Introduction to Artificial Intelligence. Problem Solving and Search
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1 Introduction to rtificial Intelligence Problem Solving and Search ernhard eckert UNIVERSITÄT KOLENZ-LNDU Summer Term eckert: Einführung in die KI / KI für IM p.1
2 Outline Problem solving Problem types Problem formulation Example problems asic search algorithms. eckert: Einführung in die KI / KI für IM p.2
3 Problem solving Offline problem solving cting only with complete knowledge of problem and solution Online problem solving cting without complete knowledge Here Here we are concerned with offline problem solving only. eckert: Einführung in die KI / KI für IM p.3
4 Example: Travelling in Romania Scenario On holiday in Romania; currently in Flight leaves tomorrow from ucharest Goal e in ucharest Formulate problem States: various cities ctions: drive between cities Solution ppropriate sequence of cities e.g.:, Sibiu, Fagaras, ucharest. eckert: Einführung in die KI / KI für IM p.4
5 Example: Travelling in Romania Oradea Neamt Zerind Iasi Sibiu Fagaras Vaslui Timisoara Rimnicu Vilcea Lugoj Pitesti Mehadia Urziceni Hirsova Dobreta raiova ucharest Giurgiu Eforie. eckert: Einführung in die KI / KI für IM p.5
6 Problem types Single-state problem observable (at least the initial state) deterministic static discrete Multiple-state problem partially observable (initial state not observable) deterministic static discrete ontingency problem partially observable (initial state not observable) non-deterministic. eckert: Einführung in die KI / KI für IM p.6
7 Example: vacuum-cleaner world Single-state 1 2 Start in: 5 Solution: [right, suck] 3 4 Multiple-state Start in: Solution: [right, suck, left, suck] right suck 4 8 left 3 7 suck 7. eckert: Einführung in die KI / KI für IM p.7
8 Example: vacuum-cleaner world 1 2 ontingency Murphy s Law: suck can dirty a clean carpet Local sensing: dirty/not dirty at location only Start in: Solution: [suck, right, suck] suck 5 7 right 6 8 suck 6 8 Improvement: [suck, right, if dirt then suck] (decide whether in 6 or 8 using local sensing). eckert: Einführung in die KI / KI für IM p.8
9 Single-state problem formulation Defined by the following four items 1. Initial state Example: 2. Successor function S Example: S gozerind Zerind gosibiu Sibiu 3. Goal test Example: x ucharest (explicit test) nodirt x (implicit test) 4. Path cost (optional) Example: sum of distances, number of operators executed, etc.. eckert: Einführung in die KI / KI für IM p.9
10 Single-state problem formulation Solution sequence of operators leading from the initial state to a goal state. eckert: Einführung in die KI / KI für IM p.10
11 Selecting a state space bstraction Real world is absurdly complex State space must be abstracted for problem solving (bstract) state Set of real states (bstract) operator omplex combination of real actions Example: Zerind represents complex set of possible routes (bstract) solution Set of real paths that are solutions in the real world. eckert: Einführung in die KI / KI für IM p.11
12 Example: The 8-puzzle Start State Goal State States ctions Goal test Path cost integer locations of tiles left, right, up, down = goal state? 1 per move. eckert: Einführung in die KI / KI für IM p.12
13 Example: Vacuum-cleaner L R L R S S L R R L R R L L S S S S L R L R S S States ctions Goal test integer dirt and robot locations left, right, suck, noop not dirty? Path cost 1 per operation (0 for noop). eckert: Einführung in die KI / KI für IM p.13
14 Example: Robotic assembly P R R R R R States real-valued coordinates of robot joint angles and parts of the object to be assembled ctions Goal test Path cost continuous motions of robot joints assembly complete? time to execute. eckert: Einführung in die KI / KI für IM p.14
15 Tree search algorithms Offline Simulated exploration of state space in a search tree by generating successors of already-explored states function TREE-SERH( problem, strategy) returns a solution or failure initialize the search tree using the initial state of problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solution else expand the node and add the resulting nodes to the search tree end. eckert: Einführung in die KI / KI für IM p.15
16 Tree search: Example Sibiu Timisoara Zerind Fagaras Oradea Rimnicu Vilcea Lugoj Oradea. eckert: Einführung in die KI / KI für IM p.16
17 Tree search: Example Sibiu Timisoara Zerind Fagaras Oradea Lugoj Rimnicu Vilcea Oradea. eckert: Einführung in die KI / KI für IM p.16
18 Tree search: Example Sibiu Timisoara Zerind Fagaras Oradea Rimnicu Vilcea Lugoj Oradea. eckert: Einführung in die KI / KI für IM p.16
