SQA Advanced Higher Computing Unit 3a: Artificial Intelligence

Transcription

1 SCHOLAR Study Guide SQA Advaned Higher Computing Unit 3a: Artifiial Intelligene David Bethune Heriot-Watt University Andy Cohrane Heriot-Watt University Ian King Heriot-Watt University Interative University Edinburgh EH12 9QQ, United Kingdom.

2 First published 2005 by Heriot-Watt University Copyright 2005 Heriot-Watt University Members of the SCHOLAR Forum may reprodue this publiation in whole or in part for eduational purposes within their establishment providing that no profit arues at any stage, Any other use of the materials is governed by the general opyright statement that follows. All rights reserved. No part of this publiation may be reprodued, stored in a retrieval system or transmitted in any form or by any means, without written permission from the publisher. Heriot-Watt University aepts no responsibility or liability whatsoever with regard to the information ontained in this study guide. SCHOLAR is a programme of Heriot-Watt University and is published and distributed on its behalf by Interative University. British Library Cataloguing in Publiation Data Interative University SCHOLAR Study Guide Unit 3a: Computing 1. Computing ISBN Typeset by: Interative University, Wallae House, 1 Lohside Avenue, Edinburgh, EH12 9QQ. Printed and bound in Great Britain by Graphi and Printing Servies, Heriot-Watt University, Edinburgh. Part Number

3 i Contents 1 Problem Representation Prior Knowledge and Revision Introdution The mathes problem Re-arranging bloks Trees and graphs Sliding numbers problem The jugs problem Route finding Real world problems AND/OR graphs Summary End of Topi test Searh Strategies Prior Knowledge and revision Introdution Exhaustive searh Heuristi Searh The Minimax Proedure Summary End of Topi test Knowledge Representation Prior Knowledge and Revision Introdution Semanti Nets Frames The relationship between semanti nets and frames Prolog Summary End of Topi test Further with Prolog Prior Knowledge and Revision Introdution How Prolog solves a query Reursion Inheritane and semanti nets

4 ii CONTENTS 4.6 List proessing Summary End of Topi Test Rule-based Systems Prior Knowledge and Revision Introdution Expert Systems IF..THEN rules Inferening in rule-based systems Certainty Fators Summary End of Topi Test Appliations of AI - Vision and Language Systems Prior Knowledge and Revision Introdution Computer Vision Natural Language Understanding Speeh Reognition The stages of Natural Language Understanding (NLU) Dealing with ambiguity A simple grammar Summary End of Topi test Appliations of AI - Robotis and Mahine Learning Prior Knowledge and Revision Introdution Robotis Mahine Learning Summary End of Topi test Glossary 151 Answers to questions and ativities Problem Representation Searh Strategies Knowledge Representation Further with Prolog Rule-based Systems Appliations of AI - Vision and Language Systems Appliations of AI - Robotis and Mahine Learning

5 Aknowledgements Thanks are due to the members of Heriot-Watt University s SCHOLAR team who planned and reated these materials, and to the many olleagues who reviewed the ontent. Programme Diretor: Professor R R Leith Series Editor: Professor J Cowan Subjet Diretors: Professor P John (Chemistry), Professor C E Beevers (Mathematis), Dr P J B King (Computing), Dr P G Meaden (Biology), Dr M R Steel (Physis), Dr C G Tinker (Frenh) Subjet Authors: Biology: Dr J M Burt, Ms E Humphrey, Ms L Knight, Mr J B MCann, Mr D Millar, Ms N Randle, Ms S Ross, Ms Y Stahl, Ms S Steen, Ms N Tweedie Chemistry: Mr M Anderson, Mr B Bease, Dr J H Cameron, Dr P Johnson, Mr B T MKerhar, Dr A A Sandison Computing: Mr I E Aithison, Dr P O B Holt, Mr S MMorris, Mr B Palmer, Ms J Swanson, Mr A Weddle Engineering: Mr J Hill, Ms H L Jakson, Mr H Laidlaw, Professor W H Müller Frenh: Mr M Fermin, Ms B Guenier, Ms C Hastie, Ms S C E Thoday Mathematis: Mr J Dowman, Ms A Johnstone, Ms O Khaled, Mr C MGuire, Ms J S Paterson, Mr S Rogers, Ms D A Watson Physis: Mr J MCabe, Mr C Milne, Dr A Tookey, Mr C White Learning Tehnology: Dr W Austin, Ms N Beasley, Ms J Benzie, Dr D Cole, Mr A Crofts, Ms S Davies, Mr A Dunn, Mr M Holligan, Dr J Liddle, Ms S MConnell, Mr N Miller, Mr N Morris, Ms E Mowat, Mr S Niol, Dr W Nightingale, Mr R Pointon, Mr D Reid, Dr R Thomas, Dr N Tomes, Ms J Wang, Mr P Whitton Cue Assessment Group: Ms F Costigan, Mr D J Fiddes, Dr D H Jakson, Mr S G Marshall SCHOLAR Unit: Mr G Toner, M G Cruse, Ms A Hay, Ms C Keogh, Ms B Laidlaw, Mr J Walsh Media: Mr D Hurst, Mr P Booth, Mr G Cowper, Mr C Gruber, Mr D S Marsland, Mr C Niol, Mr C Wilson Administration: Ms L El-Ghorr, Dr M King, Dr R Rist, We would like to aknowledge the assistane of the eduation authorities, olleges, teahers and students who helped to plan the SCHOLAR programme and who evaluated these materials. Grateful aknowledgement is made for permission to use the following material in the SCHOLAR programme: To the Sottish Qualifiations Authority for permission to use Past Papers assessments. The finanial support from the Sottish Exeutive is gratefully aknowledged. All brand names, produt names, logos and related devies are used for identifiation purposes only and are trademarks, registered trademarks or servie marks of their respetive holders.

