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Exam on Thursday

Exam on Thursday. Review session Tuesday, 7pm, DH 2210 A very brief outline of the material will be shown and you will be given time to ask questions about the material Review notes modified this morning. Problem Solving. What do we have so far?. Basic biology of the nervous system

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Exam on Thursday

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  1. Exam on Thursday • Review session • Tuesday, 7pm, DH 2210 • A very brief outline of the material will be shown and you will be given time to ask questions about the material • Review notes modified this morning

  2. Problem Solving

  3. What do we have so far? • Basic biology of the nervous system • Motivations • Senses • Learning • Perception • Memory • Thinking and mental representations

  4. What do we have so far? • All of these topics give a basic sense of the structure and operation of our mind • General architecture of mind • What kinds of tasks does our mind engage in? • Language • Problem Solving • Decision Making • Others

  5. What are problems? • Everyday experiences • How to get to the airport? • How to study for a quiz, complete a paper, and finish a lab before recitation? • Domain specific problems • Physics or math problems • Puzzles/games • Crossword, anagrams, chess

  6. General Questions • Why study problem solving? • Studying a task will help us understand the system performing the task • Its something that everyone engages in daily • What can our findings about the mind tell us about problem solving? • Mental representations • Memory constraints • Perception

  7. Early findings • Zeigarnik effect, 1927 • Participants were given a set of problems to solve • On some problems, they were interrupted before they could finish the problem • Participants were given a surprise recall test • They remembered many more of the interrupted problems than the uninterrupted ones

  8. Early Findings • Luchins water jug experiment, 1942 • Participants were given a series of water jug problems • Example: You have three jugs, A holds 21 quarts, B holds 127, C holds 3. Your job is to obtain exactly 100 quarts from a well • Solution is B – A – 2C • Participants solved a series of these problems all having the same solution

  9. Early Findings • Luchins water jug experiment, 1942 • New problem: Given 23, 49, and 3 quart jugs. Goal is to get 20 quarts. • Given 28, 76, and 3 quart jugs, obtain 25 quarts • Some failed to solve, others took a very long time • Mental set • People who solved series of problems using one method tended to over apply that method to new similar appearing problems • Even when other methods were easier or where the learned method no longer could solve the problem

  10. Early Findings • Duncker’s candle problem, 1945 • Problem: Find a way to fix a candle to the wall and light it without wax dripping on the floor. • Given: Candle, matches, and a bow of thumbtacks • Solution: Empty the box, tack it to the wall, place candle on box • Have to think of the box as something other than a container • People found the problem easier to solve if the box was empty with the tacks given separately

  11. Early Findings • Functional Fixedness • Inability to realize that something known to have a particular use may also be used for performing new functions • But is this really a bad thing? • We learn and generalize from our experience in order to be more efficient in most cases • Is it really a good idea to sit around trying to figure out how many potential uses a pair of nail clippers has? • How often do mental sets and functional fixedness save time and computation?

  12. General Problem Characteristics • What characteristics do all problems share? • Start with an initial situation • Want to end up in some kind of goal situation • There are ways to transform the current situation into the goal situation • Can we have a general theory of problem solving?

  13. General Theory of Problem Solving • Newell & Simon proposed a general theory in 1972 in their book Human Problem Solving • They studied a number of problem solving tasks • Proving logic theroems • Chess • Cryptarithmetic DONALD D=5 + GERALD ROBERT

  14. General Theory of Problem Solving • Verbal Protocols • Record people as they think aloud during a problem solving task • Computational simulation • Write computer programs that simulate how people are doing the task • Yields detailed theories of task performance that make specific predictions

  15. Initial Goal o1 …………………. Initial o2 Goal Initial General Theory of Problem Solving • Problem spaces • Initial state • Goal state(s) • Operators that transform one state into another

  16. 1 2 3 An Example • Tower of Hanoi • Given a puzzle with three pegs and three discs • Discs start on Peg 1 as shown below, and your goal is to move them all to peg 3 • You can only move one at a time • You can never place a larger disc on a smaller disc

  17. 1 2 3 An Example • Tower of Hanoi problem space • Initial condition: three discs on peg 1 • Goal: three discs on peg 3 • Operators: Move a disc following the problem constraints

  18. Tower of Hanoi Taken from Zang & Norman, 1994

  19. Another example • Missionaries and cannibals problem • Six travelers must cross a river in one boat • Only two people can fit in the boat at a time • Three of them are missionaries and three are cannibals • The number of cannibals on either shore of the river can not exceed the number of missionaries

  20. ProblemSpace Taken from Jeffries et al., 1977

  21. Operators • How do we choose which operators to apply given the current state of the problem? • Algorithm • Series of steps that guarantee an answer within a certain amount of time • Heuristic • General rule of thumb that usually leads to a solution

