=Paper=
{{Paper
|id=None
|storemode=property
|title=Towards Agent-Oriented Knowledge Base Maintenance for Description Logic ALCN
|pdfUrl=https://ceur-ws.org/Vol-710/paper34.pdf
|volume=Vol-710
|dblpUrl=https://dblp.org/rec/conf/maics/UstymenkoS11
}}
==Towards Agent-Oriented Knowledge Base Maintenance for Description Logic ALCN==
https://ceur-ws.org/Vol-710/paper34.pdf
Towards Agent-Oriented Knowledge Base Maintenance for
Description Logic ALCN
Stanislav Ustymenko1 and Daniel G. Schwartz2
Meritus University, School of Information Technology,
1
30 Knowledge Park Drive (Suite 301), Fredericton, New Brunswick, Canada E3C 2R2 ,
sustymenko@meritusu.ca
2
Department of Computer Science, Florida State University,
Tallahassee, Florida, USA
schwartz@cs.fsu.edu
Abstract formulation, belief revision postulates are stated in terms of
Artificial agents functioning in the Semantic Web are to be potentially infinite belief sets (although work has been done
capable of getting knowledge from diverse sources. This to address this issue). We believe that belief revision is a
implies the capability to continuously update their mature paradigm that can be valuable source for important
knowledge bases. New stream reasoning concepts make this insight. However, there is a need for formal approaches that
need even more pressing. Semantic Web ontologies are address the practical challenges more directly.
commonly represented using description logic knowledge New research direction under the tentative title “stream
bases. We propose an agent architecture with such features,
reasoning” [6] emerged within the Semantic Web
utilizing a Dynamic Reasoning System (DRS). This
explicitly portrays reasoning as a process taking place in community. It explicitly deals with reasoning over rapidly
time and allows for manipulating inconsistent knowledge changing and time-dependent data in a way that can deliver
bases. We sketch a procedure for user-directed ontology answers to the user while they are still relevant. Stream
debugging. This same mechanism can be used for reasoning is defined as “the new multi-disciplinary approach
automated belief revision. We identify important research which will provide the abstractions, foundations, methods,
directions that may benefit from this approach. and tools required to integrate data streams and reasoning
systems” [7]. Della Valle et al. [5] write: “Stream-reasoning
research definitely needs new theoretical investigations that
1 Introduction go beyond data-stream management systems, event-based
systems, and complex event processing. Similarly, we must
go beyond current existing theoretical frameworks such as
The Semantic Web (SW) [4] is a common name of a family
belief revision and temporal logic”. Currently, there is no
of technologies extending the Web with rich, machine-
consensus on a logic formalism appropriate for stream
interpretable knowledge. The SW retains the massively
reasoning. There is an obvious practical need for such
decentralized nature of current World Wide Web, with an
formalism to be able to integrate current, description logic-
unlimited number of knowledge sources identifiable by
based Semantic Web standards.
unique URIs. It supports rich metadata annotation, including
Dynamic Reasoning Systems (DRS) [17] provide a formal
expressive ontology languages. Description Logics (DLs) [2]
framework for modeling the reasoning process of an artificial
emerged as leading formalism for knowledge representation
agent that “explicitly portrays reasoning as an activity that
and reasoning on the Semantic Web.
takes place in time”. It sidesteps the logical omniscience
Once widely implemented, the Semantic Web will support
assumption of the classical AGM framework and has means
intelligent software agents that will work with massive,
of working with inconsistent knowledge bases by keeping
decentralized ontologies, while other agents modify them in
track of a proposition's derivation path. The DRS framework
possibly inconsistent ways. Agents will need a way to absorb
has been shown to support non-monotonic reasoning in a
new knowledge in a timely fashion, all the while protecting
natural way.
the consistency of their knowledge bases, or, alternatively,
A DRS can be defined for any language. DLs present a
be able to draw useful inferences from inconsistent premises.
challenge in that they do not have explicit derivation rules.
