Alpha-beta pruning and linear regression problem

Computer Science

Algorithm

Question Description

1) Use the alpha-beta pruning for the "1-2 steal marbles ", for 13 marbles.

Initially, there are 13 marbles on the board.

One of the players can choose to remove 1 or 2 marbles leaving 12 or 11, after that the other player can do the same, choosing to take again one or two marbles from the board. The process continue until there is only one marble in the board. The player who wins is the one the leaves the last marble on the board. (For example: If there are 3 marbles and it's my turn , then I will choose to remove 2 to leave one in the board to win)

Comment the results.

2) given the pairs (time, price) (1,3) ,(2,5), (3,8) use linear regression to find the line ax+b that approximate these values. Use that line to calculate the price when the time is 4.

Please do not plagiarize.

Alpha-beta section 5.3 in P167


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Game playing Chapter 6 Chapter 6 1 Outline ♦ Games ♦ Perfect play – minimax decisions – α–β pruning ♦ Resource limits and approximate evaluation ♦ Games of chance ♦ Games of imperfect information Chapter 6 2 Games vs. search problems “Unpredictable” opponent ⇒ solution is a strategy specifying a move for every possible opponent reply Time limits ⇒ unlikely to find goal, must approximate Plan of attack: • Computer considers possible lines of play (Babbage, 1846) • Algorithm for perfect play (Zermelo, 1912; Von Neumann, 1944) • Finite horizon, approximate evaluation (Zuse, 1945; Wiener, 1948; Shannon, 1950) • First chess program (Turing, 1951) • Machine learning to improve evaluation accuracy (Samuel, 1952–57) • Pruning to allow deeper search (McCarthy, 1956) Chapter 6 3 Types of games deterministic chance perfect information chess, checkers, go, othello backgammon monopoly imperfect information battleships, blind tictactoe bridge, poker, scrabble nuclear war Chapter 6 4 Game tree (2-player, deterministic, turns) MAX (X) X X X MIN (O) X X X X X O X X O X X O X O MAX (X) MIN (O) TERMINAL Utility X O ... X O X ... ... ... ... ... X O X O X O X O X O O X X X O X O X X X O O ... −1 0 +1 X X Chapter 6 5 Minimax Perfect play for deterministic, perfect-information games Idea: choose move to position with highest minimax value = best achievable payoff against best play E.g., 2-ply game: 3 MAX A1 A2 A3 3 MIN A 11 3 A 12 12 2 A 21 A 13 8 2 2 A 31 A 22 A 23 4 6 14 A 32 A 33 5 2 Chapter 6 6 Minimax algorithm function Minimax-Decision(state) returns an action inputs: state, current state in game return the a in Actions(state) maximizing Min-Value(Result(a, state)) function Max-Value(state) returns a utility value if Terminal-Test(state) then return Utility(state) v ← −∞ for a, s in Successors(state) do v ← Max(v, Min-Value(s)) return v function Min-Value(state) returns a utility value if Terminal-Test(state) then return Utility(state) v←∞ for a, s in Successors(state) do v ← Min(v, Max-Value(s)) return v Chapter 6 7 Properties of minimax Complete?? Chapter 6 8 Properties of minimax Complete?? Only if tree is finite (chess has specific rules for this). NB a finite strategy can exist even in an infinite tree! Optimal?? Chapter 6 9 Properties of minimax Complete?? Yes, if tree is finite (chess has specific rules for this) Optimal?? Yes, against an optimal opponent. Otherwise?? Time complexity?? Chapter 6 10 Properties of minimax Complete?? Yes, if tree is finite (chess has specific rules for this) Optimal?? Yes, against an optimal opponent. Otherwise?? Time complexity?? O(bm) Space complexity?? Chapter 6 11 Properties of minimax Complete?? Yes, if tree is finite (chess has specific rules for this) Optimal?? Yes, against an optimal opponent. Otherwise?? Time complexity?? O(bm) Space complexity?? O(bm) (depth-first exploration) For chess, b ≈ 35, m ≈ 100 for “reasonable” games ⇒ exact solution completely infeasible But do we need to explore every path? Chapter 6 12 α–β pruning example 3 MAX 3 MIN 3 12 8 Chapter 6 13 α–β pruning example 3 MAX 2 3 MIN 3 12 8 2 X X Chapter 6 14 α–β pruning example 3 MAX 2 3 MIN 3 12 8 2 X X 14 14 Chapter 6 15 α–β pruning example 3 MAX 2 3 MIN 3 12 8 2 X X 14 14 5 5 Chapter 6 16 α–β pruning example 3 3 MAX 2 3 MIN 3 12 8 2 X X 14 14 5 5 2 2 Chapter 6 17 Why is it called α–β ? MAX MIN .. .. .. MAX MIN V α is the best value (to max) found so far off the current path If V is worse than α, max will avoid it ⇒ prune that branch Define β similarly for min Chapter 6 18 The α–β algorithm function Alpha-Beta-Decision(state) returns an action return the a in Actions(state) maximizing