Problem Solving Issues and Methods
Overview Foundation Issues in Cognitive Science
All Intelligent Behavior Can Be Described as Problem Solving
Programs Can Be Written To Search Problem Spaces
The Computational Complexity is O(bn
) where b is the branching factor and n is the number of operators in the solution path
For interesting problems, b and n can be large
The Psychology 0f Problem Solving and Skilled Performance
Common Themes
Fundamental Role of Similarity
Task Orientation
Interactions of Basic Processes to Generate Action
Learning
Computer Simulation Models
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Problem Solving Issues and Methods
Artificial Intelligence
Dealing with O(bn
)
Learning the Hard Way About Values of b and n
Chess, Math, etc. b = 10 to 30, n = 10 to 40
Scene and Language Understanding, etc. b and n Much Larger
Algorithms verses Heuristic Methods
Heuristic Search Methods
Reduce b by “ignoring” alternatives
Best First Search, Means-Ends Analysis
Look ahead (chess, etc)
Planning by Abstraction (reduce both b and n)
Problem Reduction
Changes in Representation (reduce both b and n)
Learning (reduce b to 1 in the limit)
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Problem Solving Issues and Methods
Problem Space HypothesisThe fundamental organizational unit of all human goal-oriented symbolic activity is the problem space.
Assumption:
Fundamental process underlying intelligent action is Search
Alternative:
Language comprehension and knowledge-based inferences
PROBLEM SPACE
knowledge states
operators that generate new knowledge states
a sequence of operators describes a path
PROBLEM
a set of initial states
a set of goal states
a set of path constraints
the problem is to find a path from a start state to a goal state that satisfies the path constraints
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Problem Solving Issues and Methods
Simple CasesTower of Hanoi
Possible Disk Configuration Generated by Legal Moves
Water Jug Problem
Gets More Complex Quickly
“Right” Representation and Insight Problems
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Problem Solving Issues and Methods
SEARCH CONTROLDecide to quit the problem
Decide if a goal state has been produced
Select a state from the stock to be the current state
Select an operator to be the current operator
Decide to save the new state just produced by an operator
Operate Within the Following Cycle
1. Select a state; Select an operator
2. Apply operator to a state producing a new state
3. Decide if a goal state; decide to quit; decide to save a new state
Search control depend on know that is immediately available
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Problem Solving Issues and Methods
Resource and Capacity LimitsSerial Action:
At most one problem space operator can be performed at one time
Problem solving will consider on one move at a time
In the problem space, maybe several moves in the external word
Example: A move a two disk stack in the tower of Hanoi
Finite Stock:
The subject has a limited number of states (the stock) available to become the current state.
(i.e. humans can and will only consider a limited number of alternatives)
Search Control:
Use only immediately available knowledge
Multiple problem spaces, e.g. an operator selection space
Time course of behavior: 5 to 15 second per state
Grain size of analysis: very detailed in comparison to most psychological models
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Problem Solving Issues and Methods
Search Control II (Reduce b)Search Methods
Generate and Test
Heuristic Search: Depth, Breadth, or Best First
Hill Climbing*
Mean-End Analysis
Operator Subgoaling
Planning
Evaluation Functions (Computers)
Distance To Goal
Likelihood that State Is On Solution Path
Weighted Average of Desirable Properties
…
Evaluation Functions (Human)
Similarity of Appearance to Goal
Similarity of Meaning to Goal
Knowledge that This State Is On Solution Path
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Problem Solving Issues and Methods
Means-Ends AnalysisGeneral Problem Solver
Newell, Shaw, and Simon (1968)
Difference Reduction
Several Kinds of Differences Between Goal and Current State
Ends (goals and subgoals)
Set Up Goals and Subgoals to Reduce Differences
Means (Operators)
Operators Effect Some Differences and Not Others
Table of Connections
Examples
Tower of Hanoi
Algebra
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Problem Solving Issues and Methods
Monkey and BananasTop Goal = (Transform the Initial-Object into the Desired-Object)
Initial-Object =(Monkey’s-Place = Place-1, Box’s-Place = Place-2, Contents-of-Monkey’s-Hand = Empty)
Desired-Object =(Monkey’s-Place = On-Box, Box’s-Place = Under-Bananas, Contents-of-Monkey’s-Hand = Bananas)
Operators
Walk, Move-Box, Climb, Get-Bananas
Preconditions
Difference Ordering
Table of Connections
Monkey-Place Box-Place Monkey-Hand
Walk X
Move-Box X X
Climb X
Get-Bananas X
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Problem Solving Issues and Methods
Trace of GPS Solving Problem1 Transform Initial-Obj into Desired-Obj
2 Reduce Contents-of-Monkey’s Hand Diff On Initial-Obj
3 Apply Get-Bananas on Initial-Obj
4 Reduce Location-of-Box Diff On Initial-Obj
5 Apply Move-Box to Under-Bananas On Initial-Obj
6 Reduce Location-of-Monkey Diff On Initial-Obj
7 Apply Monkey Walk to Location of Box on Initial-Obj
(Monkey’s-Place = Place-2, Box’s-Place = Place-2,
Contents-of-Monkey’s-Hand = Empty)
8 Apply Move-Box to Under-Bananas
(Monkey’s-Place = Under-Bananas, Box’s-Place = Under-Bananas,
Contents-of-Monkey’s-Hand = Empty)
9 Apply Get-Bananas to Current-Obj
10 Reduce Location-of-Monkey Diff
11 Apply Climb to Current-Obj
12 Apply Get-Bananas to Current-Obj
13 Transform Initial-Obj into Desired-Obj
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Problem Solving Issues and Methods
Psychology of Problem SolvingMajor Traditions
Problem Taxonomies
Theoretical and Empirical Methodologies
Levels (Kinds) Of Theoretical Analyses
Understanding and Search
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Problem Solving Issues and Methods
Major Research Traditions
Cognitive Action
Account for Complex Action Sequences
General behavior theory
(Thorndike, Hull, Skinner, Tolman, Staats, .)
