Truth maintenance

1
Truth maintenance

• Uncertain reasoning I

2
Uncertain reasoning basic concepts

• In the following lectures we will develop some
  theories of uncertain reasoning, to be employed
  to control the behaviour of NPCs in computer
  games.
• These theories are founded on a few mathematical
  concepts, which we summarise in this lecture.

3
Basic concepts

• Universe of discourse (aka Frame of discernment)
  arbitrary finite set O. Intuitively, the universe
  of discourse characterises all relevant values of
  a variable that summarises an agent's views of
  the world. O is assumed to be finite to simplify
  mathematical characterisation (the more common
  assumption is to make O countable).

4
Basic concepts

• Example if we are dealing with ship's locations,
  we can have
• O open-sea, 12-mile-zone, 3-mile-zone,
  canal, refueling-dock, loading-dock

5
Basic concepts

• An event will be a subset of a universe of
  discourse. Typically, an agent will be able to
  collect evidences about the world, e.g. through
  perceptions, and from those evidences it will
  need to infer an event, which then will be used
  to determine its course of actions.

6
Basic concepts

• Sometimes it may be useful to move to more
  detailed (more refined) or to less detailed
  (coarser) universes of discourses.
• Sometimes it may be useful to link evidences to
  tuples of events in more detailed accounts of
  the existing possibilities of occurrences, which
  are then characterised as products of universes
  of discourse.
• We now set the grounds to deal with these
  possibilities.

7
Basic concepts

• Refinement a universe T is a refinement of a
  universe O if there is a mapping ? 2O?2T such
  that
• ?(?) ? for all ? ? O,
• If ? ? ?' then ?(?) n ?(?') ,
• U?(?) ? ? O T,
• ?(A U?(?) ? ? A.
• ? is called a refinement mapping.
• O is called a coarsening of T.

8
Basic concepts

• ExampleT open-sea, 12-mile-zone,
  3-mile-zone, canal, refueling-dock,
  loading-dockO at-sea, not-at-sea?(at-sea)
  open-sea, 12-mile-zone, 3-mile-zone?(not-
  at-sea) canal, refueling-dock, loading-dock

9
Basic concepts

• Outer reduction given O, T a refinement of O and
  ? a corresponding refinement mapping, we define
  ?' 2T?2O as?'(A) ? ? O ?(?) n A ?
  .This is called the outer reduction induced by
  ?.

10
Basic concepts

• Intuitively, given a universe T and a coarsening
  of this universe O, the outer reduction indicates
  for each element ? ? T all elements ? ? O such
  that ? ? ?(?).

11
Basic concepts

• M 1, ..., m is an index set.
• U O1xO2x...xOm, in which the Oi are universes
  of discourse, is called a general universe. The
  elements of U are tuples of the form (?1, ?2,
  ..., ?m), where ?i ? Oi.
• If N i1, ..., in ? M, we define de
  sub-universe UN Oi1xOi2x...xOin. By definition,
  if N then U e, where e denotes the
  empty tuple.

12
Basic concepts

• Pointwise projection let U be a general universe
  with index set M. Let C, S and T be subsets of M
  such that T S U C and S n C . Then we
  define p'(TS) UT ? US asp'(TS) (uT) uS, where
  for each ?i ? uS and for each ?i ? uT we have
  that ?i ?i. By definition, if S then
  p'(TS) (uT) e.

13
Basic concepts

• Projection let U, M, C, S and T be as before.
  Then we define ?'(TS) 2UT ? 2US as?'(TS) (AT)
  uS ? US there is at least one uT ? AT p'(TS)
  (uT) uS.
• Projection is a special case of outer reduction!

14
Basic concepts

• If projection is a special case of outer
  reduction, and if outer reduction is related to a
  refinement mapping, then what is the refinement
  mapping associated to a projection, so that UT is
  a refinement of US?

15
Basic concepts

• Cylindrical extension let U, M, C, S and T be as
  before. Then we define?(TS) 2US ? 2UT as?(TS)
  (AS) uT ? UT p'(TS) (uT) ? AS.
• Cylindrical extension is a special case of
  refinement, to which projection is the
  corresponding outer extension.

16
Basic concepts

• Some properties relating cylindrical extensions
  and projections let U be a general universe with
  index set M, and let S, T and C be subsets of M
  such that S ? C ? T. Then, for all A ? UT, B ?
  US
• ?'(CS)(?'(TC)(A)) ?'(TS)(A)
• ?(TC)(?(CS)(B)) ?(TS)(B)
• ?'(TS)(?(TS)(B)) B
• ?(TS)(?'(TS)(A)) A

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