Chapter 3: State Space Search

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Chapter 3 State Space Search

• AI uses state space search to find solutions to
  problems
• State space search uses state space graphs to
  represent problems
• this representation may be used instead of or in
  addition to representation in predicate calculus
• Graph theory allows us to analyze the structure
  and complexity of problems and search strategies
• A graph is a set of nodes and a set of arcs that
  connect pairs of nodes
• In state space search, a node in a graph
  represents a state of the world or a state in the
  solution of a problem
• An arc represents a transition between states
• a chess move (arc) changes the game board from
  one state to another
• a robot arm moving a block (arc) changes the
  blocks world from one state to another
• a rule (arc) like If its raining, take an
  umbrella, changes the state of the known world

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Euler and the Bridges of Konigsberg

• Problem Can we walk through the city of
  Konigsberg and cross each of its seven bridges
  exactly once?

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Representing the Problem as a Graph

• This graphical representation allowed Euler to
  consider the degree of a node
• the degree of a node is the number of arcs
  entering it
• The walk is impossible for any graph unless it
  has exactly zero or two nodes of odd degree
• with no odd degree nodes, you can start and end
  at the same node
• with two odd degree nodes, you can start at one
  and end at the other
• The predicate calculus representation does not
  let us see this principle!

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Graph Definitions
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More Graph Definitions
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The Finite State Machine

• A finite state machine (FSM) is one type of
  graphical representation used to show how
  transitions from one state to another can be made
• In AI, they are primarily used in natural
  language processing
• Outside of AI, they are used in compiler
  construction to recognize valid syntax and in
  circuit design for building counters and flip
  flops
• Informally, you begin at some initial state of
  the world, and you transition through other
  states based on input received
• In AI, you usually want to transition into a goal
  state
• In other applications (for example, a counter),
  you may just want to continue cycling through
  states indefinitely
• The following example FSM accepts strings of
  a,b,c,d containing abc

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State Space

• A state space is a more general graphical
  representation than a FSM

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State Space Graph for Tic Tac Toe
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Tic Tac Toe

• The graph for tic tac toe is a directed acyclic
  graph (DAG)
• The start state (usually drawn at the top) is an
  empty board
• A goal state is any state with three Xs in a row
• The path from the start state to a goal state
  tells the moves needed to win the game
• There are 9! (362,880) different paths through
  the graph
• Exhaustive search is a strategy that simply looks
  at all the paths to find the best one
• This might work for tic tac toe, but for larger
  problems with factorial or exponential
  complexity, heuristics are needed
• Chess has 10120 possible game paths

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The 8-Puzzle

• This puzzle is based on a childrens game in
  which there are 15 tiles to arrange in numerical
  order. (The original is called the 15-Puzzle.)
• A start state has the tiles scrambled there are
  many possible start states, each with its own
  graph
• To represent the problem as simply as possible,
  think of moving the blank
• The state space graph is directed, but unlike in
  tic tac toe, there can be cycles
• Solving the puzzle means finding a path from the
  start state to the goal state

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Partial State Space for the 8-Puzzle
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Traveling Salesperson Problem

• This is a classical problem in which the goal is
  for a salesperson to visit several cities and
  return home in the most efficient order
• Exhaustive search has complexity (N-1)! for N
  cities

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Strategies to Make Search Tractable

• Branch and bound is a technique in which branches
  are pruned if the cost of a partial path exceeds
  the lowest cost path found so far
• This is faster and can not eliminate any good
  paths
• This may not be fast enough for large N
• A heuristic approach is finding a path by always
  choosing to visit the nearest unvisited city
• This is very fast but is not guaranteed to be
  optimal in terms of cost
• Often, it is good enough

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Data-Driven and Goal-Driven Search Strategies

• We do not always search in the same way. There
  are different strategies, depending on the
  problem at hand.
• In data-driven search (forward chaining), we
  start with facts and a set of rules for changing
  states
• Rules are applied to obtain new facts, in an
  iterative cycle, until a path to a goal state is
  found (or we fail)
• In goal-driven search (backward chaining), we
  start with our goal (or hypothesis) and work
  backwards
• We consider the subgoals that could have led to
  the goal, the subgoals that could have led to
  those subgoals, and so on, until we get to the
  initial facts (or we fail)
• One strategy or the other may be more efficient
  or better suited to a particular problem, or the
  strategies may be combined to solve the same
  problem

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Choosing a Search Strategy

• Data-driven search may be best if
• Most of the initial data is given in the problem
  statement
• Ex PROSPECTOR analyzes geographic data to
  determine where ore deposits may be found
• There are a lot of potential goals, but facts
  limit which goals apply
• Ex DENTRAL finds the molecular structure of
  organic compounds using mass spectrographic data
  the structure depends on the data
• An initial goal or hypothesis is difficult to
  formulate
• Ex Theres no way for DENDRAL to guess what the
  structure might be initially
• Ex We may be interested in any type of finding,
  as in data mining applications (e.g., grocery
  store shopping pattern analysis)

