Problem Solving Lectures
Cognitive Psychology
Problem Solving And Skill
Problem Solving
• Knowledge-Based Behavior
• Declarative Knowledge (Facts, Episodic Memory)
Skill Acquisition
• Transition from Knowledge-Based to Skill-based
• Procedural Knowledge
How do we use knowledge/memory to achieve our goals?
• Bridge from Memory to Problem Solving and Skill
• Search
Problem Solving Important Component of
Learning
Planning Future Courses of Action
Scientific Discovery
Modern Research On Problem Solving
Artificial Intelligence
Problem Solving as Search
Herbert Simon and Alan Newell
Page 1
Problem Solving Lectures
Cognitive Psychology
A Theory of Action
Goals:
What We Want to Happen
The Central Role of Goals in Problem Solving
Execution:
What We Do To The World
Memory for the Required Actions
Possibility of Inferring Them
Evaluation
Comparing What Happened With What We Wanted To
Happen
The Roles of Attention, Comprehension, and Memory
The World
Page 2
Problem Solving Lectures
Cognitive Psychology
Problem Solving
Fundamental Part of Intelligent Behavior
When ever we don’t know how to reach a goal
Finding a lost article
Homework problems
Planning for a day, life, etc.
Learning new skills
Scientific discovery
Creativity
Many test of intelligence are problem-solving tasks
Professions
Outline of Lectures
(1) Understanding the problem
(2) Problem solving-methods
(3) Why problem solving is hard
Page 3
Problem Solving Lectures
Cognitive Psychology
Types of Problems Used in Studies of
Problem Solving
Insight Problems
Two String Problem
Candle Stick Problem
One critical step, the insight
Gestalt Psychologists
Intermediate steps hidden
Puzzles
Water Jug Problems
River Crossing Problems
Tower of Hanio
Limited background knowledge
Visible intermediate steps
Physics, Algebra, Geometry, etc. Problems
Background knowledge required
Visible intermediate steps
Design Problems
Page 4
Problem Solving Lectures
Cognitive Psychology
Research Methods
Just Record Success or Failure
Not very useful
Need to try to make intermediate steps visible
Record Visible Intermediate Steps
Number of steps, error, etc. time to solution
Verbal Protocols!
Think aloud
Computer Simulations
Incorporate theory into program
Compare behavior of program to subjects
Artificial Intelligence
Branch of computer science
Famous Programs
General Problem Solver (GPS)
Deep Blue
Page 5
Problem Solving Lectures
Cognitive Psychology
Water Jug Problems and Other Puzzles?
Why use such tasks to study problem solving?
Can by solved by novices
History
Can manipulate problem structure to test theories
Detours
Why build computer simulation models?
Make sure that the theory actually works
Compute predictions to compare with human data
Page 6
Problem Solving Lectures
Cognitive Psychology
Atwood and Polson (1976) Theory
Three Stage Move Selection Process
1. Pick moves that lead to states that are similar to the
goal
2. New moves (Use of LTM)
3. Best move or random (STM limits)
Memory
Very simplified model
Page 7
Problem Solving Lectures
Cognitive Psychology
Water Jugs
Figure 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
12.03
11.84
7.44
6.66
Mean
StD
Page 8
Problem Solving Lectures
Cognitive Psychology
Task Environments
(Kinds of Problem Solving Tasks)
Closed Problems
Specific Goal
Well Defined Moves
Laboratory and School Tasks
Puzzles
Mathematics and Physics Problems
Games
Open Problems
Poorly Specific Goal
Ill Defined Moves
Real World Problems
Design
Real Science
“Plan a happy life”
“Solve the population problem”
Page 9
Problem Solving Lectures
Cognitive Psychology
Major Parts of Problem Solving Task
1) Finding the problem!
2) Understanding the problem!
What is the goal? (Goal State)
What is given?
(Start State)
What can be done? (Rules, Actions, Moves, Operations)
Can the problem be broken up into simpler problems?
Problem Solving as Understanding
3) Solving the problem!
Problem Spaces
Search!
Search Methods (Heuristics)
Problem Solving as Search
Page 10
Problem Solving Lectures
Cognitive Psychology
Understanding is Hard
The Right Representation is Hard to Find
Mutilated-checkerboard
The Two String Problem
The Candle Problem
Word Problems
Students can easily perform required manipulations
Cannot build correct problem representation
Mary is 10 years younger than twice Susan’s age. Five
years from now, Mary will be 8 years older than Susan’s
age at that time. How old are Mary and Susan?
Word Arithmetic Problems for 1st and 2nd Graders
Sets: Whole-Set, Part-Set-1, Part-Set-2
There are 5 birds and 3 worms. How many more birds
are there than worms?
There are 5 birds and 3 worms. How many birds will not
get a worm?
Page 11
Problem Solving Lectures
Cognitive Psychology
Search is Hard!!!
