Problem Solving as State Space Search. Assignments

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1 Problem olving as tate pace earch rian. Williams 6.0- ept th, 00 lides adapted from: 6.0 Tomas Lozano Perez, Russell and Norvig IM rian Williams, Fall 0 ssignments Remember: Problem et #: Java warm up Out last Wednesday, ue this Wednesday, eptember th Reading: Today: olving problems through search [IM] h..- Wednesday: symptotic nalysis Lecture Notes of 6.06J; Recurrences, Lecture Notes of 6.0J. rian Williams, Fall 0

2 Recap - ourse Objectives. Understand the major types of agents and architectures: goal-directed vs utility-based deliberative vs reactive model-based vs model-free. Learn the modeling and algorithmic building blocks for creating agents: Model problem in an appropriate formal representation. pecify, analyze, apply and implement reasoning algorithms to solve the problem formulations. rian Williams, Fall 0 Recap gent rchitectures Mission oals Locate in World Monitor & iagnosis Plan Execute Map Plan Routes Maneuver and Track Functions: Robust, coordinated operation + mobility It egins with tate pace earch! rian Williams, Fall 0

3 Problem olving as tate pace earch Problem Formulation (Modeling) Problem solving as state space search Formal Representation raphs and search trees Reasoning lgorithms epth and breadth-first search rian Williams, Fall 0 Most gent uilding lock Implementations Use earch Robust Operations: ctivity Planning iagnosis Repair cheduling Resource llocation Mobility: Path Planning Localization Map uilding ontrol Trajectory esign rian Williams, Fall 0 6

4 Example: Outpost Logistics Planning rian Williams, Fall 0 7 Image produced for N by John Frassanito and ssociates. an the astronaut get its supplies safely across a Lunar crevasse? Rover stronaut oose rain stronaut + item allowed in the rover. oose alone eats rain alone eats oose Early I: What are the universal problem solving methods? imple Trivial rian Williams, Fall 0 8

5 Problem olving as tate pace earch Formulate oal tate stronaut,, oose & rain below crevasse. Formulate Problem tates stronaut,, oose & rain above or below the crevasse. Operators Move: stronaut drives rover and or 0 items to other side of crevasse. Initial tate stronaut,, oose & rain above crevasse. enerate olution equence of Operators (or tates) Move(goose,astronaut), Move(astronaut),... rian Williams, Fall 0 9 oose rain stronaut stronaut oose rain rain stronaut oose stronaut oose rain oose rain oose stronaut stronaut rain rian Williams, Fall 0 0

6 6 rian Williams, Fall 0 stronaut oose rain rain stronaut oose stronaut oose rain oose stronaut rain stronaut rain oose stronaut oose rain oose rain stronaut stronaut oose rain rain stronaut oose stronaut oose rain oose stronaut rain oose rain stronaut stronaut rain oose stronaut oose rain stronaut oose rain stronaut oose rain rian Williams, Fall 0 stronaut oose rain rain stronaut oose stronaut rain oose oose stronaut rain stronaut oose rain stronaut oose rain rain stronaut oose stronaut oose rain oose stronaut rain oose rain stronaut stronaut oose rain stronaut rain oose stronaut oose rain oose rain stronaut stronaut oose rain stronaut oose rain

7 Formulation Example: 8-Puzzle tart oal tates: Operators: Initial and oal tates: integer location for each tile N?? move empty square up, down, left, right as shown above rian Williams, Fall 0 Languages for Expressing tates and Operators for omplex Tasks waggert & Lovell assemble emergency rig for pollo lithium hydroxide unit. Image source: N. rian Williams, Fall 0 7

8 Example: TRIP Planning Language Initial state: (and (hose a) (clamp b) (hydroxide-unit c) (on-table a) (on-table b) (on-table c) (clear a) (clear b) (clear c) (empty arm)) goal (partial state): (and (connected a b) (connected b c))) Operators precondition: (and (clear hose) (on-table hose) (empty arm)) pickup hose effect: (and (not (clear hose)) (not (on-table hose)) (not (empty arm)) (holding arm hose))) rian Williams, Fall 0 Problem: Find a Route from home to MIT tates? Operators?, Initial and oal tate? 6 Mapuest, Navigation Technologies. ll rights reserved. This content is excluded from our reative ommons license. For more information, see 8

