1. CSE473 Winter 1998 1 02/04/98 State-Space Search Administrative –Next topic: Planning. Reading, Chapter 7, skip 7.3 through 7.5 –Office hours/review after class today, Thursday 2:30 Last time –informed search, satisficing and optimizing (A*) This time –adversarial (game-tree) search –introduction to Planning

2. CSE473 Winter 1998 2 Search in Adversarial Games Non-adversarial game: you make a sequence of moves, and at the end you get a payoff depending on the state you are in –games of perfect information: deterministic moves (FreeCell) –games against nature: you make a move, “nature” changes the world same as perfect information if nature is perfectly predictable, but more generally probabilistic (stochastic next state generator) but, we assume that nature is dispassionate: her choice of move is not meant to minimize your payoff –adversarial games: you make a move, then an opponent makes a move, then both get a payoff (possibly negative) both you and opponent are attempting to maximize an individual payoff function often maximizing one means minimizing the other –zero-sum game perfect information: everybody knows all payoff functions

3. CSE473 Winter 1998 3 Example: The Game of Chicken You Him What is your optimal strategy if: actions are chosen simultaneously you get to choose first

4. CSE473 Winter 1998 4 General Approach to Game Playing by Search Expand the tree some fixed number of moves Apply a heuristic evaluation function to the (incomplete) state Apply MINIMAX to compute the best first move Example: TIC-TAC-TOE –players are MAX (drawing X’s) and MIN (drawing O’s) –e(p) is  if p is a win for MAX -  if p is a win for MIN (number of available rows/columns/diagonals for MAX) - (number of available rows/columns/diagonals) for MIN)

5. CSE473 Winter 1998 5 MINIMAX search, cutoff depth = 2 X XX XXXXX XX XXXXX O OOO O O O OOO OO 6-5=1 5-5=0 10 12 0 0 -2 1 1 MAX MIN MAX

6. CSE473 Winter 1998 6 Early Pruning: The ALPHA-BETA Procedure The previous algorithm (implicitly) –generate the tree –evaluate the leaves –backup to generate the optimal first action Interleaving evaluation with generation means that some paths Cache partial evaluation information at each node –A MAX node has an  value which is the best (greatest) choice so far. It can never decrease. –A MIN node has a  value which is the best (least) choice so far. It can never increase.

7. CSE473 Winter 1998 7 Cached Values MAX MIN MAX  =10  =10  =4  =4 MIN MAX  =-1  =-1  =3  =3

8. CSE473 Winter 1998 8 Two sorts of pruning Search can be discontinued below any MIN node having a  value less than or equal to the  value of any of its MAX node ancestors. Search can be discontinued below any MAX node having an  value greater than or equal to the  value of any of its MIN node ancestors This can have an order-of-magnitude impact on the search –provided you choose the first alternative(s) well!

9. CSE473 Winter 1998 9 State-Space Search: Summary A very abstract characterization of problem solving –non-deterministic graph search An interesting split between domain-dependent and domain- independent aspects of the process –the domain-independent part can be a library Extensions to optimizing, adversarial search, continuous spaces Disadvantages –the “direction” of the search may be wrong (progression versus regression) –the domain-independent components are “black boxes” perhaps state generation, goal recognition could be further automated

10. CSE473 Winter 1998 10 Planning: The “Neutral” Problem Description Inputs –a set of states S = {s 1, s 2,..., s n } –a set of actions A={a 1, a 2,..., a m } each action is a partial function a i : S  S –a unique initial state s i –a goal region G  S Output –a sequence of actions such that b k (... b 3 (b 2 (b 1 (s i ))...)  G

11. CSE473 Winter 1998 11 Planning as Search Search: can easily implement a planner using the standard search code/algorithms But we would like to –have a declarative representation for states and actions ease in specification (move generator, goal checker) could support explanation and learning tasks –exploit the goal better using a regression algorithm we believe fan-out is worse than fan-in –further exploit the nature of the goal goal is a conjunction of subgoals common solution technique is “divide and conquer” –to solve G = G 1 ^G 2 ^..., solve the G i subgoals separately, and merge the solutions

12. CSE473 Winter 1998 12 Planning States and Operators Example: –goal is to be at B and fuel tank full –truck is currently at A and fuel tank half –A and B are connected –you can only refuel at B State: –S0 = { at(TRUCK, A), fuel(HALF), connected(A,B), refuel-at(B) } –everything is false unless explicitly stated true

13. CSE473 Winter 1998 13 States versus State Descriptions A state is a set of formulas that describes a single state of the world –by convention, we include only positive formulas and assume everything else is false We also need to represent sets of states –the goal is to be at B and have half a tank of gas, which describes a set of states –there might be other formulas that describe the world, but we don’t care what state they are A state description is a set of formulas that describes a set of states –both positive and negative formulas are allowed in the set –any formula not mentioned is a “don’t care”