Problem Solving in Common Lisp

1
Problem Solving in Common Lisp

• Common Lisp
• J.E. Spragg
• May 1997

2
The Problem Space Model

• People seem to handle adequately the notion of
  physical space.
• It seems natural to extend our notions of
  physical space and use it as a topological
  metaphor for problems and problem solving.
• The problem space model of problems and problem
  solving makes use of the ideas of location,
  shape, and distance that are so useful in
  reasoning about physical space.

3
Problem Space

• A problem space is a set of states that represent
  distinct configurations of the objects and
  relationships of the domain.
• And a set of operators which define how to move
  between states.
• Operators typically have parameters.

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Defining a Problem Space

• Defining a problem in a given problem space
  requires two things
• an initial state
• a goal
• The initial state is a distinguished state that
  represents the starting point within the space.
• The goal is a specification of the subset of the
  problem space which could serve as a solution to
  the problem.

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Solving Problems

• Solving a problem in this model is accomplished
  by finding a sequence of operators which, when
  applied to the initial state, allows one to reach
  a state satisfying the goal criterion.
• In the Blocks World example, the solution is a
  sequence of transformations that represent a
  legal plan.

6
Search

• It is easy to see why this is an attractive model
  of problem solving.
• To solve a problem, all we have to do is define
  the problem space and then unleash a
  general-purpose search engine on it.

7
Classic Problem Solving Systems

• The design of a classic problem solver consists
  of two parts
• an interface for user-supplied problem spaces,
  and a
• search engine.

8
Interface

• The interface requires the ability to manipulate
  states and operators.
• What kinds of manipulations are needed?
• Goal detection Ascertain whether a given state
  satisfies the goal criterion.
• State identity Detect when two descriptions of
  states refer to the same state.
• State display Produce a human readable
  description of a given state.

9

• The importance of goal detection is obvious.
• State identity is important because we make no
  progress by re exploring already examined states.
• Displaying states allows us to observe and debug
  system behaviour.

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Expanding a State

• Identify what operators are available.
• Determine whether a given operator is applicable
  to a particular state.
• Given a state and an operator applicable to it,
  ascertain all the ways the operator can be
  instantiated on that state.
• Figure out what new state results from applying
  an instantiated operator to a state.