9 Problem: ompensating for Error Online Policy, π(v) e, dictates how to act in all states. Policy π corresponds to a shortest path tree from all vertices to the destination. rian Williams, Fall 09 7 How do we Map Path Planning to tate pace earch? tart position Vehicle translates, but no rotation oal position 8 9

10 . reate onfiguration pace tart position Vehicle translates, but no rotation Idea: Transform to equivalent problem of navigating a point. oal position 9. Map From ontinuous Problem to raph earch: reate Visibility raph tart position oal position 0 0

11 . Map From ontinuous Problem to raph earch: reate Visibility raph tart position oal position tart position. Find hortest Path (e.g., *) oal position

12 Resulting olution tart position oal position Visibility raph is a Kind of Roadmap tart position What are the strengths / weaknesses of roadmaps? What are some other types of roadmaps? oal position

13 Voronoi iagrams Lines equidistant from pace obstacles Roadmaps: pproximate Fixed ell rian Williams, Fall 0 6

14 Roadmaps: pproximate Fixed ell rian Williams, Fall 0 7 Roadmaps: pproximate Variable ell rian Williams, Fall 0 8

15 Roadmaps: Exact ell ecomposition rian Williams, Fall 0 9 How do we handle large state spaces? tart position RRT, T a oal position RRT, T b Rapid exploring Random Trees rian Williams, Fall 0 0

16 Problem olving as tate pace earch Problem Formulation (Modeling) Problem solving as state space search Formal Representation raphs and search trees Reasoning lgorithms epth and breadth-first search rian Williams, Fall 0 Problem Formulation: raph Operator Edge tate Vertex h Head Vertex of Edge irected raph (one-way streets) t Tail Vertex of Edge neighbors (adjacent) Undirected raph (two-way streets) rian Williams, Fall 0 6

17 Problem Formulation: raph In egree () b Out egree () egree () b irected raph (one-way streets) Undirected raph (two-way streets) rian Williams, Fall 0 Problem Formulation: raph trongly connected graph irected path between all vertices. onnected graph Path between all vertices. omplete graph ll vertices are adjacent. b b a d c e e ub graph ubset of vertices edges between vertices in ubset a c d irected raph (one-way streets) lique complete subgraph (ll vertices are adjacent). Undirected raph (two-way streets) rian Williams, Fall 0 7

18 pecifying a raph: = <V, E> Notation: <a, b, n> an ordered list of elements a, b {a, b, n} an unordered set of distinguished elements. b Vertices V = {a, b, c, d, e} a d c e Edges E = {<a, b>, <a, c>, <b, e>, <c, b>, <c,d>, <e, d>} rian Williams, Fall 0 Examples of raphs oston an Fran Roadmap Wash L allas Planning ctions (graph of possible states of the world) Put on Put on Put on Put on Put on rian Williams, Fall 0 6 8

19 Formalizing tate pace earch Input: search problem = <g,, > where graph g = <V, E>, start vertex in V, and goal vertex in V. Output: simple path P = <, v, > in g from to. rian Williams, Fall 0 7 imple Paths of raph g = <V, E> <,,, > start end (directed) path P of graph g is a sequence of vertices <v, v n > in V such that each successive pair <v i,v i+ > is a (directed) edge in E simple path is a path that has no cycles. cycle is a subpath where start = end (i.e., repeated vertices). rian Williams, Fall 0 8 9

20 Problem olver earches through all imple Paths rian Williams, Fall 0 9 earch Tree enotes ll imple Paths Enumeration is: omplete ystematic ound rian Williams, Fall 0 0 0

