Structures and Strategies For Space State Search

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Chapter 3 Structures and Strategies For Space
State Search
Contents

• Graph Theory
• Strategies for Space State Search
• Using the Space State to Represent Reasoning with
  the Predicate Calculus

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The city of Königsberg

• Leonhard Euler
• Problem if there is a walk around the city that
  crosses each bridge exactly once?

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Representations

• Predicate calculus connect(X, Y, Z)
• connect(i1, i2, b1) connect(i2, i1, b1)
• connect(rb1, i1, b2) connect(i1, rb1, b2)
• connect(rb1, i1, b3) connect(i1, rb1, b3)
• connect(rb1, i2, b4) connect(i2, rb1, b4)
• connect(rb2, i1, b5) connect(i1, rb2, b5)
• connect(rb2, i1, b6) connect(i1, rb2, b6)
• connect(rb2, i2, b7) connect(i2, rb2, b7)
• Graph theory
• Nodes
• Linkes
• Easy proof the walk is impossible since all
  nodes have odd degrees

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Graph of the Königsberg bridge system
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A labeled directed graph
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A rooted tree, exemplifying family relationships
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Finite State Machine (FSM)
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Flip Flop FSM
(a) The finite state graph for a flip flop
and (b) its transition matrix.
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Finite State Accepting Machine

• Deterministic FSM transition function for any
  input value to a state gives a unique next state
• Probabilistic FSM the transition function
  defines a distribution of output states for each
  input to a state

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String Recognition

• The finite state graph
• The transition matrix for string recognition
  example

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State Space and Search
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State Space of the 8-Puzzle

• generated by move blank operations
• ? -- up
• ? -- left
• ? -- down
• ? -- left

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The travelling salesperson problem

• Find the shortest path for the salesperson to
  travel, visiting each city and returning to the
  starting city

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Search for the travelling salesperson problem.
Each arc is marked with the total weight of all
paths from the start node (A) to its endpoint.
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An instance of the travelling salesperson problem
with the nearest neighbour path in bold. Note
this path (A, E, D, B, C, A), at a cost of 550,
is not the shortest path. The comparatively high
cost of arc (C, A) defeated the heuristic.
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Strategies for State Space Search

• Data-driven search forward chaining
• Begin with the given facts and a set of legal
  rules for changing states
• Apply rules to facts to produce new facts
• Repeat rules application until finding a path
  that satisfies the goal condition
• Goal-driven search backward chaining
• Begin with the goal and a set of facts and legal
  rules
• Search rules that generate this goal
• Determine conditions of these rules ? subgoals
• Repeat until all conditions are facts

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Data-driven Search
State space in which data-directed search prunes
irrelevant data and their consequents and
determines one of a number of possible goals.
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Goal-driven Search
State space in which goal-directed search
effectively prunes extraneous search paths.
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Search and Backtrack

• Search find a path
• Backtrack when the path is dead, try others
• Backtrack to the most recent node on the path
  having unexamined siblings
• Continue toward to a new path
• Like a recursion
• Implemented in Prolog as an internal mechanism

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Backtrack algorithm
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Backtracking search of a hypothetical state space
space.
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A trace of backtrack on the previous graph
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Depth-First and Breadth-First Search

• Determine the order of nodes (states) to be
  examined
• Depth-first search
• When a state is examined, all of its children and
  their descendants are examined before any of its
  siblings
• Go deeper into the search space where possible
• Breadth-first search
• When a state is examined, all of its children are
  examined after any of its siblings
• Explore the search space in a level-by-level
  fashion

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Graph for search examples
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The breadth-first search algorithm
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A trace of breadth-first search
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The graph at iteration 6 of breadth-first search.
States on open and closed are highlighted
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Breadth-first search of the 8-puzzle, showing
order in which states were removed from open
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The depth-first search algorithm
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A trace of depth-first search
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The graph at iteration 6 of depth-first search.
States on open and closed are highlighted
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Depth-first search of 8-puzzle with a depth bound
of 5
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Comparison between breadth- and depth-first search

• Breadth-first
• Always find the shortest path to a goal
• High branching factor -- Combinatorial explosion
• Depth-first
• More efficient
• May get lost

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State Space Representation of Logical Systems

• Representation
• Logical expressions as states
• Inference rules as links
• Correctness
• Soundness and completeness of predicate calculus
  inference rules guarantee the correctness of
  conclusions
• Theorem Proof
• State space search

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State space graph of the propositional calculus

• Letters as nodes
• Implications as links
• q?p
• r?p
• v?q
• s?r
• t?r
• s?u

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And/or graph

• Or separate
• And -- connected
• And/or graph of expression q ? r ? p
• And/or graph of the expression q ? r ? p

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And/or graph of a set of propositional calculus
expressions.
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And/or graph of part of the state space for
integrating a function
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The facts and rules of this example are given as
English sentences followed by their predicate
calculus equivalents
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The solution subgraph showing that Fred is at the
museum.
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Rules for a simple subset of English grammar are
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And/or graph for the grammar. Some of the nodes
(np, art, etc) have been written more than once
to simplify drawing the graph.
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And/or graph searched by the financial advisor.
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Parse tree for the sentence The dog bites the
man.