Thursday, January 12, 2006
Additional reading:
Cory Casanave's paper on Data
Access
work on ontology for biological signal pathways
[127]
ß parallel discussion in
generative methodology bead thread
[127]
ß parallel discussion on Rosen
complexity
On the meaning of “axiom”
Earlier
part of this discussion
[342], [343], [344], [345], [346], [347], [348], [349], [350]
Original Communication [353]
Response
index [355]
Communication from Andrea Proli in Italy
Footnotes by Paul Prueitt
Hello, I have made some considerations about this issue and I would like to share them with you.
I think that it is not wrong to consider as "axioms" many "things" in Semantic Web Languages, but in order to clarify my arguments I need to draw a generalization of the Semantic Web setting. So, I give the following definitions.
1) Suppose we have a decidable base language S over a finite alphabet. S contains syntactically well-formed formulae, which in our case could be thought of as statements.
2) Let's say an S-interpretation is a partial function from S to {T, F} assigning truth values to formulae, and call the set of all such interpretations I(S). Let's define the signature of an interpretation function f as its domain: sig(f) = dom(f). Obviously sig(f) is a subset of S.
3) Now, let's define a 'semantic language' L (any other term is ok, I'm just trying to provide a basis for my arguments) as a pair L=<C(L),S(L)>, where S(L) is the language containing syntactically well-formed formulas (like S above) and C(L) is a set of 'semantic conditions'.
4) Semantic conditions are functions from I(S(L)) to {T, F}, and an interpretation i is 'L- admissible' iff every semantic condition in C(L) maps i into T. Denote the set of all L-admissible interpretations as V(L).
5) Given two semantic languages L' and L'', let's say that L' is a 'semantic extension' of L'' iff C(L') is a superset of C(L''), which trivially implies that V(L') is a subset of V(L'').
6) An interpretation i L- satisfies a formula (statement) p iff (1) sig(i) is a superset of I(S (L)) and (2) i(p)=T. A set of formulae X L-entails a set of formulae Y iff every L-admissible interpretation in V(L) L-satisfying all formulae in X also L-satisfies all formulae in Y.
7) Given a semantic language L, a subset A of S(L), containing axiomatic formulae (statements), defines an L-theory T(A) in this way: p belongs to T(A) iff p is L- entailed by A.
8) Finally, a 'semantic framework' is obtained by fixing a common syntax language S, and by defining a sequence of semantic languages L_i=<C(L_i),S>, i=1,..,n where L_i+1 is a semantic extension of L_i for every i. A simple monotonicity result can be deduced by the given definitions: if a formula p is L'-entailed by a formula q, and L'' is a semantic extension of L', then p is also L''- entailed by q (this is also what happens in the Semantic Web framework).
In this setting, RDF is both a semantic framework and a (basic) semantic language: it fixes the common syntax, and it establishes a core set of trivial semantic conditions over interpretations (the framework is then completed by RDF Schema, OWL and so on which are semantic extensions of RDF - not that true, but still a good approximation).
So, formulae (statements) in an L-theory A (if L is OWL, then A is an ontology) are axioms of the theory T(A) so it is not wrong to say that every statement in an ontology is an 'axiom'. However, as we know, in order to keep the same syntax as RDF, those that are commonly known as 'axioms' in Description Logics are encoded in OWL-DL as many RDF statements: in this case, many 'axioms' become a single 'axiom'... so? So, it is not wrong to denote both type of things as 'axioms', because indeed they are, just it is important to clarify the context. By the way, using the name 'statement' to denote axioms of any L-theory with L being a semantic extension of RDF (so, even an OWL ontology) solves the ambiguity.
Another ambiguity issue is resolved by considering again the nature of RDF, being both a framework and a base language. If L is a Semantic Web language (included RDF, which defines some elementary semantic condition on interpretations), then the set C(L) is not empty and, due to this fact, the empty set of formulae (statements) could L-entail a non-empty set of formulae (statements). Such statements are called, in the RDF Semantics formalization, 'axiomatic triples'. The name is justified by the fact that those statements, together with a proper set of inference rules which are straightforwardly obtained by semantic conditions, allow to 'syntactically' perform entailment like in a traditional axiomatic theory. By the way, if we take into account this observation, then we can recognize that the term 'axiomatic triple' is somewhat different from 'axiom' and I think this difference is enough to communicate the nuance in meaning.
So, if you want one single answer to the question 'if I say 'axiom' in the Semantic Web, what do I mean?', then I'm afraid there is not such an answer, and the reason is that an 'axiom' could be a statement, it could be a DL axiom, it could be an axiomatic triple, because indeed all these things *are* axioms, if taken in their proper context.
It remains true that specific names could be used to specifically denote such different kinds of axioms.
Thank you, and sorry for being (again) so long to express my arguments.
Best regards,
Andrea Proli