State Space Search

State Space representation of a problem is a graph

Nodes correspond to problem states Arcs correspond to steps in a solution

process One node corresponds to an initial

state One node corresponds to a goal state

Solution Path

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

Search Algorithm

Finds a solution path through a state space

The Water Jug Problem

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

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}

Rules1. 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

fill the 4 gallon jug from the 3 gallon jug(x,y) (4, y – (4 – x))

6. if (x+y) >= 3 and x > 0Fill the 3 gallon jug from the 4 gallon jug(x,y) (x –(3 – y), 3))

7. if (x+y) <= 4 and y > 0 Pour the 3 gallon jug into the 4 gallon jug: (x,y) (x+y), 0)

8. if (x+y) <= 3 and x > 0pour the 4 gallon jug into the 3 gallon jug: (x,y) (0, x + y)

Is there a solution path?

Initial State: (0,0)Goal State: (2,N)

Breadth First Search

(0,3)

(0,3)

(4,0)

(4,3) (1,3)(3,0)

(0,3)

1

2

2

6 7

etc

Depth First

(3,0)

(3,3)

(0,3)

(4,0)

(4,3)

(0,0)

12

3

7

2

Etc. and without visiting already visited states

Backward/Forward Chaining

Search can proceed1. From data to goal2. From goal to dataEither could result in a successful

search path, but one or the other might require examining more nodes depending on the circumstances

Data to goal is called forward chaining for data driven search

Goal to data is called backward chaining or goal driven search

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

Object

Reduce the size of the search space

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

Use Data Driven If

– Most data is given at the outset– Only a few ways to use the facts– Difficult to form an initial hypothesis

Water Jug: could go either way, but the presence of only a few rules, data existing in only a handful of states, and the difficulty of forming a hypothesis suggests data driven

Said another way: initial data constrains search