020498 StateSpace Search Administrative Next topic Planning Reading

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 1 CSE 473 Winter 1998

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 (Free. Cell) – 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 2 CSE 473 Winter 1998

Example: The Game of Chicken Him You What is your optimal strategy if: • actions are chosen simultaneously • you get to choose first 3 CSE 473 Winter 1998

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) 4 CSE 473 Winter 1998

MINIMAX search, cutoff depth = 2 1 MAX -1 -2 MIN X X O 1 X MAXO 6 -5=1 5 X O 5 -5=0 X X X O 1 X O 0 1 X MIN OX 2 O X O MIN -1 O 0 X O -1 X 0 O X -2 -1 CSE 473 Winter 1998

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. 6 CSE 473 Winter 1998

Cached Values =10 =-1 MIN MAX =4 =10 MIN MAX 7 =4 =-1 MAX MAX =3 CSE 473 Winter 1998

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! 8 CSE 473 Winter 1998

State-Space Search: Summary • A very abstract characterization of problem solving – non-deterministic graph search • An interesting split between domain-dependent and domainindependent 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 9 CSE 473 Winter 1998

Planning: The “Neutral” Problem Description • Inputs – a set of states S = {s 1, s 2, . . . , sn} – a set of actions A={a 1, a 2, . . . , am} • each action is a partial function ai: S S – a unique initial state si – a goal region G S • Output – a sequence of actions <b 1, b 2, . . . , bk> such that bk(. . . b 3(b 2(b 1(si)). . . ) G 10 CSE 473 Winter 1998

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 11 • goal is a conjunction of subgoals • common solution technique is “divide and conquer” – to solve G = G 1^G 2^. . . , solve the Gi subgoals separately, and merge the solutions CSE 473 Winter 1998

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: – S 0 = { at(TRUCK, A), fuel(HALF), connected(A, B), refuel-at(B) } – everything is false unless explicitly stated true 12 CSE 473 Winter 1998

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” 13 CSE 473 Winter 1998