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7 1 Topi 1 Problem Representation Contents 1.1 Prior Knowledge and Revision Introdution Problem Abstration Symboli Representation Searh tehniques The mathes problem Review of Mathes Problem Creating a searh tree Re-arranging bloks Trees and graphs Sliding numbers problem The jugs problem Introduing the Water Jugs problem The Water Jugs - formal representation The Water Jugs - tree representation The Water Jugs - graph representation Route finding Real world problems AND/OR graphs Summary End of Topi test Learning Objetives After studying this topi, you should be able to: Explain the terms problem abstration and problem representation Desribe problems in terms of initial state, goal state, transitions and onstraints Represent transitional states using state spae graphs, AND/OR graphs, trees and prodution rules Apply problem representation tehniques to a variety of simple AI problems

8 2 TOPIC 1. PROBLEM REPRESENTATION 1.1 Prior Knowledge and Revision Before studying this topi, you should have studied the mathes problem, a simple game in whih the player an move mathes between 3 piles to try to get the piles to be equal in size. The only rules are that moving mathes must double the number of mathes in the pile, and that no move may leave an empty pile. You should know how the possible moves in this game an be represented by a searh tree. You should also know that to find a solution to the mathes problem, it is possible to searh the nodes of the tree systematially. You should be able to desribe two searh methods - depth-first and breadth-first searh, and apply them to the mathes problem. You should already know that the tehniques of searhing a tree to find solutions an be applied to many AI problems, but that in many ases the tree rapidly beomes too big to searh. This problem is known as ombinatorial explosion. Revision Q1: The mathes problem is a lassial AI problem, in whih a) the player moves mathes between 3 piles b) the player has to find mathing pairs ) a set of football mathes have to be arranged d) you have to figure out a way of lighting a fire Q2: Whih of the following is a legal move in the mathes problem? a) (10 3 5) (10 5 3) b) (10 3 5) (10 6 2) ) (10 3 5) (8 5 5) d) (10 3 5) (10 0 8) Q3: A searh tree: a) is a programming language used to solve AI problems b) is a lassial AI problem ) is a method of representing the possible states of simple problems d) an be used to solve every AI problem Q4: Combinatorial explosion ours when: a) there is more than one solution to a problem b) there is more than one start state for a problem ) the rules of a problem are not learly defined d) there are many options at eah stage in the problem 1.2 Introdution Congratulations on hoosing to study the Artifiial Intelligene optional unit of Advaned Higher Computing. You will build on what you already know from the orresponding unit at Higher level, taking many of the topis you have already studied to a more advaned

9 1.2. INTRODUCTION 3 and detailed level. In this first sub-topi, you will study a number of fairly simple lassial AI problems, and onsider ways that these problems an be represented and then solved. Many problems in AI an be solved systematially, using a 3 stage proess: Problem abstration Symboli representation Searh strategy Problem Abstration Problem abstration is the proess of defining the problem learly and unambiguously. This inludes defining the initial or start state, defining the goal or end state, and defining the onstraints that apply. For example, if the problem was to write a hess-playing omputer program, the start state would be the initial layout of the 32 hess piees on the board. This might be represented in the omputer as a 2-dimensional array. The end or goal state would be a definition of the end of a game. It would be neessary to somehow represent the fat that the loser s King was unable to esape, or that one player had resigned. The onstraints would be the rules of hess - how the piees an move. In the rest of this Topi, we will study a number of simpler problems, and see some different ways in whih the problem an be abstrated Symboli Representation Having defined the start and goal states, and the onstraints that apply, the next stage is to onsider the transitional states, all the possible positions between the start and goal states, and how they relate to eah other. In the game of hess the number of transitional states is virtually infinite, and this poses a huge problem for AI. You have already ome aross this at Higher level - the term used to desribe this type of situation is ombinatorial explosion. Many real world problems suffer from ombinatorial explosion, and AI has developed some tehniques for dealing with it. In this Topi, we will hoose simpler problems, where the number of transitional states is manageable. We will study a number of methods of representing these problems, inluding State spae graphs Trees AND/OR graphs Prodution rules Eah method has its advantages and disadvantages, and we will see that some are more appropriate to one problem than another.