Min-Value(Result(a, state)) function Max-Value(state, α, β) returns a utility value inputs: state, current state in game α, the value of the best alternative for max along the path to state β, the value of the best alternative for min along the path to state if Terminal-Test(state) then return Utility(state) v ← −∞ for a, s in Successors(state) do v ← Max(v, Min-Value(s, α, β)) if v ≥ β then return v α ← Max(α, v) return v function Min-Value(state, α, β) returns a utility value same as Max-Value but with roles of α, β reversed Chapter 6 19 Properties of α–β Pruning does not affect final result Good move ordering improves effectiveness of pruning With “perfect ordering,” time complexity = O(bm/2) ⇒ doubles solvable depth A simple example of the value of reasoning about which computations are relevant (a form of metareasoning) Unfortunately, 3550 is still impossible! Chapter 6 20 Resource limits Standard approach: • Use Cutoff-Test instead of Terminal-Test e.g., depth limit (perhaps add quiescence search) • Use Eval instead of Utility i.e., evaluation function that estimates desirability of position Suppose we have 100 seconds, explore 104 nodes/second ⇒ 106 nodes per move ≈ 358/2 ⇒ α–β reaches depth 8 ⇒ pretty good chess program Chapter 6 21 Evaluation functions Black to move White to move White slightly better Black winning For chess, typically linear weighted sum of features Eval(s) = w1f1(s) + w2f2(s) + . . . + wnfn(s) e.g., w1 = 9 with f1(s) = (number of white queens) – (number of black queens), etc. Chapter 6 22 Digression: Exact values don’t matter MAX MIN 2 1 1 2 2 20 1 4 1 20 20 400 Behaviour is preserved under any monotonic transformation of Eval Only the order matters: payoff in deterministic games acts as an ordinal utility function Chapter 6 23 Deterministic games in practice 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 sixgame match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply. Othello: human champions refuse to compete against computers, who are too good. Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves. Chapter 6 24 Nondeterministic games: backgammon 0 25 1 2 3 4 5 6 24 23 22 21 20 19 7 8 9 10 11 12 18 17 16 15 14 13 Chapter 6 25 Nondeterministic games in general In nondeterministic games, chance introduced by dice, card-shuffling Simplified example with coin-flipping: MAX 3 CHANCE −1 0.5 MIN 2 2 0.5 0.5 4 4 7 0.5 0 4 6 −2 0 5 −2 Chapter 6 26 Algorithm for nondeterministic games Expectiminimax gives perfect play Just like Minimax, except we must also handle chance nodes: ... if state is a Max node then return the highest ExpectiMinimax-Value of Successors(state) if state is a Min node then return the lowest ExpectiMinimax-Value of Successors(state) if state is a chance node then return average of ExpectiMinimax-Value of Successors(state) ... Chapter 6 27 Nondeterministic games in practice Dice rolls increase b: 21 possible rolls with 2 dice Backgammon ≈ 20 legal moves (can be 6,000 with 1-1 roll) depth 4 = 20 × (21 × 20)3 ≈ 1.2 × 109 As depth increases, probability of reaching a given node shrinks ⇒ value of lookahead is diminished α–β pruning is much less effective TDGammon uses depth-2 search + very good Eval ≈ world-champion level Chapter 6 28 Digression: Exact values DO matter MAX 2.1 DICE 1.3 .9 MIN .1 2 2 .9 3 2 3 .1 1 3 1 21 .9 4 1 4 40.9 20 4 20 .1 30 20 30 30 .9 1 1 .1 400 1 400 400 Behaviour is preserved only by positive linear transformation of Eval Hence Eval should be proportional to the expected payoff Chapter 6 29 Games of imperfect information E.g., card games, where opponent’s initial cards are unknown Typically we can calculate a probability for each possible deal Seems just like having one big dice roll at the beginning of the game∗ Idea: compute the minimax value of each action in each deal, then choose the action with highest expected value over all deals∗ Special case: if an action is optimal for all deals, it’s optimal.