The Semantic Web – WEEK 8: Proof in FOL continued

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The “Layer Cake” Model – [From Rector & Horrocks Semantic Web cuurse]

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

The Semantic Web

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.

1.

Find the set of all pairs of clauses in W that can resolve, and resolve them 2.

3.

4.

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 The Semantic Web

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. Fred’s dad can’t waltz and Dave can do whatever fred can’t 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 can’t jive.

Answer is on web http://scom.hud.ac.uk/scomtlm/cam326/logic/logic.html

See section on resolution refutation 2. Try out the RR theorem prover in /local/public/cam326/tp/ The Semantic Web