Problem solving by search
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1 Problem solving by search based on tuart ussel s slides ( February 27, 2017, E533KUI - Problem solving by search 1
2 Outline Problem-solving agents Problem types Problem formulation Example problems asic search algorithms February 27, 2017, E533KUI - Problem solving by search 2
3 estricted form of general agent: Problem-solving agents function imple-problem-olving-gent( percept) returns an action static: seq, an action sequence, initially empty state, some description of the current world state goal, a goal, initially null problem, a problem formulation state Update-tate(state, percept) if seq is empty then goal Formulate-Goal(state) problem Formulate-Problem(state, goal) seq earch( problem) action ecommendation(seq, state) seq emainder(seq, state) return action Note: this is offline problem solving; solution executed eyes closed. Online problem solving involves acting without complete knowledge. February 27, 2017, E533KUI - Problem solving by search 3
4 Example: omania On holiday in omania; currently in rad. Flight leaves tomorrow from ucharest Formulate goal: be in ucharest Formulate problem: states: various cities actions: drive between cities Find solution: sequence of cities, e.g., rad, ibiu, Fagaras, ucharest February 27, 2017, E533KUI - Problem solving by search 4
5 71 Oradea Example: omania Neamt Zerind 75 rad Timisoara Dobreta 151 ibiu 99 Fagaras 80 imnicu Vilcea ugoj Mehadia 120 Pitesti ucharest raiova 90 Giurgiu 87 Iasi Urziceni Vaslui Hirsova 86 Eforie February 27, 2017, E533KUI - Problem solving by search 5
6 Problem types Deterministic, fully observable = single-state problem gent knows exactly which state it will be in; solution is a sequence Non-observable = conformant problem gent may have no idea where it is; solution (if any) is a sequence Nondeterministic and/or partially observable = contingency problem percepts provide new information about current state solution is a contingent plan or a policy often interleave search, execution Unknown state space = exploration problem ( online ) February 27, 2017, E533KUI - Problem solving by search 6
7 Example: vacuum world ingle-state, start in #5. olution?? February 27, 2017, E533KUI - Problem solving by search 7
8 Example: vacuum world ingle-state, start in #5. olution?? [ight, uck] onformant, start in {1,2,3,4,5,6,7,8} e.g., ight goes to {2,4,6,8}. olution?? February 27, 2017, E533KUI - Problem solving by search 8
9 Example: vacuum world ingle-state, start in #5. olution?? [ight, uck] onformant, start in {1,2,3,4,5,6,7,8} e.g., ight goes to {2,4,6,8}. olution?? [ight, uck, ef t, uck] ontingency, start in #5 Murphy s aw: uck can dirty a clean carpet ocal sensing: dirt, location only. olution?? February 27, 2017, E533KUI - Problem solving by search 9
10 Example: vacuum world ingle-state, start in #5. olution?? [ight, uck] onformant, start in {1,2,3,4,5,6,7,8} e.g., ight goes to {2,4,6,8}. olution?? [ight, uck, ef t, uck] ontingency, start in #5 Murphy s aw: uck can dirty a clean carpet ocal sensing: dirt, location only. olution?? [ight, if dirt then uck] February 27, 2017, E533KUI - Problem solving by search 10
11 problem is defined by four items: initial state e.g., at rad ingle-state problem formulation successor function (x) = set of action state pairs e.g., (rad) = { rad Zerind,Zerind,...} goal test, can be explicit, e.g., x = at ucharest implicit, e.g., N odirt(x) path cost (additive) e.g., sum of distances, number of actions executed, etc. c(x,a,y) is the step cost, assumed to be 0 solution is a sequence of actions leading from the initial state to a goal state February 27, 2017, E533KUI - Problem solving by search 11
12 electing a state space eal world is absurdly complex state space must be abstracted for problem solving (bstract) state = set of real states (bstract) action = complex combination of real actions e.g., rad Zerind represents a complex set of possible routes, detours, rest stops, etc. For guaranteed realizability, any real state in rad must get to some real state in Zerind (bstract) solution = set of real paths that are solutions in the real world Each abstract action should be easier than the original problem! February 27, 2017, E533KUI - Problem solving by search 12
