A COMPUTATIONAL MODEL OF TOWER OF HANOI PROBLEM SOLVING By Sashank Varma Dissertation Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Psychology May, 2006 Nashville, Tennessee Approved: Professor Timothy P. McNamara Professor Susan R. Goldman Professor John D. Bransford Professor Gordon D. Logan Professor David C. Noelle UMI Number: 3230587 3230587 2006 Copyright 2006 by Varma, Sashank UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 All rights reserved. by ProQuest Information and Learning Company. Copyright © 2006 by Sashank Varma All Rights Reserved iii To my father, Sreekanth iv TABLE OF CONTENTS Page DEDICATION iii LIST OF TABLES vii LIST OF FIGURES ix Chapter I. INTRODUCTION 1 II. THE TOH TASK 5 Task Definition 5 Solution Strategies 8 Goal Recursion 8 The Sophisticated Perceptual Strategy 10 The Simple Perceptual Strategy 12 Other Strategies 14 III. REVIEW OF THE EMPIRICAL LITERATURE 17 Dimensions of Variation 19 Behavioral Studies of Normal Adults 23 Ruiz (1987) 24 Anderson, Kushmerick, and Lebiere (1993) 26 Carpenter, Just, and Shell (1990) 31 Behavioral Studies of Patients with Frontal Lobe Lesions 34 Goel and Grafman (1995) 34 Morris, Miotto, Feigenbaum, Bullock, and Polkey (1997a) 39 Morris, Miotto, Feigenbaum, Bullock, and Polkey (1997b) 41 Neuroimaging Studies of Normal Adults 44 Fincham, Carter, van Veen, Stenger, and Anderson (2002) 44 Anderson, Albert, and Fincham (2005) 50 IV. EXISTING TOH MODELS 54 ACT-R Models 55 Anderson, Kushmerick, and Lebiere (1993) 55 v Anderson and Lebiere (1998) 56 Altmann and Trafton (2002) 57 Anderson, Albert, and Fincham (2005) 58 3CAPS Models 59 Just, Carpenter, and Hemphill (1996) 60 Goel, Pullara, and Grafman 61 A Connectionist Model 62 V. THE 4CAPS COGNITIVE ARCHITECTURE 64 Why a Separate Architectural Level? Why 4CAPS? 66 Historical Development 67 Operating Principles 68 VI. A MODEL OF FRONTO-PARIETAL INTERACTION 74 Newell and Simon’s Theory of Problem Solving 75 Shallice’s Theory of Executive Function 78 A Theoretical Synthesis 81 Schemas 82 The SAS 83 The Contention Scheduler 84 VII. THE TOH MODEL 85 The Spatial Centers 89 LH-Spatial 89 RH-Spatial 91 The Executive Centers 93 LH-Executive 94 Graded, Unary Preferences 94 Design Decisions 95 Heuristics 97 Selection and Suppression 99 RH-Executive 101 Collaborative Processing 104 VIII. BEHAVIORAL MEASURES OF NORMAL YOUNG ADULTS 108 Individual Move Times 109 Number of Moves 121 Summary 125 vi IX. BEHAVIORAL MEASURES OF LESION PATIENTS 127 Morris, Miotto, Feigenbaum, Bullock, and Polkey (1997a) 128 Morris, Miotto, Feigenbaum, Bullock, and Polkey (1997b) 130 Goel, Grafman, and Pullara (2001) 131 X. NEUROIMAGING MEASURES OF NORMAL YOUNG ADULTS 144 Fincham, Carter, van Veen, Stenger, and Anderson (2002) 145 Anderson, Albert, and Fincham (2005) 146 Behavioral Data 147 Neuroimaging Data 152 XI. GENERAL DISCUSSION 165 The Nature of Goals 167 The Nature of Selection 170 The Reunification of Problem Solving and Executive Function 172 Degrees of Freedom as Design Decisions, not Free Parameters 173 Validating the 4CAPS Cognitive Architecture 176 XII. FUTURE DIRECTIONS 178 Additional Populations 178 Additional Tasks, Brain Areas, and Functions 181 Learning 184 Appendix A. MODEL SOURCE CODE 187 B. ANNOTATED SIMULATION TRACE 236 C. PARAMETER SETTINGS FOR ALL SIMULATIONS 270 ENDNOTES 274 REFERENCES 276 vii LIST OF TABLES Table Page 1. Average individual move times for Ruiz (1987) 26 2. Overall solution time and number of moves for Anderson et al. (1993) 27 3. Average individual move times for Anderson et al. (1993) 31 4. Planning times and execution times for Morris et al. (1997a) 41 5. Planning times and execution times for Morris et al. (1997b) 44 6. Average individual move times for Fincham et al. (2002). **ESTIMATED** 47 7. Average individual move times for Anderson et al. (2005) 52 8. Synthesis of the Soar theory of problem solving and Shallice’s theory of executive function 82 9. Design decisions of the model 86 10. Productions of the model 88 11. Weights of the heuristic productions 109 12. Correlations between individual move times and the 48 model variants 111 13. Correlation between