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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(rb1, i1, b2)
connect(rb1, i1, b3)
connect(rb1, i2, b4)
connect(rb2, i1, b5)
connect(rb2, i1, b6)
connect(rb2, i2, b7)
Graph theory
connect(i2,
connect(i1,
connect(i1,
connect(i2,
connect(i1,
connect(i1,
connect(i2,
i1, b1)
rb1, b2)
rb1, b3)
rb1, b4)
rb2, b5)
rb2, b6)
rb2, b7)
– 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
(a)The finite state graph
(b)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.”
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