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Lecture 2 Solving Problems by Searching by Marco Chiarandini

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1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem

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Lecture 2 Solving Problems by Searching Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Slides by Stuart Russell and Peter Norvig
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Last Time Agents are used to provide a consistent viewpoint on various topics in the field AI Essential concepts: Agents intereact with environment by means of sensors and actuators. A rational agent does “the right thing” ≡ maximizes a performance measure è PEAS Environment types: observable, deterministic, episodic, static, discrete, single agent Agent types: table driven (rule based), simple reflex, model-based reflex, goal-based, utility-based, learning agent 2
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Structure of Agents Agent = Architecture + Program Architecture operating platform of the agent computer system, specific hardware, possibly OS Program function that implements the mapping from percepts to actions This course is about the program, not the architecture 3
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 5
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 6
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline ♦ Problem-solving agents ♦ Problem types ♦ Problem formulation ♦ Example problems ♦ Basic search algorithms 7
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Problem-solving agents Restricted form of general agent: function Simple-Problem-Solving-Agent( 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-State(state,percept) if seq is empty then goal ← Formulate-Goal(state) problem ← Formulate-Problem(state,goal) seq ← Search( problem) action ← Recommendation(seq,state) seq ← Remainder(seq,state) return action Note: this is offline problem solving; solution executed “eyes closed.” Online problem solving involves acting without complete knowledge. 8
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 9
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 10
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem State space Problem formulation A problem is defined by five items: 1. initial state e.g., “at Arad” 2. actions defining the other states, e.g., Go(Arad) 3. transition model res(x, a) e.g., res(In(Arad), Go(Zerind)) = In(Zerind) alternatively: set of action–state pairs: {h(In(Arad), Go(Zerind)), In(Zerind)i, . . .} 4. goal test, can be explicit, e.g., x = “at Bucharest” implicit, e.g., NoDirt(x) 5. 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 11
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Selecting a state space Real world is complex ⇒ state space must be abstracted for problem solving (Abstract) state = set of real states (Abstract) action = complex combination of real actions e.g., “Arad → Zerind” represents a complex set of possible routes, detours, rest stops, etc. For guaranteed realizability, any real state “in Arad” must get to some real state “in Zerind” (Abstract) solution = set of real paths that are solutions in the real world Each abstract action should be “easier” than the original problem! Atomic representation 12
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Vacuum world state space graph Example R L S S S S R L R L R L S S S S L L L L R R R R states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: Left, Right, Suck, NoOp transition model??: arcs in the digraph goal test??: no dirt path cost??: 1 per action (0 for NoOp) 13
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example: The 8-puzzle 2 Start State Goal State 5 1 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.) transition model??: effect of the actions goal test??: = goal state (given) path cost??: 1 per move [Note: optimal solution of n-Puzzle family is NP-hard] 14
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example: robotic assembly R R R P R R 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 15
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Problem types Deterministic, fully observable, known, discrete =⇒ state space 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”) 16
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example: vacuum world State space, start in #5. Solution?? [Right, Suck] Non-observable, 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 17
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example Problems Toy problems vacuum cleaner agent 8-puzzle 8-queens cryptarithmetic missionaries and cannibals Real-world problems route finding traveling salesperson VLSI layout robot navigation assembly sequencing 18
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 19
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Objectives Formulate appropriate problems in optimization and planning (sequence of actions to achive a goal) as search tasks: initial state, operators, goal test, path cost Know the fundamental search strategies and algorithms uninformed search breadth-first, depth-first, uniform-cost, iterative deepening, bi- directional informed search best-first (greedy, A*), heuristics, memory-bounded Evaluate the suitability of a search strategy for a problem completeness, optimality, time & space complexity 20
