Action Model for Dynamic Description Logic
Abdenour Bouzouane1,2, Bruno Bouchard2, Sylvain Giroux2
1
Université du Québec à Chicoutimi
555, boul. de l'Université,
Chicoutimi (Québec), Canada
[email protected]
Abstract
This paper describes a formal framework of action
concepts for description logic. We focus on defining the
semantic of the action based on state-transition model.
This framework constitutes a first step towards a more
expressive planning language which introduces the
dynamic into the description logic. The concrete case
being used as validation for the action model proposed is
plan recognition…..
1. Introduction
Recent developments in networking technology,
especially, the semantic web services that combine the
description logic with service descriptions based on plan
specification languages have brought forth an increasing
interest for dynamic description logic, i.e., reasoning
about actions at different levels of abstraction enables a
system to reason more efficiently about how different
actions relate [1][15]. Several knowledge representation
and reasoning languages for plan generation have been
developed such as STRIPS [10], PDDL[12] which use
formulas of first-order languages to describe states, are not
sufficiently expressive to express ontologies of actions
allowing taxonomic reasoning and abstraction of planning
tasks in space of states of problems resolution. Taxonomic
reasoning has largely been the concern of research in
knowledge representation, particularly, in KL-ONE [8]
like system and related description formalisms, which are
mainly concerned with modelling and representing object
concepts (static entities) organised in hierarchical way
through subsumption relation [3]. Dynamic description
logic can be useful in planning problems, in the sense that
dynamics entities (action concepts) have to be integrated
into these knowledge representation languages, resulting
in planning systems that will be better equipped to address
more effectively the needs of real word planning
applications. Several approaches have dealt with the
integration of action concept into terminological
knowledge representation languages [13]. Most of these
2
Université de Sherbrooke
2500, boul. de l’Université,
Sherbrooke (Québec), Canada
{Bruno.Bouchard, Sylvain.Giroux}@usherbrooke.ca
approaches, however, developed into sophisticated
complex languages systems featuring the representation of
time on planning [2] and plan recognition methods [18],
or are more practically oriented and deal with
implementation of representations in terminological
languages [11]. For example, the CLASP system [9] is
built on top of the terminological system CLASSIC [7]
and focuses on a language for representing plans-action
subsumption. The representation of actions is still
expressed using the underlying terminological language.
The semantic of actions subsumption has largely been
unaddressed.
Despite its importance, it is striking that the literature
on the formalization of the action in Description Logic
(DL) is quite thin. Borgida [4] presents techniques for
extending terminological systems, and illustrates the
techniques by reconstructing the plan subsumption
reasoning developed in CLASP. To formally specify the
extension, it presents axioms defining the semantic of
CLASP using natural semantic rules of inference. The
limit of this solution lies on its dependence on assuming a
propositional representation of planning problems. Our
approach follows the lines of Borgida’s work on dynamic
description logic, it can be seen as an extension of that
proposed by Kemke [14], that, in our view, overlook on
the interpretation of the object concepts and the roles
when the action is performed by causing the world to
make a transition to one state to another. Conversely, our
contribution argues that the state-transition model can be
considered as a model for dynamic description logic by
presenting how the interpretation of the concepts changes
when the action is performed.
The plan of the paper is as follows. The next section
describes the action model that serves as a basis for
dynamic DL. The section 3 presents our validation case
while section 4 points to some related work. Finally the
section 5 concludes the paper.
2. Action Model
We draw on the state-transition model of action to
develop a theoretical model of the action. A state-
transition model is a pair W , A , where W is a set of
possible states of the world and A is a set of actions over
those states. An action a over a set of states W is a
binary relation a  W  W such that  w, e  a if and
only if a ( w)   e |  w, e  W W  , where w and e are
respectively the current and next states. Within this
framework, the actions are deterministic and operate on
the assertions formulas which are particular cases of first
order logic formulas [5]. If the conceptual expressions and
the assertions of the DL are used to describe facts about a
state of the world, they can be satisfiable or unsatisfiable
according to this state. Therefore, the states of the world
can
correspond
to
semantic
structures.
Let
I
w   Dom( w), (.) w  a semantic structure such that
Dom( w) is the domain of interpretation, i.e., the nonempty set of objects called individuals that exist in the
world when the world is in that state w at given time. The
I
function (.) w , referred to as interpretation function
associated with w, assigns to each concept symbol, C , a
I
subset of the domain Dom( w), i.e., C w  Dom( w) , and
to each role a subset of the domain Dom( w)  Dom( w),
such that the following equations hold:
(C * D )
(C + D )
(ШC )
Iw
Iw
Iw
C
Iw
D
C
Iw
( D
C
( and r .C )
Iw
( some r .C )
( n r )
Iw
( n r )
Iw
Iw
Iw