13 Example: vacuum world state space graph states?? actions?? goal test?? path cost?? February 27, 2017, E533KUI - Problem solving by search 13
14 Example: vacuum world state space graph states??: integer dirt and robot locations (ignore dirt amounts etc.) actions?? goal test?? path cost?? February 27, 2017, E533KUI - Problem solving by search 14
15 Example: vacuum world state space graph states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: eft, ight, uck, NoOp goal test?? path cost?? February 27, 2017, E533KUI - Problem solving by search 15
16 Example: vacuum world state space graph states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: eft, ight, uck, NoOp goal test??: no dirt path cost?? February 27, 2017, E533KUI - Problem solving by search 16
17 Example: vacuum world state space graph states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: eft, ight, uck, NoOp goal test??: no dirt path cost??: 1 per action (0 for NoOp) February 27, 2017, E533KUI - Problem solving by search 17
18 Example: The 8-puzzle tart tate Goal tate states?? actions?? goal test?? path cost?? February 27, 2017, E533KUI - Problem solving by search 18
19 Example: The 8-puzzle tart tate Goal tate states??: integer locations of tiles (ignore intermediate positions) actions?? goal test?? path cost?? February 27, 2017, E533KUI - Problem solving by search 19
20 Example: The 8-puzzle tart tate Goal tate states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test?? path cost?? February 27, 2017, E533KUI - Problem solving by search 20
21 Example: The 8-puzzle tart tate Goal tate states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test??: = goal state (given) path cost?? February 27, 2017, E533KUI - Problem solving by search 21
22 Example: The 8-puzzle tart tate Goal tate states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test??: = goal state (given) path cost??: 1 per move [Note: optimal solution of n-puzzle family is NP-hard] February 27, 2017, E533KUI - Problem solving by search 22
23 Example: robotic assembly P states??: real-valued coordinates of robot joint angles parts of the object to be assembled actions??: continuous motions of robot joints goal test??: complete assembly with no robot included! path cost??: time to execute February 27, 2017, E533KUI - Problem solving by search 23
24 Tree search algorithms asic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states) February 27, 2017, E533KUI - Problem solving by search 24
25 Tree search example rad ibiu Timisoara Zerind rad Fagaras Oradea imnicu Vilcea rad ugoj rad Oradea February 27, 2017, E533KUI - Problem solving by search 25
26 Tree search example rad ibiu Timisoara Zerind rad Fagaras Oradea imnicu Vilcea rad ugoj rad Oradea February 27, 2017, E533KUI - Problem solving by search 26
27 Tree search example rad ibiu Timisoara Zerind rad Fagaras Oradea imnicu Vilcea rad ugoj rad Oradea February 27, 2017, E533KUI - Problem solving by search 27
28 Implementation: states vs. nodes state is a (representation of) a physical configuration node is a data structure constituting part of a search tree includes parent, children, depth, path cost g(x) tates do not have parents, children, depth, or path cost! parent, action tate Node depth = 6 g = state The Expand function creates new nodes, filling in the various fields and using the uccessorfn of the problem to create the corresponding states. February 27, 2017, E533KUI - Problem solving by search 28
29 Implementation: general tree search function Tree-earch( problem, fringe) returns a solution, or failure fringe Insert(Make-Node(Initial-tate[problem]), fringe) loop do if fringe is empty then return failure node emove-front(fringe) if Goal-Test(problem, tate(node)) then return node fringe Insertll(Expand(node, problem), fringe) function Expand( node, problem) returns a set of nodes successors the empty set for each action, result in uccessor-fn(problem, tate[node]) do s a new Node Parent-Node[s] node; ction[s] action; tate[s] result Path-ost[s] Path-ost[node] + tep-ost(tate[node], action, result) Depth[s] Depth[node] + 1 add s to successors return successors February 27, 2017, E533KUI - Problem solving by search 29
30 earch strategies strategy is defined by picking the order of node expansion trategies are evaluated along the following dimensions: completeness does it always find a solution if one exists? time complexity number of nodes generated/expanded space complexity maximum number of nodes in memory optimality does it always find a least-cost solution? Time and space complexity are measured in terms of b maximum branching factor of the search tree d depth of the least-cost solution m maximum depth of the state space (may be ) February 27, 2017, E533KUI - Problem solving by search 30
31 Uninformed search strategies Uninformed strategies use only the information available in the problem definition readth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search February 27, 2017, E533KUI - Problem solving by search 31
32 Expand shallowest unexpanded node readth-first search Implementation: frontier is a FIFO queue, i.e., new successors go at end D E F G February 27, 2017, E533KUI - Problem solving by search 32
33 Expand shallowest unexpanded node readth-first search Implementation: frontier is a FIFO queue, i.e., new successors go at end D E F G February 27, 2017, E533KUI - Problem solving by search 33
34 Expand shallowest unexpanded node readth-first search Implementation: frontier is a FIFO queue, i.e., new successors go at end D E F G February 27, 2017, E533KUI - Problem solving by search 34
35 Expand shallowest unexpanded node readth-first search Implementation: frontier is a FIFO queue, i.e., new successors go at end D E F G February 27, 2017, E533KUI - Problem solving by search 35
36 Properties of breadth-first search omplete?? February 27, 2017, E533KUI - Problem solving by search 36
37 omplete?? Yes (if b is finite) Time?? Properties of breadth-first search February 27, 2017, E533KUI - Problem solving by search 37
38 Properties of breadth-first search omplete?? Yes (if b is finite) Time?? 1+b+b 2 +b b d +b(b d 1) = O(b d+1 ), i.e., exp. in d pace?? February 27, 2017, E533KUI - Problem solving by search 38
39 Properties of breadth-first search omplete?? Yes (if b is finite) Time?? 1+b+b 2 +b b d +b(b d 1) = O(b d+1 ), i.e., exp. in d pace?? O(b d+1 ) (keeps every node in memory) Optimal?? February 27, 2017, E533KUI - Problem solving by search 39
40 omplete?? Yes (if b is finite) Properties of breadth-first search Time?? 1+b+b 2 +b b d +b(b d 1) = O(b d+1 ), i.e., exp. in d pace?? O(b d+1 ) (keeps every node in memory) Optimal?? Yes (if cost = 1 per step); not optimal in general pace is the big problem; can easily generate nodes at 100M/sec so 24hrs = 8640G. February 27, 2017, E533KUI - Problem solving by search 40
41 Expand least-cost unexpanded node Uniform-cost search Implementation: frontier = queue ordered by path cost, lowest first Equivalent to breadth-first if step costs all equal omplete?? Yes, if step cost ǫ Time?? # of nodes with g cost of optimal solution, O(b /ǫ ) where is the cost of the optimal solution pace?? # of nodes with g cost of optimal solution, O(b /ǫ ) Optimal?? Yes nodes expanded in increasing order of g(n) February 27, 2017, E533KUI - Problem solving by search 41
42 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 42
43 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 43
44 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 44
45 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 45
46 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 46
47 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 47
48 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 48
49 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 49
50 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 50
51 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 51
52 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 52
53 Expand deepest unexpanded node Depth-first search Implementation: frontier = IFO queue, i.e., put successors at front D E F G H I J K M N O February 27, 2017, E533KUI - Problem solving by search 53
54 Properties of depth-first search omplete?? February 27, 2017, E533KUI - Problem solving by search 54
55 Properties of depth-first search omplete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time?? February 27, 2017, E533KUI - Problem solving by search 55
56 Properties of depth-first search omplete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first pace?? February 27, 2017, E533KUI - Problem solving by search 56
57 Properties of depth-first search omplete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first pace?? O(bm), i.e., linear space! Optimal?? February 27, 2017, E533KUI - Problem solving by search 57
