The wolf sheep cabbage problem. Search. Terminology. Solution. Finite state acceptor / state space

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1 Search The wolf sheep cabbage problem What is search? Terminology: search space, strategy Modelling Uninformed search (not intelligent ) Breadth first Depth first, some variations omplexity space and time Heuristic search Terminology You are on the bank of a river with a boat, a cabbage, a sheep, and a wolf. Your task is to get everything to the other side. 1. only you can handle the boat. only space for you and one more item 3. wolf eats sheep, sheep eats cabbage not allowed State = YSW Graph Node rc, edge directed edge Path, cycle, connected Directed Graph, cyclic graph, DG Tree: root, leaf Family: parent, child, ancestor, descendant Finite state acceptor / state space S the set of states s 0 initial state G set of goal states subset of S F (flow) transition function. Variations: F is a subset of S x S S x Name x S each move has a name. sequence of moves has a sequence of names (word). The moves can be input (in Luger, Name is called I) or output. S x S x ost each move has a cost. Find not just any solution, but the cheapest solution. S x Name x S x ost Solution sequence of states s 0, s 1,..., s n : s n in G, and (s i,s i+1 ) in F (0 i < n). sequence of moves (word)a 1,..., a n : (s i,a i+1,s i+1 ) in F (0 i < n) and s n in G. The cost of a solution is (usually) the sum of the cost of the moves. 1

2 Modelling What are the states? How are the states represented? What are the moves? How are the moves represented? What search strategy? (oming 3 lectures.) Real problems are not toy problems This is not a working approach: onstruct the search space in memory pply shortest path algorithm search space is the part of the state space that can be reached by the search strategy. The search tree is the part of the search space that is actually searched (can be unfolded as a tree). These are not data structures! Usually only parts of the search tree are present in memory during the search , 15, 39 puzzle 9!/ = !/ = 1 x !/ = 4 x 10 4 Smarter modelling model with cost model with cost Y:1 W: G:5 :10

3 Search What is search? Terminology: search space, strategy Modelling Uninformed search (not intelligent ) Breadth first Depth first, some variations omplexity space and time Heuristic search Search tree Search data structures Open nodes your to do list The order is very important losed nodes set of nodes you are done with May take too much memory to keep all nodes Parent of each node To recreate the solution after the search ost, etc. Search algorithm pattern Repeat Pick the first node N from Open N is a goal node? Done! ompute children of N Remove those that are in Open or losed dd new children of N to Open Put N in losed The solution is the path back from N, using Parent Breadth first search Search the tree level by level Open is a queue Properties Finds the shortest path omplete omplexity B = branching factor Dg = depth of Goal (shortest path to goal) Size of Open = B Dg 3

4 Depth first search Properties Go down first Open is a stack B = branching factor Dt = depth of tree Size of Open = B x Dt Size of losed = B Dt The good news No guarantees: Not complete (if search space is infinite) Not the shortest path To avoid loops To avoid double work D S D Why losed? B G D State space: Is it worth it to used losed? S B G Don t use losed in DFS! If there are loops, use the list of parents Risk of double work, but: Repeat Pick the first node N from Open N is a goal node? Done! ompute children of N Remove those that are in Open or losed dd new children of N to Open Put N in losed requires that you check losed every step! (Hashing might help a lot, but...) Depth first search To summarize Breadth first omplete Finds shortest path to goal Exponential memory Depth first without losed Not complete No guarantee Linear memory an we have the best of both? 4

5 Binding the depth Search Bounded depth first search You have an upper bound d for the shortest solution? Do a depth first search up to depth d: BDFS(d) omplete. Linear memory. Iterative deepening i= 1 Repeat BDFS(i) i= i+1 omplete. Finds shortest path to goal Linear memory. Repeated work is only 1/(B 1) What is search? Terminology: search space, strategy Modelling Uninformed search (not intelligent ) Breadth first Depth first, some variations omplexity space and time Heuristic search Heuristic search Heuristic = (Quantitative) information: good bad pproximate can be wrong sometimes! Domain/problem dependent Understand the problem Uninformed Heuristic Depth first Hill climbing Breadth first Best first search and * ppear intelligent Hill limbing Each state S has a heuristic value h(s). Higher is better! Goal: reach the state with the highest value. N = state Repeat ompute children 1,...,n of N ompute h(1),...h(n) Find maximum value h() for child If h(n) > h() then Stop (you re at a top!) else N = Hill limbing There is no going back Risks to find a local maximum In large search spaces, this can be the only feasible method (learning). Some variations Re from different (random) states Simulated annealing (probabilistically accept bad moves) 5