I
w Ј C( x ) iff x
Iw
w Ј r( x, y ) iff  x
C
Iw
Iw
,y
Iw
r
Iw
w Ј  x  iff w i x  Ј  ,  i  Dom( w)
w Ј  x  iff w i x  Ј  ,  i  Dom( w)
w Ј Ш iff w —
w Ј  *  iff w Ј  and w Ј 
w Ј  +  iff w Ј  or w Ј 
where w i x  designate a state obtained from w by
substituting i to x. The actions may not alter the set of
objects that exist in the world; that is, for every
 w, e  a , it must be the case that Dom( w)  Dom(e).
2.1. Action structure
 i  Dom ( w) |  j : (i , j )  r
Iw
I
w Ј ( x1  x2 )  iff x1 w  x2 w
Iw
 Dom ( w)

constraints do not necessarily constitute a well defined
concept in the DL sense. At this level, the state notion
used assigns interpretations to the variables in addition to
other symbols used in DL. Let w Ј  indicate that
assertion  is satisfiable or that the state w satisfies  .
Then, satisfiability of  is defined recursively.
Iw
 i  Dom ( w) |  j : (i , j )  r

  i  Dom ( w) |
Iw


I
 j C w
 j C
 i  Dom ( w) | ( j  Dom ( w) : (i , j )  r
Iw
( j  Dom ( w) : (i , j )  r
Iw
Iw

)  n
)n
Let C and D designate concept names and r a name
of a role in the sense of the DL. The subsumption relation
among objects concept is given by C subsumes D , which
I
I
is equivalent to D w  C w in state w . The actions
intervene at the factual (assertional) level relative to the
extension of the concepts. The assertion of the form C (i )
stipulates that the individual i is an instance of concept
C , and the assertion r (i, j ) indicate that the couple of
individuals (i , j ) is in the extension of r . In order to
associate an interpretation to the assertions, the function
I
(.) w is extended to individuals such that, e.g., C (i ) is
satisfied by w, and we note w Ј C (i ) if and only if
I
i C w .
The notion variable is not used in DL since a concept
indicates the set of the individuals that form the concept
and there is a logical equivalence between the semantic of
a concept (a role) and a unary (a binary) predicate. In the
case of an action, one needs to introduce free or quantified
variables to express constraints about states. These
The action a ( w) is structured in a traditional
formulation provided by the STRIPS language [10]. Each
precondition pre(a) is a conjunction of assertion
formulas concerning the conceptual objects as well as the
roles which bind these objects. The set of states in which
the action a ( w) may be performed, is given by the
domain: Dom( a )   w W  wЈ pre( a)  . That is, every
assertion which composes pre(a) must satisfy each state
w such that a ( w)  Ї .
The effects of actions pos ( a ) can be expressed by the
adding conditions of assertions described by the assertion

formulas pos ( a ), which means the addition to the
interpretation of concept or role involved in an action
a ( w) and the deletion from the interpretation of concept
or role denoted by pos (a) . For each concept symbol C
a
of assertion C ( x ), one can construct a function fC that
specifies how the interpretation for C changes when the
action is performed. Therefore, for every  w, e  a , the
new interpretation of concept C is expressed as follow:
C
Ie

 fca ( w)  i  Dom( w)
  e Ј posc ( a )

such that for each concept C , there must exist an
assertion formula posc ( a ) which satisfies the next state
e. Similarly, for each role symbol r , one can build an
a
interpretation function f r for r involved in a ( w) as:
r
Ie

a
 fr ( w)  i, j   Dom( w)
  e Ј posr (a)