58 Properties of depth-first search omplete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first pace?? O(bm), i.e., linear space! Optimal?? No February 27, 2017, E533KUI - Problem solving by search 58
59 Depth-limited search = depth-first search with depth limit l, i.e., nodes at depth l have no successors ecursive implementation: function Depth-imited-earch( problem, limit) returns soln/fail/cutoff ecursive-d(make-node(initial-tate[problem]), problem, limit) function ecursive-d(node, problem, limit) returns soln/fail/cutoff cutoff-occurred? false if Goal-Test(problem, tate[node]) then return node else if Depth[node] = limit then return cutoff else for each successor in Expand(node, problem) do result ecursive-d(successor, problem, limit) if result = cutoff then cutoff-occurred? true else if result failure then return result if cutoff-occurred? then return cutoff else return failure February 27, 2017, E533KUI - Problem solving by search 59
60 Iterative deepening search function Iterative-Deepening-earch( problem) returns a solution inputs: problem, a problem for depth 0 to do result Depth-imited-earch( problem, depth) if result cutoff then return result end February 27, 2017, E533KUI - Problem solving by search 60
61 Iterative deepening search l = 0 imit = 0 February 27, 2017, E533KUI - Problem solving by search 61
62 Iterative deepening search l = 1 imit = 1 February 27, 2017, E533KUI - Problem solving by search 62
63 Iterative deepening search l = 2 imit = 2 D E F G D E F G D E F G D E F G D E F G D E F G D E F G D E F G February 27, 2017, E533KUI - Problem solving by search 63
64 Iterative deepening search l = 3 imit = 3 D E F G D E F G D E F G D E F G H I J K M N O H I J K M N O H I J K M N O H I J K M N O D E F G D E F G D E F G D E F G H I J K M N O H I J K M N O H I J K M N O H I J K M N O D E F G D E F G D E F G D E F G H I J K M N O H I J K M N O H I J K M N O H I J K M N O February 27, 2017, E533KUI - Problem solving by search 64
65 Properties of iterative deepening search omplete?? February 27, 2017, E533KUI - Problem solving by search 65
66 omplete?? Yes Time?? Properties of iterative deepening search February 27, 2017, E533KUI - Problem solving by search 66
67 Properties of iterative deepening search omplete?? Yes Time?? (d+1)b 0 +db 1 +(d 1)b b d = O(b d ) pace?? February 27, 2017, E533KUI - Problem solving by search 67
68 Properties of iterative deepening search omplete?? Yes Time?? (d+1)b 0 +db 1 +(d 1)b b d = O(b d ) pace?? O(bd) Optimal?? February 27, 2017, E533KUI - Problem solving by search 68
69 omplete?? Yes Properties of iterative deepening search Time?? (d+1)b 0 +db 1 +(d 1)b b d = O(b d ) pace?? O(bd) Optimal?? Yes, if step cost = 1 an be modified to explore uniform-cost tree Numerical comparison for b = 10 and d = 5, solution at far right leaf: N(ID) = ,000+20, ,000 = 123,450 N(F) = ,000+10, , ,990 = 1,111,100 ID does better because other nodes at depth d are not expanded F can be modified to apply goal test when a node is generated February 27, 2017, E533KUI - Problem solving by search 69
70 ummary of algorithms riterion readth- Uniform- Depth- Depth- Iterative First ost First imited Deepening omplete? Yes Yes No Yes, if l d Yes Time b d+1 b /ǫ b m b l b d pace b d+1 b /ǫ bm bl bd Optimal? Yes Yes No No Yes February 27, 2017, E533KUI - Problem solving by search 70
71 epeated states Failure to detect repeated states can turn a linear problem into an exponential one! D February 27, 2017, E533KUI - Problem solving by search 71
72 Graph search function Graph-earch( problem, fringe) returns a solution, or failure closed an empty set fringe Insert(Make-Node(Initial-tate[problem]), fringe) loop do if fringe is empty then return failure node emove-front(fringe) if Goal-Test(problem, tate[node]) then return node if tate[node] is not in closed then add tate[node] to closed fringe Insertll(Expand(node, problem), fringe) end February 27, 2017, E533KUI - Problem solving by search 72
73 ummary Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored Variety of uninformed search strategies Iterative deepening search uses only linear space and not much more time than other uninformed algorithms Graph search can be exponentially more efficient than tree search February 27, 2017, E533KUI - Problem solving by search 73
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