AI_03_Solving Problems by Searching.pptx
- 1. LECTURE 3 Solving Problems by Searching Instructor : Yousef Aburawi Cs411 -Artificial Intelligence Misurata University Faculty of Information Technology Spring 2022/2023
- 2. Outline Problem-solving agents. Search problems and solutions. Example problems. Formulating problems. Basic search algorithms. 2
- 3. Problem-Solving Agents When the correct action to take is not immediately obvious, an agent may need to plan ahead: to consider a sequence of actions that form a path to a goal state. Such an agent is called a problem-solving agent, and the computational process it undertakes is called search. In this chapter, we consider only the simplest environments: episodic, single agent, fully observable, deterministic, static, discrete, and known. Describing one kind of goal-based agent called a problem-solving agent. 3
- 4. Problem-Solving Agents (2) Four-phase problem-solving process: Goal formulation: What are the successful world states. Problem formulation: What actions and states to reach the goal. Search: Determine the possible sequence of actions that lead to the states of known values and then choosing the best sequence that reaches the goal. Execution: Give the solution perform the actions. The state space can be represented as a graph in which the vertices are states and the directed edges between them are actions, (as shown next slide). 4
- 6. e.g. Romania (2) 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, … . 6
- 7. Search Problems and Solutions A search problem can be defined formally as follows: State space: A set of possible states that the environment can be in. The initial state that the agent starts in. A set of one or more goal states. The actions available to the agent. A transition model, which describes what each action does. An action cost function. A sequence of actions forms a path. A solution is a path from the initial state to a goal state. An optimal solution has the lowest path cost among all Solutions. 7
- 8. Example Problems A standardized problem is intended to illustrate or exercise various problem-solving methods. It can be given a concise, exact description and hence is suitable as a benchmark for researchers to compare the performance of algorithms. A real-world problem, such as robot navigation, is one whose solutions people actually use, and whose formulation is idiosyncratic, not standardized, because. for example, each robot has different sensors that produce different data. 8
- 9. Problem Formulation A problem is defined by: An initial state, e.g. Arad. Successor function: S(X)= set of action-state pairs. e.g. S(Arad)={<Arad Zerind, Zerind>,…}. initial state + successor function = state space. Goal test, can be Explicit : e.g. x=‘at Bucharest’. Implicit : e.g. checkmate(x). Path cost (additive). e.g. sum of distances, number of actions executed, … c(x,a,y) is the step cost, assumed to be >= 0. 9
- 10. Problem Formulation (2) A solution is a sequence of actions from initial to goal state. Optimal solution has the lowest path cost. 10
- 11. e.g. Vacuum World States?? 8 states: 2 locations with or without dirt. Initial state?? Any state can be initial. Actions?? {Left, Right, Suck}. Goal test?? Check whether squares are clean. Path cost?? Number of actions to reach goal. 11
- 12. e.g. 8-Puzzle 12 States?? The location of each tile. Initial state?? Any state can be initial. Actions?? {Left, Right, Up, Down}. Goal test?? Check whether goal configuration is reached. Path cost?? Number of actions to reach goal.
- 13. Search Algorithms A search algorithm takes a search problem as input and returns a solution, or an indication of failure. We will study algorithms that superimpose a search tree over the state-space graph, forming various paths from the initial state, trying to find a path that reaches a goal state. Each node in the search tree corresponds to a state in the state space and the edges in the search tree correspond to actions. The root of the tree corresponds to the initial state of the problem. 13
- 14. State Space Vs. Search Tree A state is a (representation of) a physical configuration. The state space describes the (possibly infinite) set of states in the world, and the actions that allow transitions from one state to another. The search tree describes paths between these states, reaching towards the goal. The search tree may have multiple paths to (and thus multiple nodes for) any given state, but each node in the tree has a unique path back to the root (as in all trees). 14
- 15. Search Tree A node is a data structure belong to a search tree. A node has a parent, children, … , and includes path cost, depth, … Here node= <state, parent-node, action, path-cost, depth> FRINGE = contains generated nodes which are not yet expanded. White nodes with black outline. 15
- 16. Search Tree (2) 16 The root node of the search tree is at the initial state, Arad. We can expand the node, by considering the available ACTIONS for that state, using the RESULT function to see where those actions lead to, and generating a new node (called a child node or successor node) for each of the resulting states. Each child node has Arad as its parent node.
