# Artificial Intelligence Search I

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1 Content: Search I Artificial Intelligence Search I Lecture 3 Introduction Knowledge, Problem Types and Problem Formulation Search Strategies General Search Problem 4 Criteria for Evaluating Search Strategies Blind Search Strategies: BFS, UCS, DFS, DLS, IDS, BDS Comparing Search Strategies Class Activity 1: Various Blind Searches workouts for CMU Traffic problem Introduction to Search Introduction to Search: Popular Classical Problem Domains Search is one of the most powerful approaches to problem solving in AI Search is a universal problem solving mechanism that Systematically explores the alternatives Finds the sequence of steps towards a solution Problem Space Hypothesis (Allen Newell, SOAR: An Architecture for General Intelligence.) All goal-oriented symbolic activities occur in a problem space Search in a problem space is claimed to be a completely general model of intelligence 8-puzzle Tower of Hanoi Missionaries and Cannibals Water Jug Vacuum World Wumpus World Block World Travelling Salesperson Maze Crossword Puzzle Crypt-arithmetic Wheel of Fortune Chess, Bridge, etc Knowledge & Problem Types Vacuum World Domain as an illustration Knowledge & Problem Types 4 Problem Types Let the world be consist of only 2 locations - Left and Right Box Each location may contain dirt The agent may be in either box There are 8 possible states The agent can have 3 possible actions - Left, Right and Suck Fig 4.1 The 8 possible states of a Vacuum World Single-state problems exact state known effects of actions known eg. In vacuum world, if the initial state is 5, to achieve the goal, do action sequence [Right, Suck] Multiple-state problems one of a set of states effects of actions known eg. In vacuum world, where there is no sensors, the agent knows that there are 8 initial states, it can be calculated that an action of Right will achieve state {2, 4, 6, 8} and the agent can discover that the action sequence [Right, Suck, Left, Suck] is guaranteed to each the goal Contingency problems limited sensing conditional effects of actions More complex algorithms involving planning eg 1. In vacuum world, adding a simple Sense_Dirt, to use before the action Suck. eg 2. Most of us keep our eyes open while walking, Exploration problems execution reveals states needs to experiment in order to survive Hardest task faces by an intel-agent. eg 1. Mars Pathfinder eg 2. Robot World Cup - Robocup

2 Knowledge & Problem Types State Space for Vacuum World as a SS & MS problem Defining a Search Problem State Space: described by an initial space and the set of possible actions available (operators). A path is any sequence of actions that lead from one state to another. Fig 4.2 Single State Search Space Multiple-State Search Space Fig 4.3 Goal test: applicable to a single state problem to determine if it is the goal state. Path cost: relevant if more than one path leads to the goal, and we want the shortest path. (1) Vacuum World as a Single-state problem (2) Vacuum World as a Multiple-state problem Fig 4.4 Fig 4.5 Initial State: one of the 8 states shown above. Operators move Left, move Right, Suck. Goal Test: no dirt in any square. Path cost: each action costs 1. State sets: subsets of state 1-8 shown in Fig 4.1 or Fig 4.2 Operators move Left, move Right, Suck. Goal Test: all states in state set have no dirt. Path cost: each action costs 1. (3) 8-puzzle problem (4) 8-queens problem Initial State: The location of each of the 8 tiles in one of the nine squares Operators: blank moves (1) Left (2) Right (3) Up (4) Down Goal Test: state matches the goal configuration Fig 4.6 Path cost: each step costs 1, total path cost = no. of steps Initial State: Any arrangement of 0 to 8 queens on board. Operators: add a queen to any square. Fig 4.7 Goal Test: 8 queens on board, none attacked. Path cost: not applicable or Zero (because only the final state counts, search cost might be of interest).

3 (5) Cryptarithmetic Real-world Problems Route Finding - computer networks, automated travel advisory systems, airline travel planning. VLSI Layout - A typical VLSI chip can have as many as a million gates, and the positioning and connections of every gate are crucial to the successful operation of the chip. Fig 4.8 Initial State: a cryptarithmetic puzzle with some letters replaced by digits. Operators: replace all occurences of a letter with a non-repeating digit. Goal Test: puzzle contains only digits, and represents a correct sum. Path cost: not applicable or 0 (because all solutions equally valid). Robot Navigation - rescue operations Mars Pathfinder - search for Martians or signs of intelligent lifeforms Time/Exam Tables Search Strategies General Search Problem General Search Problem Criteria for evaluating search strategies Blind (un-informed) search strategies Breadth-first search Uniform cost search Depth-first search Depth-limited search Iterative deepening search Bi-directional search Comparing search strategies Heuristic (informed) search strategies Fig 4.9 Fig 4.10 Criteria for Evaluating Search Strategies BS1. Breadth-First Search Each of the search strategies are evaluated based on: Completeness: is the strategy guaranteed to find a solution when there is one? Time complexity: how long does it take to find a solution Space complexity: how much memory does it need to perform the search? Optimality: does the strategy find the highest-quality solution when there are several solutions? One of the simplest search strategy Time and Space complex Cannot be use to solve any but the smallest problem, see next page for a simulation. Time complexity: b d Space complexity: b d (b - branching factor, d - depth) Fig 4.11 Breadth-first search tress after 0, 1, 2, and 3 node expansions (b=2, d=2)

