L18ProblemsShort - Lecture Outline Issues(a Representing...

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1 Lecture Outline Lecture Outline Issues: (a) Representing problems (b) Methods & common flaws in problem solving (c) Expertise 1. Problems and problem representation Well-structured vs. ill-structured problems Stages in problem solving The importance of problem representation 2. Common flaws in problem solving Analogies Hindrances to forming appropriate representations 3. Problem solving methods Algorithms and heuristics Heuristics: Hill climbing, means-ends analysis, working backward 4. Expertise Very domain specific (chess study) Power law of practice Characteristics of expertise
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2 A Problem consists of some initial state in which a person begins and a goal state that is to be attained, plus a non-obvious way of getting from the first to the second. Initial State Goal State Methods Problem Problem
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3 Types of problems Types of problems Well-structured (Well-defined) Well-structured (Well-defined) Completely specified starting conditions, goal state, and methods for achieving the goal. » Geometry proofs Ill-structured (Ill-defined) Ill-structured (Ill-defined) Some aspects are not completely specified. » Finding the perfect mate. » Choosing a career. » Writing the best novel.
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4 Stages of Problem Solving Stages of Problem Solving Polya (1957) Form a representation Construct a plan Execute plan Checking/Evaluation Reformulate
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5 The price of a notebook is four times that of a The price of a notebook is four times that of a pencil. The pencil costs $.30 less than the pencil. The pencil costs $.30 less than the notebook. What is the price of each? notebook. What is the price of each? Problem Solving Concepts: Initial & Goal states. Intermediate States. Representation of problem. Operators: actions that move between states Problem Space: Whole range of possible states and operators, only some of which will lead to goal state
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6 n = 4p, and p = n - 30 Initial state n = and p = Goal state Substitute for p Divide by 4 Divide by 30 Problem space whole range of possible states and operators, only some of which will lead to the goal state The price of a notebook is four times that of a pencil. The pencil costs $.30 less than the notebook. What is the price of each? Substitute for n Subtract 4p from both sides: p = 4p - 30 -3p = -30 Divide, substitute Operators Intermediate states 40 10
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7 O O O O O O O O O The Nine Dot Problem The Nine Dot Problem There are 9 dots arranged in a square below. Using no more than 4 lines, connect all the dots without lifting your pencil.
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8 Nine Dot Problem Nine Dot Problem Initial & Goal States well defined Operators: Four connected lines Representation: Graphical layout Problem space: all possible lines you can draw O O O O O O O O O
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9 O O O O O O O O O The Nine Dot Problem: The Nine Dot Problem: Thinking Outside the Box Thinking Outside the Box
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10 Problem Representation Problem Representation For many problems, the representation may make it easier or harder to solve.
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