10 4 TOPIC 1. PROBLEM REPRESENTATION Searh tehniques Having represented the problem fully, the final stage is to apply a searh tehnique. You have already met depth-first and breadth-first searhing in Higher Topi 6. In Topi 2 of Advaned Higher, we will review these tehniques and study some other heuristi tehniques whih an be applied to ombat ombinatorial explosion. 1.3 The mathes problem Review of Mathes Problem The first problem we will review is the mathes problem that was desribed in Higher Topi 6. Introduing the problem You are given 24 mathes, arranged in 3 piles as shown below: 11 mathes 7 mathes 6 mathes This is the start state. It an be written as (11,7,6). The aim of the game is to move mathes to form 3 equal piles. We an write that the goal state is (8,8,8). The rules are as follows: any move must double the number of mathes in the destination pile; no move may result in a pile ending up with 0 mathes. In priniple, there are 6 possible operations at any time: 1. move mathes from the left pile to the middle pile (LtoM); 2. move mathes from the left pile to the right pile (LtoR); 3. move mathes from the middle pile to the left pile (MtoL); 4. move mathes from the middle pile to the right pile (MtoR); 5. move mathes from the right pile to the left pile (RtoL);

11 1.3. THE MATCHES PROBLEM 5 6. move mathes from the right pile to the middle pile (RtoM). However, at any time, only some of these moves are possible, depending on the number of mathes in eah pile. Q5: What are the 3 possible moves from the start state? Q6: Why are RtoM, RtoL and MtoL not legal moves from this starting state? Creating a searh tree The 3 legal moves ould be written down like this: 1. (11,7,6) using LtoM beomes (4,14,6) 2. (11,7,6) using LtoR beomes (5,7,12) 3. (11,7,6) using MtoR beomes (11,1,12) These moves an be shown on a diagram alled a searh tree, as follows: 11, 7, 6 LtoM LtoR MtoR 4, 14, 6 5, 7, 12 11, 1, 12 None of the states we have reahed is the goal state (8,8,8), so we need to apply the operators again to eah of the three states reahed. Note the use of problem abstration - the start state and the goal states have been defined. The onstraints have been larified (the 2 rules). Then a symboli representation has been hosen. Eah state is represented by 3 numbers (inside brakets); the initial state is (11,7,6) and the goal state is (8,8,8). The possible operations are represented by the operators LtoM, LtoR, MtoL, MtoR, RtoL and RtoM. Finally, a tree has been drawn to show the transitional states obtained by applying these operators. The tree represents the state spae. The state spae is all the possible states that an be ahieved by applying the rules to the initial state. Drawing a tree to represent the mathes problem For this ativity, run the mathes problem with 12 mathes. Make the initial state (7,2,3) and the goal state (4,4,4).

12 6 TOPIC 1. PROBLEM REPRESENTATION Construt a tree down to the 3rd level. You should find a solution! A tree or searh tree is a useful method of representing the state spae for a problem, espeially where the number of possible states is fairly limited. 1.4 Re-arranging bloks Let s now look at another similar problem. We have three bloks A, B and C in a stak and we require to re-arrange them to produe a partiular stak, e.g. C A B A B C Initial state Goal state We an only move one blok at a time by plaing it on the table or on top of another blok, e.g. we ould plae blok C on the table and then plae A on top of C. It should be lear that we an plae the top blok on the table then plae the seond blok on the table so forming three staks eah of one blok and then assemble a stak in whatever order we require. This gives rise to the following simple solution for the arrangement above: C A A A B B B C B C B A C A C Note that this illustration only shows one possible way of re-arranging the bloks.

13 1.4. RE-ARRANGING BLOCKS 7 We an use the following symboli representation for this problem: C A A A B B B C B C B A C A C Example Problem: How many possible arrangements of the bloks are there? arrangements separately, for example:) (don t ount equivalent = Solution: You should have found 13 different possible arrangements. Here are the different possible arrangements in a diagram, with arrows showing whih arrangements an be hanged into another by one move. Notie that the arrows are 2- way or bi-diretional, as (unlike the mathes problem) every possible move is reversible.

14 8 TOPIC 1. PROBLEM REPRESENTATION A B C A C B B AC C AB C B A B AC ABC et C AB B C A A CB A CB C A B B A C This diagram is an example of a state spae graph. It is similar to a tree, but: it has no single root node there are ross branhes some (in this ase, all) of the arrows are bi-diretional Q7: Using the diagram above, show the stages required to make the following rearrangement of bloks:

15 1.5. TREES AND GRAPHS 9 C B A A C B 1.5 Trees and graphs We used a tree to represent the state spae of the mathes problem, and a graph to represent the state spae of the bloks problem. For many problems, both representations are possible. A tree an be onverted into a graph (by notiing where idential nodes appear, and replaing them with a single node). Here are 3 representations of the mathes problem: Tree: LtoM LtoR RtoM LtoM LtoR MtoR RtoL RtoM LtoM LtoM LtoR MtoR Graph 1: (replaing the two idential (2 4 6) nodes with a single node) LtoM LtoR RtoM LtoM MtoR LtoR RtoL LtoM RtoM LtoM LtoR MtoR

16 10 TOPIC 1. PROBLEM REPRESENTATION It is possible to go a stage further, if we deide that the atual positions are irrelevant (e.g. if we say that (2 4 6) is effetively the same state as (4 6 2). Making this assumption, the graph an beome: LtoM LtoR RtoM MtoR RtoL LtoM LtoM It is also possible to onvert a graph into a tree. You ould try onverting the state spae graph for the blok problem into a tree, but it would beome very large and need to be spread out on a large sheet of paper. Generally a state spae graph is more ompat than a tree. However, trees are important when we onsider searh strategies in Topi Sliding numbers problem Another favourite problem of artifiial intelligene is the eight puzzle in whih eight numbered tiles an slide horizontally and vertially in a 3x3 frame. One ell is empty and any adjaent tile an slide into the spae. The puzzle starts with an initial position of the tiles and a stated goal position. The objet is to manoeuvre the tiles to reah the goal position. Initial state Goal state