6 Best first search Each state S has a heuristic value. Lower is better! Open is sorted best first hooses the globally best step (hill climbing the locally best step) 4 Best first search 3 5 Best first search Best first search The heuristic estimate How do we get a good estimate? What is a good estimate? lose to the actual value... Over or underestimate? lgorithm Best first search, using the following f: For a node n, estimate f(n) [distance from to goal via n] = g(n) [length of shortest path to n found so far] + h(n) [heuristic: guessed distance from n to goal]

7 The lamp over the bridge example Y:1 W: G:5 :10 h(n) = if L then sum of YWG else sum of YWG + 4 ompare f(n) [estimated distance from to goal via n] = g(n) [length of shortest path to n found so far] + h(n) [heuristic: guessed distance from n to goal] f*(n) [shortest distance from to goal via n] = g*(n) [length of shortest path to n] + h*(n) [shortest distance from n to goal] lgorithm * lgorithm * Best first search, f(n) = g(n) + h(n), h(n) h*(n) for all nodes n. h is optimistic h=5 S G 3 B h=0 h=3 S : B: G:8 G:9 When we close the goal, we have found the shortest path. We have found the goal G(9) not optimal yet. is closed now. When we close G(8) we know it s optimal. Proof (informal) Suppose the path is not optimal. Let n be the first node on the optimal path that is not closed. The path found to n is optimal: g(n) = g*(n). f(g) = g(g)+h(g) = g(g) length of path to G found f*(n) = g*(n)+h*(n) = g(n)+h*(n) length of optimal path g(n)+h(n) = f(n) n should have been closed before G. Monotonicity (consistency) Between two adjacent nodes, their heuristic value differs not more than their distance. and h(g) = 0. Monotonic along a path, estimates go up Monotonic For every node that is closed, we have the shortest path Monotonic Optimistic

8 Example (not optimistic) Example (not monotonic) S h=1 8 h=4 B h=10 5 h=1 G h=0 Open losed S(1) (1) B(1) S(1) (15) B(1) S(1) (1) G(1) B(1) S(1) (1) (15) B(1) G(1) The high estimate for B blocks the shortest path. S h=1 8 h=4 B 5 h=1 G h=0 h=10 Open S(1) (1) B(1) (15) B(1) B(1) G(0) (13) G(0) G(18) losed S(1) S(1) (1) S(1) (1) (15) S(1) (1) B(1) was closed, but reopened when a shorter path was found. S(1) (1) B(1) (13) G(18)... Example (monotonic) Proofs S h=1 8 h=4 B 5 h=1 G h=0 h= Open S(1) (1) B(13) B(13) (15) (13) G(18) losed S(1) S(1) (1) S(1) (1) B(13) S(1) (1) B(1) G(18)... long a path, estimates go up. n f(n)=g(n)+h(n) d g(m) = g(n) + d h(n) h(m) d m f(m) = g(m)+h(m) f(m) = g(m) + h(m) g(n)+d + h(n) d = f(n) Proofs Proofs For every node that is closed, we have the shortest path. When we close a node n, all nodes with a lower estimate are closed (and since estimates along a path go up) all nodes on the shortest path to n are closed Monotonic Optimistic Take the optimal path from n to G: n=n 1,n,...,n k =G. ompare the sequences 0 = h*(g), h*(n k ),..., h*(n ), h*(n). 0 = h(g), h(n k ),..., h(n ), h(n). The first sequence actual distances has bigger steps than the second heuristic differences h*(n) h(n). 8

9 The heuristic estimate How do we get a good estimate? domain dependent Over or underestimate? underestimate general idea: ignore some constraints What is a good estimate? h(n) = 0? lose to the actual value... Informedness Informedness For all nodes n: h 1 (n) h (n) h*(n) h is more informed than h 1 Result: h expands fewer nodes than h Example: h 1 (n) = # tiles out of place h (n) = distance a les must move = 14 h*(n) = Which nodes are expanded? If h is monotonic: all nodes n with f(n) = g*(n)+h(n) < f(g) and some nodes where f(n)=f(g) If h is higher, fewer nodes are expanded. Trade off: computing smarter h takes more time. Iterative deepening * (ID*) utoff = h(s) Repeat run DFS, expand node n only if f(n) utoff utoff = lowest f value of open nodes Until goal node is expanded 9