∗ GIB, current best bridge program, approximates this idea by 1) generating 100 deals consistent with bidding information 2) picking the action that wins most tricks on average Chapter 6 30 Example Four-card bridge/whist/hearts hand, Max to play first 6 6 8 7 4 2 3 9 8 6 6 4 2 9 7 6 6 7 6 3 4 2 3 4 9 2 6 7 3 6 6 7 4 3 Chapter 6 0 31 Example Four-card bridge/whist/hearts hand, Max to play first MAX 6 6 8 7 MIN 4 2 9 3 MAX 6 6 8 7 MIN 4 2 9 8 3 8 6 6 4 2 6 6 4 2 7 9 9 6 6 7 4 2 3 7 6 6 7 6 3 4 2 3 4 3 9 9 6 2 2 6 4 7 6 3 6 7 3 6 7 4 3 6 6 4 0 7 0 3 Chapter 6 32 Example Four-card bridge/whist/hearts hand, Max to play first MAX 6 6 8 7 MIN 4 2 9 3 MAX 6 6 8 7 MIN 4 2 9 8 8 3 6 6 4 2 6 6 4 2 9 9 7 6 6 7 6 3 4 2 3 4 7 6 6 7 6 3 4 2 3 4 9 9 2 2 6 7 6 3 6 7 6 7 4 3 6 3 6 4 MAX 6 6 8 7 MIN 4 2 9 3 8 6 6 4 2 7 9 3 6 9 6 7 4 2 3 6 2 4 6 7 3 6 7 0 3 6 6 0 7 4 3 6 7 4 3 −0.5 −0.5 Chapter 6 33 Commonsense example Road A leads to a small heap of gold pieces Road B leads to a fork: take the left fork and you’ll find a mound of jewels; take the right fork and you’ll be run over by a bus. Chapter 6 34 Commonsense example Road A leads to a small heap of gold pieces Road B leads to a fork: take the left fork and you’ll find a mound of jewels; take the right fork and you’ll be run over by a bus. Road A leads to a small heap of gold pieces Road B leads to a fork: take the left fork and you’ll be run over by a bus; take the right fork and you’ll find a mound of jewels. Chapter 6 35 Commonsense example Road A leads to a small heap of gold pieces Road B leads to a fork: take the left fork and you’ll find a mound of jewels; take the right fork and you’ll be run over by a bus. Road A leads to a small heap of gold pieces Road B leads to a fork: take the left fork and you’ll be run over by a bus; take the right fork and you’ll find a mound of jewels. Road A leads to a small heap of gold pieces Road B leads to a fork: guess correctly and you’ll find a mound of jewels; guess incorrectly and you’ll be run over by a bus. Chapter 6 36 Proper analysis * Intuition that the value of an action is the average of its values in all actual states is WRONG With partial observability, value of an action depends on the information state or belief state the agent is in Can generate and search a tree of information states Leads ♦ ♦ ♦ to rational behaviors such as Acting to obtain information Signalling to one’s partner Acting randomly to minimize information disclosure Chapter 6 37 Summary Games are fun to work on! (and dangerous) They illustrate several important points about AI ♦ perfection is unattainable ⇒ must approximate ♦ good idea to think about what to think about ♦ uncertainty constrains the assignment of values to states ♦ optimal decisions depend on information state, not real state Games are to AI as grand prix racing is to automobile design Chapter 6 38 Artificial Intelligence A Modern Approach Third Edition PRENTICE HALL SERIES IN ARTIFICIAL INTELLIGENCE Stuart Russell and Peter Norvig, Editors F ORSYTH & P ONCE G RAHAM J URAFSKY & M ARTIN N EAPOLITAN RUSSELL & N ORVIG Computer Vision: A Modern Approach ANSI Common Lisp Speech and Language Processing, 2nd ed. Learning Bayesian Networks Artificial Intelligence: A Modern Approach, 3rd ed. Artificial Intelligence A Modern Approach Third Edition Stuart J. Russell and Peter Norvig Contributing writers: Ernest Davis Douglas D. Edwards David Forsyth Nicholas J. Hay Jitendra M. Malik Vibhu Mittal Mehran Sahami Sebastian Thrun Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo   Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2016 The rights of Stuart J. Russell and Peter Norvig to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Authorized adaptation from the United States edition, entitled Artificial Intelligence: A Modern Approach, Third Edition, ISBN 9780136042594, by Stuart J. Russell and Peter Norvig published by Pearson Education © 2010. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 9 8 7 6 5 4 3 2 1 ISBN 10: 1292153962 ISBN 13: 9781292153964 Printed and bound in Malaysia   For Loy, Gordon, Lucy, George, and Isaac — S.J.R. For Kris, Isabella, and Juliet — P.N. This page intentionally left blank Preface Artificial Intelligence (AI) is a big field, and this is a big book. We have tried to explore the full breadth of the field, which encompasses logic, probability, and continuous mathematics; perception, reasoning, learning, and action; and everything from microelectronic devices to robotic planetary explorers. The book is also big because we go into some depth. The subtitle of this book is “A Modern Approach.” The intended meaning of this rather empty phrase is that we have tried to synthesize what is now known into a common framework, rather than trying to explain each subfield of AI in its own historical context. We apologize to those whose subfields are, as a result, less recognizable. New to this edition This edition captures the changes in AI that have taken place since the last edition in 2003. There have been important applications of AI technology, such as the widespread deployment of practical speech recognition, machine translation, autonomous vehicles, and household robotics. There have been algorithmic landmarks, such as the solution of the