Building Control Algorithms For State Space Search

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Chapter 6 Building Control Algorithms For State
Space Search
Contents

• Recursion-Based Search
• Production Systems
• The Blackboard Architecture for Problem Solving

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Recursive Search

• Recursive search
• A recursive step procedure calls itself
• A terminating condition
• Depth-first recursive search algorithm

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Recursive Search with Global Variables
Global variables open and closed
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Pattern-Driven Reasoning

• Problem
• Given a set of assertions (predicate expressions)
• Determine whether a given goal is a logical
  consequence of the given set of assertions
• Solution
• Use unification to select the implications
  (rules) whose conclusions match the goal
• Unify the goal with the conclusion of the rule
• Apply the substitutions throughout the rule
• Transform the rule premise into a new subgoal
• If the subgoal matches a fact, terminate
• Otherwise recur on the subgoal
• Recursive algorithm next page

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Pattern-driven Reasoning
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Some Issues

• The order of assertions
• Logical connectives in the rule premises
• Logical negation

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A production system. Control loops until working
memory pattern no longer matches the conditions
of any productions.
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Trace of a simple production system.
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The 8-puzzle as a production system
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The 8-puzzle searched by a production system with
loop detection and depth-bound.
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The Knights Tour Problem

• Problem find a series of legal moves in which
  the knight lands on each square of the chessboard
  exactly once
• Legal moves of a chess knight.

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A 3 x 3 chessboard with move rules for the
simplified knight tour problem.
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Production rules for the 3 x 3 knight problem.
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A production system solution to the 3 x 3
knights tour problem.
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Control Algorithms

• The general recursive path definition
• ?X path(X,X)
• ?X,Ypath(X,Y) ? ?Zmove(X,Z) ? path(Z,Y)
• The revised path definition to avoid infinite
  loop
• ?X path(X,X)
• ?X,Ypath(X,Y) ? ?Zmove(X,Z) ? ?(been(Z)) ?
  assert(been(Z)) ? path(Z,Y)

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The recursive path algorithm as production system.
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A Production System in Prolog

• Farmer, wolf, goat, and cabbage problem
• A farmer with his wolf, goat, and cabbage come to
  the edge of a river they wish to cross. There is
  a boat at the rivers edge, but, of course, only
  the farmer can row. The boat also can carry only
  two things, including the rower, at a time. If
  the wolf is ever left alone with the goat, the
  wolf will eat the goat similarly if the goat is
  left alone with the cabbage, the goat will eat
  the cabbage. Devise a sequence of crossings of
  the river so that all four characters arrives
  safely on the other side of the river.
• Representation
• state(F, W, G, C) describes the location of
  Farmer, Wolf, Goat, and Cabbage
• Possible locations are e for east, w for west,
  bank
• Initial state is state(w, w, w, w)
• Goal state is state(e, e, e, e)
• Predicates opp(X, Y) indicates that X and y are
  opposite sides of the river
• Facts
• opp(e, w).
• opp( w, e).

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Sample crossings for the farmer, wolf, goat, and
cabbage problem.
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Portion of the state space graph of the farmer,
wolf, goat, and cabbage problem, including unsafe
states.
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Production Rules in Prolog

• Unsafe states
• unsafe(state(X, Y, Y, C)) - opp(X, Y).
• unsafe(state(X, W, Y, Y)) - opp(X, Y).
• Move rules
• move(state(X, X, G, C), state(Y, Y, G, C))) -
  opp(X, Y), not(unsafe(state(Y, Y, G, C))),
  writelist(farms takes wolf, Y, Y, G, C).
• move(state(X, W, X, C), state(Y, W, Y, C)) -
  opp(X, Y), not(unsafe(state(Y, W, Y, C))),
  writelist(farmers takes goat, Y, W, Y,C).
• move(state(X, W, G, X), state(Y, W, G, Y)) -
  opp(X, Y), not(unsafe(state(Y, W, G, Y))),
  writelist(farmer takes cabbage, Y, W, G, Y).
• move(state(X, W, G, C), state(Y, W, G, C))
  -opp(X, Y), not(unsafe(state(Y, W, G, C))),
  writelist(farmer takes self, Y, W, G, C).
• move(state(F, W, G, C), state(F, W, G, C)) -
  writelist(Backtrack from , F, W, G, C), fail.
• Path rules
• Path(Goal, Goal, Stack) - write(Solution Path
  Is ), nl, reverse_print_stack(Stack).
• Path(State, Goal, Stack) - move(State, Next),
  not(member_stack(Next, Stack)), stack(Next,
  Stack, NewStack), path(Next, Goal, NewStack), !.
• Start rule
• Go(Start, Goal) - empty_stack(EmptyStack),
  stack(Start, EmptyStack, Stack), path(Start,
  Goal, Stack).
• Question
• ?- go(state(w, w, w, w), state(e, e, e, e)

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Data-driven search in a production system.
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Goal-driven search in a production system.
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Bidirectional search missing in both directions,
resulting in excessive search.
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Bidirectional search meeting in the middle,
eliminating much of the space examined by
unidirectional search.
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Major advantages of production systems for
artificial intelligence

• Separation of Knowledge and Control
• A Natural Mapping onto State Space Search
• Modularity of Production Rules
• Pattern-Directed Control
• Opportunities for Heuristic Control of Search
• Tracing and Explanation
• Language Independence
• A Plausible Model of Human Problem-Solving

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Blackboard architecture

• Extend production systems
• Separate productions into modules
• Each module is an agent -- knowledge source
• A single global structure -- blackboard