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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 State Space

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**State Space Solution Path**

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

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**State Space Search Algorithm**

Finds a solution path through a state space State Space

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**The Water Jug Problem Die Hard: With a Vengeance (1995)**

(Bruce Willis, Samuel L. Jackson, Jeremy Irons) The Water Jug Problem

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**A Slight Variation Suppose we have An empty 4 gallon jug**

A source of water A task: put 2 gallons of water in the 4 gallon jug A Slight Variation

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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} Representation

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**Rules if x < 4, fill x : (x,y) (4,y)**

if y < 3, fill y : (x,y) (x,3) if x > 0, empty x : (x,y) (0,y) if y > 0, empty y : (x,y) (x,0) 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) if (x+y) >= 3 and x > 0 : (x,y) (x –(3 – y), 3)) Fill the 3 gallon jug from the 4 gallon jug if (x+y) <= 4 and y > 0 : (x,y) (x+y), 0) Pour the 3 gallon jug into the 4 gallon jug if (x+y) <= 3 and x > 0 : (x,y) (0, x + y) pour the 4 gallon jug into the 3 gallon jug Rules

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**5 & 6 Redux if (x+y) >= 4 and y > 0 : (x,y) (4, y – (4 – x))**

fill the 4 gallon jug from the 3 gallon jug if (x+y) >= 3 and x > 0 : (x,y) (x –(3 – y), 3)) Fill the 3 gallon jug from the 4 gallon jug 4-X 3-Y 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). 5 & 6 Redux

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**if (x+y) <= 4 and y > 0 if (x+y) <= 3 and x > 0**

Pour the 3 gallon jug into the 4 gallon jug: (x,y) (x+y), 0) if (x+y) <= 3 and x > 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) Is there a solution path?

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**Breadth First Search (0,0) 1 2 (0,3) (4,0) (0,3) 6 7 2 (3,0) 6 (4,3)**

(1,3) Breadth First Search etc

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**Depth First (0,0) 1 (4,0) 2 3 (4,3) 7 (0,3) (3,0) 2 (3,3)**

Etc. and without visiting already visited states

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**Backward/Forward Chaining**

Search can proceed From data to goal 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 Backward/Forward Chaining

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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 Examples

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**Reduce the size of the search space**

Object

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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 Use Goal Driven

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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 Use Data Driven

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Basic Problem Solving Search strategy Problem can be solved by searching for a solution. An attempt is to transform initial state of a problem into some.

Basic Problem Solving Search strategy Problem can be solved by searching for a solution. An attempt is to transform initial state of a problem into some.

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