17 1.7. THE JUGS PROBLEM 11 Sliding Squares Graph Construt a graph to represent the first 3 moves of this sliding numbers problem. Here is how the tree might look after the first move (there are 3 options - slide the 4, the 1, or the 3 into the spae; Initial state Missionaries and Cannibals Look at the web site Read through the problem. How are the initial state and final states defined? What are the rules and onstraints of the problem? Draw a tree to represent all the possible transitional states. From your tree, identify the "best" solution. 1.7 The jugs problem Introduing the Water Jugs problem The water jugs problem is another of the lassi puzzles of artifiial intelligene. Here is the problem. You have a 4 litre jug and a 3 litre jug, a supply of water, and a

18 12 TOPIC 1. PROBLEM REPRESENTATION drain. How an you measure exatly 2 litres of water? Here are the rules: You an fill either jug from the water soure pour water from one jug into the other empty a jug down the drain Solving the water jugs problem 5 min Figure out a solution to the water jugs problem. If there are other students in your lass, ompare solutions. Is there more than one solution? Whih is the best solution? The Water Jugs - formal representation The two jugs an be represented by an ordered pair (a,b) where a = 0, 1, 2, 3 or 4, the number of litres in the larger jug, while b = 0, 1, 2 or 3, the number of litres in the smaller jug. Hene our initial state is (0,0) and the goal state is (2,0). There are several options at eah stage. Here they are: Condition New State Desription 1 (a,b) if a 4 (4,b) Fill the 4l jug 2 (a,b) if b 3 (a,3) Fill the 3l jug 3 (a,b) if a 0 (0,b) Empty the 4l jug on the ground 4 (a,b) if b 0 (a,0) Empty the 3l jug on the ground 5 (a,b) if a+b = 4 and b 0 (4,b-4+a) 6 (a,b) if a+b = 3 and a 0 (a-3+b,3) Pour water from the 3l jug to fill the 4l jug Pour water from the 4l jug to fill the 3l jug 7 (a,b) if a+b = 4 and b 0 (a+b,0) Empty the 3l jug into the 4l jug 8 (a,b) if a+b = 3 and a 0 (0,a+b) Empty the 4l jug into the 3l jug These operations, desribed in this formal mathematial way, are alled prodution rules. By writing them in this form, they an more easily be onverted into omputer programs. For example, rule 1 ould be written in the form: IF a 4 THEN a=4. Water Jugs Prodution rules Write eah of the 8 prodution rules listed above in the form: IF ondition(s) THEN ation(s).

19 1.7. THE JUGS PROBLEM The Water Jugs - tree representation As before, we an use a searh tree where at eah level we list all the possible states that an be produed from the state at the level above. These branhes are generated by applying the prodution rules above. Here is the start of the searh tree for the water jugs problem: (0,0) (4,0) (0,3) (4,3) (0,0) (1,3) (4,3) (0,0) (3,0) Figure 1.1: The Water Jugs - tree representation Q8: Copy the searh tree shown in Figure 1.1 and add another level. What are the problems in ontinuing to add levels to this tree? What else do you notie about the entries in the tree? Take one branh of the tree (you should reognise whih one to use) and extend it so that you obtain the goal state that we require with 2 litres in the larger jug The Water Jugs - graph representation You would notie that : the tree beomes very large very quikly and most of it need never be searhed for the goal state; nodes on the tree are repeated on eah path by simply reversing the previous hange, e.g. (0,0) (4,0) (0,0); nodes on the tree an be generated by different paths but the atual path used does not matter, e.g. (4,3) an be reahed by filling the 4l jug then the 3l jug or vie versa.

20 14 TOPIC 1. PROBLEM REPRESENTATION A state spae graph solves the dupliation problems of searh trees. Here is the start of the searh graph for the water jugs problem: (0,0) (4,0) (0,3) (1,3) (4,3) (3,0) Q9: Copy the searh graph above and develop the bottom three nodes to another level. A Variation on the Water Jugs Problem There are three jugs on a table. The largest one is full and ontains 8 litres of wine. The other jugs are empty but an hold 5 litres and 3 litres respetively. The problem is to divide the wine so that there are 4 litres of wine in the 8 litre jug and 4 litres of wine in the 5 litre jug. This time there is a limited supply of wine and obviously it annot be thrown away. Using one of the strategies disussed above, produe a solution to the problem where the first step is to fill the 5 litre jug. Now produe a seond solution where the first step is to fill the 3 litre jug. Test the two solutions by heking that the three jugs always hold 8 litres among them. Compare these two solutions for effiieny. 1.8 Route finding Route finding is another lassial AI problem. Here is a simple map of the roads linking five towns that we will all A, B, C, D and E. The numbers on eah line indiate the distane between that pair of towns. This is another lassi problem in artifiial intelligene, where a travelling salesman starts and finishes his visits to the other towns at town A and the task is to find the shortest route for the salesman to travel.

21 1.8. ROUTE FINDING 15 A B E D C You should reognise that this map is really a form of state spae graph. We an now onstrut a searh tree of all the routes starting and finishing at A and, for eah branh of the tree, add up the distane travelled. A 30 B C D E C D E D E C E C D E D E C D C A A A A A A et. The figures 165, 205 and 210 are the totals for the first three routes.

22 16 TOPIC 1. PROBLEM REPRESENTATION Travelling salesman Complete the searh tree. By inspeting the searh tree for the 5 towns, write down the number of possible routes that start at A and finish at A. If we extended the problem to 6 towns, how many routes would there be? If the problem is extended to 50 towns, whih phrase would desribe the situation that arises? 1.9 Real world problems We have looked at methods of representing problems, in partiular the use of trees and searh graphs to represent the state spae for a problem. These methods work well for well-defined problems with a limited number of rules and ations. They do not work so well when applied to "real life problems". For example, onsider what the searh tree would look like for the following map of flights:

23 1.10. AND/OR GRAPHS 17 or for route finding through ity streets like this: It would be humanly impossible to draw the tree to represent this. However, if the data an be represented in a way that the omputer an proess, then the task of searhing the map/graph/tree an in priniple be done by a omputer. Even a problem like playing hess, where there are usually at least 20 possible moves at eah turn, an be handled using searh trees. However, even with modern fast proessors, and the use of parallel proessing to allow multiple branhes to be searhed simultaneously, other tehniques are required to ut down the searh spae. This will be explored in topi 2.3 (heuristi searh). These problems have a very large but finite searh spae. Many real life problems have an infinite searh spae. There are an infinite number of branhes from eah node of the tree. Most human ativity is like this. In order to takle real-life problems, simplifiations have to be made to redue the state spae to a finite size AND/OR graphs In the searh trees and graphs that we have studied so far eah branh is an OR branh. In other words we an go down one branh or down one of the others. Another method of symboli representation alled an AND/OR graph is used for problems that an be split into independent sub-problems. OR branhes are shown as before but AND branhes are shown with an ar joining the branh lines.

24 18 TOPIC 1. PROBLEM REPRESENTATION OR AND An example of a problem that splits into independent sub-problems is one of finding routes between start and finish points that must go through some intermediate point or points. You may have ome aross Autoroute Express or similar software that an be used to plan road journeys. Typially, you enter the start and finish points and also plaes that you want to go through on the journey. You also enter details about the speed that you want to travel at and the sort of route that you want to follow, e.g. seni or motorway. The program then searhes for routes and displays the shortest route; the fastest route; the route using motorways most; the route using B roads most and other possibilities that you have seleted. Here is a system of roads between our start (S), our finish (F) and going through an intermediate point (I). S A C B I X F Z Y We an split the problem of finding routes from S to F into the sub-problems of finding routes from S to I and from I to F. Similarly, routes from S to I an be split into the sub-problems of finding routes from S to A and from A to I and so on. Eah road would be alloated ertain properties suh as its distane and its quality whih would be ombined to assign a ost for that setion of the route. We would then searh for the route with a partiular minimum ost suh as the shortest distane or the fastest route or the route on the best quality roads.

25 1.11. SUMMARY 19 Here is part of the AND/OR graph for this route finding problem: Q10: Copy and omplete the above AND/OR graph Summary The first stage in solving a problem is problem abstration - identifying the start and goal states, rules and onstraints The seond stage is symboli representation - finding a way of representing the state spae State spae an be represented using trees or graphs Prodution rules define the legal ways of moving from one transitional state to another AND/OR graphs an be used where a problem an be divided into a number of independent sub-problems Real life problems may have an infinite state spae, so need to be simplified before they an be solved End of Topi test Q11: The 3 stage proess whih an be applied to many AI problems is: a) problem abstration symboli representation searh strategy b) symboli representation problem abstration searh strategy ) analysis design testing d) tree prodution rules state spae graph!

26 20 TOPIC 1. PROBLEM REPRESENTATION Q12: Whih of the following is a legal move in the mathes problem? " " " " a) (12 8 4) (8 12 4) b) (12 8 4) ( ) ) (12 8 4) (4 16 4) d) (12 8 4) ( ) Q13: This diagram: LtoM LtoR RtoM LtoM LtoR MtoR RtoL RtoM LtoM LtoM LtoR MtoR a) is a graph b) is a tree ) is a prodution rule d) is an AND/OR graph #

27 1.12. END OF TOPIC TEST 21 Q14: This diagram: LtoM LtoR RtoM MtoR RtoL LtoM LtoM a) is a graph b) is a tree ) is a prodution rule d) is an AND/OR graph Q15: An AND/OR graph an be used to represent: a) a problem that an be divided into independent sub-problems b) route finding problems only ) any real-life problem d) prodution rules $

28 22 TOPIC 1. PROBLEM REPRESENTATION %

29 & & & & & & & 23 Topi 2 Searh Strategies Contents 2.1 Prior Knowledge and revision Introdution Exhaustive searh Breadth-first Depth-first Comparing Breadth and Depth first searhing Heuristi Searh Hill Climbing Problems with Hill Climbing Route finding Best-first Disadvantages of heuristi searh A* The Minimax Proedure Summary End of Topi test Learning Objetives After studying this topi, you should be able to: Desribe and exemplify depth-first and breadth-first searh on a searh tree Define and exemplify a heuristi Define and exemplify a ost/evaluation funtion Explain the advantages of heuristi searh over exhaustive searh Desribe and exemplify the following searh tehniques: Hill-limbing Best-first searh A* Desribe the advantages and disadvantages of different searh tehniques Desribe and exemplify the minimax proedure in the ontext of game-playing

30 24 TOPIC 2. SEARCH STRATEGIES 2.1 Prior Knowledge and revision Before studying this topi, you should know (from Topi 1) that many problems in AI an be solved by a systemati approah, involving problem abstration, symboli representation and appliation of a searh strategy. You should also know (from Topi 1) that the state spae of many problems an be represented using a tree, state spae graph or an AND/OR graph. The nodes of the tree or graph represent transitional states, and the operations between states an be represented by prodution rules. You should also know (from Higher) that the state spae of a problem an be searhed by exhaustive searh tehniques, inluding breadth-first and depth-first searh. This works well for some simple problems, but in many ases the number of branhes from eah node leads to the ombinatorial explosion. Even with large powerful omputers, and the use of parallel proessing, the searh tree simply gets too large for an exhaustive searh to be made. You should know (from Higher) that heuristis an be used to redue the searh spae, so that solutions may be found more quikly. When we study the searh methods, we will see that the algorithms to desribe them make use of staks and queues. You should know about these data strutures from your study of Software Development at Advaned Higher level. Revision Q1: Exhaustive searh a) is more effiient than heuristi searh b) is the only way to searh a searh tree ) means searhing every possible node of a searh tree d) means searhing from the root node, one level at a time Q2: Breadth-first searh a) is a form of heuristi searh b) requires baktraking to avoid infinite loops ) is sometimes alled depth-first searh d) is a form of exhaustive searh Q3: If the following tree were searhed using depth-first searh, until a solution was found at F, the order of visiting the nodes would be: A B C D E F G H I J K '

31 2.2. INTRODUCTION 25 a) ABCDEF b) ABF ) ABEBF d) ABCDF Q4: Heuristi searh a) redues the searh spae and time b) uses random guessing to find the solution ) is used by humans but annot be applied by omputers d) is a type of exhaustive searh Q5: A data struture where new items are added to the front (or top) and items are also removed from the front (or top), is alled a a) queue b) array ) tree d) stak 2.2 Introdution In Topi 1 we onsidered various methods of representing problems, inluding tress and graphs. One a problem has been represented in this way, it is possible to apply standard searh tehniques to find solutions. There are two main types of searhing. These are exhaustive searhing and heuristi searhing. You have studied exhaustive searh before at Higher, but it is so important to many areas of AI researh, that we will revise the 2 main types, before exploring heuristis in more depth. 2.3 Exhaustive searh Exhaustive searhing means exploring every possibility in a systemati way. If the problem is represented as a searh tree, it means testing every node of the tree, beginning at the root node (the start state), and ontinuing until a solution (goal state) is found. Exhaustive searh an be used to solve AI problems, and is the basis for the behaviour of many Expert Systems and the Prolog language. In "real life" we don t often use exhaustive searh, as it is slow and time-onsuming. If you have lost your keys, you probably won t searh methodially through every plae in every room of the house, starting at the ground floor and ontinuing room by room to the atti. However, if you were a polie offier looking for evidene you might take that approah. There are two ways of going about an exhaustive searh. In the example above, you ould take a quik look in eah room first, then go bak and look on the top surfaes in eah room. If you still haven t found the keys, you might go bak and look in any (

32 ) ) ) 26 TOPIC 2. SEARCH STRATEGIES drawers and upboards in eah room, and so on. This is alled "breadth-first" searh. An alternative is to start in one room, look on all the surfaes, then look in the drawers and upboards, then under all the furniture. Only after having looked in every possible plae in the first room, would you go on to look in the next room. This is "depth-first" searh. In pratie, intelligent behaviour would be to think about where you have been most reently, and look there first. Next you might look in plaes where you have found your keys before. Only if that fails, start searhing more methodially. You would be using heuristis. We will study several important heuristi methods in topi 2.4, but first we will remind ourselves of the 2 main types of exhaustive searh - breadth-first searh and depth-first searh Breadth-first This searh starts at the root of the tree and searhes all possible nodes at that level, only searhing the next level one all nodes at the previous level have been heked. The following diagram indiates the order in whih the nodes will be tested: m 1 k 2 i 3 n 4 b 5 h 6 u 7 v 8 9 f 10 * t 11 r 12 d 13 x 14 z 15 * * A formal algorithm for breadth-first searh an be written as: 1. Start with queue = [initial state] and found = FALSE 2. While queue not empty and not found remove first node X from the queue if X is a goal then found = TRUE find all suessor nodes of X and put them on the end of the queue Apply this algorithm to the tree shown above, to onvine yourself that it is orret. *

33 2.3. EXHAUSTIVE SEARCH 27 Stak X found ommentary Step 1 [m] false stak ontains m, the initial state Step 2.1 [ ] m false Remove m from queue Step 2.2 [ ] m false Is it a goal? no Step 2.3 [k, i, n] m false Add m s suessors - k, i and n Step 2.1 [i, n] k false Remove k from queue Step 2.2 [i, n] k false Is it a goal? no Step 2.3 [i, n, b, h, u] k false Add k s suessors - b, h and u Step 2.1 [n, b, h, u] i false Remove i from queue Step 2.2 [n, b, h, u] i false Is it a goal? no Step 2.3 [n, b, h, u, v] i false Add i s suessor - v Step 2.1 [b, h, u, v] n false Remove n from queue Step 2.2 [b, h, u, v] n false Is it a goal? no Step 2.3 [b, h, u, v, n false Add n s suessor - an f, f] Step 2.1 [h, u, v,, f] Step 2.2 [h, u, v,, f] Step 2.3 [h, u, v,, f] b false Remove b from queue b false Is it a goal? no b false Add b s suessor - none Step 2.1 [u, v,, f] h false Remove h from queue Step 2.2 [u, v,, f] h false Is it a goal? no Step 2.3 [u, v,, f] h false Add h s suessor - none Step 2.1 [v,, f] u false Remove u from queue Step 2.2 [v,, f] u false Is it a goal? no Step 2.3 [v,, f] u false Add u s suessors - none Step 2.1 [, f] v false Remove v from queue Step 2.2 [, f] v false Is it a goal? no Step 2.3 [, f, t, r, d] v false Add v s suessors - t, r and d Step 2.1 [f, t, r, d] false Remove from queue Step 2.2 [f, t, r, d] true Is it a goal? Yes, so stop! Depth-first This searh starts at the root of the tree and works down the left-hand branh of the tree onsidering eah node until it reahes a leaf or terminal node. If this is not a goal state then it goes bak up and tries the next available path down. This searh, as its name suggests, tries to get as deep as possible as fast as possible. The following diagram +

34 ,,, 28 TOPIC 2. SEARCH STRATEGIES indiates the order in whih the nodes will be tested (goal states are marked with an asterisk): m 1 k 2 i 6 n 11 b 3 h 4 u 5 v 7 12 f 13 * t 8 r 9 d 10 x 14 z 15 * * A formal algorithm for depth-first searh an be written as: 1. Start with stak = [initial state] and found = FALSE 2. While stak not empty and not found remove first node X from the stak if X is a goal then found = TRUE find all suessor nodes of X and add them to the stak Notie that it is idential to the breadth-first algorithm, exept that suessor nodes are added to the top of a stak, rather than to the end of a queue. Apply this algorithm to the tree shown above, to onvine yourself that it is orret. -

35 2.3. EXHAUSTIVE SEARCH 29 Stak X found ommentary Step 1 [m] false stak ontains m, the initial state Step 2.1 [ ] m false Remove m from stak Step 2.2 [ ] m false Is it a goal? no Step 2.3 [k, i, n] m false Add m s suessors - k, i and n Step 2.1 [i, n] k false Remove k from stak Step 2.2 [i, n] k false Is it a goal? no Step 2.3 [b, h, u, i,n] k false Add k s suessors - b, h and u Step 2.1 [h, u, i,n] b false Remove b from stak Step 2.2 [h, u, i,n] b false Is it a goal? no Step 2.3 [h, u, i,n] b false Add b s suessor - none Step 2.1 [u, i,n] h false Remove h from stak Step 2.2 [u, i,n] h false Is it a goal? no Step 2.3 [u, i,n] h false Add h s suessor - none Step 2.1 [i,n] u false Remove u from stak Step 2.2 [i,n] u false Is it a goal? no Step 2.3 [i,n] u false Add u s suessor - none Step 2.1 [n] i false Remove i from stak Step 2.2 [n] i false Is it a goal? no Step 2.3 [v, n] i false Add i s suessor - v Step 2.1 [n] v false Remove v from stak Step 2.2 [n] v false Is it a goal? no Step 2.3 [t, r, d, n] v false Add v s suessors - t, r and d Step 2.1 [r, d, n] t false Remove t from stak Step 2.2 [r, d, n] t true Is it a goal? Yes, so stop! Comparing Breadth and Depth first searhing a) The two searh methods may enounter different solutions first. For example, depth-first searh finds solution t after 8 nodes while breadth-first searh finds solution after 9 nodes. Depth-first always finds the first solution from the left while breadth-first finds the shallowest solution first.... b) Depth-first searh requires less memory as only nodes on the urrent path are stored. For example, when searhing to t, depth-first searh must store nodes m, i and v and when the solution is found then the path m i v t is returned as a solution. Breadth-first searh requires more memory as it must store the whole tree generated so far. For example, when searhing to t, breadth-first searh must store all the paths used so far [m, k, b], [m, k, h], [m, k, u], [m, i, v], [m, n ], and /

36 30 TOPIC 2. SEARCH STRATEGIES [m, n, f] and when the solution is found then the path m i v t is returned as a solution. ) By hane, depth-first searh may find a solution very quikly if it is deep at the left-hand side of the tree and so it an stop if only one solution is required. Breadth-first searh must searh all the previous levels before finding a deep solution. d) Depth-first searh seems to have all the advantages but... Depth-first searh may get into an infinite loop if we do not do extra programming to ensure that previously generated states are not searhed again. For example, we must eliminate the reverse of the previous hange. Breadth-first searh will not get into an infinite loop. e) Depth-first searh may find a very long solution deep in a left-hand side branh of the tree when a short solution exists elsewhere in the tree. Breadth-first searh is guaranteed to find a solution if it exists and this solution will be the solution with the minimum number of steps. Humans often use a ombination of both tehniques to try to get the best of eah. A Frenh mathematiian, Henri Poinaré, desribed searhing for solutions to mathematial problems as follows: Imagine that you are in an underground avern trying to get bak to the surfae and that there are passages running off in many different diretions but you an only see a few feet along eah passage. The born mathematiian always selets a passage that runs a long way if, indeed, it is not the solution passage. There are artifiial intelligene methods that ombine depth-first and breadth-first searhing. One suh method is alled branh-and-bound. Prolog uses the depth-first searh strategy. You should be familiar with this from earlier work that you have done in Prolog where depth-first searhing an be seen by using the language s trae faility when it is attempting to satisfy a query. A problem assoiated with both depth-first and breadth-first searh methods is ombinatorial explosion. For non-trivial problems, the number of possible solutions that need to be heked an beome unaeptably high. An example is to ompare two games: For the sliding number puzzle, there are only 2, 3 or 4 moves possible at eah stage. It is possible to draw a omplete searh tree for the problem, and for a omputer to searh it exhaustively until a solution is found. Its searh tree would look something like: 1

37 2.4. HEURISTIC SEARCH 31 A game of hess, on the other hand, has 20 possible 1st moves. The seond player an then play any of 20 responses. That means 20 x 20 (whih is 400) nodes at level 3 of the searh tree. At the next level, there are over 8000 possible positions, and at the next almost 1/4 million! The searh tree beomes unmanageable very quikly: To avoid this problem of ombinatorial explosion, heuristi methods must be applied. Q6: Whih method uses less memory? a) Breadth-first b) Depth-first Q7: Whih method is guaranteed to find the best solution? a) Breadth-first b) Depth-first Q8: Whih method may get into an infinite loop? a) Breadth-first b) Depth-first 2.4 Heuristi Searh The world hess hampion may look ahead 6 moves, but he doesn t look at every possible move - otherwise his opponent would probably be dead by the time he had onsidered all 4000 trillion possibilities. Instead, he knows, from what he has been 2

38 TOPIC 2. SEARCH STRATEGIES taught, and from his experiene, that some moves are more likely to be helpful than others. He has "rules of thumb" whih quikly redue the number of moves he needs to onsider: he knows that there is not muh point looking at moves whih would allow his major piees to be aptured; he also knows that it is best to build up a strong group of piees towards the entre of the board; he knows that some piees are more valuable than others. Using "rules of thumb" like these, he an quikly narrow down the possible options to a muh smaller number, and he an then searh through these (in his imagination) in an exhaustive way probably using a ombination of breadth-first and depth-first searhing. For example, he may be in a situation where there are 43 legal moves, but applying his "rules of thumb" he knows that only 4 of them are likely to very useful. He may onsider the likely situation 1 move beyond eah of these 4 possibilities (breadth first searh). One of them looks partiularly promising, so he tries to think ahead what are the most likely series of moves from that position (depth-first searh). This may lead to a dead end, so he baktraks, and looks in depth at another possibility, and so on. Similarly, we use heuristis in everyday onversation. When someone asks you a question, you don t onsider every possible word or sentene whih you might use as a reply. You hoose from a restrited set of options of "sensible" responses. How do you know what responses are sensible? You have built up experiene over the years, and you know that ertain responses are more likely to lead to a suessful outome than others. However, heuristis are not always helpful. Sometimes you an win a game of hess by setting a trap for the other player. For example, you might put your queen (a very powerful piee) into a plae where it an be taken by your opponent. This does not look like a promising move, and a heuristi might suggest this move is not worth onsidering. However, if your opponent does take your queen, this might make a surprise hekmate possible, and you win the game. A good hess player has to be able to spot moves like that whih break the usual "rules of thumb". The whole idea of a "rule of thumb" is that it usually works, but not always. Similarly, in onversation. You might have a rule of thumb whih says "never shout at a teaher". There might be a situation when you would need to overrule this rule, for example if the teaher was about to step out in front of a fast-moving vehile. (You would, wouldn t you?). Let s return to the mathes problem. This time we will start with 24 mathes in piles of 11, 7 and 6. The goal state is (8,8,8). The searh tree would look like this: 4

39 HEURISTIC SEARCH 33 11, 7, 6 4, 14, 6 5, 7, 12 11, 1, 12 8, 10, 6 8, 12, 2 4, 8, 12 10, 2, 12 10, 7, 7 5, 14, 5 10, 2, 12 22, 1, 1 11, 2, 11 2, 10, 12 16, 2, 6 8, 4, 12 8, 4, 12 8, 8, 8 4, 16, 4 If you experiment, you will find that states with 4 mathes in any of the piles are more likely to lead to a solution. States with 8 mathes in any pile are even better. These are "rules of thumb" or heuristis. This ould be programmed by giving eah node a value using the following formula: Value = (number of 8s x 2) + (number of 4s). So, for example: value(11,7,6)=0; value(8,14,2)=2; value(4,8,12)=3. Go bak to your mathes problem breadth-first searh tree, and write the value beside eah node. Here is how the tree should now look, with the values of the evaluation funtion shown in eah node, instead of the atual arrangements of mathes: The formula you have used here is an example of a ost/evaluation funtion. This is one of the ways of making a heuristi more than a "rule of thumb". It beomes a pratial tool that an be used to guide the searh towards a solution, avoiding wasting system time and resoures searhing unfruitful branhes. If the searh algorithm " at eah level hoose the node with the highest value" is used, it will lead diretly to the solution. Of ourse, if the heuristi or ost/evaluation funtion is figured out (as in this ase) by a human, then simply programmed into the system, the system annot really be desribed as "intelligent". Real intelligene would be displayed when the omputer system an develop its own heuristis, and improve them in the light of experiene. This leads into a very omplex and diffiult field of researh alled Mahine Learning. 6