1CS 188: Artificial IntelligenceFall 2007Lecture 5: CSPs II9/11/2007Dan Klein – UC BerkeleyMany slides over the course adapted from either Stuart Russell or Andrew MooreToday More CSPs Applications Tree Algorithms Cutset Conditioning Local Search2Reminder: CSPs CSPs: Variables Domains Constraints Implicit (provide code to compute) Explicit (provide a subset of the possible tuples) Unary Constraints Binary Constraints N-ary ConstraintsExample: 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 assignment3Example: 3-SAT Variables: Domains: Constraints:Implicitly conjoined (all clauses must be satisfied)CSPs: Queries Types of queries: Legal assignment (last class) All assignments Possible values of some query variable(s) given some evidence (partial assignments)4Example: N-Queens Formulation 3: Variables: Domains: Constraints:Reminder: 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 immediately 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 after each assignment, but before recursing A pre-filter, not search!5Arc Consistency [DEMO]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?6K-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 kthnode. 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 n-consistency! (e.g. path consistency)7Problem 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/secTree-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(dn) This property also applies to logical and probabilistic reasoning: an important example of the relation between syntactic restrictions and the complexity of reasoning.8Tree-Structured CSPs Choose a variable as root, ordervariables from root to leaves suchthat every node’s parent precedesit in the ordering For i = n : 2, apply RemoveInconsistent(Parent(Xi),Xi) For i = 1 : n, assign Xiconsistently 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 message passing)9Nearly Tree-Structured CSPs Conditioning: instantiate a variable, prune its neighbors' domains Cutset conditioning: instantiate (in all ways) a set of variables suchthat the remaining constraint graph is a tree Cutset size c gives runtime O( (dc) (n-c) d2 ), very fast for small cTypes of Problems Planning problems: We want a path to a solution (examples?) Usually want an optimal path Incremental formulations Identification problems: We actually just want to know what the goal is (examples?) Usually want an optimal goal Complete-state formulations Iterative improvement algorithms10Iterative Algorithms for CSPs Hill-climbing, simulated annealing 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., hillclimb with h(n) = total number of violated constraintsExample: 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 attacks11Performance 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 ratioCSP 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 practice12Local Search Methods Queue-based algorithms keep fallback options (backtracking) Local search: improve what you have until you can’t make it better Generally much more efficient (but incomplete)Hill Climbing Simple,
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