individual move times of the human participants and the model averaged for each value of the five design decisions 113 14. Average individual move times of the human participants and the model for the 4-disk problems of Anderson et al. (1993) 117 15. Average individual move times of the human participants and the model for the 5-disk problems of Ruiz (1987) and Anderson et al. (1993) 119 16. Number of moves required by human participants and the six model variants for the 4-disk problems of Anderson et al. (1993) 123 17. Number of moves required by human participants and the six model variants for the 5-disk problems of Anderson et al. (1993) 124 viii 18. Correlation between the number of moves required by human participants and the model averaged for each value of the top-goal-moves-only and suppress-old-states decisions 124 19. Average number of moves required by human participants and the six model variants to solve the 4-Disk and 5-Disk problems of Anderson et al. (1993) 125 20. Resource capacities of the model centers used to simulate Goel et al. (2001) 133 21. Correlation between the proportion of the Goel et al. (2001) problems solved by normal and lesioned patients and models for each value of the top-goal-moves-only decision 136 22. Correlation between the number of moves required to solve the Goel et al. (2001) problems by normal and lesioned patients and models for each value of the top-goal-moves-only decision 139 23. Correlation between the overall solution time required to solve the Goel et al. (2001) problems by normal and lesioned patients and models for each value of the top-goal-moves-only decision 142 24. Correlation between the activation time series observed in each brain region and predicted by the corresponding model center for the Anderson et al. (2005) problem, averaged for each value of the top-goal-moves-only and suppress-old-states decisions 156 25. Correlation between the activation time series observed in each brain region and that predicted by the corresponding model center for the Anderson et al. (2005) problem 160 26. Correlation between the activation time series observed in the left frontal region and predicted by each Executive center for the Anderson et al. (2005) problem, broken down separately for non-chunked and chunked moves 163 ix LIST OF FIGURES Figure Page 1. The five classes of TOH problem 7 2. Individual move times for the 5-disk problems of Ruiz (1987) 25 3. Individual move times for the 4-disk problems of Anderson et al. (1993) 29 4. Individual move times for the 5-disk problems of Anderson et al. (1993) 30 5. Error rates for Carpenter et al. (1990) 33 6. Proportion of problems solved in the allotted time (two minutes) for Goel et al. (2001) 36 7. Number of moves required for Goel et al. (2001) 37 8. Overall solution time for Goel et al. (2001) 38 9. Number of moves (above minimum) required for Morris et al. (1997a). **ESTIMATED** 40 10. (a) Number of moves required for the 6-move problems of Morris et al. (1997b). **ESTIMATED** (b) Number of moves required for the 7-move problems of Morris et al. (1997b). **ESTIMATED** 43 11. Individual move times for Fincham et al. (2002). **ESTIMATED** 46 12. R. DLPFC activations for Fincham et al. (2002). **ESTIMATED** 48 13. Bilateral parietal activations for Fincham et al. (2002). **ESTIMATED** 49 14. Individual move times for Anderson et al. (2005) 51 15. Activation time series for the left frontal and left parietal regions for Anderson et al. (2005) 53 16. Levels of the 4CAPS TOH model 65 17. Newell’s (1990) Soar theory of problem solving 76 18. Shallice’s (1982) theory of executive function 79 [...]