21 think of a tree as a family tree earch Trees tree T is a directed graph, such that there exists exactly one undirected path between any pair of vertices. In degree of each vertex is rian Williams, Fall 0 earch Trees Node (Vertex) Root ranch (Edge) Leaf think of a tree as a family tree rian Williams, Fall 0

22 earch Trees Parent iblings hild think of a tree as a family tree rian Williams, Fall 0 earch Trees ncestors think of a tree as a family tree rian Williams, Fall 0

23 earch Trees escendants think of a tree as a family tree rian Williams, Fall 0 Problem olving as tate pace earch Problem Formulation (Modeling) Problem solving as state space search Formal Representation raphs and search trees Reasoning lgorithms epth and breadth-first search rian Williams, Fall 0 6

24 lasses of earch lind epth-first ystematic exploration of whole tree (uninformed) readth-first until the goal is found. Iterative-eepening Optimal * Use path length measure. Find (informed) ranch&ound shortest path. Heuristic Hill-limbing Use heuristic measure of goodness (informed) est-first of a node. eam rian Williams, Fall 0 7 epth First earch (F) Local Rule: fter visiting node Visit its children before its siblings Visit its children left to right rian Williams, Fall 0 8

25 readth First earch (F) Local Rule: fter visiting node Visit its siblings, before its children Visit its children left to right rian Williams, Fall 0 9 Elements of lgorithm esign lgorithm escription: (Today) stylized pseudo code, sufficient to analyze and implement the algorithm (implementation next Wednesday). lgorithm nalysis: (Wednesday & Monday) Time complexity: how long does it take to find a solution? pace complexity: how much memory does it need to perform search? oundness: when a solution is returned, is it guaranteed to be correct? ompleteness: is the algorithm guaranteed to find a solution when there is one? rian Williams, Fall 0 0

26 Problem olving as tate pace earch Problem Formulation (Modeling) Formal Representation Reasoning lgorithms generic search algorithm description epth-first search example Handling cycles readth-first search example rian Williams, Fall 0 earch tates: olve <g = <V, E>,, > using tate pace earch ll simple paths <, v> in g starting at Initial tate: <> Operator: Extend a path <, v> to <, v, u> for each <v, u> in E oal: call u a child of v simple path <,, > in g rian Williams, Fall 0 6

27 olve <g = <V, E>,, > using tate pace earch How do we maintain the search state? n ordering on partial paths yet to be expanded (called a queue ). How do we perform search? Repeatedly:. elect next partial path from.. Expand it.. dd expansions to. Terminate when goal is found. rian Williams, Fall 0 imple earch lgorithm: Preliminaries partial path from to is listed in reverse order, e.g., <,, > The head of a partial path is its most recent visited node, e.g.,. The is a list of partial paths, e.g. (<,, >, <,, > >. rian Williams, Fall 0 7

28 imple earch lgorithm Let be a list of partial paths, be the tart node and be the oal node.. Initialize with partial path <>. If is empty, fail. Else, pick a partial path N from. If head(n) =, return N (goal reached!). Else: a) Remove N from b) Find all children of head(n) and create a one-step extension of N to each child c) dd all extended paths to d) o to step rian Williams, Fall 0 Problem olving as tate pace earch Problem Formulation (Modeling) Formal Representation Reasoning lgorithms generic search algorithm description epth-first search example Handling cycles readth-first search example rian Williams, Fall 0 6 8

29 epth First earch (F) Idea: fter visiting node Visit its children left to right (or top to bottom) Visit its children before its siblings ssume we remove the first element of, Where to do we add the path extensions? rian Williams, Fall 0 7 imple earch lgorithm Let be a list of partial paths, be the start node and be the oal node.. Initialize with partial path <>. If is empty, fail. Else, pick a partial path N from. If head(n) =, return N (goal reached!). Else: a) Remove N from b) Find all children of head(n) and create a one-step extension of N to each child c) dd all extended paths to d) o to step rian Williams, Fall 0 8 9

