The Semantic Web

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The Semantic Web WEEK 8 Proof in FOL continued
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The Layer Cake Model From Rector Horrocks
Semantic Web cuurse
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Recap

• Resolution is a powerful rule of inference. When
  used in refutation mode it can be a COMPLETE
  proof procedure
• Resolution is based on important ideas /
  techniques
• Unification
• Translation to Clausal Form

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Martians Example Revisited

• Deep Space 1 travels to Mars and observes many
  things about the Martians, including the fact
  that some seem very hostile towards humans.
  Concrete observations are as follows
• (a) All green Martians have antennae.
• (b) A Martian is friendly to humans if all of its
  children have antennae.
• (c) A Martian is green if at least one of its
  parents is green.
• On its way back from Mars the robot is hotly
  pursued
• by a spacecraft containing green Martians only.
  Should the robot
• suspect it is being attacked? Or can the robot
  reason with its observations to answer the
  question Are all green Martians friendly?''
• and hence avert an inter-planetary conflict.

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Systematic Proof Procedure

• Given a set of clauses W ( premises negated
  query clauses) we need to find null the empty
  clause, indicating a contradiction.
• Find the set of all pairs of clauses in W that
  can resolve, and resolve them
• C child clauses from step 1
• W W U C
• If null is in W finish, else Goto 1.

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Recall Algorithmic Properties..

• A problem is decidable if there is an algorithm
  which can always be trusted to give the correct
  answer in finite time.
• A problem is of f(n) complexity class if given
  any instance of a problem of size n it will
  take time/space f(n) to solve it.

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Problems..

• 1. Resolution (and FOL) is only SEMI-DECIDEABLE.
• That is, if you know that
• Wff1 - Wff2
• Then eventually RR will prove it
• BUT if not the procedure may go on and on
• 2. Proving Wff1 - Wff2 is of exp(n) time
  complexity in general, where n is the size of the
  Wff set.

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Problems..

• FOL is thought (by some) to be too powerful for
  the ontology/proof level of the Semantic Web (a
  contentious point). There are Syntax Conventions
  for FOL eg the KIF the Knowledge
  Interchange Format
• Biggest problems are
• No efficient proof procedures
• No built-in structure for representing classes

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Summary

• Resolution Refutation is a complete proof
  procedure but is intractable in general. Prolog
  uses an efficient version of RR.
• FOL is perhaps too unrestricted for use in the
  Semantic Web
• Execises
• 1. Dave and Fred are members of a dancing club in
  which no member can both waltz and jive. Freds
  dad cant waltz and Dave can do whatever fred
  cant do. If a child can do something, then their
  parents can do it also. Prove that there is a
  member of the dancing club who cant jive.
• Answer is on web http//scom.hud.ac.uk/scomtlm/cam
  326/logic/logic.html
• See section on resolution refutation
• 2. Try out the RR theorem prover in
  /local/public/cam326/tp/