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State Space Search I Chapter 3 The Basics

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State Space Problem Space is a Graph ◦Nodes: problem states ◦Arcs: steps in a solution process ◦One node corresponds to an initial state ◦One node corresponds to a goal state

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State Space Solution Path An ordered sequence of nodes from the initial state to the goal state

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State Space Search Algorithm Finds a solution path through a state space

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The Water Jug Problem Die Hard: With a Vengeance (1995) Die Hard (Bruce Willis, Samuel L. Jackson, Jeremy Irons)

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A Slight Variation Suppose we have An empty 4 gallon jug An empty 3 gallon jug A source of water A task: put 2 gallons of water in the 4 gallon jug

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Representation State Space Node on the graph is an ordered pair (x,y) ◦X is the contents of the 4 gallon jug ◦Y is the contents of the 3 gallon jug Intitial State: (0,0) Goal State: (2,N) N ε {0, 1, 2, 3}

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Rules 1. if x < 4, fill x : (x,y) (4,y) 2. if y < 3, fill y : (x,y) (x,3) 3. if x > 0, empty x : (x,y) (0,y) 4. if y > 0, empty y : (x,y) (x,0) 5. if (x+y) >= 4 and y > 0 : (x,y) (4, y – (4 – x)) fill the 4 gallon jug from the 3 gallon jug (see next slide) 6. if (x+y) >= 3 and x > 0 : (x,y) (x –(3 – y), 3)) Fill the 3 gallon jug from the 4 gallon jug (see next slide) 7. if (x+y) 0 : (x,y) (x+y), 0) Pour the 3 gallon jug into the 4 gallon jug 8. if (x+y) 0 : (x,y) (0, x + y) pour the 4 gallon jug into the 3 gallon jug

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5 & 6 Redux 5. if (x+y) >= 4 and y > 0 : (x,y) (4, y – (4 – x)) fill the 4 gallon jug from the 3 gallon jug 6. if (x+y) >= 3 and x > 0 : (x,y) (x –(3 – y), 3)) Fill the 3 gallon jug from the 4 gallon jug 3-Y 4-X If x is the amount in the 4 gallon, 4-X is the amount necessary to fill it. This amount has to be subtracted from the 3 gallon jug (where the water came from).

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1. if (x+y) 0 Pour the 3 gallon jug into the 4 gallon jug: (x,y) (x+y), 0) 2. if (x+y) 0 pour the 4 gallon jug into the 3 gallon jug: (x,y) (0, x + y)

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Is there a solution path? Initial State: (0,0) Goal State: (2,N)

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Breadth First Search (0,3) (0,0) (4,0) (4,3)(1,3) (3,0) (0,3) etc 6

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Depth First ( 3,0 ) ( 3,3 ) ( 0,3 ) ( 4,0 ) ( 4,3 ) ( 0,0 ) Etc. and without visiting already visited states

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Backward/Forward Chaining Search can proceed 1. From data to goal 2. From goal to data Either could result in a successful search path, but one or the other might require examining more nodes depending on the circumstances

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Data to goal is called forward chaining for data driven search Goal to data is called backward chaining or goal driven search

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Examples Water jug was data driven Grandfather problem was goal driven To make water jug goal driven: ◦Begin at (2,y) ◦Determine how many rules could produce this goal ◦Follow these rules backwards to the start state

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Object Reduce the size of the search space

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Use Goal Driven if ◦Goal is clearly stated ◦Many rules match the given facts For example: the number of rules that conlude a given theorem is much smaller than the number that may be applied to the entire axiom set

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Use Data Driven If ◦Most data is given at the outset ◦Only a few ways to use the facts ◦Difficult to form a goal (i.e., hypothesis) For example: DENDRAL, an expert system that finds molecular structure of organic compounds based on spectrographic data. There are lots of final possibilities, but only a few ways to use the initial data Said another way: initial data constrains search

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