Problem Solving and Search: Understanding Agents and Algorithms - Prof. Thad Starner, Study notes of Computer Science

Download Problem Solving and Search: Understanding Agents and Algorithms - Prof. Thad Starner and more Study notes Computer Science in PDF only on Docsity! Problem solving and search Chapter 3 Chapter 3 1 Reminders Assignment 0 due 5pm today Assignment 1 posted, due 2/9 Section 105 will move to 9-10am starting next week Chapter 3 2 Example: Romania On holiday in Romania; currently in Arad. Flight leaves tomorrow from Bucharest Formulate goal: be in Bucharest Formulate problem: states: various cities actions: drive between cities Find solution: sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest Chapter 3 5 Example: Romania Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Vaslui Iasi Rimnicu Vilcea Bucharest 71 75 118 111 70 75 120 151 140 99 80 97 101 211 138 146 85 90 98 142 92 87 86 Chapter 3 6 Problem types Deterministic, fully observable =⇒ single-state problem Agent knows exactly which state it will be in; solution is a sequence Non-observable =⇒ conformant problem Agent 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”) Chapter 3 7 Example: vacuum world Single-state, start in #5. Solution?? [Right, Suck] Conformant, start in {1, 2, 3, 4, 5, 6, 7, 8} e.g., Right goes to {2, 4, 6, 8}. Solution?? [Right, Suck, Left, Suck] Contingency, start in #5 Murphy’s Law: Suck can dirty a clean carpet Local sensing: dirt, location only. Solution?? 1 2 3 4 5 6 7 8 Chapter 3 10 Example: vacuum world Single-state, start in #5. Solution?? [Right, Suck] Conformant, start in {1, 2, 3, 4, 5, 6, 7, 8} e.g., Right goes to {2, 4, 6, 8}. Solution?? [Right, Suck, Left, Suck] Contingency, start in #5 Murphy’s Law: Suck can dirty a clean carpet Local sensing: dirt, location only. Solution?? [Right, if dirt then Suck] 1 2 3 4 5 6 7 8 Chapter 3 11 Single-state problem formulation A problem is defined by four items: initial state e.g., “at Arad” successor function S(x) = set of action–state pairs e.g., S(Arad) = {〈Arad→ Zerind, Zerind〉, . . .} goal test, can be explicit, e.g., x = “at Bucharest” implicit, e.g., NoDirt(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 A solution is a sequence of actions leading from the initial state to a goal state Chapter 3 12 Example: vacuum world state space graph R L S S S S R L R L R L S SS S L L LL R R R R states??: integer dirt and robot locations (ignore dirt amounts etc.) actions?? goal test?? path cost?? Chapter 3 15 Example: vacuum world state space graph R L S S S S R L R L R L S SS S L L LL R R R R states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: Left, Right, Suck, NoOp goal test?? path cost?? Chapter 3 16 Example: vacuum world state space graph R L S S S S R L R L R L S SS S L L LL R R R R states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: Left, Right, Suck, NoOp goal test??: no dirt path cost?? Chapter 3 17 Example: The 8-puzzle 2 Start State Goal State 51 3 4 6 7 8 5 1 2 3 4 6 7 8 5 states??: integer locations of tiles (ignore intermediate positions) actions?? goal test?? path cost?? Chapter 3 20 Example: The 8-puzzle 2 Start State Goal State 51 3 4 6 7 8 5 1 2 3 4 6 7 8 5 states??: integer locations of tiles (ignore intermediate positions) actions??: move blank left, right, up, down (ignore unjamming etc.) goal test?? path cost?? Chapter 3 21 Example: The 8-puzzle 2 Start State Goal State 51 3 4 6 7 8 5 1 2 3 4 6 7 8 5 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?? Chapter 3 22 Tree search algorithms Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a. expanding states) function Tree-Search( 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 Chapter 3 25 Tree search example Rimnicu Vilcea Lugoj ZerindSibiu Arad Fagaras Oradea Timisoara AradArad Oradea Arad Chapter 3 26 Tree search example Rimnicu Vilcea LugojArad Fagaras Oradea AradArad Oradea Zerind Arad Sibiu Timisoara Chapter 3 27 Implementation: general tree search function Tree-Search( problem, fringe) returns a solution, or failure fringe← Insert(Make-Node(Initial-State[problem]), fringe) loop do if fringe is empty then return failure node←Remove-Front(fringe) if Goal-Test(problem,State(node)) then return node fringe← InsertAll(Expand(node,problem), fringe) function Expand(node, problem) returns a set of nodes successors← the empty set for each action, result in Successor-Fn(problem,State[node]) do s← a new Node Parent-Node[s]← node; Action[s]← action; State[s]← result Path-Cost[s]←Path-Cost[node] + Step-Cost(node,action, s) Depth[s]←Depth[node] + 1 add s to successors return successors Chapter 3 30 Search strategies A strategy is defined by picking the order of node expansion Strategies 