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Searching for Solutions Traversal of some search space from the initial state to a goal state legal sequence of actions as defined by operators The search can be performed on On a search tree derived from expanding the current state using the possible operators Tree-Search algorithm A graph representing the state space Graph-Search algorithm 22
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Search: Terminology 23
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example: Route Finding 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 24
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Tree search example Lugoj Arad Arad Oradea Rimnicu Vilcea Zerind Arad Sibiu Arad Fagaras Oradea Timisoara 25
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem General tree search 26
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Implementation: states vs. nodes A state is a (representation of) a physical configuration A node is a data structure constituting part of a search tree includes state, parent, action, path cost g(x) States do not have parents, children, depth, or path cost! 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 State Node depth = 6 g = 6 state parent, action The Expand function creates new nodes, filling in the various fields using the Transition Model of the problem to create the corresponding states. 27
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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(State[node], action, result) Depth[s] ← Depth[node] + 1 add s to successors return successors 28
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Search strategies A strategy is defined by picking the order of node expansion 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) 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 path 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 ∞) 29
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 30
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 Bidirectional Search 31
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Breadth-first search Expand shallowest unexpanded node (shortest path in the frontier) A B C D E F G A B C D E F Implementation: fringe is a FIFO queue, i.e., new successors go at end 32
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of breadth-first search Complete?? Yes (if b is finite) Optimal?? Yes (if cost = 1 per step); not optimal in general Time?? 1 + b + b2 + b3 + . . . + bd + b(bd − 1) = O(bd+1 ), i.e., exp. in d Space?? bd−1 + bd = O(bd ) (explored + frontier) Space is the big problem; can easily generate nodes at 100MB/sec so 24hrs = 8640GB. 33
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Uniform-cost search Expand first least-cost path (Equivalent to breadth-first if step costs all equal) Implementation: fringe = priority queue ordered by path cost, lowest first, Complete?? Yes, if step cost ≥ Optimal??Yes—nodes expanded in increasing order of g(n) Time?? # of nodes with g ≤ cost of optimal solution, O(b1+dC∗ /e ) where C∗ is the cost of the optimal solution Space??# of nodes with g ≤ cost of optimal solution, O(b1+dC∗ /e ) 34
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Depth-first search Expand deepest unexpanded node A B C D E F G H I J K L M N O Implementation: fringe = LIFO queue, i.e., put successors at front 35
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of depth-first search Complete?? No: fails in infinite-depth spaces, or spaces with loops Modify to avoid repeated states along path ⇒ complete in finite spaces Optimal?? No Time?? O(bm ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space?? O(bm), i.e., linear space! 36
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 37
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 38
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Iterative deepening search 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 O I A B C D E F G H I J K L M N O 39
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of iterative deepening search Complete?? Yes Optimal?? Yes, if step cost = 1 Can be modified to explore uniform-cost tree Time?? (d + 1)b0 + db1 + (d − 1)b2 + . . . + bd = O(bd ) Space?? O(bd) Numerical comparison in time for b = 10 and d = 5, solution at far right leaf: NIDS) = 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 Iterative lenghtening not as successul as IDS 40
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Bidirectional Search Search simultaneously (using breadth-first search) from goal to start from start to goal Stop when the two search trees intersects 41
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Difficulties in Bidirectional Search If applicable, may lead to substantial savings Predecessors of a (goal) state must be generated Not always possible, eg. when we do not know the optimal state explicitly Search must be coordinated between the two search processes. What if many goal states? One search must keep all nodes in memory 42
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 bd1+C∗ /e bm bl bd Space bd+1 bd1+C∗ /e bm bl bd Optimal? Yes∗ Yes No No Yes∗ 43
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 Graph search can be exponentially more efficient than tree search 46
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 47