Several approaches have been proposed to model
Instead, DLs rely on inference algorithms to accomplish
knowledge evolution over time. One of the most well-
common reasoning tasks. One of the basic tasks is checking
researched formalisms is belief revision [9, 10], specifically
subsumption of concepts.
the classic AGM framework [1, 9]. Substantial efforts have
The goal of this paper is to present the DRS framework as a
been extended to apply this approach to description logics
suitable formalism for Semantic Web reasoning. To this end,
[13, 14, 19, 20], and the work is ongoing. However, the
we give an instance of DRS capable of building a concept
belief revision framework does not explicitly address
subsumption hierarchy for a well-known description logic.
knowledge evolution in time. Also, in its original
We believe it to be an important foundation for research on and individual names to elements of I . The function
belief dynamics for Semantic Web agents. Section 2 of this I
paper contains a brief formal introduction to Description . extends to arbitrary concept definition in a rather
Logics and the necessary definitions. Section 3 discusses intuitive way (for details, see [2], chapter 2). A concept is
Dynamic Reasoning Systems. Section 4 describes a DRS and unsatisfiable if for any interpretation I , C I =∅ .
agent reasoning process for deriving explicit subsumption Description Logic knowledge bases consist of two
hierarchies from description logic ALCN terminology. Short components: a TBox, a set of statements about concepts, and
abstract of this work appears in [20]. Finally, in Section 5, an ABox, a set of assertions about individuals. In general, a
we draw some conclusions and discuss directions for future TBox T contains general concept inclusion axioms of the
research. form C⊆ D (inclusion axiom). The pair of axioms
C⊆ D , D⊆C is abbreviated C≡ D (equality
axiom). An interpretation I satisfies an axiom C⊆ D
2 Description Logics if C I ⊆ D I . Interpretation I satisfies a TBox T
if it satisfies every axiom in T .
Languages for any description logic contain concept names, A definition is an equality axiom with an atomic concept
role names, and individual names. Below, we will use on the left hand side. A TBox is a terminology if it consists of
uppercase A and B for concept names, uppercase definitions and no concept name is defined more than once.
letters R , P for role names, and lowercase x , y , z A concept name is a defined name if it appears on the left
for individual names. hand side of the axiom and a base name if it doesn't. A
DL languages combine role and concept names into definition is in the extended form if only base concept names
concept definitions. Concepts of a description logic AL [16] appear on the right hand side. A terminology is definitorial if
are defined as follows: every definition has exactly one extended form (not counting
equivalent syntactic variants). In further discussion, we
assume that our TBoxes are definitorial terminologies. Under
C,D A | (atomic concept) this condition, we can assume, wlog, that definitions contain
| (universal concept) no cycles.
An ABox contains assertions regarding individual names.
⊥ | (bottom concept) These include concept assertions C a and role
¬A | (atomic negation) assertions R a , b. An interpretation I satisfies (or
C∩ D | (intersection)
is a model of) C a if a I ∈C I and it satisfies
R a , b. if a I , b I ∈ R I . Finally, I satisfies an
∀ R.C | (value restriction) assertion (or an ABox A ) with respect of a TBox
∃ R. (limited existential T if it is a model of both an assertion (or an ABox) and
quantification) the TBox.
An ontology of concepts can be expressed using a DL. The
More expressive DLs extend AL by the following term ontology is often applied either to a TBox or to a full DL
constructs: knowledge base. We will occasionally use ontology in the
former sense.
Indication Syntax Name
U C∪ D union 3 Dynamic Reasoning Systems
E ∃ R.C full existential quantification
The classical (propositional) notion of belief set [e.g., 9]
N n R , n Rnumber restriction models it as an (often infinite) set of formulas of the
C full negation underlying logical language. In our view, a belief set should
¬C
be finite and should represent the agent’s knowledge and
beliefs at a given point in time. Moreover, each formula in
The commonly used DL ALCN extends AL with full such a belief set should contain information indicating how if
negation and number restriction. In the following sections, was obtained and whether it has been used in subsequent
we will restrict ourselves to ALCN. deductions, thereby enabling both backtracking and forward
An Interpretation of a DL is a structure I = I , . I , chaining through reasoning paths for so-called “reason
maintenance”.