19 Implementation: States vs. nodes State (representation of) a physical configuration Node data structure constituting part of a search tree (includes parent, children, depth, path cost, etc.) parent State 5 4 Node depth = g = state children. eckert: Einführung in die KI / KI für IM p.17
20 Implementation of search algorithms function TREE-SERH( problem, fringe) returns a solution or failure fringe INSERT(MKE-NODE(INITIL-STTE[problem]),fringe) loop do if fringe is empty then return failure node REMOVE-FIRST(fringe) if GOL-TEST[problem] applied to STTE(node) succeeds then return node else fringe INSERT-LL(EXPND(node, problem), fringe) end fringe State Expand queue of nodes not yet considered gives the state that is represented by node creates new nodes by applying possible actions to node. eckert: Einführung in die KI / KI für IM p.18
21 Search strategies Strategy Defines the order of node expansion Important properties of strategies completeness time complexity space complexity optimality does it always find a solution if one exists? number of nodes generated/expanded maximum number of nodes in memory does it always find a least-cost solution? Time and space complexity measured in terms of b d maximum branching factor of the search tree depth of a solution with minimal distance to root m maximum depth of the state space (may be ). eckert: Einführung in die KI / KI für IM p.19
22 Uninformed search strategies Uninformed search Use only the information available in the problem definition Frequently used strategies readth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search. eckert: Einführung in die KI / KI für IM p.20
23 readth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue D E F G. eckert: Einführung in die KI / KI für IM p.21
24 readth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue D E F G. eckert: Einführung in die KI / KI für IM p.21
25 readth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue D E F G. eckert: Einführung in die KI / KI für IM p.21
26 readth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue D E F G. eckert: Einführung in die KI / KI für IM p.21
27 readth-first search: Example Romania. eckert: Einführung in die KI / KI für IM p.22
28 readth-first search: Example Romania Zerind Sibiu Timisoara. eckert: Einführung in die KI / KI für IM p.22
29 readth-first search: Example Romania Zerind Sibiu Timisoara Oradea. eckert: Einführung in die KI / KI für IM p.22
30 readth-first search: Example Romania Zerind Sibiu Timisoara Oradea Oradea Rimnicu Fagaras Vilcea Lugoj. eckert: Einführung in die KI / KI für IM p.22
31 readth-first search: Properties omplete Yes (if b is finite) Time 1 b b 2 b 3 b d b b d 1 O b d 1 Space O b d 1 i.e. exponential in d keeps every node in memory Optimal Yes (if cost = 1 per step), not optimal in general Disadvantage Space is the big problem (can easily generate nodes at 5M/sec so 24hrs = 430G). eckert: Einführung in die KI / KI für IM p.23
32 Romania with step costs in km Oradea 71 Neamt Zerind Timisoara 111 Lugoj 70 Mehadia Dobreta Sibiu 99 Fagaras 80 Rimnicu Vilcea 97 Pitesti ucharest 90 raiova Giurgiu 87 Iasi Urziceni Vaslui Hirsova 86 Eforie Straight line distance to ucharest 366 ucharest 0 raiova 160 Dobreta 242 Eforie 161 Fagaras 178 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 98 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374. eckert: Einführung in die KI / KI für IM p.24
33 Uniform-cost search Idea Expand least-cost unexpanded node (costs added up over paths from root to leafs) Implementation fringe is queue ordered by increasing path cost Note Equivalent to depth-first search if all step costs are equal. eckert: Einführung in die KI / KI für IM p.25
34 Uniform-cost search. eckert: Einführung in die KI / KI für IM p.26
35 Uniform-cost search Zerind Sibiu Timisoara. eckert: Einführung in die KI / KI für IM p.26
36 Uniform-cost search Zerind Sibiu Timisoara Oradea. eckert: Einführung in die KI / KI für IM p.26
37 Uniform-cost search Zerind Sibiu Timisoara Oradea Lugoj. eckert: Einführung in die KI / KI für IM p.26
38 Uniform-cost search: Properties omplete Yes (if step costs positive) Time Space Optimal # of nodes with past-cost less than that of optimal solution # of nodes with past-cost less than that of optimal solution Yes. eckert: Einführung in die KI / KI für IM p.27
39 Depth-first search Idea Expand deepest unexpanded node Implementation fringe is a LIFO queue (a stack), i.e. successors go in at front of queue Note Depth-first search can perform infinite cyclic excursions Need a finite, non-cyclic search space (or repeated-state checking). eckert: Einführung in die KI / KI für IM p.28
