Sep 03, 2003
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- Recap psychic search or heuristic search
	- Create the evaluation function f(n) as
	  f(n) = g(n) + h(n)
	- called A* if h(n) has some properties

- What do we do when we don't have a psychic
	- as long as your h(n) does not overestimate h*(n),
	  you are fine! (you get completeness and optimality)
		- AMAZING, ISN'T IT?

- One simple way to ensure that h(n) <= h*(n)
	- make h(n) = 0
		- reduces to BFS/UCS, which is complete and optimal

- A good heuristic is one
	- that is as close to h*(n) as possible (from below)
	- but doesn't take too long to compute
		- after all, the psychic could just be a fast
	          supercomputer that actually does BFS, finds
		  the answer, and comes back and gives you an
		  estimate of the cost!

- Special cases of evaluation functions
	- If g(n) = 0, you are only relying on the psychic
		- "greedy search"; does not guarantee optimality
	- If h(n) = 0, you are only relying on the past
		- reduces to BFS and UCS

- Heuristics for the 8-puzzle problem	
	- h1(n): number of misplaced tiles
	- h2(n): Manhattan distance

- Both share the essential characteristics of a heuristic
	- i.e., never overestimate path cost to a goal

- How do we get heuristics such as this?
	- an automatic way by "relaxing" problem specs

- Motto: Heuristic cost for original problem
	= Optimal solution cost for a relaxed problem

- Example of relaxation
	- People can fly
		(to compute time taken to travel 	
		from point A to point B)
	- tiles can magically go into their
	  correct position

- Automatic discovery of heuristics: 8-puzzle problem
	- formulate it as:
	  MOVE(x,y,z) <- ON(x,y), FREE(z), NEXTTO(y,z)
	- Tile "x" can be moved from y to z if x is originally
	  on y, z is free, and z is adjacent to y

- Relax it in different ways:
	- MOVE(x,y,z) <- ON(x,y)
	- can move a tile to anywhere
	- leads to h1(n) = #misplaced tiles

- Another
	- MOVE(x,y,z) <- ON(x,y), NEXTTO(y,z)
	- can move a tile to an adjacent position
	- leads to h2(n) = Manhattan distance

- Yet Another
	- MOVE(x,y,z) <- ON(x,y), FREE(z)
	- can swap a tile for the empty position
	- leads to h3(n) = #swaps

- How good are heuristics?
	- h3(n) is better than h2(n) is better than h1(n)
	- all approach but never cross h*(n)
		- some are even sometimes equal to h*(n)
		  	- when?