where I is a nonempty set called domain and . I is
To this end, in [17] there was defined the notion of a
an interpretation function that maps concept names to dynamic reasoning system (DRS), which explicitly portrays
subsets of a domain, role names to subsets of I × I , reasoning as an activity that takes place in time. This is
obtained from the conventional notion of formal logical For a given agent, let us denote the agent’s belief set at
system by lending special semantic status to the concept of a time step i by i . Let 0=∅ . Thus the agent
derivation path (i.e., a proof). Introduction of new knowledge
initially has no knowledge or beliefs. Then, given i , for
or beliefs into the path occurs in two ways: either new
propositions are added in the form of axioms, or some , i≥0 , i1 can be obtained in any of the following
propositions are derived from earlier ones by means of an ways:
inference rule. In either case, the action is regarded as 1. A new formula is received from an outside source,
occurring in a discrete time step, and the new proposition is
2. A formula is derived from some formulas in i
labeled with a time stamp (an integer) indicating the step at
which this occurred. Moreover, for propositions entered into by means of an inference rule,
the path as a result of rule applications, the label additionally 3. A formula in i has its status changed from on
contains a record of which inference rule was used and to off.
which propositions were employed as premises.
Changing a formula’s status from on to off occurs during a
At any given time, the contents of the path is regarded is
reason maintenance process that is invoked whenever an
being the sum total of the agent’s knowledge and beliefs as
insatisfiability, i.e., a definition of the form A≡⊥ is
of that time. Thus we here take this path as being the agent’s
belief set as of that time. entered into the agent’s belief set. The objective of reason
This is to be contrasted with other systems of belief maintenance is to remove this insatisfiability.
revision, which assume that the agent additionally knows all This has two phases. First one starts back tracking from the
the logical consequences of the basic belief set. Such systems insatisfiability, using the from lists in the formula labels,
are said to exhibit “logical omniscience.” For an in-depth looking for the “culprit” formulas that occurred earlier and
analysis of this issue, together with a manner of addressing which led to the inconsistency. A decision then must be made
it, see the paper by Fagin, Halpern, Moses, and Vardi [8]. to turn the status of at least one of these formulas to “off”.
For complete details of the notion of a DRS, please see Then one forward chains from this formula, using the to lists,
[S97]. A brief outline is as follows. A labeled formula is to find all formulas whose derivations stemmed from the
culprit formula, and likewise turns their status to “off”. This
defined as a pair P , where P ∈ L , where will include the inconsistent formula that triggered the reason
L is a logical language, and the label is an ordered maintenance process.
4-tuple (index, from, to, status), where: Which culprit formula to deactivate can be determined by
the various culprits’ degrees of belief, to wit, remove the one
1. index is a non-negative integer, the index, that is least believed. In case the culprits are all believed
representing the formulas position in the belief set. equally, one can be chosen at random. Alternatively, an agent
2. from is a from list, containing information about can remove the culprit formula that is the least important
how the formula came to be entered into the belief according to some reasonable criteria. One such criteria is a
set. Either it was received from an outside source cumulative belief level of formulas derived from the culprit.
(obtained from some other agent or through This criteria provides a finite version of the AGM epistemic
interaction with its environment), in which case the entrenchment relation.
from list contains the token rec, or it was derived This model of agent-oriented reasoning reflects that view
from some formulas occurring earlier in the belief that, at any given time, the agent’s beliefs may harbor an
set, in which case the from list contains the name of inconsistency, but the agent does not become aware of this
the derivation rule and the indexes of the formulas unless an inconsistent formula is explicitly entered into its
used as premises in the derivation. The types of belief set.
formulas that can be received are understood to This, in our opinion, is a realistic model of natural human
include both axioms of the propositional calculus reasoning. Humans can comfortably maintain inconsistent
and statements about the agents environment beliefs for long periods of time without ever realizing this.
