Problem Solving

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Problem Solving

• May 4, 2005
• Brandon Beltz

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Lecture Outline

• Defining problem solving
• goals, problem space, operators
• Solving novel problems
• Types of operators
• brute force, hill climbing, working backward,
  means-end analysis
• Memory and problem solving
• Background knowledge, analogy, functional
  fixedness
• The role of expertise
• Differences between experts and novices
• Becoming an expert
• What can nonexperts tell us about problem
  solving?

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Problem Solving Overview

• In decision making we select one choice from
  multiple choices.
• We must implement that choice through action.
• Sometimes the actions to take are not obvious,
  which can be a problem.
• Hence we encounter problem solving.

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Problem Solving Overview

• Problem When a person wants something but
  doesnt immediately know what actions he/she can
  take to get that something.

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Studying Problem Solving

• Most interesting problems take minutes or more
  to solve.
• Thus, speed (RTs) and accuracy are not usually
  helpful for studying problem solving.
• Verbal protocols more common
• An audio transcription and analysis of subjects
  verbalizations as they solve problems.

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Defining the problem

• Goal End solution to the problem.
• Well-defined problems explicitly specify the
  goal.
• Examples?
• Ill-defined problems only vaguely specify the
  goal.
• Examples?

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Defining the problem

• Problem Space Includes the initial state,
  intermediate states, and goal states of the
  problem.
• Also includes the problem solvers knowledge at
  each of these steps.

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Defining the problem

• Operators The set of legal actions that can be
  performed during problem solving.

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Defining the problem
Initial State
Goal/ end state
Goal/ end state
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Studying Problem Solving The Tower of Hanoi

• Tower of Hanoi puzzle
• Well defined problem within cognitive science.

Image http//www.viterbo.edu/personalpages/faculty
/DWillman/
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Studying Problem Solving The Tower of Hanoi
Puzzle

• Goal Move all three rings from the left peg to
  the right peg
• Operators (rules)
• You can only move one ring at a time
• You cannot put a larger ring on top of a smaller
  ring

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Problem Space Tower of Hanoi
Initial State
Operators
Image http//www.viterbo.edu/personalpages/faculty
/DWillman/
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How do people solve novel problems?

• The key is the selection of operators.
• i.e. What actions do we choose?
• Different types of operator selection
• Brute force search
• Hill climbing
• Working backwards
• Means end analysis

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Brute Force Search

• Exhaustive (serial or parallel) search through
  the problem space.
• Search through all subgoals and operators to find
  the best path to the goal.
• Contrast with
• satisficing
• A form of optimizing?

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Brute Force Search

• Advantage relatively easy to apply
• e.g. crossword puzzle, garden tool _ake
• Disadvantage combinatorial explosion
• The number of states in the problem space
  increase dramatically with moderate increases in
  attributes of the problem.
• e.g. Chess

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Operations in Chess

• Chess attributes
• 2 players
• 64 spaces
• 32 pieces
• Each piece has several moves available each turn
• Number of moves per game

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Operations in Chess

• In the beginning there are a lot of potential
  moves.
• Over time, the number of possible moves goes down
  as pieces are removed from the board.
• There are approximately 1018 potential moves
  during an entire game.
• The problem space is immense- combinatorial
  explosion!

If you thought about each move for 1 second, it
would take 30,000,000,000 years!
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Operations in Chess

• Computer players / artificial intelligence
• Modern computers have the ability to conduct
  brute force searches.
• Human players
• Must rely on strategies and heuristics.

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Hill Climbing Heuristic

• Search for an operator that will take you to a
  state in the problem space that appears to be
  closer to the goal than you are now
• Missionaries and cannibals problem

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Missionaries and Cannibals Problem

• 3 missionaries (M) and 3 cannibals (C)
• All 6 must cross the river in the boat.
• Max 2 people can ride in the boat at a time
• Cannibals can never outnumber missionaries on
  either side of the river.

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Missionaries and Cannibals Solution
1. 2. 3. 4.
5. 6. 7. 8.
9. 10. 11. 12.
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Missionaries and Cannibals Solution

• According to the hillclimbing heuristic, people
  should have extreme difficulty performing step 7
• It involves a step backward downhill!

5. 6. 7. 8.
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Working backward heuristic

• Begin at the goal state of the problem and try to
  work back to the starting state.
• Useful when the goal state is known but the
  initial state is not known.
• e.g. What would I need to do to become mayor?

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Types of operators

• Hill climbing and working backward have a limited
  range of application as most problems require
  moving backward and forward.

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Means-End Analysis

• A formalized problem-solving heuristic that uses
  a set of rules about when to work forward or
  backward.
• Also indicates when and how to set subgoals.

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Means-End Analysis

• Sequence of steps
• Compare the current state with the goal state. If
  no difference, problem solved
• If there is a difference, set a subgoal to solve
  that difference. With more than one difference,
  solve the largest difference
• Select an operator that will solve the difference
  identified in Step 2
• If the operator can be applied, do it. If not,
  set a new subgoal to reach a state to allow the
  application of the subgoal
• Return to Step 1 with the new goal set in Step 4

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Means-End Analysis

• Think about writing a paper for class
• What is the difference between my current state
  (no paper written) and the goal state (paper
  written)?
• I need a paper topic!
• Set a goal to reduce the difference
• What operator reduces the difference?
• A sheet of paper containing my brainstorming
  ideas
• Apply the operator if I can, if not set a
  subgoal.
• I need a piece of paper to write down my ideas!
• What is the difference between my current state
  (need piece of paper to write down ideas) and the
  goal state (paper written)?
• Repeat until goal is accomplished

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Types of operators

• Means-End analysis has been successfully
  implemented on computer with the General Problem
  Solver program.
• However, there are a variety of ways to select
  operators to accomplish goals.
• Different problem types are more conducive for
  different methods of selecting operators.