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Review: 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] applied to State(node) succeeds return node fringe ← InsertAll(Expand(node, problem), fringe) A strategy is defined by picking the order of node expansion 48
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Informed search strategy Informed strategies use agent’s background information about the problem map, costs of actions, approximation of solutions, ... best-first search greedy search A∗ search local search (not in this course) Hill-climbing Simulated annealing Genetic algorithms Local search in continuous spaces 49
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Best-first search Idea: use an evaluation function for each node – estimate of “desirability” ⇒ Expand most desirable unexpanded node Implementation: fringe is a queue sorted in decreasing order of desirability Special cases: greedy search A∗ search 50
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Romania with step costs in km Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to Bucharest 0 160 242 161 77 151 241 366 193 178 253 329 80 199 244 380 226 234 374 98 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 51
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Greedy search Evaluation function h(n) (heuristic) = estimate of cost from n to the closest goal E.g., hSLD(n) = straight-line distance from n to Bucharest Greedy search expands the node that appears to be closest to goal 52
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Greedy search example Arad 366 53
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of greedy search Complete?? No–can get stuck in loops, e.g., from Iasi to Fargas Iasi → Neamt → Iasi → Neamt → Complete in finite space with repeated-state checking Optimal?? No Time?? O(bm ), but a good heuristic can give dramatic improvement Space?? O(bm ) 54
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem A∗ search Idea: avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost to goal from n f(n) = estimated total cost of path through n to goal A∗ search uses an admissible heuristic i.e., h(n) ≤ h∗ (n) where h∗ (n) is the true cost from n. (Also require h(n) ≥ 0, so h(G) = 0 for any goal G.) E.g., hSLD(n) never overestimates the actual road distance Theorem: A∗ search is optimal 55
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem A∗ search example Zerind Arad Sibiu Arad Timisoara Sibiu Bucharest Rimnicu Vilcea Fagaras Oradea Craiova Pitesti Sibiu Bucharest Craiova Rimnicu Vilcea 418=418+0 447=118+329 449=75+374 646=280+366 591=338+253 450=450+0 526=366+160 553=300+253 615=455+160 607=414+193 671=291+380 56
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Optimality of A∗ (standard proof) Suppose some suboptimal goal G2 has been generated and is in the queue. Let n be an unexpanded node on a shortest path to an optimal goal G1. G n G2 Start f(G2) = g(G2) since h(G2) = 0 g(G1) since G2 is suboptimal ≥ f(n) since h is admissible Since f(G2) f(n), A∗ will never select G2 for expansion 57
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Optimality of A∗ (more useful) Lemma: A∗ expands nodes in order of increasing f value∗ Gradually adds “f-contours” of nodes (cf. breadth-first adds layers) Contour i has all nodes with f = fi, where fi fi+1 O Z A T L M D C R F P G B U H E V I N 380 400 420 S 58
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Astar vs. Depth search 59
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of A∗ Complete?? Yes, unless there are infinitely many nodes with f ≤ f(G) Optimal?? Yes—cannot expand fi+1 until fi is finished A∗ expands all nodes with f(n) C∗ A∗ expands some nodes with f(n) = C∗ A∗ expands no nodes with f(n) C∗ Time?? Exponential in [relative error in h × length of sol.] Space?? Keeps all nodes in memory 60
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Proof of lemma: Consistency A heuristic is consistent if n c(n,a,n’) h(n’) h(n) G n’ h(n) ≤ c(n, a, n0 ) + h(n0 ) If h is consistent, we have f(n0 ) = g(n0 ) + h(n0 ) = g(n) + c(n, a, n0 ) + h(n0 ) ≥ g(n) + h(n) = f(n) I.e., f(n) is nondecreasing along any path. 61
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Admissible heuristics E.g., for the 8-puzzle: h1(n) = number of misplaced tiles h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) 2 Start State Goal State 5 1 3 4 6 7 8 5 1 2 3 4 6 7 8 5 h1(S) =?? h2(S) =?? 62
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Admissible heuristics E.g., for the 8-puzzle: h1(n) = number of misplaced tiles h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) 2 Start State Goal State 5 1 3 4 6 7 8 5 1 2 3 4 6 7 8 5 h1(S) =?? 6 h2(S) =?? 4+0+3+3+1+0+2+1 = 14 63
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Dominance If h2(n) ≥ h1(n) for all n (both admissible) then h2 dominates h1 and is better for search Typical search costs: d = 14 IDS = 3,473,941 nodes A∗ (h1) = 539 nodes A∗ (h2) = 113 nodes d = 24 IDS ≈ 54,000,000,000 nodes A∗ (h1) = 39,135 nodes A∗ (h2) = 1,641 nodes Given any admissible heuristics ha, hb, h(n) = max(ha(n), hb(n)) is also admissible and dominates ha, hb 64
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Relaxed problems Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution Key point: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem 65