game of checkers. And there has been a great deal of theoretical progress, particularly in areas such as probabilistic reasoning, machine learning, and computer vision. Most important from our point of view is the continued evolution in how we think about the field, and thus how we organize the book. The major changes are as follows: • We place more emphasis on partially observable and nondeterministic environments, especially in the nonprobabilistic settings of search and planning. The concepts of belief state (a set of possible worlds) and state estimation (maintaining the belief state) are introduced in these settings; later in the book, we add probabilities. • In addition to discussing the types of environments and types of agents, we now cover in more depth the types of representations that an agent can use. We distinguish among atomic representations (in which each state of the world is treated as a black box), factored representations (in which a state is a set of attribute/value pairs), and structured representations (in which the world consists of objects and relations between them). • Our coverage of planning goes into more depth on contingent planning in partially observable environments and includes a new approach to hierarchical planning. • We have added new material on first-order probabilistic models, including open-universe models for cases where there is uncertainty as to what objects exist. • We have completely rewritten the introductory machine-learning chapter, stressing a wider variety of more modern learning algorithms and placing them on a firmer theoretical footing. • We have expanded coverage of Web search and information extraction, and of techniques for learning from very large data sets. • 20% of the citations in this edition are to works published after 2003. • We estimate that about 20% of the material is brand new. The remaining 80% reflects older work but has been largely rewritten to present a more unified picture of the field. vii viii Preface Overview of the book NEW TERM The main unifying theme is the idea of an intelligent agent. We define AI as the study of agents that receive percepts from the environment and perform actions. Each such agent implements a function that maps percept sequences to actions, and we cover different ways to represent these functions, such as reactive agents, real-time planners, and decision-theoretic systems. We explain the role of learning as extending the reach of the designer into unknown environments, and we show how that role constrains agent design, favoring explicit knowledge representation and reasoning. We treat robotics and vision not as independently defined problems, but as occurring in the service of achieving goals. We stress the importance of the task environment in determining the appropriate agent design. Our primary aim is to convey the ideas that have emerged over the past fifty years of AI research and the past two millennia of related work. We have tried to avoid excessive formality in the presentation of these ideas while retaining precision. We have included pseudocode algorithms to make the key ideas concrete; our pseudocode is described in Appendix B. This book is primarily intended for use in an undergraduate course or course sequence. The book has 27 chapters, each requiring about a week’s worth of lectures, so working through the whole book requires a two-semester sequence. A one-semester course can use selected chapters to suit the interests of the instructor and students. The book can also be used in a graduate-level course (perhaps with the addition of some of the primary sources suggested in the bibliographical notes). Sample syllabi are available at the book’s Web site, aima.cs.berkeley.edu. The only prerequisite is familiarity with basic concepts of computer science (algorithms, data structures, complexity) at a sophomore level. Freshman calculus and linear algebra are useful for some of the topics; the required mathematical background is supplied in Appendix A. Exercises are given at the end of each chapter. Exercises requiring significant programming are marked with a keyboard icon. These exercises can best be solved by taking advantage of the code repository at aima.cs.berkeley.edu. Some of them are large enough to be considered term projects. A number of exercises require some investigation of the literature; these are marked with a book icon. Throughout the book, important points are marked with a pointing icon. We have included an extensive index of around 6,000 items to make it e ...
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