This set of interpretation functions for concepts and
roles symbols through the state-transitions provides an
appropriate framework for characterizing an action in DL.
The effect of an action is defined as the difference
between the sets of interpretations of the symbols of the
concepts and the roles. Each assertion formulas
posc ( a ) of state-transition for concept C may be
expressed in terms of two others assertions formula


posc ( a ) and posc ( a ) which, respectively, describe the
additions and deletions conditions of interpretation of
each concept C . Therefore, if an action a ( w) is
performed in state w , the interpretation of C in next state
e is given by the following formulation:
C
Ie
 Iw

I

I
 C  ( posc ( a )) e  ( posc (a )) e

I

I
if ( posc ( a )) e  ( posc ( a )) e  
I

C w
elsewhere

Therefore, the expression posc ( a ) of state-transition can


be rewritten as ( posc ( a ) +  y C ( y )) *  posc ( a ))
meaning that the concept C remains satisfied after the
action a ( w) has been performed if and only if the action

makes it satisfied through posc ( a ) or else C was
satisfiable because there exists at least an individual y of
concept C and that the action a ( w) does not make it

unsatisfiable through  posc ( a ) . The same holds for a
role r concerned by the action a( w).
2.2 Subsumption of actions
We now define the subsumption relationship that
organizes actions concepts into taxonomy. Let a and b
two actions, we say, informally, that a subsumes1 b if
the action a is satisfiable with all states where b is
satisfied. This is formalized as follow and we note
b a a , the subsumption among these actions:
b
a
a   w, e  b : ( w Ј pre(b )  w Ј pre( a )) 




(e Ј pos (b)  e Ј pos ( a )) 
(e Ј pos (b)  e Ј pos ( a ))

The assertion formula posc ( a ) denotes the set of
individuals that will be added to the interpretation of C in
the new state e when the action a ( w) is performed and is
given by:

( posc ( a ))
Ie



 i  Dom( w)
  e Ј posc ( a ) .

The assertion formula posc ( a ) designates the set of
individuals that will be deleted from the interpretation of
C in the new state e when the action a ( w) is performed
and defined as:

( posc ( a ))
Ie



 i  Dom( w)
  e Ј posc ( a ) .
Hence, for every action performed, the individuals will
change type of concept throughout execution process.
Therefore, the semantic of action a ( w) in DL can be
reformulated as follow:



  w Ј pre( a ) 
 f a ( w)   i  Dom ( w)


 c



e Ј ( posc ( a ) +  y C ( y )) *  posc ( a ) 


a ( w)  
  i , j  Dom ( w)2 


  w Ј pre ( a ) 


 f ra ( w)  




e Ј ( posr ( a ) +  x, y r ( x, y )) *  posr ( a ) 