Modern cognitive theory-e.g., Rule-based models of skill acquisition
(Newell, Simon, Anderson, .....)
Cognitive Representation
Mental Representations That Generate Action Sequences
Gestalt psychology
(Duncker, Katona, Kohler, Wertheimer, ...)
Modern research on representation
(Greeno, Kintsch, Simon, ....)
Modern Research On Problem Solving Attempts to Synthesize the Two Traditions
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Problem Solving Issues and Methods
Problem TaxonomiesTask Orientation
How are various tasks related?
Well-Structured (Closed) Problems
Puzzles
Instructional Problems
Characteristics of ...
Explicit goal
Known operators
Known values of b and n, often small!
Ill-Structured (Open) Problems
Design
Real-Life” Problems
Characteristics of ...
b and n large or indefinite!
Ill-Structured Characteristics of Well-Structured Problems (Simon)
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Problem Solving Issues and MethodsDecomposition of an Ill-Structured Problem Into
A Collection of Well Structured Problems
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Problem Solving Issues and Methods
Theoretical and Empirical Methodologies
Explicit Process Models
Computer Simulations
Production Systems (Rule-Based)
Verbal Protocol Analysis
Comparisons between Novices and Experts
Problems designed for the instruction of novices
Common task done by both experts and novices
Task done by experts
Problem Solving as an Arena to Test General Theories
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Problem Solving Issues and Methods
Levels (Kinds) Of Theoretical AnalysesDecomposition into “Higher-Level” Processes
Preparation, Insight, Creativity, Incubation, Set,Functional Fixedness, Brain Storming, ...
Demonstrations of ....
Process Models
Rules
Elementary Information Processes
Description verses Explanation ...
Demonstrations verses Explanations ...
Continuum Hypothesis (Simon)
Solution of Ill-structured problems by reductionto a collection of well-structured problems
Creativity and scientific discovery can beexplained using the same processes used to solve well-structured problems
E.g., Insight is just a memory process (Recognition)
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Problem Solving Issues and Methods
Understanding and Search
Problem Solving as
Understanding (Comprehension)
(Wertheimer, Hayes and Simon, Kintsch, ....)
Search
(Newell, Simon, AI Literature...)
Problem Space Hypothesis
Understanding-Search
Increasing Importance of Understand Processes
Multiple Problem Spaces
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Problem Solving Issues and Methods
Weak Methods In Human
Generate and Test
Means-Ends Analysis
Difference Reduction-Similarity
Operator Subgoaling
Problem Decomposition (Reduction)
Planning by Abstraction
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Problem Solving Issues and Methods
Difference Reduction-SimilarityA Form of Hill Climbing
In Many Simple Problems It Turns Into:
Select moves by comparing consequences of each move from the current state with goal.
Pick move that leads to state that is "closer" (more similar to ) the goal.
Atwood and Polson (1976) Water Jug Problems
Similarity of Descriptions
Select moves by comparing descriptionsof each available action from the current state with a description of the goal.
Learning to use computers, phone mail, and other complex systems.
Lewis and Polson (1990) Label following
CoLiDeS (Kitajima, Blackmon, Polson, In Press)
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Problem Solving Issues and Methods
Atwood and Polson (1976)Perfect Example of Problem Space HypothesisStates
Legal configurations of water in each jugOperators
Legal pouring operationsSearch Control Knowledge
Decide to quit the problemQuit if succeedQuit when told by the experimenter
Decide if goal state has been producedSelect an operator to be the current operator
Prefer operators that lead to states that are similar to the goal state
Prefer operators that lead to new statesDo not prefer operators that lead to old statesReject operators that lead to the immediately
preceding stateAlways select operators that lead to the
goal state
Select operator at random if no other basis for preferred move
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Problem Solving Issues and Methods
Brief Review of Polson and JeffriesThree Stage Move Selection Process
1. Means-Ends (Similarity, Evaluation Function)
2. New moves (Use of LTM)
3. Best move or random (STM limits)
Memory
Very simplified model
Parameters
Description of random process
Simplifying assumptions
Individual differences/noise in decision processes
Constraints on parameter values
Constant across some kinds of manipulations
Vary in a “lawful” way for other kinds of manipulations
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Problem Solving Issues and Methods
Water JugsFigure showing (8,5,3) problem
Test of means-ends assumption
(8,5,3) Vs (24,21,3)
Goodness of fit
Observed and Predicted Means and Standard Deviations
(8, 5, 3) Observed Predicted
Mean 24.90 23.69
StD 14.75 15.31
(24, 21, 3) Observed Predicted
Mean 12.03 11.84
StD 7.44 6.66
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Problem Solving Issues and Methods
Working BackwardsGeometry
Novices Solving Physics Problems
Planning problems
Paint the ladder and ceiling green
Define new subgoals
Monkey and the Bananas
Trivial (for humans)
Huge search space
Important Problem in the Early History of AI
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Problem Solving Issues and Methods
Knowledge
Use of Knowledge to:
* Build Effective Representations
* Control Search
Search Control
Basic Operations (Actions)
Knowledge of Individual Steps
Rule-based representation
Transfer implications
Strategic Knowledge
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Problem Solving Issues and Methods
Strategic Knowledge
Beyond Simple Puzzle Like Problems
Text Book Problems in Geometry, Algebra, Physics
Ill-structured Problems like Design
Interaction between problem representation and search methods
Specialized knowledge required by a given search method
Strategic Knowledge, Constructions, Etc.
Solution Schemata
Problem Classification
Problem Representation
Expertise
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