Tuesday, February 21, 2006
The Second School of Semantic Science
ontologyMapping Glass Bead Games
On the limits of the OWL standard à [184]
Reading material [1]
Reading material [2]
Reading material [3]
Summary of the discussion up to this point à [186]
Differences between information and theory
Communication to Protégé forums and others from Paul S Prueitt.
Earlier
statement from PSP: Theorems proved by mathematicians are not like theorems in logic
proved by computer programs.
Comment: I don't think so. Mathematics is founded on logics. Set theory (I assume you are talking about Zermelo/Fraenkel set theory with the Axiom of Choice here) is a first-order theory, which is given a standard model-theoretic, denotational semantics. Theorems are valid well-formed formulas that are deducible from the axioms by entailment, and entailment is defined in terms of interpretation, and so on... that is standard model theory. Description Logics are also a fragment of first-order logics, with a standard model-theoretic semantics; thus, theorems are exactly the same things, entailment is exactly the same thing, interpretation is the same thing, etc.
Respectfully,
Mathematics is founded on logics. Some will suggest that the history of mathematics and studies into the foundations of mathematics will shed light on the incompleteness of the founding of mathematics on logics.
Differences and similarities can be stated in a scholarly venue. Perhaps this scholarly discussion should be part of the common liberal education that information technologists are required to know? I suggest this only because so many foundational issues related to knowledge representation seem to be unknown to the knowledge engineering community. It is for this reason that we have developed the proposal that the US federal government re-program 20% of the federal support for computer science in the funding of a new academic discipline in the natural sciences.
Logics is often about asserted truths. Mathematics is really about the formation of the language of science. I am not sure that I can properly put the position forward, particularly in an ontology forum. I will try to be brief. The issue of the extension of mathematics to ontological science is a complex one.
Some of us are trying to figure out how to extend, or layer on top of, Hilbert mathematics in some way that admits to an ontological difference between a formal system (an abstraction developed via induction and symbol formation) and natural systems.
The critical issue has to do with how natural systems maintain the sense of wholeness, or self, while at the same time physical laws govern the expression of that natural system. Historically, there has been an expressed sense in theoretical physics that mathematics is "used" sometimes informally to give description to physical law without "deriving the physical law from mathematics". There is a loose-ness in how the tool is used. To a large degree, this sense defines the profound differences between applied and pure mathematics. But the issues in biological, social theory, and yes even business process modeling may be more critical than what is well understood in theoretical physics.
To understand these issues might be essential to the development of Information Models that are implemented in a publicly accessible collection of standard finite state machines to be used in a future web of information (The “Second School” calls this web the “anticipatory web”, Tim Berners-Lee calls this “Semantic Web”.) We suggest that these standards be part of negotiated engines to be installed as federated interchange services such as proposed in the OASIS FERA standard. The negotiation of which interchange machines are selected, within a business or cultural context, requires human-in-the-loop decision-making process. The negotiation is to be human centric so that the limitations of current logic based ontology can be understood, by human users, and the adopted interchange machine used with this understanding fully in mind.
Mathematics and logic both seem to have some degree of limitation in some specific regards, and perhaps the limitations are not precisely the same for mathematics as for logic. The excellent work on description logics plays around with sets of categories of assertions, the assertion that properties of a class can be transitive to subclasses as one example. But one also has as “axioms” in descriptive logics, anything that the designer him or her self chooses to assert about required properties or constraints on properties. “All humans have exactly two arms”, might be one of these.
Even in biological taxonomy, the transitive property is at best only partially descriptive of the phenomenon expressed by "animals" as a super class of "horses". Building non-monotonic assertions so that the class-subclass vocabulary of OWL works in the description of (say) reactants in biological cell signal pathway expression can be questioned both on theoretical grounds and via the observations related to improper implicit assertions.
We are all understanding this better now and looking for a way forward. We celebrate the work that has been done that allows us to see what is left to do.
A practical problem arises in the esoteric nature of the notation and literature, given that scientist and even businessmen are attempting to build real world systems from incomplete theoretical development. This is shifting sand. The practical problem is reflected in the usability problems that one can observe with the Stanford developed software, Protégé.
I want to pick back up on the properties that might differentiate the four types of informational organization and various types of computational processes that might be applied. It seems that here we have an opportunity to find firm footing and immediate economic benefit related to using the Codd-type relational model when it makes sense, and using one of the other three information model standards when the theory tells us that openness/flexibility or usability is an issue.
> 1) RDBMS
> 2) Object Oriented or UML
> 3) descriptive logic based ontology
> 4) n-ary ontology
If one accepts the notion that there are specific limitations to any formalism then it is possible to extend the stability of the relational database to address the limitations that have been found through experience:
1) inflexibility to changes in data model after system is deployed.
2) absence of standard data models for common descriptors.
both of these limitations need not have any inference such as is developed using description logics. What is needed is measurement of fidelity of the model to real world needs. Do you agree?
The first issue is partially resolved with XML, and with input to relational databases from XML, or with interchanges between XML and Object Oriented Information Models. But this partial resolution might not be a complete resolution, as reflected in the incompleteness of the description logics notion of an Open World Assumption. There are important concepts about "openness" which require a complete reconstruction of all aspects of "logic" so as to produce a "situational logic".
The link above talks about Soviet era applied semiotics (and what is called quasi axiomatic theory). But some scholars believe that the "openness of the real natural world" is far more "complex" than what can be represented completely in any formalism of the type developed so far (Penrose, Rosen, and others), including quasi axiomatic theory.
How relevant are these opinions to the development of web services in the context of B-2-B or G-2-G transaction spaces? This should at least be an issue that is talked about sometimes.
If so then the conversion of OWL constructions to RDBMS schema has the effect of fixing a more dynamic modeling exercise, using the description logic based assertions. A similar conversion of Object Oriented Information Models to RDBMS schema has a similar effect. The process of fixing the model and the information is properly described as moving from a Partial Open World Assumption to a Closed World Assumption, as Andrea as discussed in this forum. This process can be mediated by human-to-human negotiations resulting in a legal agreement to instantiate a federated set of finite state machines in the form of RDBMS or perhaps n-ary repositories.