4 BS1. Breadth-First Search (cont) BS1. Breadth-First Search (cont) Time and Memory requirements for a breadth-first search. Fig 4.12 The figures shown assume (1) branching factor b=10; (2) 1000 nodes/second; (3) 100 bytes/node Fig 4.13 Breadth-first Tree Search (Numbers refer to order visited in search) BS2. Uniform Cost Search BS2. Uniform Cost Search (cont) BFS finds the shallowest goal state. Uniform cost search modifies the BFS by expanding ONLY the lowest cost node (as measured by the path cost g(n)) The cost of a path must never decrease as we traverse the path, ie. no negative cost should in the problem domain Time complexity:b d Space complexity: b d Fig 4.14 A route finding problem. (a) The state space, showing the cost for each operator. (b) Progression of the search. Each node is labelled with a numeric path cost g(n). At the final step, the goal node with g=10 is selected

5 BS3. Depth-First Search BS3. Depth-First Search (cont) DFS always expands one of the nodes at the deepest level of the tree. The search only go back once it hits a dead end (a nongoal node with no expansion) DFS have modest memory requirements, it only needs to store a single path from root to a leaf node. Using the sample simulation from Fig 4.12, at depth d=12, DFS only requires 12 kilobytes instead of 111 terabytes for a BFS approach. For problems that have many solutions, DFS may actually be faster than BFS, because it has a good chance of finding a solution after exploring only a small portion of the whole space. One problem with DFS is that it can get stuck going down the wrong path. Many problems have very deep or even infinite search trees. DFS should be avoided for search trees with large or infinite maximum depths. It is common to implement a DFS with a recursive function that calls itself on each of its children in turn. Completeness: No Time complexity: b m Space complexity: bm Optimality: No (b-branching factor, m-max depth of tree) BS3. Depth-First Search (cont)) Fig 4.15 Depth-first Tree Search BS4. Depth-Limited Search Practical DFS DLS avoids the pitfalls of DFS by imposing a cutoff on the maximum depth of a path. However, if we choose a depth limit that is too small, then DLS is not even complete. The time and space complexity of DLS is similar to DFS., if l >= d Time complexity: b l Space complexity: bl Optimality: No (b-branching factor, l-depth limit)

6 BS4. Depth-Limited Search (cont) BS5. Iterative Deepening Search The hard part about DLS is picking a good limit. IDS is a strategy that sidesteps the issue of choosing the best depth limit by trying all possible depth limits: first depth 0, then depth 1, the depth 2, and so on. In effect, it combines the benefits of DFS and BFS. Fig 4.16 Depth-first search trees for binary search tree. Same problem as Fig 4.15 It is optimal and complete, like BFS and has modest memory requirements of DFS. Depth limit, dl = 2 BS5. Iterative Deepening Search (cont) BS5. Iterative Deepening Search (cont) IDS may seem wasteful, because so many states are expanded multiple times. For most problems, however, the overhead of this multiple expansion is actually rather small. IDS is the preferred search method when there is a large search space and the depth of the solution is not known. Time complexity: b d Space complexity: bd Fig 4.17 Four iterations of iterative deepening search on a binary tree BS6. Bi-directional Search BS6. Bi-directional Search (cont) Search forward from the Initial state And search backwards from the Goal state.. Stop when two meets in the middle. Time complexity: b d/2 Space complexity: b d/2 Fig 4.18 A schematics view of a bi-directional BFS that is about to succeed, when a branch from the start node meets a branch from the goal node

7 Search Comparing Blind Search Strategies Bi-directional Search Initial State Final State Fig 4.19 * Time complexity: O(b d/2 ) * Space complexity: O(b d/2 ) O(b d ) vs. O(b d/2 )? with b=10 and d=6 results in 1,111,111 vs. 2,222. d / 2 d Comparison of 6 search strategies in terms of the 4 evaluation criteria set forth in Criteria for Evaluating Search Strategies b - branching factor; d is the depth of the solution; m is the maximum depth of the search tree; l is the depth limit Class Activity 1: Various Blind Searches workouts for CMU Traffic problem Traffic Graph (raw) Tree Representation Class Activity: Various Blind Searches workouts for CMU Traffic problem Traffic Graph (raw) Group A, B, C, D to use BFS approach Group D, E, F, G to use DFS approach BFS with start state A and goal state N DFS with start state A and goal state N IDS with start state A and goal state O Bi-DS with start state A and goal state O What s in Store for next Lecture Constrained Satisfaction Search Heuristic Search Methods Definition of a Heuristic Function Best First Search An example - Sliding Tiles Greedy Search A* Search Class Activity 1: DFS, BFS, BDS, UCS and GS workout for Romania Map problem Heuristics for an 8-puzzle problem Iterative Improvement Algorithms Hill-climbing Simulated Annealing End of Lecture 3 Good Day.

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