(sometimes distinguished as “logical” and But once they become consciously aware of a
”nonlogical” axioms). contradiction, they typically rethink their position and modify
3. to is a to list, containing the indexes of all formulas their beliefs so that the contradiction is removed.
in the belief set for which the given formula served
as a premise in the indexed formula’s derivations.
4. status is a status indicator, taking values on or off,
indicating whether the belief represented by the
formula is currently held, i.e., whether the formula
may or may not be used in any future derivations.
Whenever a formula is initially entered into the
belief set, its status is on.
1. Local copy of the ontology, expressed as an ALCN
TBox. This ontology consists of ALCN definitions
Controller
Environment that occur in the derivation path.
2. A subsumption tree of concept names.
The latter can be used to support both browsing and user
querying on both a TBox and an ABox. The user has a
preference for satisfiable ontologies, so the agent has to
detect and remove unsatisfiable concepts. Thus, our DRS
needs to support 2 types of DL reasoning:
DRS 1. Check if a defined concept A is satisfiable
2. Deduce atomic subsumption, that is, a statement of
the form A⊆ B ,
Derivation Derivation
path rules
where A, B are concept names.
To construct the DRS, we first note that if A and B are
defined by axioms A≡C , B≡ D , where C, D are
concept definition, then A⊆ B iff C ⊆D. Second, note
that C⊆ Diff C∪¬ D is unsatisfiable.
Fig.1 Reasoning agent employing a Dynamic Reasoning So both our reasoning tasks would require checking
System satisfiability of concepts. We are using a generic tableau-
based satisfiability algorithm [2, 3].
The reasoning agent (Fig. 1) uses a Dynamic Reasoning Now we can build our dynamic reasoning system. First, We
System to reach conclusions that help advance the agent's
define the language, L. The symbols of L are the same as the
goals. A controller directs DRS behavior to steer it to such
conclusions. The controller performs the following actions: symbols of logic ALCN. We use A, B for concept names
occurring in the incoming statements and A', B' for the names
1. Receive information from the outside environment. introduced by the agents. The formulas of L are the
The information can come from a human user, other following:
agents, or be harvested by an agent through sensors. 1. Equivalence statements of the form A≡C ,
The latter can get information from any external where A is a concept name and C is concept
data source. definition. Without loss of generality, we assume all
2. Enter information, as a “nonlogical” axioms concept definitions are in negation normal form, i.e.
expressed in language L, into the DRS's inference negation only occurs in front of concept names.
path. 2. Atomic subsumption statements of the form
3. Apply an inference rule.
4. Act to remove insatisfiability, by invoking belief
A⊆ B , where A, B are concept names. These
represent arcs of the subsumption tree the agent is
revision procedure described above.
building.
The agent performs these actions in the order dictated by the 3. TBox assertions C a , Ra ,b , where C is a
agent's and environment's current state, presumably in a concept, R is a role, and a,b are individual constants
manner that would advance its goals. In the following, we 4. Explicit inequality assertions x≠ y , where
are constructing an agent that would accept an ontology in
x , y are individual names.
the form of TBox definitions and construct a subsumption
hierarchy of concept names implicit in this ontology. Then we define inference rules. Implicitly, every rule that
modifies a concept definition also puts the result into
negation normal form. The inference rules will be:
4 Dynamic Reasoning for DL ALCN 1. Substitution: from A≡C and B≡D infer
A≡ E , where E is C with all
A Dynamic Reasoning System is a model for knowledge occurrences of B replaced by D . For this
base and reasoning process for artificial agent that assists a treatment we assume that our TBox does not contain
user. We describe an agent that extracts ontological cycles in definitions. By repeatedly applying this
knowledge from the Web and uses it to support a user's rule, we obtain an extension of definition for A
browsing and querying activities. To this end, an agent that only contain ground concept names on the right
maintains two information stores: side.