  22. Algorithm Examples • Columnar algorithm for addition • Add the ones column • Carry if necessary • Add the next column, etc. • People don’t have a simple algorithm for solving most problems 4 6 2+ 2 34 8 5

  23. Heuristics • Hill climbing • Just use the operator which moves you closer to the goal no matter what • What about problems where you have to first move away from the goal in order to get to it? • Fractionation and Subgoaling • Break the problem into a series or hierarchy of smaller problems

  24. Heuristics • Working Backwards from the goal • Works well if there are fewer branchings going from the goal to the initial state • Only works if you can reverse the operators

  25. Heuristics • Means-ends analysis • Always choose an operator that reduces the difference between your current state and the goal state • Tests for their applicability of the operator on the current problem state • Adopts subgoals if there is no move that will take you to the goal in one step • Must have a difference-operator table or its equivalent • Tells you what operator(s) to use given the current difference between the state of the problem and the goal

  26. Simple Example • Difference-operator table Operators Differences

  27. First AI programs • Newell & Simon • Logic Theorist (LT) • LT completed proofs for a number of logic theorems • General Problem Solver (GPS) • GPS incorporated means-ends analysis, capable of solving a number of problems • Planning problems • Cryptarithmetic • Logic proofs

  28. Limitations of GPS • What about problems where there is no explicit test for a goal state? • Well-defined problems have a clearly defined goal state • Ill-defined problems don’t have a clearly defined goal state • GPS and other AI programs work only on well-defined problems

  29. Examples of ill-defined problems • Engineering Design • Architecture • Painting • Sculpture • How to run a business? • A number of other creative or difficult tasks that people engage in

  30. Limits of AI? • Can AI programs be applied to ill-defined problems? • AARON • Program created by Harold Cohen • Produces paintings using a number of heuristics and general conceptions of aesthtics

  31. Art by AARON

  32. What makes problems hard? • Large problem spaces are usually harder to search than small ones • Compare playing tic-tac-toe to chess • What factors from our architecture of mind play a role in determining how hard a problem is? • Memory constraints • Memory contents • Types of mental representations we use

  33. 1 2 3 Memory constraints • Kotovsky, Hayes, & Simon, 1985 • Work on isomorphs of the Tower of Hanoi • An isomorph of a problem is one in which the structure of the problem space is the same but the appearance of the problem is different • Remember the Tower of Hanoi?

  34. Isomorphs Taken from Kotovsky, Hayes, & Simon, 1985

  35. Isomorphs Taken from Kotovsky, Hayes, & Simon, 1985

  36. Results of Isomorphs Adapted from Kotovsky, Hayes, & Simon, 1985

  37. Memory constraints • In the original Tower of Hanoi and in the condition with monster models there was an external memory aid • Change problems are harder than move problems • Takes more processing to assess whether a change is valid than it does for a move • Spatial proximity of the information • Working with unchanging discs (stable representation) vs. changing discs

  38. Contents of Memory • Does the contents of memory influence how easy a problem is? • Knowledge rich problems • Require domain knowledge to answer, physics problems • Knowledge lean problems • Can use a general problem solving method to solve, don’t need a lot of domain knowledge

  39. Expertise • Physics (Simon et al., 1980) • Physics experts approach physics problems differently than do novices • Chess (Chase & Simon, 1973) • Given a mid-game chessboard, grandmasters can reconstruct it almost perfectly after studying it for only 5 seconds • Novices can only place 3-5 pieces correctly after the same amount of study • However, if the pieces are randomly placed on the board, novices and experts perform at the same level

  40. Knowledge in Chess • Why do experts and novices perform differently? • Experts have more knowledge and experience • But the organization of this knowledge is crucial • Experts can chunk the chess board into meaningful units that are already in memory • Novices have no such chunking mechanism • Random placement of pieces eliminates this chunking from an expert’s performance

  41. Mental Representations • Insight problems • Insight is a seemingly sudden understanding of a problem or strategy that aids in solving the problem • Sometimes require a change in mental representation before the problem can be solved

  42. Mutilated Checkerboard • Place dominoes on the mutilated checkerboard until it is entirely covered Taken from Kaplan & Simon, 1990

  43. Mutilated Checkerboard • Subjects had difficulty solving this problem • Average of 38 minutes • Requires parity to be part of the representation Taken from Kaplan & Simon, 1990

  44. Learning in Problem Solving • Can knowledge learned on one problem be transferred to another problem? • Sometimes, if people notice a similarity between the source and target problems • How do people map knowledge from a source problem to a target problem • Analogy

  45. Analogy • Classic example (Gick & Holyoak, 1983) • Army problem • Cancer problem • Mapping between the two leads to a solution for the cancer problem

  46. Conclusions • Problem solving is an everyday activity • We can use findings from problem solving to further our understanding of the mind and its processes • We can use our knowledge of the mind’s structure and operation to understand elements of problem solving • What are some methods of problem solving? • Why are some problems harder than others?

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