State Space Search

1
State Space Search
Thinking about Algorithm abstractly
2
Structures and Strategies for State Space Search

• Introduction
• Graph Theory
• Structures for state space search
• state space representation of problems
• 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
• Using the State Space to Represent Reasoning with
  the Predicate Calculus
• State Space Description of a Logical System
• AND/OR graphs
• Examples and Applications

3
Introduction (1)

• 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?
• State space search is the tool for answering
  these questions.

4
Introduction (2)

• A graph consists of nodes and a set of arcs or
  links connecting pairs of nodes.
• Nodes are used to represent discrete states.
• A configuration of a game board. (tic-tac-toe)
• Arcs are used to represent transitions between
  states.
• Legal moves of a game
• Leonhard Euler invented graph theory to solve the
  bridge of Königsberg problem
• Is there a walk around the city that crosses each
  bridge exactly once.

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State space representation of Tic-tac-toe
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Königsberg Bridge System (1)
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Königsberg Bridge System (2)

• 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.
• 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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Definition of Graph(1)

• A graph consists of nodes and arcs
• 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
• 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
• 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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Definition of Graph (2)

• 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).
• A path that contains any node more than once is
  said to contain a cycle or loop.
• A tree is a graph in which there is a unique path
  between every pair of nodes
• Two nodes in a graph are said to be connected if
  a path exists that includes them both.

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State Space Representation of Problems(1)

• 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.
• 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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State Space Representation of Problems(2)

• 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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Tic-tac-toe (1)

• The set of states are all different
  configurations of Xs and Os that the game can
  have. N
• 39 ways to arrange blank, X, O in nine spaces.
• Arcs(steps) are generated by legal moves of the
  game, alternating between placing an X and an O
  in an unused location A
• The start state is an empty board. S
• The goal state is a board state having three Xs
  in a row, column, or diagonal. GD

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Tic-tac-toe (2)
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Tic-tac-toe (3)

• There are no cycles in the state space because
  the directed arcs of the graph do not allow a
  move to be undone.
• 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 Opponents
  possible wining lines
• Chess has 10120 possible game paths

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Possible Heuristic for Tic-tac-toe
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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
• The goal stateunlike tic-tac-toe, cycles are
  possible in the 8-puzzle.
• Applicable heuristics

17
The 8 puzzle (2)
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Possible Heuristic for 8-puzzle
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The Traveling Salesperson(1)

• 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.
• The goal description requires a complete circuit
  with minimum cost.
• The complexity of exhaustive search is (N-1)!,
  where N is the number of cities
• Techniques for reducing the search complexity.
• Branch and Bound, The Nearest neighbor.

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The Traveling Salesperson(2)
21
The Traveling Salesperson(3)

• 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.
• The Nearest Neighbor technique
• Go to the closest unvisited city.
• (A E D B C A) is not the shortest path

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The Traveling Salesperson(4)
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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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Data-Driven and Goal-Driven Search (2)

• 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.
• Confirming or denying the statement I am a
  descendant of Thomas Jefferson.
• Assume that Thomas Jefferson was born about 250
  years ago and that 25 years per generation and
  that 3 children per family
• I -gt Jefferson (backward) 210 ancestors
• Jefferson -gt I (forward) 310 nodes of family

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Goal-Driven Search is Suggested(1)

• A goal or hypothesis is given in the problem
  statement or can easily be formulated.
• Mathematics theorem prover, diagnostic systems
• Early selection of a goal can eliminate most
  branches, making goal-driven search more
  effective in pruning the space.
• Mathematics theorem prover.
• Problem data are not given but must be acquired.
• A medical diagnosis program.