2. Subsumption test introduction: from
A≡C , B≡ D infer A '≡C∩¬D , where
A ' is a previously-unused agent-generated { yi ≠ y j∨1i≤ jn1} derive A≡⊥ ,
concept name. where x , y 1,. .. , y n1 are individual names,
3. From A≡C , B≡ D and A '≡⊥ , provided
R is a role name and n0 .
that name A ' was introduced using rule 2 with
A≡C , B≡ D as premises, derive A⊆ B . Finally, rule 14 derives a subsumption axiom, using reduction
to unsatisfiability:
The following rules 4-10 are added to enable tableau-
based consistency checks. These are derived from the 14. From A≡C , B≡D , A1≡C ∪¬D
transformation rules listed in [2], p. 81. Individual names and A 1 ≡⊥ , derive A⊆ B
x , y , z , ... are unique names generated by the agent.
All TBox statements are derived from the same ABox A Dynamic Reasoning System based on language L and rules
statement (that is undergoing satisfiability check) 1-14 is capable of supporting an agent that builds an explicit
subsumption hierarchy. We will now describe a controller
A≡C 0 :
that can achieve this goal.
An agent starts with an empty derivation path and empty
4. From A≡C 0 , infer C o x 0 , if no ABox subsumption hierarchy. It will receive TBox definitions from
statements were inferred from A≡C 0 . the user. To start the hierarchy, before receiving the first
5. From A≡C 0 and C 1∩C 2 x , infer axiom, the controller will enter a root concept, R≡ , as
a first formula in the derivation path and R as a root node
C 1 x and C 2 x , if any one of them is
in the hierarchy.
not yet inferred.
Upon entering a new axiom of the form A≡C , the
6. From A≡C 0 and C 1∪C 2 x , infer controller will perform the following actions:
C 1 x or C 2 x , if neither of them is
inferred yet.
1. Derive an expanded definition of A by
repeatedly employing Rule 1 until the right side of
7. From A≡C 0 and ∃R.C x , infer
the resulting definition contains no defined concept
C y and R x , y , where y is a new names.
generated name, if no z exists such that 2. Test satisfiability of A using Rules 4-13. If it is
C z and R x , z are already derived. unsatisfiable, flag it for a belief revision procedure
8. From A≡C 0 , ∀ R.C x and 3. Expand all (extended) definitions that depend on
using Rule 1. Test the affected concepts'
R x , y infer C y , if not already
satisfiability, flagging for a belief revision process if
derived.
unsatisfiable. Update the hierarchy of concepts
9. From A≡C 0 and n R x , infer affected by this step, testing subsumption by using
R x , y1 , ... , R x , y n and ( y i≠ y j , Rules 2-14.
and R x , y , unless 4. Place A into its appropriate place in the
R x , z 1 ,... , R x , z n are already inferred. subsumption hierarchy, using Rules 2-14 to test
subsumption with definitions of concept names
10. From A≡C 0 and n R x , if already there.
R x , y1 , ... , R x , y n1 are in the
To test satisfiability by employing Rules 3-13, an agent
derivation path and y i≠ y j is not in the path for follows a tableau-based algorithm. Details of the appropriate
some i≠ j : replace all occurrences of y i algorithm, with discussion of termination and complexity,
with y j . can be found in [2].
Rules 6 and 10 are non-deterministic: for a given ABox, they
The following rules 11-13 detect inconsistency in TBoxes can be applied in finitely many different ways, leading to
built using rules 4-10. As above, TBox statements are finitely many ABox'es. The concept is satisfiable if at least
derived from A≡⊥ : one such ABox is consistent. Each ABox is a branch in the
satisfiablilty algorithm. The controller may handle branches
11. From A≡C 0 and ⊥ x , derive A≡⊥ , by setting the belief status of statements on inactive branches
where x is any individual name. to off. In practice, it may be useful to remove such statements
12. From A≡C 0 , A1 x and ¬ A1 x , from the path to save space.
We did not specify the details of modifying subsumption
derive A≡⊥ , where x is any individual hierarchy on steps 3 and 4. In principle, the controller may
name and A1 is any concept name. simply search the existing hierarchy starting at the root,
13. From A≡C 0 , n R x , set testing the concept in question's subsumption with each node.
{R x , y i ∨1in1} and This is a natural and decidable procedure that will result in
set
the correct hierarchy. Studying the complexity of such an
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