LEARNING SPATIAL REASONING Jack Gelfand Center for the Study of Brain, Mind and Behavior Department of Psychology

LEARNING SPATIAL REASONING Computer Game Playing Game Playing and Pattern-Based Reasoning Organization of the visual system –Multi-stream hierarchy –Form perception –Motion perception Elements of perceptual organization –Gestalt figural organization –Popout phenomena Learning New Spatial Concepts - Spatial Concept Formation Languages Structural Features and Functional Features

Readings Epstein, Gelfand and Lock, Constraints, 2, 239 (1998). Gelfand et al., Proceedings of the Joint Conference on Information Systems (1998). Epstein, Gelfand and Lesniak, Computational Intelligence, 12, 199 (1996).

WHY PLAY GAMES WITH COMPUTERS? From the Artificial Intelligence and Cognitive Psychology Point of View –Games are excellent testbeds as they: Have well-defined rules generating a large search space Easily represented in a computer Easy to test Computers can compete with humans at some games but not others From the Game Playing Point of View –Can make the game much more enjoyable to play –New levels of analysis

Search The brute force approach of search has been highly effective in games such as Checkers and Chess. –Checkers Chinook (World Champion) –Chess Best programs can hold their own with the best humans. Deep Blue II –move generation and evaluation in hardware –parallel search in software

EXHAUSTIVE SEARCH From the starting position 1.Generate every legal move for player 1. 2.For each legal move of player 1 generate every legal move for player 2. 3.Repeat steps 1 & 2 until the game reaches a definitive result.

PROBLEM WITH EXHAUSTIVE SEARCH Not practical –A player in chess has, on average, 36 legal moves. –A game could take 45 moves to reach a conclusion (underestimate). –Total number of positions = –There is only ~10 81 atoms in the universe Couldn’t store all the positions in computer the size of the universe.

EVALUATION FUNCTIONS Assign a value for each factor contributing to the worth of a position. Add up the terms Search positions based upon the values

Searching the Game Tree MAX MIN MAX This is the Minimax Algorithm

Improving Minimax The Minimax Algorithm has various improvements that are used in practice. –Alpha-Beta –Principle Variation Search (PVS) –Transposition Tables –Killer Move Heuristics At best they can halve the work of the search.

Computer Chess Deep Blue II –256 dedicated chess processors generate moves evaluate positions –Search process in software (PVS) –Database of opening sequences –Databases of endgame sequences Deep Blue II can evaluate 200 million positions per second (3 billion in 3 minutes). Deep Blue II can hold its own with the best players in the world, but it is not invincible!

WHAT IS WRONG WITH THIS PICTURE?

THERE ARE AS MANY POSSIBLE GAME STATES IN CHESS AS ATOMS IN THE UNIVERSE. THERE IS ABOUT 20 X 6 FEET OF SPACE FOR CHESS BOOKS IN THE LIBRARY. WHAT’S WRONG?

MOST OF THE GAME STATES IN CHESS ARE IRRELEVANT. HUMANS HAVE AN EXTREMELY COMPACT WAY OF REPRESENTING THE SALIENT CONCEPTS IN CHESS.

LEARNING NEW REPRESENTATIONS THROUGH EXPERIENCE Statement of problem or early experience does not necessarily provide optimal representation. People acquire optimal representations gradually. Often problems are stated in terms of local relationships. Experts utilize global spatial heuristics acquired through performing the task. –CHESS - Control of the center of the board –Othello - Control of edges –Go - Shape and thickness of zones

Vertical and Horizontal Control on the Chess Board

Diagonal Control on the Chess Board

HIERARCHICAL ORGANIZATION OF THE HUMAN VISUAL SYSTEM Multiple streams of processing System of feature hierarchies

RECEPTIVE FIELD OF A CORTICAL VISUAL NEURON

CONVERGENT PROJECTIONS IN THE VISUAL FEATURE HIERARCHY

NEURONS IN THE HIGHEST VISUAL FORM RECOGNITION AREAS OF CORTEX RESPOND TO COMPLEX STIMULI

PERCEPTUAL ORGANIZATION

URGE TO ORGANIZE

GESTALT FIGURAL GROUPING Forms or objects composed of elements Organization of elements into perceptual objects involves an active construction process Gestalt researchers studied the way in which these elements tend to become formlike or object like perceptions GESTALT LAWS OF PERCEPTUAL ORGANIZATION Works for sounds as well Gestalt thinking was widely applied but became discredited because it lacked an underlying model. More modern neural models can account for these mechanisms.

