CS 188 Artificial Intelligence Spring 2007 Lecture 8 Logical Agents I 2 8 2007 Srini Narayanan ICSI and UC Berkeley Many slides over the course adapted from Dan Klein Stuart Russell or Andrew Moore Announcements Concurrent Enrollment Assignment 1 Solutions up Note on notational variants Non Zero Sum Games Similar to minimax Utilities are now tuples Each player maximizes their own entry at each node Propagate or back up nodes from children 1 2 6 4 3 2 6 1 2 7 4 1 5 1 1 1 5 2 7 7 1 5 4 5 Stochastic Single Player What if we don t know what the result of an action will be E g In solitaire shuffle is unknown In minesweeper don t know where the mines are max Can do expectimax search Chance nodes like actions except the environment controls the action chosen Calculate utility for each node Max nodes as in search Chance nodes take average expectation of value of children Later we ll learn how to formalize this as a Markov Decision Process average 8 2 5 6 Stochastic Two Player E g backgammon Expectiminimax Environment is an extra player that moves after each agent Chance nodes take expectations otherwise like minimax Game Playing State of the Art Checkers Chinook ended 40 year reign of human world champion Marion Tinsley in 1994 Used an endgame database defining perfect play for all positions involving 8 or fewer pieces on the board a total of 443 748 401 247 positions Chess Deep Blue defeated human world champion Gary Kasparov in a six game match in 1997 Deep Blue examined 200 million positions per second used very sophisticated evaluation and undisclosed methods for extending some lines of search up to 40 ply Othello human champions refuse to compete against computers which are too good Go human champions refuse to compete against computers which are too bad In go b 300 so most programs use pattern knowledge bases to suggest plausible moves Logical Agents Reflex agents find their way from Arad to Bucharest by dumb luck Chess program calculates legal moves of its king but doesn t know that no piece can be on 2 different squares at the same time Logic Knowledge Based agents combine general knowledge current percepts to infer hidden aspects of current state prior to selecting actions Crucial in partially observable environments Outline Knowledge based agents Wumpus world example Logic in general models and entailment Propositional Boolean logic Equivalence validity satisfiability Inference Knowledge bases 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 Ask itself what to do answers should follow from the KB Agents 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 KB and algorithms that manipulate them A simple knowledge based agent 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 Wumpus World PEAS description Performance measure gold 1000 death 1000 1 per step 10 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 Stench Breeze Glitter Bump Scream Actuators Left turn Right turn Forward Grab Release Shoot Wumpus world characterization Fully 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 Exploring the Wumpus World 1 The KB initially contains the rules of the environment 2 1 1 The first percept is none none none none none Move to safe cell e g 2 1 3 2 1 Breeze indicates that there is a pit in 2 2 or 3 1 4 Return to 1 1 to try next safe cell Exploring the Wumpus World 4 1 2 Stench in cell wumpus is in 1 3 or 2 2 YET not in 1 1 Thus not in 2 2 or stench would have been detected in 2 1 Thus wumpus is in 1 3 Thus 2 2 is safe because of lack of breeze in 1 2 Thus pit in 3 1 Move to next safe cell 2 2 Exploring the Wumpus World 5 2 2 Detect nothing Move to unvisited safe cell e g 2 3 6 2 3 Detect glitter smell breeze Thus pick up gold Thus pit in 3 3 or 2 4 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 2 y is a sentence x2 y is not a sentence x 2 y is true iff the number x 2 is no less than the number y x 2 y is true in a world where x 7 y 1 x 2 y is false in a world where x 0 y 6 Entailment Entailment means that one thing follows from another KB Knowledge base KB entails sentence if and only if 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 x y 4 entails 4 x y Entailment is a relationship between sentences i e syntax that is based on semantics Schematic perspective If KB is true in the real world then any sentence derived from KB by a sound inference procedure is also true in the real world 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 a sentence if is true in m M is the set of all models of Then KB iff M KB M E g KB Giants won and Reds won Giants won Entailment in the wumpus world Situation after detecting nothing in 1 1 moving right breeze in 2 1 Consider possible models for KB assuming only pits 3 Boolean choices 8 possible models Wumpus models Wumpus models KB wumpus world rules observations Wumpus models KB wumpus world rules observations 1 1 2 is safe KB 1 proved by model checking Wumpus models KB wumpus world rules observations Wumpus models KB wumpus world rules observations 2 2 2 is safe KB 2 Inference Procedures KB i sentence can be derived from KB by procedure i Soundness i is sound if whenever KB i it is also true that KB no wrong inferences but maybe not all true statements can be derived Completeness i is complete if whenever KB it is also …
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