CS 188 Artificial Intelligence Spring 2007 Lecture 7 CSP II and Adversarial Search 2 6 2007 Srini Narayanan ICSI and UC Berkeley Many slides over the course adapted from Dan Klein Stuart Russell or Andrew Moore Summary Consistency Basic solution DFS backtracking Add a new assignment Check for violations Forward checking Pre filter unassigned domains after every assignment Only remove values which conflict with current assignments Arc consistency We only defined it for binary CSPs Check for impossible values on all pairs of variables prune them Run or not after each assignment before recursing A pre filter not search Limitations of Arc Consistency After running arc consistency Can have one solution left Can have multiple solutions left Can have no solutions left and not know it What went wrong here K Consistency Increasing degrees of consistency 1 Consistency Node Consistency Each single node s domain has a value which meets that node s unary constraints 2 Consistency Arc Consistency For each pair of nodes any consistent assignment to one can be extended to the other K Consistency For each k nodes any consistent assignment to k 1 can be extended to the kth node Higher k more expensive to compute You need to know the k 2 algorithm Strong K Consistency Strong k consistency also k 1 k 2 1 consistent Claim strong n consistency means we can solve without backtracking Why Choose any assignment to any variable Choose a new variable By 2 consistency there is a choice consistent with the first Choose a new variable By 3 consistency there is a choice consistent with the first 2 Lots of middle ground between arc consistency and nconsistency e g path consistency Iterative Algorithms for CSPs Greedy and local methods typically work with complete states i e all variables assigned To apply to CSPs Allow states with unsatisfied constraints Operators reassign variable values Variable selection randomly select any conflicted variable Value selection by min conflicts heuristic Choose value that violates the fewest constraints I e hill climb with h n total number of violated constraints Example 4 Queens States 4 queens in 4 columns 44 256 states Operators move queen in column Goal test no attacks Evaluation h n number of attacks Performance of Min Conflicts Given random initial state can solve n queens in almost constant time for arbitrary n with high probability e g n 10 000 000 The same appears to be true for any randomly generated CSP except in a narrow range of the ratio Example Boolean Satisfiability Given a Boolean expression is it satisfiable Very basic problem in computer science Turns out you can always express in 3 CNF 3 SAT find a satisfying truth assignment Example 3 SAT Variables Domains Constraints Implicitly conjoined all clauses must be satisfied CSPs Queries Types of queries Legal assignment All assignments Possible values of some query variable s given some evidence partial assignments Problem Structure Tasmania and mainland are independent subproblems Identifiable as connected components of constraint graph Suppose each subproblem has c variables out of n total Worst case solution cost is O n c dc linear in n E g n 80 d 2 c 20 280 4 billion years at 10 million nodes sec 4 220 0 4 seconds at 10 million nodes sec Tree Structured CSPs Theorem if the constraint graph has no loops the CSP can be solved in O n d2 time Compare to general CSPs where worst case time is O d n This property also applies to logical and probabilistic reasoning an important example of the relation between syntactic restrictions and the complexity of reasoning Tree Structured CSPs Choose a variable as root order variables from root to leaves such that every node s parent precedes it in the ordering For i n 2 apply RemoveInconsistent Parent Xi Xi For i 1 n assign Xi consistently with Parent Xi Runtime O n d2 why Tree Structured CSPs Why does this work Claim After each node is processed leftward all nodes to the right can be assigned in any way consistent with their parent Proof Induction on position Why doesn t this algorithm work with loops Note we ll see this basic idea again with Bayes nets and call it belief propagation Nearly Tree Structured CSPs Conditioning instantiate a variable prune its neighbors domains Cutset conditioning instantiate in all ways a set of variables such that the remaining constraint graph is a tree Cutset size c gives runtime O dc n c d2 very fast for small c CSP Summary CSPs are a special kind of search problem States defined by values of a fixed set of variables Goal test defined by constraints on variable values Backtracking depth first search with one legal variable assigned per node Variable ordering and value selection heuristics help significantly Forward checking prevents assignments that guarantee later failure Constraint propagation e g arc consistency does additional work to constrain values and detect inconsistencies The constraint graph representation allows analysis of problem structure Tree structured CSPs can be solved in linear time Iterative min conflicts is usually effective in practice Games Motivation Games are a form of multi agent environment What do other agents do and how do they affect our success Cooperative vs competitive multi agent environments Competitive multi agent environments give rise to adversarial search a k a games Why study games Games are fun Historical role in AI Studying games teaches us how to deal with other agents trying to foil our plans Huge state spaces Games are hard Nice clean environment with clear criteria for success Game Playing Axes Deterministic or stochastic One two or more players Perfect information can you see the state Want algorithms for calculating a strategy policy which recommends a move in each state Deterministic Single Player Deterministic single player perfect information Know the rules Know what actions do Know when you win E g Freecell 8 Puzzle Rubik s cube it s just search Slight reinterpretation Each node stores the best outcome it can reach This is the maximal outcome of its children Note that we don t store path sums as before After search can pick move that leads to best node lose win lose Deterministic Two Player E g tic tac toe chess checkers Minimax search A state space search tree Players alternate Each layer or ply consists of a round of moves Choose move to position with highest minimax value best achievable utility against best play Zero sum games One player maximizes result The other minimizes result
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