30 epth-first Pick first element of ; dd path extensions to front of () rian Williams, Fall 0 9 imple earch lgorithm Let be a list of partial paths, be the tart node and be the oal node.. Initialize with partial path <>. If is empty, fail. Else, pick a partial path N from. If head(n) =, return N (goal reached!). Else: a) Remove N from b) Find all children of head(n) and create a one-step extension of N to each child c) dd all extended paths to d) o to step rian Williams, Fall

31 epth-first Pick first element of ; dd path extensions to front of () rian Williams, Fall 0 6 epth-first Pick first element of ; dd path extensions to front of () ( ) dded paths in blue rian Williams, Fall 0 6

32 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) dded paths in blue rian Williams, Fall 0 6 imple earch lgorithm Let be a list of partial paths, Let be the start node and Let be the oal node.. Initialize with partial path <>. If is empty, fail. Else, pick a partial path N from. If head(n) =, return N (goal reached!). Else: a) Remove N from b) Find all children of head(n) and create a one-step extension of N to each child c) dd all extended paths to d) o to step rian Williams, Fall 0 6

33 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) dded paths in blue rian Williams, Fall 0 6 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) dded paths in blue rian Williams, Fall 0 66

34 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) dded paths in blue rian Williams, Fall 0 67 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) dded paths in blue rian Williams, Fall 0 68

35 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) dded paths in blue rian Williams, Fall 0 69 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) rian Williams, Fall 0 70

36 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) rian Williams, Fall 0 7 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 6 ( )( ) rian Williams, Fall 0 7 6

37 epth-first Pick first element of ; dd path extensions to front of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 6 ( )( ) rian Williams, Fall 0 7 imple earch lgorithm Let be a list of partial paths, Let be the start node and Let be the oal node.. Initialize with partial path <>. If is empty, fail. Else, pick a partial path N from. If head(n) =, return N (goal reached!). Else: a) Remove N from b) Find all children of head(n) and create a one-step extension of N to each child c) dd all extended paths to d) o to step rian Williams, Fall 0 7 7

38 Problem olving as tate pace earch Problem Formulation (Modeling) Formal Representation Reasoning lgorithms generic search algorithm description epth-first search example Handling cycles readth-first search example rian Williams, Fall 0 7 Issue: tarting at and moving top to bottom, will depth-first search ever reach? rian Williams, Fall

39 epth-first Effort can be wasted in more mild cases () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) 6 ( )( ) visited multiple times Multiple paths to, & How much wasted effort can be incurred in the worst case? rian Williams, Fall 0 77 How o We void Repeat Visits? Idea: Keep track of nodes already visited. o not place expanded path on if head is a visited node. oes this maintain correctness? ny goal reachable from a node that was visited a second time would be reachable from that node the first time. oes this always improve efficiency? Visits only a subset of the original paths, such that each node appears at most once at the head of a visited path. rian Williams, Fall

40 How o We Modify The imple earch lgorithm? Let be a list of partial paths, Let be the tart node and Let be the oal node.. Initialize with partial path <> as only entry;. If is empty, fail. Else, pick some partial path N from. If head(n) =, return N (goal reached!). Else a) Remove N from b) Find all children of head(n) and create a one-step extension of N to each child c) dd to all the extended paths d) o to step rian Williams, Fall 0 79 imple earch lgorithm Let be a list of partial paths, Let be the start node and Let be the oal node.. Initialize with partial path <> as only entry; set Visited = {}. If is empty, fail. Else, pick some partial path N from. If head(n) =, return N (goal reached!). Else a) Remove N from b) Find all children of head(n) not in Visited and create a one-step extension of N to each child c) dd to all the extended paths d) dd children of head(n) to Visited e) o to step rian Williams, Fall