Problem Solution: Find path from start state to goal state that
satisfies the rules
VERY Large Number of Possible Paths
Example
A Problem That Takes 10 Moves to Solve
10 Different Moves From Each State
10,000,000,000 Possible Paths
Number of Possible Games of Chess:
10 Followed by 40 Zeros
Search Methods (Problem Solving Approaches)
Generate and Test (Trial and Error)
Difference Reduction-Similarity
Means-Ends Analysis
Simple Puzzles
High Schools Math and Science Problems
Computer Chess
Scientific Discovery
Page 12
Problem Solving Lectures
Cognitive Psychology
Problem Solving As Search
Understanding Before Search
Understand What Goal Is
Know All of Moves
Know About Detours
Search Methods
Generate and Test (Trial and Error)
Difference Reduction-Similarity
Working Backwards
Means-Ends Analysis
Simple Puzzles
High Schools Math and Science Problems
Computer Chess
Scientific Discovery
Page 13
Problem Solving Lectures
Cognitive Psychology
Search is Guided By Goals
Goals
Something Person Is Trying to Accomplish
Examples
Solve Puzzle
Record TV Program Using VCR
Write Paper Using Work Processor
Win Game
Design Beautiful House That Does Not Cost To
Much
Behavior is Guided by Goals
Solve Hard Problems by Breaking Down into Simpler
Problems
Subgoal Decomposition
Focused On Trying to Discover Correct Sequence of Moves
Page 14
Problem Solving Lectures
Cognitive Psychology
Generate And Test
Problem solvers adhering to the generate-and-test paradigm
use two basic modules.
> One module, the generator, enumerates possible
solutions.
> The second, the tester, evaluates each proposed
solutions, either accepting or rejecting that solution.
All Depends on Properties of Generator
A powerful intelligent Generator will only produce a few
"good" solutions including the correct one.
Evolution is an example of Generate-and-Test
All controversies about Darwin and the application of
evolutionary ideas focus on the claim that the generator
is NOT intelligent.
Blind trial and error
Page 15
Problem Solving Lectures
Cognitive Psychology
Difference Reduction-Similarity
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.
“Measure” difference between current situation and goal
Pick more that you think will lead you closer to the goal
Water Jug Problems
Atwood and Polson (1976)
Learning to use computers, phone mail, and other complex
systems.
The Cognitive Walkthrough (Wharton, Lewis, Rieman, &
Polson)
Label following
Find information on the Web
The Scent of Information
Hill Climbing
Page 16
Problem Solving Lectures
Cognitive Psychology
Working Backwards
Geometry
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
Page 17
Problem Solving Lectures
Cognitive Psychology
Means-Ends Analysis
General 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
Page 18
Problem Solving Lectures
Cognitive Psychology
Monkey and Bananas
Top 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
Walk
X
Move-Box
X
Climb
X
Box-Place
Get-Bananas
Monkey-Hand
X
X
Page 19
Problem Solving Lectures
Cognitive Psychology
Trace of GPS Solving Problem
1 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
Page 20
Problem Solving Lectures
Cognitive Psychology
Simple Algebra Problems
Problem: Solve ax - b = x for x
Goal State: Term x on the left side of the equation; all other
terms on the right
Possible Operators
Add x to both sides
Subtract x from both sides
Multiply x on both sides
Divide through x on both sides
Add a to both sides
Subtract a from both sides
Multiply a on both sides
Divide through a on both sides
Add b to both sides
Subtract b from both sides
Multiply b on both sides
Divide through b on both sides
Factor something on one side
...
Page 21
Problem Solving Lectures
Cognitive Psychology
Means-Ends Applied to Simple Algebra
Problems
Problem: Solve ax - b = x for x
Goal State: Term x on the left side of the equation; all other
terms on the right
1. Current state: ax - b = x
Differences: a on left, b on left, x on right.
Apply operator: Add b to both sides
2. Current state: ax = x + b
Differences: a on left, x on right.
Apply operator: Subtract x from both sides
3. Current state: ax - x = b
Differences: a on left, x on left twice.
Apply operator: Factor x on left
4. Current state: x(a - 1) = b
Differences: (a-1) on left.
Apply operator: Divide both sides by (a-1).
5. Current state: x = b/(a-1)
Differences: None
Page 22
Problem Solving Lectures
Cognitive Psychology
Factors That Effect Problem Solving
When Means-Ends Does Not Work
Detours
When the label following strategy fails
Mental Set
Transfer
Doing it the hard way(?)
Functional Fixedness
Insight Problems
Incubation
Page 23
Problem Solving Lectures
Cognitive Psychology
Expertise
Experts Build Better Representations
Schemata, Problem types
Elaborating on Initial States (Planning)
Programming
Writing
Soviet political economy (Voss et al, 1983)
Experts are More Effective Searchers
Metacognitive Skills
Allocating Time to Planning
Error Recovery
Control of Search
MEMORY
Chess Players (De Groot, 1966)
Many other domains
Remembering where you have been!!!
Page 24