I

I
We note ( posc ( a )) e  ( posc ( a )) e  
which
implies that there does not exist individuals who can
simultaneously satisfy assertions formulas of additions and
deletions. Hence,
 prec (a) —posc (a) * posc (a ) .
If b a a , we have Dom(b)  Dom(a), there must
exist a state wb  Dom(b) , such that wb  Dom( a ),
hence wb Ј pre( a ). The same holds for the postcondition
of these actions by using the co-domain (range) given as
follows: CoDom(a)   w W  wЈ pos ( a)  . Therefore,
we will also say that a plan p1 given as sequence of
actions b1 ,..., bn , is subsumed by p2  a1 ,...,am if and
only if, for each action a i , there must exist a
corresponding action b j , such that b j a ai .
3. Validation
The DOMUS2 lab consists of a standard apartment
(kitchen, living room, dining hall, bedroom, and
bathroom) augmented with sensors, smart tags (RFID),
location and identification systems for objects and people,
audio and video devices, etc. This apartment is used for
research on smart homes, ubiquitous computing and
mobile computing. For instance, research projects explore
how to provide pervasive cognitive assistance to people
suffering from cognitive deficits (Alzheimer disease, head
traumas, schizophrenia, etc.). In these projects, one
difficult issue is to recognize activities of daily living
1 The subsumption relation among action concepts will be indicated by
a .
2 The DOMUS lab is sponsored by the Natural Sciences and
Engineering Research Council of Canada (NSERC), the Canadian
Foundation for Innovation (CFI).
(ADL) from inhabitant basic actions and events resulting
from these actions. Once observed actions are integrated
to infer the current ADL a user has in mind, the assistant
has to decide
 if help is needed,
 the right moment to provide assistance,
 how to assist (simply giving pieces of advices or
performing required actions at user place),
 Which devices to use to assist among the available
effectors (light, TVs, micros, speakers, PDAs, etc.)…
 how to take into account the user profile, in particular
user’s specific cognitive deficits and abilities.
So DOMUS lab and cognitive assistance will serve as
validation means for our theoretical model for reasoning
upon actions.
Therefore in a smart home, assistant agents and the
inhabitant (the user) have to share authority and control.
According to a given ADL, assistant agents can get
control of the decision making process and act according
to their viewpoints, for instance turning off the stove.
Agents and user go alternate from controller to observer.
As observer, they seem passive and take no decision or
actions. However they build and update their viewpoint on
the situation. In particular, they will use our model to
classify actions and events and integrates them into plans
to infer on-going ADLs, or to detect failures.
A viewpoint is formalized as a function performing
plan recognition based on actions observed in an
environment where direct communication between the
agents and the user is not possible. The result of this
function is an ontology of actions organized in a lattice.
For a given state of the world, that is the context, if the
viewpoint of an agent has a non empty lower bound, said
otherwise the state does not contain contradiction, and
then observed actions are part of a plan that has been
recognised by the agent, forming a potential situation for
assistance. In such a case, the agent can switch from the
observer mode to the controller mode and then perform
assistive actions eventually required.
A simple scenario is when a paraplegic on a
wheelchair has to go out late at night. He goes towards the
entrance, turns the handle, and goes out. The agent, that is
in observer mode, updates its trace of actions and its state
of the world: sensors indicates the handle was turned,
sensors indicate the door is open. When the agent detects
these transformations of the world through sensors, it
conceptualize them in an action structure. It builds in fact
a viewpoint on the progress of activities. This is achieved
through the classification of actions stored in its trace.
Before performing assistive actions, the system must build
a coherent viewpoint able to integrate the low-level sensed
actions into a high-level abstract activity, here opening the
front door to exit the house. This viewpoint is a
terminological basis describing the state of world w at a
given time. Once this step completed successfully, the
model for sharing authority enables the agent to take an
assistive initiative, in the present case to shut off the
apartment lights. To formalize this example, let suppose
the agent has observed the change TurnDoorHandle in the
state of the environment. The viewpoint of the agent on
this action will be to recognized and to build an
ontolology of actions subsuming this one, as shown in the
following example:
OpenDoor   pre :  d Door ( d ) * Close ( d )

pos : Open ( d )

pos : Close( d )
TurnDoorHandle   pre :  d , h, s Door ( d ) * Handle ( h ) * HandleSenser ( s )
placeOn ( d , h ) * placeOn ( h, s ) * Activated ( s ) * Close ( d )
* Close( h )

pos : Open ( d ) * Open( h) * Unactivated ( s )

