CS188 Spring 2010 Section 2: CSP s

Transcription

1 CS188 Spring 2010 Section 2: CSP s 1 Campus Layout ( ) You are asked to determine the layout of a new, small college. The campus will have three structures: an administration building (A), a bus stop (B), a classroom (C), and a dormitory (D). Each building must be placed somewhere on the grid below. The following constraints must be satisfied: (i) The bus stop (B) must be adjacent to the road. (ii) The administration building (A) and the classroom (C) must both be adjacent to the bus stop (B). (iii) The classroom (C) must be adjacent to the dormitory (D). (iv) The administration building (A) must not be adjacent to the dormitory (D). (v) The administration building (A) must not be on a hill. (vi) The dormitory (D) must be on a hill or adjacent to the road. (vii) All buildings must be in different grid squares. Here, adjacent means that the buildings must share a grid edge, not just a corner. (a) Let the variables A, B, C, and D each range over the set of locations on the grid. Express the description above as unary and binary constraints over these variables. Implicit statements in precise but evocative notation such as different(x,y) are acceptable. 1. adjacent(b, road) 2. adjacent(a,b) 3. adjacent(c,b) 4. adjacent(c,d) 5. NOT adjacent(a,d) 6. NOT on(a, hill) 7. on(d, hill) OR adjacent(d, road) 8. different(x, Y ) if X Y for X, Y {A, B, C, D} 1

2 (b) Cross out eliminated values to show the domains of all variables after unary constraints and arc consistency have been applied (but no variables have been assigned). A [ (1,3) (2,2) (2,3) ] B [ (1,3) (2,3) ] C [ (1,2) (1,3) (2,2) (2,3) ] D [ (1,2) (1,3) (2,1) ] (c) Cross out eliminated values to show the domains of the variables after B = (1, 3) has been assigned and arc consistency has been rerun. A [ ] B [ ] C [ ] D [ ] (d) Give a solution for this CSP or state that none exist. A=3 B=6 C=5 D=4 2

3 2 Trains ( ) A train scheduler must decide when trains A, B and C should depart. Once a train departs, it moves one space along its track each hour (in discrete jumps) until it arrives at its destination platform. Each train can depart at 1, 2 or 3 pm. The scheduler has two restrictions: All trains must leave at different times, and two trains should not both occupy crossing sections of track after any one hour time step is over. Note that train A is two spaces long. Also note that the collision constraint is enforced only at the conclusion of every hour - time is discrete in this problem. Start A Goal A B C B C a) Describe the constraint satisfaction problem that, when solved, will tell the train scheduler when each train should depart. Let the variables A, B and C represent the departure times of the three trains. 1. variables and domains: A, B, C 1, 2, 3 2. different times constraints: A B, B C, A C 3. intersection constraints: A + 1 B, A + 1 C, A + 2 C, B C equivalent intersection constraints: B > A + 1 or A > B, C > A + 2 or A + 1 > C, B C + 1 b) Draw the constraint graph for the CSP you defined. A fully connected graph with A, B and C. A hypergraph connecting all three nodes with one arc is okay. c) After selecting A = 2, cross out all values for B and C eliminated by forward checking. A B C Forward checking eliminates all values from B and C that are not compatible with A = 2. Forward checking does not eliminate B = 1, even though it conflicts with C = 1. That s the job of arc consistency. 3

4 d) Cross out all values eliminated by arc consistency before assigning any variables. A B C C = 3 is eliminated because it is not compatible with A {1, 2, 3}. A = 1 is eliminated because it is not compatible with C {1, 2}. B = 3 is eliminated because it is not compatible with A {2, 3}. B = 2 is eliminated because it is not compatible with C {1, 2}. C = 1 is eliminated because it is not compatible with B {1}. A = 2 is eliminated because it is not compatible with C {2}. e) After selecting A = 2, cross out all values for B and C eliminated by arc consistency. A B C Any answer with the domain of either B or C fully crossed out is acceptable. f) Describe the execution of backtracking search using forward checking and the minimum remaining values (MRV) and least constraining values (LCV) heuristics. Specifically, in what order are the variables assigned and what values do they take? Start by assigning variable A. You may not need to fill all the lines below: (1) variable A is assigned value 3. 3 is the least constraining value. (2) variable B is assigned value 1. B and C both have two values remaining and degree 1 (they constrain exactly one other unassigned variable). B = 1 and C = 2 are least constraining. (3) variable C is assigned value 2. Forward checking leaves only one option. Note: Lines (2) and (3) may be switched. 4

5 3 Search as a CSP Consider the following generic search problem formulation with finitely many states: States: there are d + 2 states: {s s, s g } {s 1,..., s d } Initial state: s s Successor function: Succ(s) generates at most b successors Goal test: s g is the only goal state Step cost: each step has a cost of 1 (a) Suppose an optimal solution exists with cost n. What is the tightest upper bound on n in terms of the quantities defined above? The optimal solution is acyclic, and will not visit any state twice; thus, n d + 1. (b) Suppose we must solve this search problem using BFS, but with limited memory. Specifically, assume we can only store k states during search. Give an upper bound on n for which the search will fit in the available memory (do not worry about off-by-one errors here, but give the tightest bound possible). Note that in the worst case, the distinction between tree and graph search is not important here. b n = k, so n log b (k) (c) Would any other search procedure allow problems with substantially deeper solutions to be solved? Either argue why not, or give a method along with an improved bound on n. DFS tree search will find deeper solutions (if very slowly), and iterative deepening will even find shallowest and therefore optimal solutions, all with the lower memory requirements of DFS. These methods would require O(nb) memory, giving a bound of k/b. It is also acceptable to claim that no graph search method gives a better bound, because all must store a potentially exponential closed list. 5

6 (d) If we knew the exact value of n, we could formulate a CSP whose complete assignment specifies an optimal solution path (X 0, X 1..., X n ) for this search problem. Give binary and/or unary constraints which guarantee that a satisfying assignment is a valid solution to the original search problem. Variables: X 0, X 1,..., X n Domains: Dom(X i ) = {s s, s g } {s 1,..., s d } i {0, 1,..., n} Constraints: X 0 = s s, X i Succ(X i 1 ) i {1,..., n}, X n = s g (e) Assume the branching factor b is much less than d. How can the successor function be used to efficiently enforce the consistency of an arc X i X i 1? (Reminder: Enforcing the consistency of this arc prunes values from the domain of X i, not X i 1.) The consistent values in the domain of X i are Succ(x i 1 ) for values x i 1 in the domain of X i 1. This allows the consistency of an arc to be checked / enforced in O(db) time, rather than O(d 2 ). (f) After reducing the domains of any variables with unary constraints, suppose we then make all arcs X i X i 1 consistent, processed in order from i = 1 to n. Next, we try to assign variables in reverse order, from X n to X 0, using backtracking DFS. Why is this a particularly good variable ordering? Because the tree-structured CSP is directionally arc-consistent, there is always a value for X i 1 that is consistent with the value chosen for X i ; thus, the assignment search will not backtrack. 6