Military Communications and Information Technology: A Trusted ...

206 Military Communications and Information Technology...
evaluated. The relation Rn is the interpretation. Combining universe and
interpretation results in the model of the language.
• Definition 3: A sentence σ is an assertion that can be assigned the Boolean value
true or false. A language is generated by a set of its elementary sentences
and using its logical operators.
These three initial definitions give us the tools to understand what an interpretation
of truth is in model theory. Clearly distinguishing between the language,
the model of the language, and the interpretation of truth within the model of the languages
helps to better address interoperability challenges. The next set of definitions
allows for an unambiguous representation of truth using these constructs.
• Definition 4: Let Σ be a set of sentences. U is a model of Σ whenever U⊩σ
for each σΣ. This is written as U⊩Σ. Σ is satisfiable if and only if there
is a structure U for which U⊩Σ.
• Definition 5: A theory T is a set of sentences. If T is a theory and σ is a sentence
then we write T⊩σ whenever we have that for all U we can show that
if U⊩T then U⊩σ. We define σ to be a consequence of T. A theory is defined
to be closed whenever it contains all consequences.
• Definition 6: If U is a model of L then we define the theory of the model
U, named ThU, as the set of all sentences of L which are true in U, or
ThU={σL: U⊩σ}.
• Definition 7: If ΣT fulfills that Σ⊩σ for every σT, in other words Σ⊩T,
then Σ is a set of axioms of the theory T.
Requirement sets, models, and simulation are all formal languages that shall
express the same truth. Using model theory, this can now be captured that sentences
of each model must be satisfiable under all other models, or we have different
versions of truth at the same time in our distributed application. In order for two
systems to be interoperable they have represent the same theory of the common
model. What is true in one interpretation shall be true in an alternative interpretation.
If something is wrong in one interpretation, it shall be wrong in the others
as well. If this is not the case, we will run in mistakes and ambiguities when such
systems are executed side by side.
We need to significantly broaden our understanding of data: in model theory,
everything in the universe of a language is a datum. As such, all of them need to be
taken into consideration as metadata within the future interoperability theory. This
vision of interoperability as enabling the consistent representation of truth in two
interoperable systems cannot be reached by the current standards, as they only
address syntactic and semantic issues, but they do not even touch the pragmatic
domain of interoperation.
As a first step, the research team described the LCIM formally to allow for addressing
all aspects of semiotics in a consistent way. This actually allows extending
the LCIM towards and Interoperability Maturity Model. Using symbols and the interpretation
of these symbols mapping them to appropriate domains, the following