Cognitive Modeling. Tower of Hanoi: Description. Tower of Hanoi: The Task. Lecture 5: Models of Problem Solving. Frank Keller.

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1 Cognitive Modeling Lecture 5: Models of Problem Solving Frank Keller School of Informatics University of Edinburgh January 22, Reading: Cooper (2002:Ch. 4). Frank Keller Cognitive Modeling 1 Frank Keller Cognitive Modeling 2 : Description : The Task There are three pegs. The left peg has a number of disks on it, decreasing in size as they go up the peg. The task is to move the tower of disks from the left peg to the right peg: by moving disks, one at a time, from one peg to another; subject to the constraint that no disk can be placed on top of a smaller disk. The task can be made more difficult by increasing the number of disks (while keeping just three pegs). with three disks: initial state goal state Frank Keller Cognitive Modeling 3 Frank Keller Cognitive Modeling 4

2 Simon (1975) Series of sub-problems: 1 move the largest disk to the right peg; 2 move intermediate-sized disk to the right peg; 3 move the smallest disk to the right peg. Solve sub-problems in order: move largest disk to the right peg: achieve a state where this can be solved in one move (i.e., no other disks on it, no disks on right peg). Now move two-disk tower from left to middle peg: easier version of initial problem; the same principles used to solve it. Problem space for four disk version is 81 states can be done in 15 moves; few achieve this first time. Simon (1975) analyzed possible solution strategies and identified four classes of strategy: problem decomposition strategy (see above); two simpler strategies that move disks (rather than towers), moves triggered by perceptual features of the changing state; a strategy of rote learning. Strategies have different properties in terms of generalization to larger numbers of disks and processing requirements. Frank Keller Cognitive Modeling 5 Frank Keller Cognitive Modeling 6 Anzai and Simon (1979) Selection Without Search One subject, four attempts at five-disk version: analyzed verbal protocols: 1 Initially little sign of strategy, moving disks using simple constraints avoid backtracking, avoid moving same disk twice in a row. 2 By third attempt had a sophisticated recursive strategy, with sub-goals of moving disks of various sizes to various pegs. 3 Final attempt, strategy evolved further, with sub-goals involving moving pyramids of disks. Developed adaptive production system to simulate acquisition and evolution of strategies. The basic procedure: at each stage in the solution process: enumerate the possible moves; evaluate those moves; select the move with the highest evaluation; apply the selected move; if the goal state has not been achieved, repeat the process. This approach can be applied to any well-specified problem (Newell and Simon 1972). Frank Keller Cognitive Modeling 7 Frank Keller Cognitive Modeling 8

3 Selection Without Search Representing the Current State We require: one buffer to hold the current state; one buffer to hold the representation of operators; and one process to manipulate buffer contents. Each disk may be represented as a term: disk(size,peg,position) With this representation, the initial state of the five disk problem might be represented as: disk(30,left,5) disk(40,left,4) disk(50,left,3) disk(60,left,2) disk(70,left,1) Frank Keller Cognitive Modeling 9 Frank Keller Cognitive Modeling 10 Initializing the Current State Operator Proposal Current State should initially contain one term for each disk: In the operator proposal phase, we propose moving the top-most disk on any peg to any other peg: IF not operator(anymove,anystate) is in Possible top disk on peg(size,peg1) other peg(peg1,peg2) THEN add operator(move(size,peg1,peg2),possible) to Possible Some possible operators may violate task constraints. Frank Keller Cognitive Modeling 11 Frank Keller Cognitive Modeling 12

