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
n
The Computational Complexity is O(b ) 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
Page 1
Problem Solving
Issues and Methods
Artificial Intelligence
n
Dealing with O(b )
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 Hypothesis
The fundamental organizational unit of all human goaloriented 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 Cases
Tower 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 CONTROL
Decide 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 Limits
Serial 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 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 8
Problem Solving
Issues and Methods
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
Monkey-Hand
X
Get-Bananas
X
Page 9
Problem Solving
Issues and Methods
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 10
Problem Solving
Issues and Methods
Psychology of Problem Solving
Major Traditions
Problem Taxonomies
Theoretical and Empirical Methodologies
Levels (Kinds) Of Theoretical Analyses
Understanding and Search
Page 11
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 theorye.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 Taxonomies
Task 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 Methods
Decomposition 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
Page 15
Problem Solving
Issues and Methods
Levels (Kinds) Of Theoretical Analyses
Decomposition 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 reduction
to a collection of well-structured problems
Creativity and scientific discovery can be
explained 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
Page 17
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-Similarity
A 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 descriptions
of 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 Hypothesis
States
Legal configurations of water in each jug
Operators
Legal pouring operations
Search Control Knowledge
Decide to quit the problem
Quit if succeed
Quit when told by the experimenter
Decide if goal state has been produced
Select 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 states
Do not prefer operators that lead to old states
Reject operators that lead to the immediately
preceding state
Always select operators that lead to the
goal state
Select operator at random if no other
basis for preferred move
Page 20
Problem Solving
Issues and Methods
Brief Review of Polson and Jeffries
Three 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
Page 21
Problem Solving
Issues and Methods
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
24.90
23.69
14.75
15.31
(24, 21, 3)
Observed
Predicted
Mean
12.03
11.84
7.44
6.66
Mean
StD
StD
Page 22
Problem Solving
Issues and Methods
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 23
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
Page 25