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KU NLP Artificial Intelligence1 Ch 3. Structures and Strategies for State Space Search q Introduction q Graph Theory Structures for state space search state space representation of problems q Strategies for state space search Data-Driven and Goal-Driven Search Implementing Graph Search Depth-First and Breadth-First Search Depth-First Search with Iterative Deepening q Using the State Space to Represent Reasoning with the Predicate Calculus State Space Description of a Logical System AND/OR graphs Examples and Applications

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KU NLP Artificial Intelligence2 Introduction(1) q Questions for designing search algorithms Is the problem solver guaranteed to find a solution? Will the problem solver always terminate? When a solution is found, is it guaranteed to be optimal? What is the complexity of the search process? How can the interpreter most effectively reduce search complexity? How can the interpreter effectively utilize a representation language? q State space search is the tool for answering these questions.

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KU NLP Artificial Intelligence3 Introduction(2) q A graph consists of nodes and a set of arcs or links connecting pairs of nodes. q Nodes are used to represent discrete states. A configuration of a game board. (tic-tac-toe, p43, tp4) q Arcs are used to represent transitions between states. Legal moves of a game Leonhard Euler invented graph theory to solve the “bridge of K ö nigsberg problem” Is there a walk around the city that crosses each bridge exactly once. (Fig. 3.1, P82, tp5)

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KU NLP Artificial Intelligence4 State space representation of Tic-tac-toe (p 43)

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KU NLP Artificial Intelligence5 K ö nigsberg Bridge System (p82)

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KU NLP Artificial Intelligence6 K ö nigsberg Bridge System q Euler focused on the degree of the nodes of the graph Even degree node has an even number of arcs joining it to neighboring nodes. Odd degree node has an odd number of arcs. q Unless a graph contained either exactly zero or two nodes of odd degree, the walk was impossible. No odd dgree node: the walk start at the first and end at the same node Two odd degree nodes: the walk could start at the first and end at the second

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KU NLP Artificial Intelligence7 Definition of Graph(1) A graph consists of nodes and arcs(Fig. 3.3, p85) A set of nodes N1, N2, …, Nn … need not be finite. A set of arcs connects pairs of nodes. A directed graph has an indicated direction for traversing each arc(Fig. 3.3, p85) If a directed arc connects Nj and Nk, then Nj is called the parent of Nk and Nk is called the child of Nj. A rooted graph has a unique node Ns from which all paths in the graph originate (Fig. 3.4, p86) A tip or leaf node is a node without children. An ordered sequence of nodes [N1, N2, N3, … Nn] is called a path of length n-1 in the graph

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KU NLP Artificial Intelligence8 Definition of Graph (2) q On a path in a rooted graph, a node is said to be an ancestor of all nodes positioned after it (to its right) as well as a descendant of all nodes before it (to its left). q A path that contains any node more than once is said to contain a cycle or loop. q A tree is a graph in which there is a unique path between every pair of nodes q Two nodes in a graph are said to be connected if a path exists that includes them both.

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KU NLP Artificial Intelligence9 State Space Representation of Problems(1) q In the state space representation of a problem, the nodes of a graph corresponds to partial problem solution states, the arcs corresponds to steps in a problem-solving process. q State space search characterize problem solving as the process of finding a solution path from the start state to a goal.

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KU NLP Artificial Intelligence10 State Space Representation of Problems(2) q Definition : STATE SPACE SEARCH A state space is represented by a four tuple [N, A, S, GD] where N is the set of states. A is the set of steps between states S is the start state(S) of the problem. GD is the goal state(S) of the problem. The states in GD are described: 1. A measurable property of the states. (winning board in tic- tac-toe) 2. A property of the path.(shortest path in traveling sales man problem) A solution path is a path through this graph from S to GD.

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KU NLP Artificial Intelligence11 Tic-tac-toe (1) q The set of states are all different configurations of Xs and Os that the game can have. [N] 3 9 ways to arrange {blank, X, O} in nine spaces. q Arcs(steps) are generated by legal moves of the game, alternating between placing an X and an O in an unused location [A] q The start state is an empty board. [S] q The goal state is a board state having three Xs in a row, column, or diagonal. [GD] q Fig II.5 (p 43, tp12)

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KU NLP Artificial Intelligence12 Tic-tac-toe (2)

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KU NLP Artificial Intelligence13 Tic-tac-toe (3) q There are no cycles in the state space because the directed arcs of the graph do not allow a move to be undone. q The complexity of the problem : 9!(362,880) different path can be generated. Need heuristics to reduce the search complexity. e.g. My possible winning lines – Opponent’s possible wining lines (Fig 4.16, p149, tp14) Chess has 10 120 possible game paths

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KU NLP Artificial Intelligence14 Possible Heuristic for Tic-tac-toe

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KU NLP Artificial Intelligence15 The 8-puzzle (1) The set of states are all different configurations of 9 tiles (9!). The legal moves are : move the blank tile up( ), right( ), down( ), and the left( ). make sure that it does not move the blank off the board. All four moves are not applicable at all times. The start state (e.g. Fig. 3.6, p90, tp16) The goal state (e.g. Fig. 3.5, p89) unlike tic-tac-toe, cycles are possible in the 8-puzzle. Applicable heuristics (Fig. 4.8, p132, tp17)

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KU NLP Artificial Intelligence16 The 8 puzzle (2)

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KU NLP Artificial Intelligence17 Possible Heuristic for 8-puzzle

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KU NLP Artificial Intelligence18 The Traveling Salesperson(1) q The goal of the problem is to find the shortest path for the salesperson to travel, visiting each city, and then returning to the starting city. q The goal description requires a complete circuit with minimum cost. q The complexity of exhaustive search is (N-1)!, where N is the number of cities(Fig 3.8, p93) q Techniques for reducing the search complexity. Branch and Bound, The Nearest neighbor.

