Re: Why must the web be monotonic ?

[just to get rid of it]

monoticity is indeed a necessity!!!

at this moment we are very glad that
we get a "no proof found" for
the russell paradox
but we are even more glad that proving
the negation of it gives a
"no proof found"

--
Jos De Roo





pat hayes <phayes@ai.uwf.edu>@w3.org on 2001-07-27 01:16:41 AM

Sent by:  www-rdf-logic-request@w3.org


To:   Enrico Franconi <franconi@cs.man.ac.uk>
cc:   "www-rdf-logic" <www-rdf-logic@w3.org>
Subject:  Re: Why must the web be monotonic ?

> > There is a way to combine the global security of monotonic reasoning
> > with the local advantages of nonmonotonic reasoning (eg when working
> > with hospital records on a website, say), which is to provide a way
> > to state the closed world assumptions explicitly. Take the
> > hospital-records example again, where you fail to find any record of
> > a patient and conclude that the person never was a patient.  That is
> > a non-monotonic inference from just the patient records, but it is
> > monotonic from the patient records PLUS an explict statement of the
> > closed-world assumption, ie the statement that the set of records is
> > exhaustive. So if we have a way to refer to a set of assertions -
> > say, all those at a certain URL, or all those which use a certain
> > namespace, or by some other means - and a way to state the
> > closed-world assumptions that are being made about this set of
> > assertions - say, they they are exhaustive with respect to all
> > queries of a certain form - then the overall reasoning can be
> > considered monotonic, even though it proceeds locally by using
> > efficient nonmonotonic methods.
>
>Well, this is the usual trick of using a monotonic logic to encode a
>theory (i.e., a KB) from a nonmonotonic logic. However, the original
>logic *remains* nonmonotonic, otherwise the encoding would be
>incorrect. So, if the original theory is nonmonotonic, then it stays
>nonmonotonic. That is, if I infer something on the ground that there
>is no patient (just because I did not find any record), then when a
>new patient is inserted the original inference becomes invalid.

That depends on what you count the inference as being from. If the
closed-world assumption is made explicit as an antecedent of the
conclusion, then when a new patient is inserted, the original
*conclusion* may be incorrect, but the *reasoning* is still valid.
The incorrectness of the conclusion is due to the fact that you now
have a false premis in the (monotonic, and hence still valid)
argument. An emerging idea in the semantic web, as I understand it,
is that proofs will be published and be publicly checkable. The key
point is that these proofs should be monotonic, not that any
particular reasoner's behaviour should be.

Of course, one can take either route. However, I would point out that
when real databases rely on such closed world assumptions in a
context where the world is changeable, they usually take pains to
provide a mechanism for recording exactly this kind of dependence.
The use of transaction time in temporal databases, for example, is
exactly the kind of reference to the particular state of the database
at the time the query was made that is necessary to keep the
reasoning globally monotonic, even though nonmonotonic techniques are
in constant use. In general, any reasoning process which needs to be
'accountable' to future queries about its validity needs to keep an
audit trail of the assumptions it was using at the time the inference
was made; and that is another way of saying that it needs to be
globally monotonic, if necessary by recording the local assumptions
that warranted the use of nonmonotonic techniques. Tort logic cannot
afford to be globally nonmonotonic.

>The
>crucial point is that the encoding of the first theory (the one
>without any patient) is NOT included in the encoding of the second
>theory (the one with a patient). So, the deductive closure of the
>first theory is not included in the deductive closure of the second
>theory: you loose exactly the inferences that came form the assumption
>that no explicit record means no existing record.

Well, that is exactly what I am suggesting should NOT happen. If
proofs of conclusions are to be published on the Web, they should be
checkable for validity, and retain validity in the face of future
changes to the world. Then,  the burden of checking the truth of the
conclusion can be transferred to that of establishing the truth of
the assumptions. But if the derivation is nonmonotonic, publishing
and checking it is pointlesss, since it can be rendered invalid by
changes in the world, and there is no way even to check whether any
such changes are of the kind that could possibly alter its validity,
since it does not contain all the assumptions on which its validity
depends. Such proofs are, literally, useless; they are intrinsically
ephemeral; they only report a kind of activity trail of the behavior
of a reasoner at some time in the past, and project no reliable
information into the future.

>So, your observation should be read as follows: a monotonic logic can
>be used to reason with a fixed KB which is (contextually)
>nonmonotonic; at each update of the nonmonotonic KB, we should
>nonmonotonically change the encoding in the monotonic logic. The
>overall reasoning remains nonmonotonic.

That is not what I would recommend. I would describe the situation as
follows: a nonmonotonic reasoner is one whose validity depends on
some assumptions which it does not explicitly state or use. Processes
which are predicated on unstated assumptions - one might call them
enthymematic processes - are often of great utility; but when
published on the Web, their conclusions should be filled out with the
missing premises on which their validity depends.

> > Right now, DAML+OIL and RDF have not entered into this area, but
> > 'rules' languages need to consider it seriously, in my view.
>
>It is quite interesting to notice that the closed world assumption
>(together with the domain closure assumption, which is also important)
>can be encoded in standard Description Logics, if nominals with the
>Unique Name Assumption are expressible. I know little of DAML+OIL, but
>I believe that it can express nominals with UNA.

Right, I believe it can also. However, to be practicable, it may be
necessary to extend the languages in some theoretically small ways
which have great practical utility, eg by being able to refer to a
set of assumptions by name rather than by ostention.

Best wishes

Pat Hayes

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Received on Thursday, 26 July 2001 20:53:11 UTC