[DL] DL ontology Query: Tree or Graph structure?

[william fitzgerald]

Dear DL gurus,

How does one view DL ontology structures? Are they Trees, Semantic Trees (ie
semantically enhanced Trees), Directed Graphs etc?

I am looking to understand the advantages of DL ontologies over representing
knowledge in Tree-structures (with AND and OR operator capabilities).

I work in the computer security domain. Currently looking at the advantages
of DL (OWL-DL) over Attack Trees [1].  "Attack trees are multi-level
diagrams consisting of one root, leaves, and children" [1]. They can also be
constrained by AND and OR operators along the branches of the tree.

"An attack tree is a tree in which the nodes represent attacks. The root
node of the tree is the global goal of an attacker. Children of a node are
refinements of this goal, and leafs therefore represent attacks that can no
longer be refined. A refinement can be conjunctive (aggregation) or
disjunctive (choice) [2]".

Each Node can also be decorated with attributes (starting at the leave node)
such as the cost or probability of an attack. Some refer to these attribute
additions as semantically enriching the tree. However, my gut feeling is its
nothing more than syntactically enriching the tree and that DL ontologies,
by providing uniform explanations of a domain, provide semantics in a more
natural and formal way. Note I have been using the term "DL ontology" rather
than "DL Tree" or "DL Graph" as I am unsure of what category a DL model
falls into.

When developing an DL model of the attack tree paradigm or any ontology
domain for that matter, one can see a Tree-like structure as a result of the
class/concept hierarchy. However from what I can see, actual
instances/individuals of the ontology seem to form a graph. So my question
is are DL ontologies Trees, Graphs or a hybrid?

Presumably a tree cannot capture "Roles" easily, like DL can. However one
could argue that concepts, roles and individuals are codified in a tree
structure (such as attack trees) only its not as obvious. For example from
the quote above in [2] one can visualise the leaf nodes as the final atoms
and so are "individuals" within an ontology. The tree root node might
possibly be seen as a "Role" in an ontology.

Instinctively I know what ontologies provide a better mechanism of formal
explanations over Tree-like structures (like the attack tree) however I am
initially finding it hard to tease out the differences of both when putting
pen to paper.  [Presumably I can't just argue that there is a more intuitive
reasoning framework and a unified language when taking the DL approach, as
one could write some constraint program to parse and understand the
"semantics" of a Tree structure (just an initial thought here!) ]

My query is just to get some direction on further reading that will make
clear what category a DL ontology fits into and thus allow me to set about
defining the advantages of using DL constrained ontologies over the old
paradigm of (attack) trees within the security domain.

[1] http://en.wikipedia.org/wiki/Attack_tree
[2] http://www.win.tue.nl/~sjouke/publications/papers/attacktrees.pdf

kind regards,
Will.
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