... source of variation between participants, enabling precise calculation of the computational demand of each problem and, when a constrained presentation paradigm is adopted, of each move The casualty of explicit instruction is ecological validity – there follows a certain artificiality to problem solving This is problematic if the goal is to investigate the acquisition of new solution strategies, as it was... that makes heavy computational demands over the course of tens of seconds (or longer) The non-executive components of problem solving are domain-specific In particular, the TOH task is visuospatial in nature (versus, say, logical deduction, which is comparatively linguistic) The visuospatial aspects of problem solving are localized in parietal areas such as intra-parietal sulcus (IPS) The account of. .. is that constrained presentation lacks ecological validity, straightjacketing natural problem solving behavior The benefits and drawbacks of constrained presentation are reversed in unconstrained paradigms: although problem solving is more ecologically valid, participants cannot be compared on individual moves because the solution sequences of different participants can differ A single study has tried... representations and processes specific to the TOH task itself In the middle is an account of the interaction between frontal and parietal areas that bridges between the other levels in a principled way The model has been evaluated against behavioral data collected from normal young adults, behavioral data collected from patients with frontal lobe lesions, and neuroimaging data collected from normal young adults... the ebb and flow of activation over the course of problem solving, which can be compared with the ebb and flow of information processing in computational models Generally speaking, studies that employ event-related designs provide richer data than studies that employ block designs Behavioral Studies of Normal Adults Most empirical investigations of TOH problem solving have studied normal adults and collected... lacking, and the likelihood that one will emerge decreases as empirical investigations grow more disconnected The primary achievement of this dissertation has been to fill this gap A new computational model has been constructed that accounts for a broad range of the new data on TOH problem solving It consists of three levels At the bottom is an architectural foundation At the top are 1 those representations... have been a number of secondary achievements as well One is a new account of how goals are organized and how they control cognition For the first forty years of the cognitive revolution, it was widely assumed that goals are organized as a stack of arbitrary depth, with only the topmost (i.e., most recent) goal guiding problem solving This assumption has been challenged in recent years (Altmann & Trafton,... Their participants spent up to three hours solving TOH problems, making thousands of moves! Given the time-on-task and the regularity of the problems (all belonged to the tower- to -tower class), it is possible that some participants memorized solution sequences However, all subsequent studies have regarded rote memorization as a nuisance strategy and have actively guarded against it by having participants... fronto-parietal interaction offered here explains how left and right DLPFC and left and right IPS collaborate in the service of high-level cognition Another achievement is methodological Every theory or model faces the problem of degrees of freedom Degrees of freedom typically take the form of numerically-valued free parameters Although the model contains such parameters, they are regarded as scientifically... each class for the case where N=3 In the standard tower- to -tower problem, all N disks are stacked on the left peg in the starting configuration and all N disks stacked on the right peg in the ending configuration There is only one standard tower- to -tower problem for each value of N The second class of problems is slightly larger: tower- to -tower problems have all N disks stacked on one peg in the starting . A COMPUTATIONAL MODEL OF TOWER OF HANOI PROBLEM SOLVING By Sashank Varma Dissertation Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the. frontal and parietal areas that bridges between the other levels in a principled way. The model has been evaluated against behavioral data collected from normal young adults, behavioral data collected. of problem solving are localized in parietal areas such as intra-parietal sulcus (IPS). The account of fronto-parietal interaction offered here explains how left and right DLPFC and left and