- 17. e.g. Romania 17
- 18. Graph Tree 18
- 19. Uninformed Search Strategies (blind search) = use only information available in problem definition. When strategies can determine whether one non-goal state is better than another informed search. Categories defined by expansion algorithm: Breadth-first search. Uniform-cost search. Depth-first search. Depth-limited search. Iterative deepening search. Bidirectional search. 19
- 20. Breadth-First Search When all actions have the same cost, an appropriate strategy is breadth-first search. The root node is expanded first, then all the successors of the root node are expanded next, then their successors, and so on. Also called best-First-Search. Implementation: fringe is a FIFO queue. 20
- 21. Breadth-First Search on a Simple Binary Tree 21
- 22. Uniform-Cost Search When actions have different costs, an obvious choice is to use best-first search where the evaluation function is the cost of the path from the root to the current node. This is called Dijkstra’s algorithm by the theoretical computer science community, and uniform-cost search by the AI community. Extension of BF-search: Expand node with lowest path cost. Implementation: fringe = queue ordered by path cost. UC-search is the same as BF-search when all step-costs are equal. 23
- 23. Depth-First Search Expands the deepest node in the frontier first. Implementation: fringe is a LIFO queue (=stack). Depth-first search is not cost-optimal; it returns the first solution it finds, even if it is not cheapest. 24
- 24. 25 Depth-First Search on a Simple Binary Tree A dozen steps (left to right, top to bottom) in the progress of a depth- first search on a binary tree from start state A to goal M.
- 25. Depth-Limited Search Depth-Limited search is Depth-First search with depth limit 𝑙. i.e. nodes at depth 𝑙 have no successors. Problem knowledge can be used. Solves the infinite-path problem. If 𝑙 < 𝑑 then incompleteness results. If 𝑙 > 𝑑 then not optimal. Time complexity: 𝑂 𝑏𝑙 Space complexity: 𝑂 𝑏𝑙 26
- 26. Iterative Deepening Search A general strategy to find best depth limit l. Goals is found at depth d, the depth of the shallowest goal-node. Iterative deepening search solves the problem of picking a good value for 𝑙 by trying all values: first 0, then 1, then 2, and so on—until either a solution is found, or the depth- limited search returns the failure value. Combines benefits of Depth-First and Breadth-First search. 27
- 27. 28 Iterative Deepening Search on a Simple Binary Tree
- 28. 29 Iterative Deepening Search on a Simple Binary Tree (2)
- 29. Bidirectional search bidirectional search simultaneously searches forward from the initial state and backwards from the goal state(s), hoping that the two searches will meet. The motivation is: 𝑏𝑑 2 + 𝑏𝑑 2 ≠ 𝑏𝑑 Check whether the node belongs to the other fringe before expansion. Space complexity is the most significant weakness. Complete and optimal if both searches are BF. The predecessor of each node should be efficiently computable. When actions are easily reversible. 30
- 30. Summary Search algorithms introduced that an agent can use to select action sequences in a wide variety of environments - as long as they are episodic, single-agent, fully observable, deterministic, static, discrete, and completely known. There are tradeoffs to be made between the amount of time the search takes, the amount of memory available, and the quality of the solution. Before an agent can start searching, a well-defined problem must be formulated. A problem consists of five parts: the initial state, a set of actions, a transition model describing the results of those actions, a set of goal states, and an action cost function. 31
- 31. Summary (2) The environment of the problem is represented by a state space graph. A path through the state space (a sequence of actions) from the initial state to a goal state is a solution. Search algorithms generally treat states and actions as atomic, without any internal structure. Search algorithms are judged on the basis of completeness, cost optimality, time complexity, and space complexity. Uninformed search methods have access only to the problem definition. Algorithms build a search tree in an attempt to find a solution. Algorithms differ based on which node they expand first: 32
- 32. Summary (3) BEST-FIRST SEARCH selects nodes for expansion using to an evaluation function. BREADTH-FIRST SEARCH expands the shallowest nodes first; it is complete, optimal for unit action costs, but has exponential space complexity. UNIFORM-COST SEARCH expands the node with lowest path cost, 𝑔(𝑛), and is optimal for general action costs. DEPTH-FIRST SEARCH expands the deepest unexpanded node first. It is neither complete nor optimal, but has linear space complexity. Depth-limited search adds a depth bound. ITERATIVE DEEPENING SEARCH calls depth-first search with increasing depth limits until a goal is found. It is complete when full cycle checking is done, optimal for unit action costs, has time complexity comparable to breadth-first search, and has linear space complexity. 33
- 33. Summary (4) BIDIRECTIONAL SEARCH expands two frontiers, one around the initial state and one around the goal, stopping when the two frontiers meet. 34
- 34. Readings Chapters 3 of Textbox. 1-35
- 35. The End
Editor's Notes
- #16: FRINGE (الأطراف)