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KU NLP Artificial Intelligence19 The Traveling Salesperson(2)

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KU NLP Artificial Intelligence20 The Traveling Salesperson(3) q Branch and Bound technique 1. Generate paths while keeping track of the best path found so far. 2. Use this value as a bound 3. As paths are constructed one city at a time, examine each partially completed path by guessing the best possible extension of the path (the branch). 4. If the branch has greater cost than the bound, it eliminates the partial path and all of its possible extensions. q The “Nearest Neighbor” technique Go to the closest unvisited city. (A E D B C A) is not the shortest path (Fig. 3.9, p93, tp21) q Other examples of State Space representation Blocks world (p 289), FWGC problem (p 622)

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KU NLP Artificial Intelligence21 The Traveling Salesperson(4)

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KU NLP Artificial Intelligence22 3.2.1 Data-Driven and Goal-Driven Search (1) Data-driven search(forward chaining) takes the facts of the problem and applies the rules and legal moves to produce new facts that lead to a goal. Goal-driven search (backward chaining) focused on the goal, finds the rules that could produce the goal, and chains backward through successive rules and subgoals to the given facts of the problem. Both problem solvers search the same state space graph. The search order and the actual number of states searched can differ.

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KU NLP Artificial Intelligence23 3.2.1 Data-Driven and Goal-Driven Search (2) q The preferred strategy is determined by the properties of the problem: complexity of the rules, “shape” of the state space, the nature and availability of the problem data. q Confirming or denying the statement “I am a descendant of 임꺽정.” Assume that 임꺽정 was born about 250 years ago and that 25 years per generation and that 3 children per family I -> 임꺽정 (backward) 2 10 ancestors 임꺽정 -> I (forward) 3 10 nodes of family

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KU NLP Artificial Intelligence24 Goal-Driven Search is Suggested(1) q A goal or hypothesis is given in the problem statement or can easily be formulated. Mathematics theorem prover, diagnostic systems q Early selection of a goal can eliminate most branches, making goal-driven search more effective in pruning the space. (Fig.3.10 p95, tp25) Mathematics theorem prover: total # of rules to produce a given theorem is much smaller than # of rules that may be applied to the entire set of axioms q Problem data are not given but must be acquired. A medical diagnosis program: Doctor order only diagnostic tests that are necessary to confirm or deny a hypothesis

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KU NLP Artificial Intelligence25 Goal-Driven Search is Suggested(2)

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KU NLP Artificial Intelligence26 Data-driven Search is Appropriate All or most of the data are given in the initial problem statement. PROSPECTOR interpreting geological data. There are only a few ways to use the facts and given information. DENDRAL finding the molecular structure of organic compounds. For any organic compound, enormous number of possible structures. The mass spectrographic data on a compound allow DENDRAL to eliminate most of possible structures. Branching factor, availability of data, and ease of determining potential goals are carefully analyzed to determine the direction of search.

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KU NLP Artificial Intelligence27 Backtracking Search Algorithm q Backtracking is a technique for systematically trying all paths through a state space. q Search begins at the start state and pursues a path until it reaches either a goal or a “dead end”. If it finds a goal, returns the solution path. If it reaches a dead end, it backtracks to the most recent node on the path q Figure 3.12 (p 98, tp28) ABEHIFJCGABEHIFJCG

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KU NLP Artificial Intelligence28 Backtracking Search Example

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KU NLP Artificial Intelligence29 Function Backtrack (1) Begin path SL:=[start]; % state list (the states in the current path) NSL:=[start]; % new state list(Nodes awaiting evaluation) % nodes whose children are not yet been generated DE:=[ ]; % dead ends (states whose descendants have failed to % contain a goal node) CS:=start; % current state while NSL is not [ ] % while there are states to be tried do begin if CS=goal then return (SL) % on success, return list of states in path if CS has no children (except nodes already on DE, SL, NSL) then begin while SL is not empty and CS=the first element of SL do begin

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KU NLP Artificial Intelligence30 Function Backtrack (2) do begin add CS to DE; remove first element from SL; % backtrack remove first element from NSL; CS:=first element of NSL; end add CS to SL; end else begin % when CS has children % the first child becomes new current state % and the rest are placed on NSL for future. place children of CS (except nodes on DE) on NSL; CS:=first element of NSL; add CS to SL; end return FAIL; end

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KU NLP Artificial Intelligence31 Function Backtrack (3)

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KU NLP Artificial Intelligence32 Ideas used in Backtrack algorithm q The list NSL is used to allow the algorithm to backtrack to any of these states. q The list DE is used to prevent the algorithm from retrying useless paths. q The list SL is used to keep track of the current solution path. q Explicit checks for membership of new states in these lists to prevent looping.

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