GESTALT PRINCIPLES OF FIGURAL ORGANIZATION

PERCEPTUAL ORGANIZATION TAKES PLACE AT MANY LEVELS X O OO The level of perceptual organization depends upon the task and the attentional state of the viewer.

POPOUT PHENOMENA

SYNCHRONICITY OF NEURONS IN VISUAL CORTEX MAY LINK THE COMPONENTS OF THE FIGURE RELATIVE TO THE GROUND

HOYLE DECISION MAKING SYSTEM

FORR - FOr the Right Reasons Linear mixture of experts Advisors - decision-making rationales Multi-tier hierarchy –Tier 1 - guarantied correct, shallow search –Higher tiers - heuristic knowledge - probably correct Susan Epstein, CUNY

This is related to a perceptron neural network, Which we will learn about later

HOYLE DECISION MAKING SYSTEM

Figure 1: (a) Spatial arrangements of game pieces processed in the algorithms described. (b) The L-shaped arrangement fitted to a game board two different ways. (c) A fitted L-shape instantiated to produce a tic-tac-toe pattern. CONCEPT FORMATION LANGUAGE

PATTERN LEARNING SYSTEM

GENERALIZING PATTERNS INTO SPATIAL CONCEPTS

Figure 5: Three learned spatial Advisors for lose tic-tac-toe, and their weights during 200 consecutive contests. The mover for each Advisor is in the current state; the pattern is matched for in the subsequent state In an  Advisor, either  = X and  = O or  = O and  = X. In an * Advisor, * is either X or O consistently. AN ALGORTITM FOR WEIGHT LEARNING ADJUSTS THE WEIGHTS OF EACH ADVISOR BASED UPON PERFORMANCE

Table 2: Average and standard deviation of performance with and without spatial orientation against three challengers. Boldface is an improvement over play without patterns at the 95% confidence level. Estimated optima are in italics.

A LEARNED SPATIAL ADVISOR AFFECTS DECISION-MAKING

STRUCTURAL FEATURES - FUNCTIONAL FEATURES Perceptual features are related to functional features through the spatial nature of the rules and the layout of the game board. The architecture of our perceptual system is filtered through our experience in the physical world This leads to visual primatives that include lines, simple geometric arrangements, contiguous space and boundaries. Pieces influence adjacent pieces. The goal of most games involves, –simple contiguous geometric arrangements - three-in-a-row –capture of contiguous space - Go –capture of pieces with contiguous space in between - chess, checkers

FLAX’S LAW The rules of the games we like to play result in configurations on the game board we like to see.

THE GAME OF GO INVOLVES THE CONTROL OF SPACE a If white plays a stone at the point a then the three black stones will be captured and removed from the board.

THE CONTROL OF SPACE

COMPLEXITY OF DECISION MAKING There is a tendency to analyze the complexity of reasoning tasks in terms of an exhaustive search of alternatives. Humans manage to function in complex reasoning domains by compartmentalization of the problem and restriction of search based upon past experience.

REORGANIZATION OF DOMAIN KNOWLEDGE WITH EXPERIENCE Chunking of rules in production systems Reorganizing mental models

DEVISE A GAME WHERE THE RULES RESULT IN GEOMETRIC ARRANGEMENTS OF PIECES OF TACTICAL OR STRATEGIC SIGNIFICANCE THAT ARE NOT EASILY PERCEIVED. DIAGRAM THE BOARD LIST THE RULES SHOW A BOARD POSITION THAT RESULTS FROM LEGAL MOVES WHICH DEMONSTRATES THIS FACT. HOMEWORK