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The Role of Expertise

• Differences between experts and novices
• Becoming an expert
• What can nonexperts tell us about problem
  solving?

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Differences between experts and novices

• Experts have more knowledge about the domain
• Demonstrated in chess with memory tasks
• Information is organized differently
• In physics problems, novices organized by surface
  features. Experts by underlying physics
  principles
• Experts might be better at selecting operators
• Originally thought to be the case
• Evidence does not support it now many examples
  in the chess expertise literature

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Expertise in Chess

• Chess has objective chess rating system (ELO
  rating) that ranks players by wins, losses,
  quality of opponents, etc.
• Grandmaster 2500
• Master 2200-2499
• Expert 2000-2199
• Class A 1800-1999
• Class B 1600-1799
• The rating system allows one to distinguish
  between chess players expertise.

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Expertise in Chess (Gobet and Simon, 1996)

• Study examining highest ranking chess players
  performance over time
• Garry Kasparov
• Highest ranked player in chess history.
• Grandmaster 2750 rating at the time of the study

Image courtesy of http//en.wikipedia.org/
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Expertise in Chess (Gobet and Simon, 1996)

• On several different occasions Kasparov played
  simultaneous chess matches (from 1985-1992)
• He usually played six opponents at a time.
• His opponents were rated as chess masters (2400
  rating)

Image courtesy http//www.clubedexadrez.com.br/
Basically, the best player in the world played
simultaneously against the best players of
different countries!
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Expertise in Chess (Gobet and Simon, 1996)

• Method of play
• One round lasted 3 minutes
• Kasparov had to make one move in six different
  games (6 moves total). 30 seconds per move
• 6 opponents only had to move once in their
  respective games. 3 minutes per move.
• Rounds continued until winner (Kasparov or
  opponent) was decided in each game.

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Logic of Study Chess(Gobet and Simon, 1996)

• Hypothesis A
• Chess experts can either use extensive look ahead
  strategy and calculate the best move
• Prediction A
• Kasparovs rating will decrease significantly due
  to reduced time to look-ahead and plan.
• Hypothesis B
• Experts rely mostly on memory and past experience
  to recognize the state of the game and choose the
  best move.
• Prediction B
• Kasparovs rating will not be decreased as much
  from the reduced time because he quickly
  recognizes the best move from his extensive chess
  knowledge.

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Results(Gobet and Simon, 1996)

• Kasparovs rating was reduced slightly to 2650
  during the simultaneous games.
• However, he still won the majority of his games
  and his chess skill was still comparable to the
  best chess grandmasters!
• What do these results tell you about expertise?

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Becoming an Expert

• Practice
• Inherent Talent

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Becoming an Expert Practice

• Ericssons (1993) definition distinguishes
  practice from play and performance.
• Subject must be motivated
• Task must be at the appropriate level
• There must be immediate corrective feedback
• Repetition of the same or similar tasks

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Becoming an Expert Practice

• Evidence comes from a variety of sources not
  typically problem solving domains.
• Ten-Year Rule The phenomenon that experts in
  almost all fields are seldom able to compete at
  the very highest levels with less than a decade
  of intense practice.

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Effects of Practice
(Ericsson et al., 1996)
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Becoming an Expert Talent

• Inherent talent
• Some suggest that talents like perfect pitch in
  music can be acquired with practice.
• (Takeuchi Hulse, 1993)
• However, studies with twins suggest that genetic
  component is greater.
• That is identical twins were closer in the degree
  of pitch talent than fraternal twins.
• (Drayna, et al., 2001)

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Becoming an Expert Summary

• Practice and talent interact
• Developmental stages in achieving expertise.

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Developmental stages in achieving expertise
(Bloom, 1985)

• Examined common conditions surrounding
  development of experts in various domains
  (sports, music, etc).
• Parents expose child to domain under playful
  conditions child shows promise.
• Parents arrange for instruction from expert who
  works well with children. Practice emphasized!
• Parents show a great deal of enthusiasm and
  provide teachers of increasing expertise as the
  child ages.
• Parents make decision to commit teenager to
  activity full time in order to get the best
  instruction.
• Usually teenager leaves home (college,
  professional academy, etc.)

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What can nonexperts tell us about problem solving?

• Working Memory Capacity
• High correlations found between working memory
  capacity and problem solving.
• In means-end analysis, one must remember several
  things at a time.
• What steps have I done, what step do I do next,
  how do I do the next step, etc.
• Verbal influences on working memory and problem
  solving
• Instructions which reduce working memory load aid
  problem solving and vice versa.
• (Gilhooly, et al., 1993 Barrouillet, 1996)

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What can nonexperts tell us about problem solving?

• Setting Subgoals
• Using subgoals can break peoples attempts to use
  memorized solutions steps.
• e.g. functional fixedness
• Comparing Problems
• Transfer to new problems can be effective if
  subjects see the deep structure of a problem.
• e.g. analogies
• However, this is difficult to teach as subjects
  tend not to transfer unless they see an explicit
  reason to do so.

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End