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Goal-Driven Search is Suggested(2)
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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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Backtracking Search Algorithm

• Backtracking is a technique for systematically
  trying all paths through a state space.
• 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
• ABEHIFJCG

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Backtracking Search Example
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Backtrack (1)

• Begin
• SLstart state list (the states in the
  current path)
• NSLstart 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)
• CSstart current state
• while NSL is not while there are states
  to be tried
• do begin
• if CSgoal
• 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 CSthe first
  element of SL
• do begin

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Backtrack (2)

• do begin
• add CS to DE
• remove first element from SL
  backtrack
• remove first element from NSL
• CSfirst 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
• CSfirst element of NSL
• add CS to SL
• end
• end
• return FAIL
• end

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Function Backtrack (3)
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Ideas used in Backtrack algorithm

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

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Depth-First and Breadth-First Search

• A search algorithm must determine the order in
  which states are examined.
• Breadth-first search explores the space
• A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P,
  Q, R, S, T, U
• Depth-first search goes deeper into the search
  space whenever this is possible.
• A, B, E, K, S, L, T, F, M, C, G, N, H, O, P, U,
  D, I, Q, J, R

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Search Example
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Breadth-First Search (1)
Procedure breadth_first_search begin open
Start closed while open ?
do begin remove leftmost state from
open, call it X if X is a goal then
return(success) else begin generate
children of X put X on closed eliminate
children of X on open or closed put remaining
children on right end of open end end
return(failure) end.
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Breadth-First Search(2)

• 1. openA closed
• 2. openB,C,D closedA
• 3. openC,D,E,F closedB,A
• 4. openD,E,F,G,H closedC,B,A
• 5. openE,F,G,H,I,J closedD,C,B,A
• 6. openF,G,H,I,J,K,L closedE,D,C,B,A
• 7. openG,H,I,J,K,L,M(as L is already on open)
  closedF,E,D,C,B,A
• 8. openH,I,J,K,L,M,N closedG,F,E,D,C,B,A
• 9. And so on until either U is found or open

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Breadth first search (3)
39
Breadth-First Search (4)

• OPEN lists states that have been generated but
  whose children have not been examined.
• CLOSED records states that have already been
  examined.
• OPEN is maintained as a queue, FIFO data
  structure.
• States are added to the right of the list and
  removed from the left
• Figure 3.14 (p.101)
• Breadth-first search is guaranteed to find the
  shortest path from the start state to the goal.
• If the path is required for a solution, we can
  store ancestor information along with each state.
• open (D,A), (E,B), (F,B), (G,C), (H,C)
• closed (C,A), (B,A), (A,nil)

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Depth-First Search (1)
Procedure depth_first_search begin open
Start closed while open ?
do begin remove leftmost state from open,
call it X if X is a goal then
return(success) else begin generate
children of X put X on closed eliminate
children of X on open or closed put remaining
children on left end of open end end
return(failure) end.
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Depth-First Search (2)

• 1. openA closed
• 2. openB,C,D closedA
• 3. openE,F,C,D closedB,A
• 4. openK,L,F,C,D closedE,B,A
• 5. openS,L,F,C,D closedK,E,B,A
• 6. openL,F,C,D closedS,K,E,B,A
• 7. openT,F,C,D closedL,S,K,E,B,A
• 8. openF,C,D closedT,L,S,K,E,B,A
• 9. openM,C,D, as L is already on closed
  closedF,T,L,S,K,E,B,A
• 10. openC,D closedM,F,T,L,S,K,E,B,A
• 11. openG,H,D closedC,M,F,T,L,S,K,E,B,A

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Depth-First Search (3)

• OPEN is maintained as a stack, or LIFO data
  structure.
• The state are both added and removed from the
  left end of OPEN.
• Figure 3.16 (p.104)
• is not guaranteed to find the shortest path.

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Depth first search (4)
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Depth-First vs. Breadth-First

• The choice depends on the specific problem being
  solved.
• Significant features include the importance of
  finding the shortest path, the branching of the
  state space, the available time and space
  resources, the average length of paths to a goal
  node, and whether we want all solutions or only
  the first solution.
• Breadth-First
• always finds the shortest path to a goal node.
• If there is a bad branching factor, the
  combinatorial explosion may prevent the algorithm
  from finding a solution.
• Depth-First
• may not waste time searching a large number of
  shallow state in the graph.
• can get lost deep in a graph missing shorter
  paths to goal.

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Depth-First Search with Iterative Deepening

• Depth-First Search with Iterative Deepening
• performs a depth-first search of the space with a
  depth bound of 1. If it fails to find a goal, it
  performs another depth-first search with a depth
  bound of 2. This continues, increasing the depth
  bound by one at each iteration.
• Best-First Search
• orders the states on the OPEN list according to
  some measure of their heuristic merit.