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Lecture 3 problem solving
This document discusses problem solving by searching. It defines the key components of well-defined problems including the initial state, actions, transition model, goal test, and path cost. It provides examples of problems that can be formulated as searches, such as the 8-puzzle, route finding, and the traveling salesperson problem. It then covers different search strategies including uninformed searches like breadth-first, depth-first, and iterative deepening as well as informed searches like greedy best-first and A* that use heuristics to guide the search.
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Lecture 3 problem solving
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- 3. Well-defined problems andsolutions • A problem can be defined formally by five components: • Initial state • The initial state that the agent starts in • Ex: In(Arad) • Actions • A description of the possible actions available to the agent • Given a particular states, ACTIONS(s) returns the set of actions that can be executed in s • Actions(s) {a1,a2,a3,…,an} • For example, from the state In(Arad), the applicable actions are{Go(Sibiu), Go(Timisoara), Go(Zerind)}.
- 4. Well-defined problems andsolutions • Transition model • A description of what each action does • a function RESULT(s,a) that returns the state that results from doing action a in state s Result (s,a) s’ • We also use the term successor to refer to any state reachable from a given state by a single action • Ex: RESULT(In(Arad),Go(Zerind)) =In(Zerind) • The initial state, actions, and transition model implicitly define the state space of the problem • Goal test • which determines whether a given state is a goal state • The agent’s goal in Romania is the singleton set{In(Bucharest)} • GoalTest(s) T|F
- 5. Well-defined problems andsolutions • Path cost • A path cost function that assigns a numeric cost to each path pathCost(ss1s2s3) n (sum of costs) • For the agent trying to get to Bucharest, time is of the essence, so the cost of a path might be its length in kilometers • The step cost of taking action a in state s to reach state s’ is denoted by c(s,a,s’) • Stepcost (s,a,s’) n (non negative number) • We assume that step costs are nonnegative • Solution quality is measured by the path cost function, and an optimal solution has the lowest path cost among all solutions
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- 7. EXAMPLE PROBLEMS • Toyproblems • The first example we examine is the vacuum world first introduced in Chapter 2 • This can be formulated as a problem as follows: • States: • The state is determined by both the agent location and the dirt locations. The agent is in one of two locations, each of which might or might not contain dirt. Thus, there are 2×22=8 possible world states. A larger environment with n locations has n·2n states. • Initial state: • Any state can be designated as the initial state. • Actions: • In this simple environment, each state has just three actions: Left, Right, and Suck. Larger environments might also include Up and Down.
- 8. EXAMPLE PROBLEMS • Transitionmodel: • The actions have their expected effects, except that moving Left in the leftmost square, moving Right in the rightmost square, and Sucking in a clean square have no effect. The complete state space is shown in Figure 3.3.
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- 10. EXAMPLE PROBLEMS • Goaltest: • This checks whether all the squares are clean. • Path cost: • Each step costs 1, so the path cost is the number of steps in the path.
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- 12. 8-puzzle • States: Astate description specifies the location of each of the eight tiles and the blank in one of the nine squares. • Initial state: Any state can be designated as the initial state. • Actions: The simplest formulation defines the actions as movements of the blank space Left, Right, Up, or Down. Different subsets of these are possible depending on where the blank is. • Transition model: Given a state and action, this returns the resulting state; for example, if we apply Left to the start state in Figure, the resulting state has the 5 and the blank switched. • Goal test: This checks whether the state matches the goal configuration shown in Figure • Path cost: Each step costs 1, so the path cost is the number of steps in the path.
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- 14. 8-queens problem • Thereare two main kinds of formulation • An incremental formulation • A complete-state formulation • The first incremental formulation • States: Any arrangement of 0 to 8 queens on the board is a state. • Initial state: No queens on the board. • Actions: Add a queen to any empty square. • Transition model: Returns the board with a queen added to the specified square. • Goal test: 8 queens are on the board, none attacked.
- 15. Real-world problems • Route-findingproblem • Consider the airline travel problems that must be solved by a travel- planning Web site
- 16. Airline travel problems •States: Each state obviously includes a location (e.g., an airport) and the current time. Furthermore, because the cost of an action (a flight segment) may depend on previous segments, their fare bases, and their status as domestic or international, the state must record extra information about these “historical” aspects. • Initial state: This is specified by the user’s query. • Actions: Take any flight from the current location, in any seat class, leaving after the current time, leaving enough time for within-airport transfer if needed. • Transition model: The state resulting from taking a flight will have the flight’s destination as the current location and the flight’s arrival time as the current time. • Goal test: Are we at the final destination specified by the user? • Path cost: This depends on monetary cost, waiting time, flight time, customs and immigration procedures, seat quality, time of day, type of airplane, frequent-flyer mileage awards, and so on.
- 17. Real-world problems • Touringproblems • Consider, for example, the problem “Visit every city in Figure at least once, starting and ending in Bucharest.” • Initial state would be In(Bucharest),Visited({Bucharest}) • A typical intermediate state would be In(Vaslui), Visited({Bucharest,Urziceni,Vaslui})
- 19. Real-world problems • Travelingsalesperson problem • The traveling salesperson problem (TSP) is a touring problem in which each city must be visited exactly once. The aim is to find the shortest tour. • A VLSI layout problem • positioning millions of components and connections on a chip to minimize area, minimize circuit delays, minimize stray capacitances, and maximize manufacturing yield • Robot navigation • Automatic assembly sequencing • protein design
- 20. SEARCHING FOR SOLUTIONS •A solution is an action sequence • First few steps in growing the search tree for finding a route from Arad to Bucharest • The root node of the tree corresponds to the initial state, In(Arad)
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- 25. SEARCHING FOR SOLUTIONS •A search tree with the initial state at the root; the branches are actions and the nodes correspond to states in the state space of the problem • leaf node, that is, a node with no children in the tree • The set of all leaf nodes available for expansion at any given point is called the frontier
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- 27. Infrastructure for searchalgorithms
- 28. Measuring problem-solving performance •Completeness: Is the algorithm guaranteed to find a solution when there is one? • Optimality: Does the strategy find the optimal solution, • Time complexity: How long does it take to find a solution? • Space complexity: How much memory is needed to perform the search?