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Memory-Bounded Heuristic Search Try to reduce memory needs Take advantage of heuristic to improve performance Iterative-deepening A∗ (IDA∗ ) SMA∗ 67
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Iterative Deepening A∗ Uniformed Iterative Deepening (repetition) depth-first search where the max depth is iteratively increased IDA∗ depth-first search, but only nodes with f-cost less than or equal to smallest f-cost of nodes expanded at last iteration was the best search algorithm for many practical problems 68
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of IDA∗ Complete?? Yes Time complexity?? Still exponential Space complexity?? linear Optimal?? Yes. Also optimal in the absence of monotonicity 69
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Simple Memory-Bounded A∗ Use all available memory Follow A∗ algorithm and fill memory with new expanded nodes If new node does not fit remove stored node with worst f-value propagate f-value of removed node to parent SMA∗ will regenerate a subtree only when it is needed the path through subtree is unknown, but cost is known 70
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Propeties of SMA∗ Complete?? yes, if there is enough memory for the shortest solution path Time?? same as A∗ if enough memory to store the tree Space?? use available memory Optimal?? yes, if enough memory to store the best solution path In practice, often better than A∗ and IDA∗ trade-off between time and space requirements 71
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Recursive Best First Search 72
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Recursive Best First Search Zerind Arad Sibiu Arad Fagaras Oradea Craiova Sibiu Bucharest Craiova Rimnicu Vilcea Zerind Arad Sibiu Arad Sibiu Bucharest Rimnicu Vilcea Oradea Zerind Arad Sibiu Arad Timisoara Timisoara Timisoara Fagaras Oradea Rimnicu Vilcea Craiova Pitesti Sibiu 646 415 671 526 553 646 671 450 591 646 671 526 553 418 615 607 447 449 447 447 449 449 366 393 366 393 413 413 417 415 366 393 415 450 417 Rimnicu Vilcea Fagaras 447 415 447 447 417 (a) After expanding Arad, Sibiu, and Rimnicu Vilcea (c) After switching back to Rimnicu Vilcea and expanding Pitesti (b) After unwinding back to Sibiu and expanding Fagaras 447 447 ∞ ∞ ∞ 417 417 Pitesti 73
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 75
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Constraint Satisfaction Problem (CSP) Standard search problem: state is a “black box”—any old data structure that supports goal test, eval, successor CSP: state is defined by variables Xi with values from domain Di goal test is a set of constraints specifying allowable combinations of values for subsets of variables Simple example of a formal representation language Allows useful general-purpose algorithms with more power than standard search algorithms 76
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Standard search formulation States are defined by the values assigned so far ♦ Initial state: the empty assignment, { } ♦ Successor function: assign a value to an unassigned variable that does not conflict with current assignment. =⇒ fail if no legal assignments (not fixable!) ♦ Goal test: the current assignment is complete 1) This is the same for all CSPs! 2) Every solution appears at depth n with n variables =⇒ use depth-first search 3) Path is irrelevant, so can also use complete-state formulation 4) b = (n − `)d at depth `, hence n!dn leaves!!!! 77
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Backtracking search Variable assignments are commutative, i.e., [WA = red then NT = green] same as [NT = green then WA = red] Only need to consider assignments to a single variable at each node =⇒ b = d and there are dn leaves Depth-first search for CSPs with single-variable assignments is called backtracking search Backtracking search is the basic uninformed algorithm for CSPs Can solve n-queens for n ≈ 25 78
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Backtracking search function Backtracking-Search(csp) returns solution/failure return Recursive-Backtracking({ },csp) function Recursive-Backtracking(assignment,csp) returns soln/fail- ure if assignment is complete then return assignment var ← Select-Unassigned-Variable(Variables[csp],assignment,csp) for each value in Order-Domain-Values(var,assignment,csp) do if value is consistent with assignment given Constraints[csp] then add {var = value} to assignment result ← Recursive-Backtracking(assignment,csp) if result 6= failure then return result remove {var = value} from assignment return failure 79
Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Summary Uninformed Search Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search Bidirectional Search Informed Search best-first search greedy search A∗ search Iterative Deepening A∗ Memory bounded A∗ Recursive best first Constraint Satisfaction and Backtracking 80

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Lecture 2 Solving Problems by Searching by Marco Chiarandini

  • 1. Lecture 2 Solving Problems by Searching Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Slides by Stuart Russell and Peter Norvig
  • 2. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Last Time Agents are used to provide a consistent viewpoint on various topics in the field AI Essential concepts: Agents intereact with environment by means of sensors and actuators. A rational agent does “the right thing” ≡ maximizes a performance measure è PEAS Environment types: observable, deterministic, episodic, static, discrete, single agent Agent types: table driven (rule based), simple reflex, model-based reflex, goal-based, utility-based, learning agent 2
  • 3. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Structure of Agents Agent = Architecture + Program Architecture operating platform of the agent computer system, specific hardware, possibly OS Program function that implements the mapping from percepts to actions This course is about the program, not the architecture 3
  • 4. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 5
  • 5. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 6
  • 6. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline ♦ Problem-solving agents ♦ Problem types ♦ Problem formulation ♦ Example problems ♦ Basic search algorithms 7
  • 7. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Problem-solving agents Restricted form of general agent: function Simple-Problem-Solving-Agent( 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-State(state,percept) if seq is empty then goal ← Formulate-Goal(state) problem ← Formulate-Problem(state,goal) seq ← Search( problem) action ← Recommendation(seq,state) seq ← Remainder(seq,state) return action Note: this is offline problem solving; solution executed “eyes closed.” Online problem solving involves acting without complete knowledge. 8
  • 8. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 9
  • 9. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 10
  • 10. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem State space Problem formulation A problem is defined by five items: 1. initial state e.g., “at Arad” 2. actions defining the other states, e.g., Go(Arad) 3. transition model res(x, a) e.g., res(In(Arad), Go(Zerind)) = In(Zerind) alternatively: set of action–state pairs: {h(In(Arad), Go(Zerind)), In(Zerind)i, . . .} 4. goal test, can be explicit, e.g., x = “at Bucharest” implicit, e.g., NoDirt(x) 5. 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 11
  • 11. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Selecting a state space Real world is complex ⇒ state space must be abstracted for problem solving (Abstract) state = set of real states (Abstract) action = complex combination of real actions e.g., “Arad → Zerind” represents a complex set of possible routes, detours, rest stops, etc. For guaranteed realizability, any real state “in Arad” must get to some real state “in Zerind” (Abstract) solution = set of real paths that are solutions in the real world Each abstract action should be “easier” than the original problem! Atomic representation 12
  • 12. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Vacuum world state space graph Example R L S S S S R L R L R L S S S S L L L L R R R R states??: integer dirt and robot locations (ignore dirt amounts etc.) actions??: Left, Right, Suck, NoOp transition model??: arcs in the digraph goal test??: no dirt path cost??: 1 per action (0 for NoOp) 13
  • 13. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example: The 8-puzzle 2 Start State Goal State 5 1 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.) transition model??: effect of the actions goal test??: = goal state (given) path cost??: 1 per move [Note: optimal solution of n-Puzzle family is NP-hard] 14
  • 14. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example: robotic assembly R R R P R R 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 15
  • 15. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Problem types Deterministic, fully observable, known, discrete =⇒ state space 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”) 16
  • 16. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example: vacuum world State space, start in #5. Solution?? [Right, Suck] Non-observable, 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 17
  • 17. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example Problems Toy problems vacuum cleaner agent 8-puzzle 8-queens cryptarithmetic missionaries and cannibals Real-world problems route finding traveling salesperson VLSI layout robot navigation assembly sequencing 18
  • 18. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 19
  • 19. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Objectives Formulate appropriate problems in optimization and planning (sequence of actions to achive a goal) as search tasks: initial state, operators, goal test, path cost Know the fundamental search strategies and algorithms uninformed search breadth-first, depth-first, uniform-cost, iterative deepening, bi- directional informed search best-first (greedy, A*), heuristics, memory-bounded Evaluate the suitability of a search strategy for a problem completeness, optimality, time & space complexity 20
  • 20. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Searching for Solutions Traversal of some search space from the initial state to a goal state legal sequence of actions as defined by operators The search can be performed on On a search tree derived from expanding the current state using the possible operators Tree-Search algorithm A graph representing the state space Graph-Search algorithm 22
  • 21. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Search: Terminology 23
  • 22. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Example: Route Finding 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 24
  • 23. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Tree search example Lugoj Arad Arad Oradea Rimnicu Vilcea Zerind Arad Sibiu Arad Fagaras Oradea Timisoara 25
  • 24. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem General tree search 26
  • 25. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Implementation: states vs. nodes A state is a (representation of) a physical configuration A node is a data structure constituting part of a search tree includes state, parent, action, path cost g(x) States do not have parents, children, depth, or path cost! 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 State Node depth = 6 g = 6 state parent, action The Expand function creates new nodes, filling in the various fields using the Transition Model of the problem to create the corresponding states. 27
  • 26. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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(State[node], action, result) Depth[s] ← Depth[node] + 1 add s to successors return successors 28
  • 27. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Search strategies A strategy is defined by picking the order of node expansion 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) 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 path 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 ∞) 29
  • 28. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 30
  • 29. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 Bidirectional Search 31
  • 30. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Breadth-first search Expand shallowest unexpanded node (shortest path in the frontier) A B C D E F G A B C D E F Implementation: fringe is a FIFO queue, i.e., new successors go at end 32
  • 31. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of breadth-first search Complete?? Yes (if b is finite) Optimal?? Yes (if cost = 1 per step); not optimal in general Time?? 1 + b + b2 + b3 + . . . + bd + b(bd − 1) = O(bd+1 ), i.e., exp. in d Space?? bd−1 + bd = O(bd ) (explored + frontier) Space is the big problem; can easily generate nodes at 100MB/sec so 24hrs = 8640GB. 33
  • 32. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Uniform-cost search Expand first least-cost path (Equivalent to breadth-first if step costs all equal) Implementation: fringe = priority queue ordered by path cost, lowest first, Complete?? Yes, if step cost ≥ Optimal??Yes—nodes expanded in increasing order of g(n) Time?? # of nodes with g ≤ cost of optimal solution, O(b1+dC∗ /e ) where C∗ is the cost of the optimal solution Space??# of nodes with g ≤ cost of optimal solution, O(b1+dC∗ /e ) 34
  • 33. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Depth-first search Expand deepest unexpanded node A B C D E F G H I J K L M N O Implementation: fringe = LIFO queue, i.e., put successors at front 35
  • 34. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of depth-first search Complete?? No: fails in infinite-depth spaces, or spaces with loops Modify to avoid repeated states along path ⇒ complete in finite spaces Optimal?? No Time?? O(bm ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space?? O(bm), i.e., linear space! 36
  • 35. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 37
  • 36. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 38
  • 37. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Iterative deepening search 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 O I A B C D E F G H I J K L M N O 39
  • 38. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of iterative deepening search Complete?? Yes Optimal?? Yes, if step cost = 1 Can be modified to explore uniform-cost tree Time?? (d + 1)b0 + db1 + (d − 1)b2 + . . . + bd = O(bd ) Space?? O(bd) Numerical comparison in time for b = 10 and d = 5, solution at far right leaf: NIDS) = 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 Iterative lenghtening not as successul as IDS 40
  • 39. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Bidirectional Search Search simultaneously (using breadth-first search) from goal to start from start to goal Stop when the two search trees intersects 41
  • 40. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Difficulties in Bidirectional Search If applicable, may lead to substantial savings Predecessors of a (goal) state must be generated Not always possible, eg. when we do not know the optimal state explicitly Search must be coordinated between the two search processes. What if many goal states? One search must keep all nodes in memory 42
  • 41. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 bd1+C∗ /e bm bl bd Space bd+1 bd1+C∗ /e bm bl bd Optimal? Yes∗ Yes No No Yes∗ 43
  • 42. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem 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 Graph search can be exponentially more efficient than tree search 46
  • 43. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 47
  • 44. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Review: 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] applied to State(node) succeeds return node fringe ← InsertAll(Expand(node, problem), fringe) A strategy is defined by picking the order of node expansion 48
  • 45. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Informed search strategy Informed strategies use agent’s background information about the problem map, costs of actions, approximation of solutions, ... best-first search greedy search A∗ search local search (not in this course) Hill-climbing Simulated annealing Genetic algorithms Local search in continuous spaces 49
  • 46. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Best-first search Idea: use an evaluation function for each node – estimate of “desirability” ⇒ Expand most desirable unexpanded node Implementation: fringe is a queue sorted in decreasing order of desirability Special cases: greedy search A∗ search 50
  • 47. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Romania with step costs in km Bucharest Giurgiu Urziceni Hirsova Eforie Neamt Oradea Zerind Arad Timisoara Lugoj Mehadia Dobreta Craiova Sibiu Fagaras Pitesti Rimnicu Vilcea Vaslui Iasi Straight−line distance to Bucharest 0 160 242 161 77 151 241 366 193 178 253 329 80 199 244 380 226 234 374 98 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 51
  • 48. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Greedy search Evaluation function h(n) (heuristic) = estimate of cost from n to the closest goal E.g., hSLD(n) = straight-line distance from n to Bucharest Greedy search expands the node that appears to be closest to goal 52
  • 49. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Greedy search example Arad 366 53
  • 50. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of greedy search Complete?? No–can get stuck in loops, e.g., from Iasi to Fargas Iasi → Neamt → Iasi → Neamt → Complete in finite space with repeated-state checking Optimal?? No Time?? O(bm ), but a good heuristic can give dramatic improvement Space?? O(bm ) 54
  • 51. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem A∗ search Idea: avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost to goal from n f(n) = estimated total cost of path through n to goal A∗ search uses an admissible heuristic i.e., h(n) ≤ h∗ (n) where h∗ (n) is the true cost from n. (Also require h(n) ≥ 0, so h(G) = 0 for any goal G.) E.g., hSLD(n) never overestimates the actual road distance Theorem: A∗ search is optimal 55
  • 52. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem A∗ search example Zerind Arad Sibiu Arad Timisoara Sibiu Bucharest Rimnicu Vilcea Fagaras Oradea Craiova Pitesti Sibiu Bucharest Craiova Rimnicu Vilcea 418=418+0 447=118+329 449=75+374 646=280+366 591=338+253 450=450+0 526=366+160 553=300+253 615=455+160 607=414+193 671=291+380 56
  • 53. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Optimality of A∗ (standard proof) Suppose some suboptimal goal G2 has been generated and is in the queue. Let n be an unexpanded node on a shortest path to an optimal goal G1. G n G2 Start f(G2) = g(G2) since h(G2) = 0 g(G1) since G2 is suboptimal ≥ f(n) since h is admissible Since f(G2) f(n), A∗ will never select G2 for expansion 57
  • 54. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Optimality of A∗ (more useful) Lemma: A∗ expands nodes in order of increasing f value∗ Gradually adds “f-contours” of nodes (cf. breadth-first adds layers) Contour i has all nodes with f = fi, where fi fi+1 O Z A T L M D C R F P G B U H E V I N 380 400 420 S 58
  • 55. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Astar vs. Depth search 59
  • 56. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of A∗ Complete?? Yes, unless there are infinitely many nodes with f ≤ f(G) Optimal?? Yes—cannot expand fi+1 until fi is finished A∗ expands all nodes with f(n) C∗ A∗ expands some nodes with f(n) = C∗ A∗ expands no nodes with f(n) C∗ Time?? Exponential in [relative error in h × length of sol.] Space?? Keeps all nodes in memory 60
  • 57. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Proof of lemma: Consistency A heuristic is consistent if n c(n,a,n’) h(n’) h(n) G n’ h(n) ≤ c(n, a, n0 ) + h(n0 ) If h is consistent, we have f(n0 ) = g(n0 ) + h(n0 ) = g(n) + c(n, a, n0 ) + h(n0 ) ≥ g(n) + h(n) = f(n) I.e., f(n) is nondecreasing along any path. 61
  • 58. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Admissible heuristics E.g., for the 8-puzzle: h1(n) = number of misplaced tiles h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) 2 Start State Goal State 5 1 3 4 6 7 8 5 1 2 3 4 6 7 8 5 h1(S) =?? h2(S) =?? 62
  • 59. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Admissible heuristics E.g., for the 8-puzzle: h1(n) = number of misplaced tiles h2(n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) 2 Start State Goal State 5 1 3 4 6 7 8 5 1 2 3 4 6 7 8 5 h1(S) =?? 6 h2(S) =?? 4+0+3+3+1+0+2+1 = 14 63
  • 60. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Dominance If h2(n) ≥ h1(n) for all n (both admissible) then h2 dominates h1 and is better for search Typical search costs: d = 14 IDS = 3,473,941 nodes A∗ (h1) = 539 nodes A∗ (h2) = 113 nodes d = 24 IDS ≈ 54,000,000,000 nodes A∗ (h1) = 39,135 nodes A∗ (h2) = 1,641 nodes Given any admissible heuristics ha, hb, h(n) = max(ha(n), hb(n)) is also admissible and dominates ha, hb 64
  • 61. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Relaxed problems Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution Key point: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem 65
  • 62. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Memory-Bounded Heuristic Search Try to reduce memory needs Take advantage of heuristic to improve performance Iterative-deepening A∗ (IDA∗ ) SMA∗ 67
  • 63. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Iterative Deepening A∗ Uniformed Iterative Deepening (repetition) depth-first search where the max depth is iteratively increased IDA∗ depth-first search, but only nodes with f-cost less than or equal to smallest f-cost of nodes expanded at last iteration was the best search algorithm for many practical problems 68
  • 64. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Properties of IDA∗ Complete?? Yes Time complexity?? Still exponential Space complexity?? linear Optimal?? Yes. Also optimal in the absence of monotonicity 69
  • 65. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Simple Memory-Bounded A∗ Use all available memory Follow A∗ algorithm and fill memory with new expanded nodes If new node does not fit remove stored node with worst f-value propagate f-value of removed node to parent SMA∗ will regenerate a subtree only when it is needed the path through subtree is unknown, but cost is known 70
  • 66. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Propeties of SMA∗ Complete?? yes, if there is enough memory for the shortest solution path Time?? same as A∗ if enough memory to store the tree Space?? use available memory Optimal?? yes, if enough memory to store the best solution path In practice, often better than A∗ and IDA∗ trade-off between time and space requirements 71
  • 67. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Recursive Best First Search 72
  • 68. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Recursive Best First Search Zerind Arad Sibiu Arad Fagaras Oradea Craiova Sibiu Bucharest Craiova Rimnicu Vilcea Zerind Arad Sibiu Arad Sibiu Bucharest Rimnicu Vilcea Oradea Zerind Arad Sibiu Arad Timisoara Timisoara Timisoara Fagaras Oradea Rimnicu Vilcea Craiova Pitesti Sibiu 646 415 671 526 553 646 671 450 591 646 671 526 553 418 615 607 447 449 447 447 449 449 366 393 366 393 413 413 417 415 366 393 415 450 417 Rimnicu Vilcea Fagaras 447 415 447 447 417 (a) After expanding Arad, Sibiu, and Rimnicu Vilcea (c) After switching back to Rimnicu Vilcea and expanding Pitesti (b) After unwinding back to Sibiu and expanding Fagaras 447 447 ∞ ∞ ∞ 417 417 Pitesti 73
  • 69. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Outline 1. Problem Solving Agents 2. Search 3. Uninformed search algorithms 4. Informed search algorithms 5. Constraint Satisfaction Problem 75
  • 70. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Constraint Satisfaction Problem (CSP) Standard search problem: state is a “black box”—any old data structure that supports goal test, eval, successor CSP: state is defined by variables Xi with values from domain Di goal test is a set of constraints specifying allowable combinations of values for subsets of variables Simple example of a formal representation language Allows useful general-purpose algorithms with more power than standard search algorithms 76
  • 71. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Standard search formulation States are defined by the values assigned so far ♦ Initial state: the empty assignment, { } ♦ Successor function: assign a value to an unassigned variable that does not conflict with current assignment. =⇒ fail if no legal assignments (not fixable!) ♦ Goal test: the current assignment is complete 1) This is the same for all CSPs! 2) Every solution appears at depth n with n variables =⇒ use depth-first search 3) Path is irrelevant, so can also use complete-state formulation 4) b = (n − `)d at depth `, hence n!dn leaves!!!! 77
  • 72. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Backtracking search Variable assignments are commutative, i.e., [WA = red then NT = green] same as [NT = green then WA = red] Only need to consider assignments to a single variable at each node =⇒ b = d and there are dn leaves Depth-first search for CSPs with single-variable assignments is called backtracking search Backtracking search is the basic uninformed algorithm for CSPs Can solve n-queens for n ≈ 25 78
  • 73. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Backtracking search function Backtracking-Search(csp) returns solution/failure return Recursive-Backtracking({ },csp) function Recursive-Backtracking(assignment,csp) returns soln/fail- ure if assignment is complete then return assignment var ← Select-Unassigned-Variable(Variables[csp],assignment,csp) for each value in Order-Domain-Values(var,assignment,csp) do if value is consistent with assignment given Constraints[csp] then add {var = value} to assignment result ← Recursive-Backtracking(assignment,csp) if result 6= failure then return result remove {var = value} from assignment return failure 79
  • 74. Problem Solving Agents Search Uninformed search algorithms Informed search algorithms Constraint Satisfaction Problem Summary Uninformed Search Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search Bidirectional Search Informed Search best-first search greedy search A∗ search Iterative Deepening A∗ Memory bounded A∗ Recursive best first Constraint Satisfaction and Backtracking 80