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 ∞) Chapter 3 31 Uninformed search strategies Uninformed strategies use only the information available in the problem definition Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search Chapter 3 32 Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end A B C D E F G Chapter 3 35 Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end A B C D E F G Chapter 3 36 Properties of breadth-first search Complete?? Chapter 3 37 Properties of breadth-first search Complete?? Yes (if b is finite) Time?? 1 + b + b2 + b3 + . . . + bd + b(bd − 1) = O(bd+1), i.e., exp. in d Space?? O(bd+1) (keeps every node in memory) Optimal?? Chapter 3 40 Properties of breadth-first search Complete?? Yes (if b is finite) Time?? 1 + b + b2 + b3 + . . . + bd + b(bd − 1) = O(bd+1), i.e., exp. in d Space?? O(bd+1) (keeps every node in memory) Optimal?? Yes (if cost = 1 per step); not optimal in general Space is the big problem; can easily generate nodes at 100MB/sec so 24hrs = 8640GB. Chapter 3 41 Uniform-cost search Expand least-cost unexpanded node Implementation: fringe = queue ordered by path cost, lowest first Equivalent to breadth-first if step costs all equal Complete?? Yes, if step cost ≥  Time?? # of nodes with g ≤ cost of optimal solution, O(bdC ∗/e) where C∗ is the cost of the optimal solution Space?? # of nodes with g ≤ cost of optimal solution, O(bdC ∗/e) Optimal?? Yes—nodes expanded in increasing order of g(n) Chapter 3 42 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front A B C D E F G H I J K L M N O Chapter 3 45 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front A B C D E F G H I J K L M N O Chapter 3 46 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front A B C D E F G H I J K L M N O Chapter 3 47 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front A B C D E F G H I J K L M N O Chapter 3 50 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front A B C D E F G H I J K L M N O Chapter 3 51 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front A B C D E F G H I J K L M N O Chapter 3 52 Properties of depth-first search Complete?? Chapter 3 55 Properties of depth-first search Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path ⇒ complete in finite spaces Time?? Chapter 3 56 Properties of depth-first search Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path ⇒ complete in finite spaces Time?? O(bm): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space?? Chapter 3 57 Depth-limited search = depth-first search with depth limit l, i.e., nodes at depth l have no successors Recursive implementation: function Depth-Limited-Search( problem, limit) returns soln/fail/cutoff Recursive-DLS(Make-Node(Initial-State[problem]),problem, limit) function Recursive-DLS(node,problem, limit) returns soln/fail/cutoff cutoff-occurred?← false if Goal-Test(problem,State[node]) then return node else if Depth[node] = limit then return cutoff else for each successor in Expand(node,problem) do result←Recursive-DLS(successor,problem, limit) if result = cutoff then cutoff-occurred?← true else if result 6= failure then return result if cutoff-occurred? then return cutoff else return failure Chapter 3 60 Iterative deepening search function Iterative-Deepening-Search( problem) returns a solution inputs: problem, a problem for depth← 0 to ∞ do result←Depth-Limited-Search( problem, depth) if result 6= cutoff then return result end Chapter 3 61 Iterative deepening search l = 0 Limit = 0 A A Chapter 3 62 Iterative deepening search l = 3 Limit = 3 A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H I J K L M N O A B C D E F G H J K L M N OI A B C D E F G H I J K L M N O Chapter 3 65 Properties of iterative deepening search Complete?? Chapter 3 66 Properties of iterative deepening search Complete?? Yes Time?? Chapter 3 67 Properties of iterative deepening search Complete?? Yes Time?? (d + 1)b0 + db1 + (d− 1)b2 + . . . + bd = O(bd) Space?? O(bd) Optimal?? Yes, if step cost = 1 Can be modified to explore uniform-cost tree Numerical comparison for b = 10 and d = 5, solution at far right leaf: N(IDS) = 50 + 400 + 3, 000 + 20, 000 + 100, 000 = 123, 450 N(BFS) = 10 + 100 + 1, 000 + 10, 000 + 100, 000 + 999, 990 = 1, 111, 100 IDS does better because other nodes at depth d are not expanded BFS can be modified to apply goal test when a node is generated Chapter 3 70 Summary of algorithms Criterion Breadth- Uniform- Depth- Depth- Iterative First Cost First Limited Deepening Complete? Yes∗ Yes∗ No Yes, if l ≥ d Yes Time bd+1 bdC ∗/e bm bl bd Space bd+1 bdC ∗/e bm bl bd Optimal? Yes∗ Yes No No Yes∗ Chapter 3 71 Repeated states Failure to detect repeated states can turn a linear problem into an exponential one! A B C D A BB CCCC Chapter 3 72