pos : Close( d ) * Close ( h ) * Activated ( s )
Door  ( and MovableObject (or Close Open ))
Handle  ( and MovableObject (or Close Open )
( all placeOn Door ))
HandleSenser  ( and Senser (or Activated Unactivated )
( all placeOn Handle ))
Door (# EntryDoor )
Handle(# EntryDoorHandle)
placeOn (# EntryDoor , # EntryDoorHandle )
HandleSenser (# EntryDoorSenser )
placeOn (# EntryDoorSenser , # EntryDoorHandle )...
The clause OpenDoor models the user action
recognized by the system from the classification of lowlevel actions. This clause subsumes the sensed action.
TurnDoorHandle because all of its preconditions are
satisfied at the state of the worlk w at time t and all its
postconditions are also satisfied in the state of the world
resulting from this action. The two actions thus define a
lattice at time t. The terminological basis is then composed
of a set of conceptual objects that synthetise the elements
of the environments that are implied in the actions stored
in the trace (Door, Handle… ) as well as the assertions
(Door(#EntryDoor)…).
This terminological basis recognize the action
OpenDoor done by the user, the agent can then switch to
controller mode and shut down the lights. It can happen
that the agent cannot build a consistent viewpoint. The
agent is then constrained to reduce its sharing of authority,
and consequently it cannot intervene in the progress of
actions inside the apartment. Moreover it may exist many
means to assist for a given context. For instance, an agent
can turn off the stove for a paraplegic that forgot to do it.
However if the person is suffering of the Alzheimer
disease, a better choice is that the agent recall to the
person he forgot to turn off the stove in order he does it
himself. In that case, doing actions at his place would
contribute to the progress of the disease [16]. In such a
case, our model of classification of actions reveals not
only useful for plan recognition of user actions through a
viewpoint, but also for the identification of alternatives of
assistance actions for a given context.
The current implementation uses the knowledge
modelling system PowerLoom [17] to conceptualise the
objects of the environment. With respect to the
representation of actions based on the theory presented,
we are developing an extension in Java of PowerLoom by
integrating the formal model described above.
4. Related work
A lot of research was done to enhance description
logic with the concept of action. In this section, we will
discuss CLASP and Kemke denotational approach.
For one, Borgida [4] proposed a formal semantic for
actions in CLASP [9]. This semantic is based on an
axiomatic specification using inference rules of the form
ў    that means that the description  is subsumed
by the description  . This specification introduces
constructors (act, seq,...) that can be applied on concepts
belonging to states of the world to specify individual
actions and sequences of actions. These constructors can
be combined with classical constructors of description
logic (and, all,…) to define complex expressions of
actions. This formalisation is limited to the sole
description of propositional aspects of states of the world,
where each state of the world is considered as an instance
of a primitive concept associated to a proposition. This
approach has two major limitations. The first one stands at
the level of the description of preconditions and
postconditions. They are limited to the declaration of
simple conjunctions of facts. The other major limitation
lies in the lack of justification with respect to the
operations for the combination of actions. For instance,
there is no guarantee that the result of the conjunction of
the actions is an action whose precondition is the
intersection of the preconditions of the two actions, and
thus the intersection of the states of the world defining
these preconditions. In other words, the intersection
between states of the world is not formally defined. The
same observations apply to the rest of the action structure.
In our opinion, the problem arise from the semantic
associated to a state of the world which is reduced to a
simple primitive concept. Moreover there is a confusion
between the notion of instance of a concept being an
individual of the world and the notion that considers a
state of the world as an instance of the aforesaid concept.
This approach does not respect the definition of concept in
description logic. In our approach, a state of the world is
considered as a semantic structure whose interpretation
domain is made of the set of objects forming a situation of
the world.
Kemke denotational approach [14] specifies the notion
of action as a transition function from one state of the
world to another. A state of the world is made of the set of
objects of the world represented in the terminological base
in conformity to the taxonomy of object concepts. The
action concept is described by 1) a set of parameters or
variables that reference objects modified by the action, 2)
formulas for preconditions and postconditions stored as
attributes of the action. These parameters are also used by
formulas expressed in first-order logic. Kemke establishes
a correspondence between these two logics increasing a
lot the expressiveness of the description logic, and solving
the limitation of the Borgida’s proposal with respect to the
propositional description of the states of the world.
However Kemke’s proposal does not address changes of
interpretation of concepts and roles when a state transition
occurs, and consequently when an action has been just
performed. More precisely, the semantic of the application
of actions is partially described. The proposed theory
relies also on an informal introduction of the state
transition model of actions. Conversely, in our approach,
we show that the state transition model can effectively be
a model that renders description logic dynamic.
5. Conclusion
Our objective is to formally redefine the main issues
surrounding the problem of formalizing the action in
description logic. It should be emphasized that this initial
framework is not meant to bring exhaustive answers to the
questions raised by the multiple problems related to the
expressivity and complexity in DL. However, it
constitutes a first step towards a more expressive planning
language. Our aim is to develop an action language based
on the classification paradigm that will give us an
opportunity to introduce the dynamics into the description
logic.
References
[1] A. Ankolenkar, M. Burstein, J.R.Hobbs, O. Lassila, D.L.
Martin, D.McDermott, S.A. McIIraith, S. Narayanan. M.
Paolucci, T.R Payne, and K. Sycara DAML-S: Semantic Markup
for Web Services. In The First International Semantic Web
Conference (ISWC), Italy, (2002)
[2] A. Artale, and E. Franconi, A Temporel Description Logic
for Reasoning about Actions and Plans, Journal of Artificial
Intelligence Research, Vol. 9, (1998), 463-506
[3] F. Baader, D. Calvanese, D. McGuiness, D. Nardi, and P.
Patel-Schneider, The Description Logic Handbook: Theories,
Implementation, and applications, