# Some Studies in Machine Learning Using the Game of Checkers. II-Recent Progress

Save this PDF as:

Size: px
Start display at page:

Download "Some Studies in Machine Learning Using the Game of Checkers. II-Recent Progress"

## Transcription

4 Figure 2 The move tree of Fig. 1 redrawn to illustrate the detailed method used to keep track of the comparison values. Board positions are lettered in the order that they are investigated and the numbers are the successive alpha values that are assigned to the boards as the investigation proceeds. case at board c) is used as the alpha value, and unless the terminal board score exceeds this alpha value, the player at board c would be ill advised to consider entering the branch leading to this terminal board. Similarly if the negative of the terminal board score does not exceed the alpha value associated with the board immediately above in the tree (in this case at board 6) then the player at bourd d will not consider this to be a desirable move. An alternate way of stating this second condition, in keeping with Mc- Carthy s usage, is to say that the negative of the alpha value associated with the board one level up the tree (in this case board 6) is the beta value which must not be exceeded by the score associated with the board in question (in this case board e). A single set of alpha values assigned to the boards in the tree thus performs a dual role, that of McCarthy s alpha as referenced by boards two levels down in the tree and, when negated, that of McCarthy s beta as referenced by boards one level down in the tree. Returning to the analysis of Fig. 2, we note that during the initial look-ahead (leading to boarde) nothing is known as to the value of the boards, consequently the assigned alpha values are all set at minus infinity (actually within the computer only at a very large negative number). When board e is evaluated, its score (4-2) is compared with the alpha at c (- w ), and found to be larger. The negative of the score (-2) is then compared with the alpha at d( - 00) and, being larger, it is used to replace it. The alpha at d is now -2 and it is unaffected by the subsequent consideration of terminal boardsfand g. When all paths from board d have been considered, the final alpha value at d is com- 604 pared with the current alpha value at board b (- 00); it is larger, so the negative of alpha at d (now + 2) is compared with the current alpha value at c (- m) and, being larger, it is used to replace the c value, and a new move from board c is investigated leading to board h and then board i. As we go down the tree we must assign an alpha value to board h. We cannot use the alpha value at board c since we are now interested in the minimum that the other side will accept. We can however advance the alpha value from board b, which in this case is still at its initial value of - a. Now when board i is evaluated at +1 this value is compared with the alpha at board c (4-2). The comparison being unfavorable, it is quite unnecessary to consider any other moves originating at board h and we go immediately to a consideration of boards j and k, where a similar situation exists. This process is simply repeated throughout the tree. On going forward the alpha values are advanced each time from two levels above and, on backing up, two comparisons are always made. When the tree is completely explored, the final alpha value on the initial board is the score, and the correct move is along the path from which this alpha was derived. The saving that results from alpha-beta pruning can be expressed either as a reduction in the apparent amount of branching at each node or as an increase in the maximum ply to which the search may be extended in a fixed time interval. With optimum ordering, the apparent branching factor is reduced very nearly to the square root of its original value or, to put it another way, for a given investment in computer time, the maximum ply is very nearly doubled. With moderately complex trees the savings can be astronomical. For example consider a situation with a A. L. SAMUEL

7 Figure An actual look-ahead move tree as printed by the computer during play J , IT n Y MOVE TREE ALPHA BETA SCORE L v OUOlO OOOlU L2,14-18 *I 0004 * I *27-24 t * az Z *0-26, b * * r24-15,10-19, ~262.1~ t , * * ~ * 4-8 * l l~rL1-17 *10-19,2-7 -(I IC * ZZ~26-17vll-16~0-26~ 8-11 OW *IO-15 OC * r t - a,~2-17,15-19.~4-15,11-1n, U ,10-171Z~lS~15-24r S , *u r v *lz-16 * 2-1, U07 *ll L r15r11-1BrZ&Z* *1C-19v17- t I s rZC *rZ8-19.1* J0045 * 5- v OU45 *11-1b *ll-l C17~21-14.I0-17~22-ltl OOLOO * I * lI.2-IE -0004ii * * ~.2~18.1'1-~12b * , r I *10-15r t LL t. 1-17r25-21~17-22~2b * * , *12-L ,24-lS , 6-9 OCUJ * *l5-18, OC , a *ll *ti ply TOTAL USAGE I ENDS bZl MACHINE LEARNING: PT. 11

11 Allowable range of values 7 5 First level tables Range of values Range of values First level Entries tables values Entries Entries Entries , 125 Entrie5 Entries Entries table 7 4 Entries Final score D Entries Entries Entries Entries Range of values \r 225 Third level table I Figure 4 A -level signature-table arrangement with 27 terms. Figure 5 Revised -level signature-table scheme with 24 terms. thus be , a negative number which would not be tabulated but the function value would be the negative of the value listed under , as it of course must be for the sum of the functions for the two sides to be zero. The results obtained with this relatively simple method of handling parameter interactions were quite encouraging and as a result a series of more elaborate studies has been made using signature procedures of varying degrees of complexity. In particular, efforts were made (1) to decrease the total number of parameters by eliminating those found to be of marginal utility, (2) to increase the range of values permitted for each parameter, initially increasing the range for certain parameters to permit 7 values (-, -2, - 1, 0, + 1, +2, +) and more recently dividing the parameters into two equal groups-one group being restricted in range to 5 values, and () to introduce a hierarchical structure of signature tables where the outputs from the first level signature tables are combined in groups and used as inputs to a set of second level tables etc. (This is illustrated in a simplified form in the cover design of this issue.) Most of the experimental work has been restricted to a consideration of the two arrangements shown in Figs. 4 and 5. These are both three-level arrangements. They differ in the degree of the correlation between parameters which is recognized and in the range of values permitted the individual parameters. Both are compromises. Obviously, the optimum arrangement depends upon the actual number of parameters that must be used, the degree to which these parameters are interrelated and the extent to which these individual parameters can be safely represented by a limited range of integers. In the case of checkers, the desired number of parameters seems to lie in the range of 20 to 0. Constraints on the range of values required to define the parameters can be easily determined but substantially nothing is known concerning the interdependencies between the parameters. A series of quite inconclusive experiments was performed in an effort to measure these interdependencies. About all that can be said is that the constraints imposed upon the permissible distribution of pieces on the board in any actual game, as set by the rules of the game and as dictated by good playing procedures, seem to produce an apparent average correlation between all parameters which is quite independent of the specific character of these parameters. The problem is further complicated by the fact that two quite opposing lines of argument can be advanced-the one to suggest that closely related terms be placed in the same subsets to allow for their interdependencies and the second to suggest that such terms be scattered 611 MACHINE LEARNING: PT. 11

17 program able to do this, then it could adopt a strategy for any particular move. If the program finally made a move that was consistent with this strategy, and if the opponent were unable to vitiate this strategy, then the program would, on the next move, again tend to adopt the same strategy. Of course, if the program had been unable to maintain an advantage by following its initial strategy, it might now find that a different strategy was indicated and it would therefore change its strategy. Nevertheless, on the average, the program might follow a given strategy for several moves in a row and so exhibit playing characteristics that would give the impression of long range planning. A possible mechanism for introducing this kind of strategic planning is provided by the signature table procedure and by the plausibility analysis. It is only necessary to view the different signature types as different strategic elements and to alter the relative weights assigned to the different signature types as a result of the plausibility analysis of the initial board situation. For this to be effective, some care must be given to the groupings of the parameters into the signature types so that these signature types tend to correspond to recognizable strategic concepts. Fortunately, the same initial-level grouping of parameters that is indicated by interdependency considerations seems to be reasonable in terms of strategies. We conclude that it is quite feasible to introduce the concept of strategy in this restricted way. For reasons of symmetry, it seems desirable to pick two signature types for emphasis, that one yielding the highest positive value and that one yielding the most negative value for the most plausible move found during the initial plausibility analysis. This procedure recognizes the fact that to the opponent, the signs are reversed and his strongest signature type will be the first player s weakest one and vice versa. The simplest way to emphasize a particular strategy is to multiply the resulting values found for the two selected signature types by some arbitrary constant before entering a subsequent stage of the analysis. A factor of 2 (with a limit on the maximum resulting value so as not to exceed the table range) seemed reasonable and this has been used for most of the experiments to date. The results to date have been disappointing, presumably because of the ineffectual arrangement of terms into usable strategic groups, and as a consequence, this method of introducing strategies has been temporarily abandoned. Conclusions While the goal outlined in Ref. 1, that of getting the program to generate its own parameters, remains as fzr in the future as it seemed to be in 1959, we can conclude that techniques are now in hand for dealing with many of the tree pruning and parameter interaction problems which were certainly much less well understood at the time of the earlier paper. Perhaps with these newer tools we may be able to apply machine learning techniques to many problems of economic importance without waiting for the long-sought ultimate solution. Acknowledgments These studies were largely carried out while the writer was at the Thomas J. Watson Research Laboratories of the IBM Corporation, and while he was a Visiting Professor at M.I.T. More recently, the work has been supported in part by Stanford University and by the Advance Research Projects Agency of the Office of the Secretary of Defense (SD-18). The IBM Corporation has continued to aid the work by supplying time on an IBM 7094 computer at their San Jose Development Laboratories. Many individuals have contributed to these studies, and in particular, Arnold Griffith of M.I.T. deserves commendation for suggesting the initial form of the signature table procedure. The continuing interest and cooperation of the officers and playermembers of the American Checker Federation has been most helpful. Received June 5, i I 61 7 MACHINE LEARNING: PT. I1

### COMP-424: Artificial intelligence. Lecture 7: Game Playing

COMP 424 - Artificial Intelligence Lecture 7: Game Playing Instructor: Joelle Pineau (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/comp424 Unless otherwise noted, all material posted

### Game-Playing & Adversarial Search MiniMax, Search Cut-off, Heuristic Evaluation

Game-Playing & Adversarial Search MiniMax, Search Cut-off, Heuristic Evaluation This lecture topic: Game-Playing & Adversarial Search (MiniMax, Search Cut-off, Heuristic Evaluation) Read Chapter 5.1-5.2,

### Game playing. Chapter 6. Chapter 6 1

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.

### Laboratory work in AI: First steps in Poker Playing Agents and Opponent Modeling

Laboratory work in AI: First steps in Poker Playing Agents and Opponent Modeling Avram Golbert 01574669 agolbert@gmail.com Abstract: While Artificial Intelligence research has shown great success in deterministic

### The Taxman Game. Robert K. Moniot September 5, 2003

The Taxman Game Robert K. Moniot September 5, 2003 1 Introduction Want to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game. The taxman game

### Game Theory and Poker

Game Theory and Poker Jason Swanson April, 2005 Abstract An extremely simplified version of poker is completely solved from a game theoretic standpoint. The actual properties of the optimal solution are

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Making Sense of the Mayhem: Machine Learning and March Madness

Making Sense of the Mayhem: Machine Learning and March Madness Alex Tran and Adam Ginzberg Stanford University atran3@stanford.edu ginzberg@stanford.edu I. Introduction III. Model The goal of our research

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Pentominoes: A First Player Win

Games of No Chance MSRI Publications Volume 29, 1996 Pentominoes: A First Player Win HILARIE K. ORMAN Abstract. This article reports on a search-based computer proof that the game of pentominoes is a first-player

### Piano Accordion vs. Chromatic Button Accordion

Piano Accordion vs. Chromatic Button Accordion Which is best, piano accordion (PA), or five row chromatic button accordion (CBA)? This is a question which is often debated in newsgroups. The question should

### Artificial Intelligence in Chess

Artificial Intelligence in Chess IA in Chess is quite complex Even the fastest computer cannot solve the chess game (cannot analyze every possible situation) There are many ways of reducing the need of

### Double Deck Blackjack

Double Deck Blackjack Double Deck Blackjack is far more volatile than Multi Deck for the following reasons; The Running & True Card Counts can swing drastically from one Round to the next So few cards

### Project and Production Management Prof. Arun Kanda Department of Mechanical Engineering Indian Institute of Technology, Delhi

Project and Production Management Prof. Arun Kanda Department of Mechanical Engineering Indian Institute of Technology, Delhi Lecture - 15 Limited Resource Allocation Today we are going to be talking about

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### Information, Entropy, and Coding

Chapter 8 Information, Entropy, and Coding 8. The Need for Data Compression To motivate the material in this chapter, we first consider various data sources and some estimates for the amount of data associated

### TD-Gammon, A Self-Teaching Backgammon Program, Achieves Master-Level Play

TD-Gammon, A Self-Teaching Backgammon Program, Achieves Master-Level Play Gerald Tesauro IBM Thomas J. Watson Research Center P. O. Box 704 Yorktown Heights, NY 10598 (tesauro@watson.ibm.com) Abstract.

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Chapter 13: Binary and Mixed-Integer Programming

Chapter 3: Binary and Mixed-Integer Programming The general branch and bound approach described in the previous chapter can be customized for special situations. This chapter addresses two special situations:

### Automatic Inventory Control: A Neural Network Approach. Nicholas Hall

Automatic Inventory Control: A Neural Network Approach Nicholas Hall ECE 539 12/18/2003 TABLE OF CONTENTS INTRODUCTION...3 CHALLENGES...4 APPROACH...6 EXAMPLES...11 EXPERIMENTS... 13 RESULTS... 15 CONCLUSION...

### Chapter 6: The Information Function 129. CHAPTER 7 Test Calibration

Chapter 6: The Information Function 129 CHAPTER 7 Test Calibration 130 Chapter 7: Test Calibration CHAPTER 7 Test Calibration For didactic purposes, all of the preceding chapters have assumed that the

### Binary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University

Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of number is a mathematical abstraction. To use numbers, we must represent

### NODAL ANALYSIS. Circuits Nodal Analysis 1 M H Miller

NODAL ANALYSIS A branch of an electric circuit is a connection between two points in the circuit. In general a simple wire connection, i.e., a 'short-circuit', is not considered a branch since it is known

### Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus

Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives

### AI and computer game playing. Artificial Intelligence. Partial game tree for Tic-Tac-Toe. Game tree search. Minimax Algorithm

AI and computer game playing Artificial Intelligence Introduction to game-playing search imax algorithm, evaluation functions, alpha-beta pruning, advanced techniques Game playing (especially chess and

### Offline sorting buffers on Line

Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

### H. The Study Design. William S. Cash and Abigail J. Moss National Center for Health Statistics

METHODOLOGY STUDY FOR DETERMINING THE OPTIMUM RECALL PERIOD FOR THE REPORTING OF MOTOR VEHICLE ACCIDENTAL INJURIES William S. Cash and Abigail J. Moss National Center for Health Statistics I. Introduction

### Measuring the Performance of an Agent

25 Measuring the Performance of an Agent The rational agent that we are aiming at should be successful in the task it is performing To assess the success we need to have a performance measure What is rational

### http://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions

A Significance Test for Time Series Analysis Author(s): W. Allen Wallis and Geoffrey H. Moore Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 36, No. 215 (Sep., 1941), pp.

### File Management. Chapter 12

Chapter 12 File Management File is the basic element of most of the applications, since the input to an application, as well as its output, is usually a file. They also typically outlive the execution

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### Compact Representations and Approximations for Compuation in Games

Compact Representations and Approximations for Compuation in Games Kevin Swersky April 23, 2008 Abstract Compact representations have recently been developed as a way of both encoding the strategic interactions

### Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### THE STATISTICAL TREATMENT OF EXPERIMENTAL DATA 1

THE STATISTICAL TREATMET OF EXPERIMETAL DATA Introduction The subject of statistical data analysis is regarded as crucial by most scientists, since error-free measurement is impossible in virtually all

### What mathematical optimization can, and cannot, do for biologists. Steven Kelk Department of Knowledge Engineering (DKE) Maastricht University, NL

What mathematical optimization can, and cannot, do for biologists Steven Kelk Department of Knowledge Engineering (DKE) Maastricht University, NL Introduction There is no shortage of literature about the

### Multiple Linear Regression in Data Mining

Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### The Epsilon-Delta Limit Definition:

The Epsilon-Delta Limit Definition: A Few Examples Nick Rauh 1. Prove that lim x a x 2 = a 2. (Since we leave a arbitrary, this is the same as showing x 2 is continuous.) Proof: Let > 0. We wish to find

### 13. *-Minimax. Introduction. Stochastic Games. Deep Search!? Perfect information deterministic games? Imperfect information games?

13. *-Minimax Introduction Perfect information deterministic games? Lots of success Imperfect information games? Hard because of missing information Subject of active research Bridge, poker Non-deterministic/stochastic

### Decision Theory. 36.1 Rational prospecting

36 Decision Theory Decision theory is trivial, apart from computational details (just like playing chess!). You have a choice of various actions, a. The world may be in one of many states x; which one

### Measurement Information Model

mcgarry02.qxd 9/7/01 1:27 PM Page 13 2 Information Model This chapter describes one of the fundamental measurement concepts of Practical Software, the Information Model. The Information Model provides

### Recent progress in additive prime number theory

Recent progress in additive prime number theory University of California, Los Angeles Mahler Lecture Series Additive prime number theory Additive prime number theory is the study of additive patterns in

### Using Earned Value, Part 2: Tracking Software Projects. Using Earned Value Part 2: Tracking Software Projects

Using Earned Value Part 2: Tracking Software Projects Abstract We ve all experienced it too often. The first 90% of the project takes 90% of the time, and the last 10% of the project takes 90% of the time

### !"!!"#\$\$%&'()*+\$(,%!"#\$%\$&'()*""%(+,'-*&./#-\$&'(-&(0*".\$#-\$1"(2&."3\$'45"

!"!!"#\$\$%&'()*+\$(,%!"#\$%\$&'()*""%(+,'-*&./#-\$&'(-&(0*".\$#-\$1"(2&."3\$'45"!"#"\$%&#'()*+',\$\$-.&#',/"-0%.12'32./4'5,5'6/%&)\$).2&'7./&)8'5,5'9/2%.%3%&8':")08';:

### OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

### Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur

Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Script for How to Win a Chinese Chess Game

Script for How to Win a Chinese Chess Game Jinzhong Niu April 25, 2004 1 Presentation Set Up Chinese chess is played on a 10 9 board, on which either player has 16 pieces. Pieces are located at the intersections

### Chess Algorithms Theory and Practice. Rune Djurhuus Chess Grandmaster runed@ifi.uio.no / runedj@microsoft.com October 3, 2012

Chess Algorithms Theory and Practice Rune Djurhuus Chess Grandmaster runed@ifi.uio.no / runedj@microsoft.com October 3, 2012 1 Content Complexity of a chess game History of computer chess Search trees

### The Rules of Sorry! (As found in the 1939 United States version of the game Sorry!)

The Rules of Sorry! (As found in the 1939 United States version of the game Sorry!) THE PACK consists of 44 cards, four each of the denominations: 1, 2, 3, 4, 5, 7, 8, 10, 11, 12 and four Sorry cards.

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### Announcement. HW#3 will be posted today. Exercise # 4 will be posted on Piazza today.

Announcement HW#3 will be posted today. Exercise # 4 will be posted on Piazza today. Operating Systems: Internals and Design Principles Chapter 7 Memory Management Seventh Edition William Stallings Modified

### IBM SPSS Direct Marketing 23

IBM SPSS Direct Marketing 23 Note Before using this information and the product it supports, read the information in Notices on page 25. Product Information This edition applies to version 23, release

### On Correlating Performance Metrics

On Correlating Performance Metrics Yiping Ding and Chris Thornley BMC Software, Inc. Kenneth Newman BMC Software, Inc. University of Massachusetts, Boston Performance metrics and their measurements are

### ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015

ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1

### Accounting Machinery of Today

Accounting Machinery of Today THE ECONOMIC development of business organization during the last fifty years can be attributed primarily to three basic influences: (1) improvements in transportation facilities,

### Implementation of a Chess Playing Artificial. Intelligence. Sean Stanek.

Implementation of a Chess Playing Artificial Intelligence Sean Stanek vulture@cs.iastate.edu Overview Chess has always been a hobby of mine, and so have computers. In general, I like creating computer

### IBM SPSS Direct Marketing 22

IBM SPSS Direct Marketing 22 Note Before using this information and the product it supports, read the information in Notices on page 25. Product Information This edition applies to version 22, release

### Elementary School Mathematics Priorities

Elementary School Mathematics Priorities By W. Stephen Wilson Professor of Mathematics Johns Hopkins University and Former Senior Advisor for Mathematics Office of Elementary and Secondary Education U.S.

### CCPM: TOC Based Project Management Technique

CCPM: TOC Based Project Management Technique Prof. P.M. Chawan, Ganesh P. Gaikwad, Prashant S. Gosavi M. Tech, Computer Engineering, VJTI, Mumbai. Abstract In this paper, we are presenting the drawbacks

### Project Risks. Risk Management. Characteristics of Risks. Why Software Development has Risks? Uncertainty Loss

Project Risks Risk Management What can go wrong? What is the likelihood? What will the damage be? What can we do about it? M8034 @ Peter Lo 2006 1 M8034 @ Peter Lo 2006 2 Characteristics of Risks Uncertainty

### Rafael Witten Yuze Huang Haithem Turki. Playing Strong Poker. 1. Why Poker?

Rafael Witten Yuze Huang Haithem Turki Playing Strong Poker 1. Why Poker? Chess, checkers and Othello have been conquered by machine learning - chess computers are vastly superior to humans and checkers

### Using Analytic Hierarchy Process (AHP) Method to Prioritise Human Resources in Substitution Problem

Using Analytic Hierarchy Process (AHP) Method to Raymond Ho-Leung TSOI Software Quality Institute Griffith University *Email:hltsoi@hotmail.com Abstract In general, software project development is often

### Gas Dynamics Prof. T. M. Muruganandam Department of Aerospace Engineering Indian Institute of Technology, Madras. Module No - 12 Lecture No - 25

(Refer Slide Time: 00:22) Gas Dynamics Prof. T. M. Muruganandam Department of Aerospace Engineering Indian Institute of Technology, Madras Module No - 12 Lecture No - 25 Prandtl-Meyer Function, Numerical

### THEORY OF CONSTRAINTS AS AN EFFECTIVE TOOL FOR SUPPLY CHAIN MANAGEMENT JOANNA NOWAKOWSKA-GRUNT 1 EWA MOROZ 2

Advanced Logistic Systems, Vol. 7, No. 1 (2013), pp. 71 78. THEORY OF CONSTRAINTS AS AN EFFECTIVE TOOL FOR SUPPLY CHAIN MANAGEMENT JOANNA NOWAKOWSKA-GRUNT 1 EWA MOROZ 2 Abstract: Supply chain management

### The Trip Scheduling Problem

The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems

### Gerry Hobbs, Department of Statistics, West Virginia University

Decision Trees as a Predictive Modeling Method Gerry Hobbs, Department of Statistics, West Virginia University Abstract Predictive modeling has become an important area of interest in tasks such as credit

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

### BIG DATA IN THE FINANCIAL WORLD

BIG DATA IN THE FINANCIAL WORLD Predictive and real-time analytics are essential to big data operation in the financial world Big Data Republic and Dell teamed up to survey financial organizations regarding

### Pascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.

Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive

### Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur

Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture No. # 11 Block Cipher Standards (DES) (Refer Slide

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### APPROACHES TO SOFTWARE TESTING PROGRAM VERIFICATION AND VALIDATION

1 APPROACHES TO SOFTWARE TESTING PROGRAM VERIFICATION AND VALIDATION Validation: Are we building the right product? Does program meet expectations of user? Verification: Are we building the product right?

### Chess Algorithms Theory and Practice. Rune Djurhuus Chess Grandmaster runed@ifi.uio.no / runedj@microsoft.com September 23, 2014

Chess Algorithms Theory and Practice Rune Djurhuus Chess Grandmaster runed@ifi.uio.no / runedj@microsoft.com September 23, 2014 1 Content Complexity of a chess game Solving chess, is it a myth? History

### Random Fibonacci-type Sequences in Online Gambling

Random Fibonacci-type Sequences in Online Gambling Adam Biello, CJ Cacciatore, Logan Thomas Department of Mathematics CSUMS Advisor: Alfa Heryudono Department of Mathematics University of Massachusetts

### Dynamic programming formulation

1.24 Lecture 14 Dynamic programming: Job scheduling Dynamic programming formulation To formulate a problem as a dynamic program: Sort by a criterion that will allow infeasible combinations to be eli minated

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows

TECHNISCHE UNIVERSITEIT EINDHOVEN Branch-and-Price Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### SCRATCHING THE SURFACE OF PROBABILITY. Robert J. Russell, University of Paisley, UK

SCRATCHING THE SURFACE OF PROBABILITY Robert J. Russell, University of Paisley, UK Scratch cards are widely used to promote the interests of charities and commercial concerns. They provide a useful mechanism

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Static IP Routing and Aggregation Exercises

Politecnico di Torino Static IP Routing and Aggregation xercises Fulvio Risso August 0, 0 Contents I. Methodology 4. Static routing and routes aggregation 5.. Main concepts........................................

### A PROOF OF "GOLDBACH'S CONJECTURE" By Roger Ellman GOLDBACH'S CONJECTURE states:

A PROOF OF "GOLDBACH'S CONJECTURE" By Roger Ellman GOLDBACH'S CONJECTURE states: Every even number greater than two can be expressed as the sum of two primes. STEP 1 - General All of the prime numbers

### Combining player statistics to predict outcomes of tennis matches

IMA Journal of Management Mathematics (2005) 16, 113 120 doi:10.1093/imaman/dpi001 Combining player statistics to predict outcomes of tennis matches TRISTAN BARNETT AND STEPHEN R. CLARKE School of Mathematical

### CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2

### Chapter 15: Dynamic Programming

Chapter 15: Dynamic Programming Dynamic programming is a general approach to making a sequence of interrelated decisions in an optimum way. While we can describe the general characteristics, the details

### Effects of Filler Traffic In IP Networks. Adam Feldman April 5, 2001 Master s Project

Effects of Filler Traffic In IP Networks Adam Feldman April 5, 2001 Master s Project Abstract On the Internet, there is a well-documented requirement that much more bandwidth be available than is used

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### Problem Solving Basics and Computer Programming

Problem Solving Basics and Computer Programming A programming language independent companion to Roberge/Bauer/Smith, "Engaged Learning for Programming in C++: A Laboratory Course", Jones and Bartlett Publishers,

### THE INFORMATION TECHNOLOGY PROJECT CHARTER

1-01-12 INFORMATION MANAGEMENT: STRATEGY, SYSTEMS, AND TECHNOLOGIES THE INFORMATION TECHNOLOGY PROJECT CHARTER John P. Murray INSIDE Gaining Project Charter Approval; Project Charter Components; Project

### A fairly quick tempo of solutions discussions can be kept during the arithmetic problems.

Distributivity and related number tricks Notes: No calculators are to be used Each group of exercises is preceded by a short discussion of the concepts involved and one or two examples to be worked out

### RELEVANT TO ACCA QUALIFICATION PAPER P3. Studying Paper P3? Performance objectives 7, 8 and 9 are relevant to this exam

RELEVANT TO ACCA QUALIFICATION PAPER P3 Studying Paper P3? Performance objectives 7, 8 and 9 are relevant to this exam Business forecasting and strategic planning Quantitative data has always been supplied