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This article delves into the concept of state space search in artificial intelligence, illustrated through the classic Water Jug Problem. The problem is represented as a graph where nodes represent states of the jugs and arcs reflect the actions taken to transition between these states. By starting from an initial state and applying specific rules for filling, emptying, and pouring between jugs, the goal is to reach a designated state with exactly 2 gallons in the 4-gallon jug. We also examine different search algorithms like Breadth-First and Depth-First Search, highlighting the effectiveness of forward and backward chaining.
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State Space Search Classic AI
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}
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 fill the 4 gallon jug from the 3 gallon jug (x,y) (4, y – (4 – x)) • if (x+y) >= 3 and x > 0 Fill the 3 gallon jug from the 4 gallon jug (x,y) (x –(3 – y), 3)) • if (x+y) <= 4 and y > 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)
Is there a solution path? Initial State: (0,0) Goal State: (2,N)
Breadth First Search (0,3) 1 2 (0,3) (4,0) (0,3) 6 7 2 (3,0) (4,3) (1,3) etc
Depth First (0,0) (4,0) 1 2 3 (4,3) 7 (0,3) (3,0) 2 (3,3) Etc. and without visiting already visited states
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
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 • 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