4 Operator Evaluation Evaluation Function In operator evaluation phase, assign numerical evaluations to all possible operators: IF operator(move,possible) is in Possible evaluate operator(move,value) THEN delete operator(move,possible) from Possible add operator(move,value(value)) to Possible Possible operators that violate task constraints receive low evaluations. Other operators receive high evaluations. On first attempt, the subject avoided backtracking and moving the same disk twice. Her evaluation function referred to her previous move. To implement this we record the prior moves. Anzai and Simon (1979) note a further aspect of the subject s first attempt at the problem: whenever she moves the small disk, it is never back to the peg it was on two moves previously. If this additional constraint is adopted, all moves are fully determined by the initial move. Frank Keller Cognitive Modeling 13 Frank Keller Cognitive Modeling 14 Operator Selection Operator Application The selection rule should fire at most once on any cycle: IF not operator(anymove,selected) is in Possible operator(move,value(x)) is in Possible not operator(othermove,value(y)) is in Possible Y is greater than X THEN add operator(move,selected) to Possible Once an operator has been selected, others can be deleted: IF exists operator(move,selected) is in Possible operator(anymove,value(v)) is in Possible THEN delete operator(anymove,value(v)) from Possible Applying an operator involves changing the current state: IF operator(move(size,frompeg,topeg),selected) is in Possible disk(size,frompeg,fromposition) is in Current State get target position(topeg,toposition) THEN delete disk(size,frompeg,fromposition) from Current State add disk(size,topeg,toposition) to Current State clear Possible Frank Keller Cognitive Modeling 15 Frank Keller Cognitive Modeling 16

5 Termination on Task Completion Properties Achieved when no disks on left or middle pegs: goal achieved :- not disk(anysize,left,anyposition) is in Current State not disk(anysize,middle,anyposition) is in Current State should only be possible when the goal has not been achieved: IF not goal achieved not operator(anymove,anystate) is in Possible top disk on peg(size,peg1) other peg(peg1,peg2) THEN add operator(move(size,peg1,peg2),possible) to Possible Properties of selection without search: selection of the first move is random; if the model selects the wrong first move, it can go off into an unproductive region of the problem space; the model will find a solution eventually, but it can be very inefficient. However: selection without search seems to be the initial strategy adopted by subjects when they are first confronted with the problem (Anzai and Simon 1979). Frank Keller Cognitive Modeling 17 Frank Keller Cognitive Modeling 18 Better Model: Goal-Directed Selection Goal Directed Selection On her third attempt to solve the, Anzai and Simon s (1979) subject explicitly mentioned goals. A goal-based strategy involves setting intermediate goals (sub-goals) and moving disks in order to achieve these goals. Sub-goals may be stacked if they cannot be achieved directly. Subgoal: move the largest blocking disk to the third peg. If this is not directly achievable: the process can be repeated, with the creation of a further subgoal to move another (yet smaller) disk to another peg. Processing is guaranteed to terminate when we get to smallest disk. Necessary to maintain a goal stack. When processed, the top-most element is removed or popped from the stack. Frank Keller Cognitive Modeling 19 Frank Keller Cognitive Modeling 20

6 A Revised Diagram Setting Possible should be a stack buffer. Current State could be a propositional buffer (for Prolog terms) or an analogue buffer (for graphical representations). IF the goal stack is empty and there is a difference between the current state and the goal state THEN set a goal to reduce that difference More formally: IF not AnyGoal is in Goal Stack biggest difference(size,targetpeg) THEN send push(move(size,targetpeg)) to Goal Stack This version of the rule concentrates first on the biggest difference between the current and goal states. Frank Keller Cognitive Modeling 21 Frank Keller Cognitive Modeling 22 Determining the Biggest Difference Setting The biggest difference concerns the largest disk that isn t on the right-most peg. Target peg for that disk is the right-most peg: biggest difference(size,right) :- disk(size,peg1,position1) is in Current State Peg1 is distinct from right not disk(othersize,peg2,position2) is in Current State OtherSize is greater than Size Peg is distinct from right If the current goal is not directly achievable, create a subgoal to achieve the current goal s preconditions: IF move(size,targetpeg) is in Goal Stack disk(size,sourcepeg,sourceposition) is in Current State not move is possible(sourcepeg,size,targetpeg) biggest blocking disk(move(sourcepeg, Size,TargetPeg),SmallerDisk) other peg(sourcepeg,targetpeg,peg2) THEN send push(move(smallerdisk,peg2)) to Goal Stack Frank Keller Cognitive Modeling 23 Frank Keller Cognitive Modeling 24

