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

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

Problem Solving: Definition A problem exists when you want to get from “ here ” (a knowledge state) to “ there ” (another knowledge state) and the path is not immediately obvious.

4 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

A Partial Problem Typology Well-defined vs. ill-defined problems: Problems where the goal or solution is recognizable--where there is a right answer (ex. a math or physics problem) vs. problems where there is no "right" answer but a range of more or less acceptable answers. Knowledge rich vs. knowledge lean problems: problems whose solution depends on specialized knowledge. Insight vs. non-insight problems--those solved "all of a sudden" vs. those solved more incrementally-- in a step by step fashion. 5

6 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

7 Some Problem Examples Tower of Hanoi Weighing problem Traveling salesman (100 cities = 100! or or each electron, 10 9 operations per sec. would take years!!) but 100,000 cities within 1% in 2 days via heuristic breakup (reduce search!) Missionaries & Cannibals Flashlight: 1, 2, 5, 10 min. walkers to cross bridge 21 link gold necklace/21 day stay Subway Problem Vases (or 3-door)

12 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 Moss et al. (2007) recent RAT results: open goals

13 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

14 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

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16 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

17 Early Findings Duncker’s candle problem

Maier’s two-string problem

19 Early Findings Functional Fixedness Inability to realize that something familiar for a particular use may also be used for 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 pliers has? How often do mental sets and functional fixedness save time and computation?

20 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?

21 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

22 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

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

24 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 123

25 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 123

26 Tower of Hanoi Taken from Zang & Norman, 1994

27 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

Problem Space

30 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

31 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

32 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 (detour problems)? Fractionation and Subgoaling Break the problem into a series or hierarchy of smaller problems

Problem Space: Subgoaling

34 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

35 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. Might have to modify operator if none can be applied in current state

36 First AI programs Newell & Simon (1956) 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

Centrality of Representation Problem space and representation Problem difficulty and representation The interaction of representation and processing limitations (problem isomorphs) 37

Representation: Example Number scrabble

39 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

40 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

41 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

42 Art by AARON

43 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

44 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? 123

45 Isomorphs Taken from Kotovsky, Hayes, & Simon, 1985

46 Isomorphs Taken from Kotovsky, Hayes, & Simon, 1985

Isomorph Difficulty 47

48 Results of Isomorphs Adapted from Kotovsky, Hayes, & Simon, 1985

49 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

Computational Model Tested understanding via a computer model that was: Goal driven, subgoaling, limited memory capable of perfect behavior except for limited working memory To see if we were in right “ballpark” To separate actions of various mechanisms to see which had the most control/influence To be able to experiment with the separate postulated mechanisms 50 From Kushmerick & Kotovsky, 1993)

Model-Human Agreement 51

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Chinese Ring Puzzle 53 From Kotovsky & Simon, 1990

55 Two-Phase Problem Solving: Exploratory & Final Path

Non-conscious problem-solving From Reber & Kotovsky, 1997

Strategy acquisition can be unconscious--

Expertise Hayes on ten year rule

Expertise: What ’ s being Learned in the Ten Years? DeGroot and Chase & Simon ’ s work on chunking and chess Estimates of knowledge base size Ericsson: idea of deliberate practice

Practice Makes Perfect! Power law of practice: T a = cP b + d

66 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

67 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

68 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

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

70 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

71 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

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

73 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 Different types of problems and different contributions to problem difficulty