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Goal-Driven Search

• Goal-driven search may be best if
• A goal or hypothesis is given in the problem
  statement
• Ex Does the patient have appendicitis?
• Many rules apply to the facts of the problem,
  leading to many possible conclusions and goals
• Ex In proving a mathematical theorem, we want to
  focus on relevant rules, not blindly apply all
  mathematical principles
• All relevant facts are not initially known
• In diagnosis or troubleshooting, you may need to
  gather additional facts
• The goal youre working on suggests facts to
  acquire
• Ex A physician will order only necessary tests
  for a patient, and a computer technician will
  perform only relevant diagnostics

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Implementing Graph Search Backtracking

• Backtracking search begins at the start state and
  systematically traverses the state space graph
  until it either finds a solution or searches the
  entire graph
• The backtracking algorithm makes use of
• CS, the current state, a node in the graph
• SL, the state list, an ordered list of nodes on
  the path currently being tried
• When a goal state is found, SL contains the path
  to that goal state from the start state
• Depending on the problem, you may just want the
  solution stored at the goal state node, or you
  may want to know the whole path
• NSL, new state list, an ordered list of nodes
  waiting to be evaluated
• DE, dead end list, a list of nodes that have been
  evaluated and shown not to lead to goal states
• This list is maintained for efficiency, so that
  no dead end node is evaluated more than once
• The algorithm, as presented, is for data-driven
  search. For goal-driven search, you could start
  with the goal state and work backwards to the
  start state.

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The Backtracking Algorithm
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Backtracking Example
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Depth-First and Breadth First Search

• The backtracking algorithm implements a kind of
  depth-first search
• Depth-first search proceeds down a graph, so that
  a nodes children are considered before its
  siblings
• Breadth-first search proceeds across a graph,
  examining one level of a graph at a time. Here,
  a nodes siblings are considered before its
  children
• The breadth-first search uses two lists
• Open is a first-in-first-out (FIFO) list, or
  queue, which lists nodes that have been
  generated, but whose children have not been
  examined
• The FIFO quality ensures that search proceeds
  across the graph
• Closed is a list of nodes which have already been
  examined
• Unlike in the backtracking algorithm, there is no
  SL state list that maintains the path from start
  to goal. If desired, bookkeeping can be added to
  track each nodes parent and trace the path.
• An advantage of breadth-first search is that,
  since it searches across levels, it is guaranteed
  to find the shortest path to a goal.
• A disadvantage is that there are BN nodes per
  level of a graph , where B is the average number
  of children per node and N is the level. Storing
  a whole level on open gives exponential space
  complexity. (Dont try this with chess!)

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Breadth First Search Algorithm
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Breadth-First Search Example
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Depth-First Search

• Depth-first search is more space efficient, but
  it is not guaranteed to find a shortest path from
  start to goal
• Depth-first search can also get stuck, if the
  graph has infinitely deep paths
• This can happen for problems where the graph is
  not set in advance, but generated a node at a
  time as the problem is worked through
• The generalized depth-first search algorithm is a
  simplification of the backtracking algorithm.

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Depth-First Search Algorithm
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Depth-First Search with Iterative Deepening

• Depth-first search with iterative deepening
  combines the best features of depth-first and
  breadth-first search
• Start with a depth-first search, but set a depth
  bound n (a limit) on how many levels down into
  the graph the search can go
• The search starts with a depth-first search with
  depth bound n 1
• This is the same as a breadth-first search that
  stops after the first level
• If a goal is not found, the depth bound is set to
  2 and the algorithm repeats
• The algorithm iterates until either a goal state
  is found or the depth bound n is reached
• This algorithm is guaranteed to find a shortest
  path solution (if one exists in the first n
  levels), like breadth-first search, and has only
  linear space complexity, like depth-first search
• Space used is Bn, where B is the average number
  of children per node
• Breadth-first search, depth-first search, and
  depth-first search with iterative deepening all
  have exponential time complexity O(Bn)
• Heuristic searches (next chapter) can be used
  when B and/or n are large

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Example of DFS with Iterative Deepening
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Representing Reasoning with State Space Graphs

• In propositional calculus, let each proposition
  be a node and each implication be a directed arc
• Then, to see if a proposition is true, just find
  a path in the graph between the unknown
  proposition and propositions that are known to be
  true
• You can use data-driven (forward) or goal-driven
  (backward) search
• You can use depth-first or breadth-first search

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And/Or Graphs

• Logical statements contain ands and ors
• The first example showed only ors
• When a proposition contains ands, arcs are linked
  together to show that both or all of them must be
  true to make the implication
• An and/or graph is a kind of hypergraph, a graph
  in which each hyperarc is formed by linking one
  or more nodes to another node

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Predicate Calculus Example
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Example of Goal-Driven Search in an And/Or Tree

• ? W location(fred, W) Wheres Fred?