7 Moving Disks and Popping Properties If the current goal is directly achievable, move the disk and pop the goal off the goal stack: IF move(size,targetpeg) is in Goal Stack disk(size,sourcepeg,sourceposition) is in Current State move is possible(sourcepeg,size,targetpeg) get target position(targetpeg,targetposition) THEN delete disk(size,sourcepeg,sourceposition) from Current State add disk(size,targetpeg,targetposition) to Current State send pop to Goal Stack Properties of goal-directed selection: selection of moves is no longer random; selection is guided by the goal of moving the largest disk that is in an incorrect position; if the goal is not directly achievable, it is recursively broken down into subgoals; efficient strategy that avoids unproductive regions of the search space. Goal-directed selection seems to be used by experienced players (evidence for learning; Anzai and Simon 1979). Frank Keller Cognitive Modeling 25 Frank Keller Cognitive Modeling 26 Setting The problem solving strategy in the previous model is known as means-ends analysis. It involves locating the largest difference between current and goal state, and selecting an operator to eliminate this difference. Not specific to the ; can be generalized to apply to a range of problems. Most people seem to have access to this general strategy. The generalized primary goal generation rule: IF not AnyGoal is in Move Stack biggest difference(difference) appropriate operator(eliminate(difference), Operator) THEN send push(operator) to Move Stack appropriate operator/2 is a condition that, given a difference between the current and desired states, returns an operator that will eliminate this difference. Frank Keller Cognitive Modeling 27 Frank Keller Cognitive Modeling 28

8 Applying Unblocked Popping Unblocked IF Operator is in Move Stack not operator is blocked(operator,blockage) operator changes state(operator,before,after) Before is in Current State THEN delete Before from Current State add After to Current State operator is blocked/2 corresponds to a generalized version of (the negation of) move is possible/3. operator changes state/3 specifies how an operator changes state elements. One operator may alter multiple state elements. If an operator is not blocked, the previous rule will apply the operator. It may fire several times. A separate rule is required to pop the operator off of the operator stack: IF Operator is in Move Stack not operator is blocked(operator,blockage) THEN send pop to Move Stack Frank Keller Cognitive Modeling 29 Frank Keller Cognitive Modeling 30 Pushing Switching to MEA If the current operator is blocked, an operator that will eliminate the blockage should be pushed onto the operator stack: IF Operator is in Move Stack operator is blocked(operator,blockage) appropriate operator(eliminate(blockage), SubOperator) THEN send push(suboperator) to Move Stack Why didn t the subject use MEA from the outset? She may have assumed a simpler solution strategy (selection without search) was sufficient. She may have lacked the knowledge of the problem space needed to perform MEA (operators and differences that they can be used to eliminate). Hypothesis: during her first attempt the subject acquired an understanding of how to decompose the problem into subgoals (moving disks of successively smaller sizes to the right peg). Evidence: the explicit mention of subgoals became more common as she gained experience with the task. Frank Keller Cognitive Modeling 31 Frank Keller Cognitive Modeling 32

9 Summary References as a case study for human problem solving ability; experiments show that human subjects gradually develop a way of decomposing the problem into subproblems; problem solving strategies: selection without search: enumerate all solutions, select the best one; goal-directed selection: decompose the problem into subgoals and solve those; generalized means-ends analysis: find the largest difference between the current state and the goal state and select an operator to eliminate it. Anzai, Y. and H. A. Simon The theory of learning by doing. Psychological Review 86: Cooper, Richard P Modelling High-Level Cognitive Processes. Lawrence Erlbaum Associates, Mahwah, NJ. Newell, Alan and Herbert A. Simon Human Problem Solving. Prentice-Hall, Englewood Cliffs, NJ. Simon, H. A The functional equivalence of problem solving skills. Cognitive Psychology 7: Frank Keller Cognitive Modeling 33 Frank Keller Cognitive Modeling 34