for each square: B, (P, P, ) B, (P, P, P 3, ). What we really want is a way to epress such a statement in one sentence for all squares, e.g. Breezy(i, j) (Pit(i, j) Pit(i+, j) Pit(i, j ) Pit(i, j+ )) First-Order Logic will allow us to do this (net lecture). Logical agents apply inference to a knowledge base to derive new information and make decisions. Basic 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 soundness: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences
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