41 Testing for the oal This algorithm stops (in step ) when head(n) =. We could have performed this test in step 6 as each extended path is added to. This would catch termination earlier and be perfectly correct for all the searches we have covered so far. However, performing the test in step 6 will be incorrect for the optimal search algorithms that we look at later. We have chosen to leave the test in step to maintain uniformity with these future searches. rian Williams, Fall 0 8 Problem olving as tate pace earch Problem Formulation (modeling) Formal Representation Reasoning lgorithms generic search algorithm description epth-first search example Handling cycles readth-first search example rian Williams, Fall 0 8

42 readth First earch (F) Idea: fter visiting node Visit its children left to right Visit its siblings, before its children ssume we remove the first element of, Where to do we add the path extensions? rian Williams, Fall 0 8 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () 6 rian Williams, Fall 0 8

43 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () 6 rian Williams, Fall 0 8 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, 6 rian Williams, Fall 0 86

44 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, 6 rian Williams, Fall 0 87 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, ( ) ( ) ( ),,,, 6 rian Williams, Fall 0 88

45 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, ( ) ( ) ( ),,,, 6 rian Williams, Fall 0 89 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, ( ) ( ) ( ),,,, ( ) ( ) ( )*,,,,, 6 * We could stop here, when the first path to the goal is generated. rian Williams, Fall 0 90

46 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, ( ) ( ) ( ),,,, ( ) ( ) ( )*,,,,, 6 * We could stop here, when the first path to the goal is generated. rian Williams, Fall 0 9 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, ( ) ( ) ( ),,,, ( ) ( ) ( )*,,,,, ( ) ( ),,,,, 6 rian Williams, Fall 0 9 6

47 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, ( ) ( ) ( ),,,, ( ) ( ) ( )*,,,,, 6 ( ) ( ),,,,, 6 ( ),,,,, rian Williams, Fall 0 9 readth-first with Visited List Pick first element of ; dd path extensions to end of Visited () ( ) ( ),, ( ) ( ) ( ),,,, ( ) ( ) ( )*,,,,, 6 ( ) ( ),,,,, 6 ( ),,,,, rian Williams, Fall 0 9 7

48 epth-first with Visited List Pick first element of ; dd path extensions to front of Visited () ( ) ( ),, ( ) ( ) ( ),,,, ( ) ( ),,,, ( ) ( ),,,,, rian Williams, Fall 0 9 For each search type, where do we place the children on the queue? epth First earch (F) epth-first: readth First earch (F) dd path extensions to front of Pick first element of readth-first: dd path extensions to back of Pick first element of rian Williams, Fall

49 What You hould Know Most problem solving tasks may be formulated as state space search. tate space search is formalized using graphs, simple paths, search trees, and pseudo code. epth-first and breadth-first search are framed, among others, as instances of a generic search strategy. ycle detection is required to achieve efficiency and completeness. rian Williams, Fall 0 97 ppendix rian Williams, Fall

50 readth-first (without Visited list) Pick first element of ; dd path extensions to end of () 6 7 rian Williams, Fall 0 99 readth-first (without Visited list) Pick first element of ; dd path extensions to end of () ( ) ( ) 6 7 dded paths in blue rian Williams, Fall

51 readth-first (without Visited list) Pick first element of ; dd path extensions to end of () ( ) ( ) ( ) ( ) ( ) 6 7 dded paths in blue rian Williams, Fall 0 0 readth-first (without Visited list) Pick first element of ; dd path extensions to end of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* 6 7 dded paths in blue rian Williams, Fall 0 0 Revisited nodes in pink * We could have stopped here, when the first path to the goal was generated.

52 readth-first (without Visited list) Pick first element of ; dd path extensions to end of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* ( ) ( ) ( ) 6 7 rian Williams, Fall 0 0 readth-first (without Visited list) Pick first element of ; dd path extensions to end of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* 6 ( ) ( ) ( ) 6 ( ) ( ) ( ) ( ) 7 rian Williams, Fall 0 0

53 readth-first (without Visited list) Pick first element of ; dd path extensions to end of () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )* 6 7 ( ) ( ) ( ) 6 ( ) ( ) ( ) ( ) 7 ( ) ( ) ( )( )( ) rian Williams, Fall 0 0

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