- 29. UNINFORMED SEARCH STRATEGIES •Uninformed search (also called blind search) • The term means that the strategies have no additional information about states beyond that provided in the problem definition
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- 32. Breadth-first search • complete—ifthe shallowest goal node is at some finite depth d, breadth-first search will eventually find it after generating all shallower nodes (provided the branching factor b is finite). • Optimal - breadth-first search is optimal if the path cost is a non decreasing function of the depth of the node. The most common such scenario is that all actions have the same cost.
- 33. Breadth-first search • Timecomplexity - Now suppose that the solution is at depth d. In the worst case, it is the last node generated at that level. Then the total number of nodes generated is If the algorithm were to apply the goal test to nodes when selected for expansion, rather than when generated, the whole layer of nodes at depth d would be expanded before the goal was detected and the time complexity would be O(bd+1).
- 34. Breadth-first search • spacecomplexity: • For any kind of graph search, which stores every expanded node in the explored set, the space complexity is always within a factor of b of the time complexity. • There will be O(bd−1)nodes in the explored set and O(bd)nodes in the frontier • so the space complexity is O(bd)
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- 36. Breadth-first search • Thememory requirements are a bigger problem for breadth-first search than is the execution time • If your problem has a solution at depth 16, then (given our assumptions) it will take about 350 years for breadth-first search (or indeed any uninformed search) to find it • Exponential-complexity search problems cannot be solved by uninformed methods for any but the smallest instances
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- 38. Uniform-cost search • Insteadof expanding the shallowest node, uniform-cost search expands the node n with the lowest path cost g(n)
- 39. Uniform-cost search • problemis to get from Sibiu to Bucharest Sibiu Rimnicu Vilcea 80 Fagaras 99
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- 44. Depth-first search • Depth-firstsearch always expands the deepest node in the current frontier of the search tree. • whereas breadth-first-search uses a FIFO queue, depth-first search uses a LIFO queue
- 46. Depth First Search(DFS) • Example 4.3 • Explore the following state space using DFS to find the traversal for reaching the goal node G.
- 47. Depth First Search(DFS)
- 48. Depth First Search(DFS) • Tree version of DFS is not complete. • nonoptimal. • Better space complexity than BFS • b number of children nodes for each node • if a goal node is at level d • we expand only bd amount of nodes • Thus space complexity of DFS is of order bd ,i.e. O(bd). • Time Complexity • Time complexity of DFS= O(b^d)
- 49. Depth First Search(DFS)
- 50. Depth-limited search • Depthlimited search is a version of depth-first search. In this case, depth search is applied for pre-decided depth such as 4. If the goal node is within the first four levels, we will be able to find a solution as in DFS. This algorithm has similar features to DFS.
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- 52. Iterative deepening depth-firstsearch • Iterative deepening search(or iterative deepening depth-first search) is a general strategy, often used in combination with depth-first tree search, that finds the best depth limit. It does this by gradually increasing the limit—first 0, then 1, then 2, and so on—until a goal is found.
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- 55. Bidirectional search • Theidea behind bidirectional search is to run two simultaneous searches—one forward from the initial state and the other backward from the goal—hoping that the two searches meet in the middle
- 56. Comparing uninformed searchstrategies
- 57. INFORMED(HEURISTIC)SEARCHSTRATEGIES • This sectionshows how an informed search strategy—one that uses problem-specific knowledge beyond the definition of the problem itself— can find solutions more efficiently than can an uninformed strategy. • The general approach we consider is called best-first search. Best-first search is an instance of the general TREE-SEARCH or GRAPH-SEARCH algorithm in which a node is selected for expansion based on an evaluation function, f(n) • Most best-first algorithms include as a component of f a heuristic function, denoted h(n) • h(n)=estimated cost of the cheapest path from the state at node n to a goal state.
- 58. Greedy best-first search •Greedy best-first search tries to expand the node that is closest to the goal, on the grounds that this is likely to lead to a solution quickly • f(n)=h(n) • we use the straight line distance heuristic, which we will call hSLD
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- 60. Greedy best-first search •A greedy best-first search using hSLD to find a path from Arad to Bucharest
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- 63. Greedy best-first search •This is not optimal • The path via Sibiu and Fagaras to Bucharest is 32 kilometers longer than the path through Rimnicu Vilcea and Pitesti • Greedy best-first tree search is also incomplete • Consider the problem of getting from Iasi to Fagaras. The heuristic suggests that Neamt be expanded first because it is closest to Fagaras, but it is a dead end • The solution is to go first to Vaslui—a step that is actually farther from the goal according to the heuristic—and then to continue to Urziceni, Bucharest, and Fagaras • The worst-case time and space complexity for the tree version is O(bm), where m is the maximum depth of the search space
- 64. A* search • Themost widely known form of best-first search is called A∗ search • It evaluates nodes by combining g(n), the cost to reach the node, and h(n), the cost to get from the node to the goal: • f(n)=g(n)+h(n) • g(n) gives the path cost from the start node to node n • h(n) is the estimated cost of the cheapest path from n to the goal • f(n)=estimated cost of the cheapest solution through n • A ∗ search is both complete and optimal
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- 72. Problem-solving works when: •Fully observable • Known • Discrete • Deterministic • Static
- 73. Summary • Well-defined problemsand solutions • Searching for solutions