problem solving and skill -...

Problem Solving Lectures Cognitive Psychology

Problem Solving And SkillProblem 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


A Theory of ActionGoals:

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


Problem SolvingFundamental 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


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


Research MethodsJust 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


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


Atwood and Polson (1976) TheoryThree 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


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


Task Environments(Kinds of Problem Solving Tasks)

Closed ProblemsSpecific Goal

Well Defined Moves

Laboratory and School Tasks

Puzzles

Mathematics and Physics Problems

Games

Open ProblemsPoorly Specific Goal

Ill Defined Moves

Real World Problems

Design

Real Science

“Plan a happy life”

“Solve the population problem”


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


Understanding is HardThe 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?


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



Problem Solving As SearchUnderstanding 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



Search is Guided By GoalsGoals

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


Generate And TestProblem 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


Difference Reduction-SimilaritySelect 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


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


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


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


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


Simple Algebra ProblemsProblem: 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

...


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


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


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!!!

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