40 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
41 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
42 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
43 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
44 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
45 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
46 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
47 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
48 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
49 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
50 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
51 Depth-first search D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.29
52 Depth-first search: Example Romania. eckert: Einführung in die KI / KI für IM p.30
53 Depth-first search: Example Romania Zerind Sibiu Timisoara. eckert: Einführung in die KI / KI für IM p.30
54 Depth-first search: Example Romania Zerind Sibiu Timisoara Oradea. eckert: Einführung in die KI / KI für IM p.30
55 Depth-first search: Example Romania Zerind Sibiu Timisoara Oradea Zerind Sibiu Timisoara. eckert: Einführung in die KI / KI für IM p.30
56 Depth-first search: Properties omplete Yes: if state space finite No: if state contains infinite paths or loops Time O b m Space O bm (i.e. linear space) Optimal No Disadvantage Time terrible if m much larger than d dvantage Time may be much less than breadth-first search if solutions are dense. eckert: Einführung in die KI / KI für IM p.31
57 Iterative deepening search Depth-limited search Depth-first search with depth limit Iterative deepening search Depth-limit search with ever increasing limits function ITERTIVE-DEEPENING-SERH( problem) returns a solution or failure inputs: problem /* a problem */ for depth result if result end 0 to do DEPTH-LIMITED-SERH( problem, depth) cutoff then return result. eckert: Einführung in die KI / KI für IM p.32
58 Iterative deepening search with depth limit 0 Limit = 0. eckert: Einführung in die KI / KI für IM p.33
59 Iterative deepening search with depth limit 1 Limit = 1. eckert: Einführung in die KI / KI für IM p.34
60 Iterative deepening search with depth limit 2 Limit = 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. eckert: Einführung in die KI / KI für IM p.35
61 Iterative deepening search with depth limit 3 Limit = 3 D E F G H I J K L M N O D E F G H I J K L M N O D E F G H I J K L M N O D E F G H I J K L M N O D E F G H I J K L M N O D E F G H I J K L M N O D E F G H I J K L M N O D E F G H I J K L M N O D E F G H I J K L M N O D E F G H I J K L M N O D E F G H J K L M N O I D E F G H I J K L M N O. eckert: Einführung in die KI / KI für IM p.36
62 Iterative deepening search: Example Romania with l 0. eckert: Einführung in die KI / KI für IM p.37
63 Iterative deepening search: Example Romania with l 1. eckert: Einführung in die KI / KI für IM p.38
64 Iterative deepening search: Example Romania with l 1 Zerind Sibiu Timisoara. eckert: Einführung in die KI / KI für IM p.38
65 Iterative deepening search: Example Romania with l 2. eckert: Einführung in die KI / KI für IM p.39
66 Iterative deepening search: Example Romania with l 2 Zerind Sibiu Timisoara. eckert: Einführung in die KI / KI für IM p.39
67 Iterative deepening search: Example Romania with l 2 Zerind Sibiu Timisoara Oradea. eckert: Einführung in die KI / KI für IM p.39
68 Iterative deepening search: Example Romania with l 2 Zerind Sibiu Timisoara Oradea Oradea Fagaras Rimnicu Vilcea. eckert: Einführung in die KI / KI für IM p.39
69 Iterative deepening search: Example Romania with l 2 Zerind Sibiu Timisoara Oradea Oradea Fagaras Rimnicu Vilcea Lugoj. eckert: Einführung in die KI / KI für IM p.39
70 Iterative deepening search: Properties omplete Yes Time d 1 b 0 db 1 d 1 b 2 b d O b d 1 Space O bd Optimal Yes (if step cost = 1) (Depth-First) Iterative-Deepening Search often used in practice for search spaces of large, infinite, or unknown depth.. eckert: Einführung in die KI / KI für IM p.40
71 omparison riterion readthfirst Uniformcost Depthfirst Iterative deepening omplete? Yes Yes No Yes Time b d 1 b d b m b d Space b d 1 b d bm bd Optimal? Yes Yes No Yes. eckert: Einführung in die KI / KI für IM p.41
72 omparison readth-first search Iterative deepening search. eckert: Einführung in die KI / KI für IM p.42
73 Summary 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. eckert: Einführung in die KI / KI für IM p.43
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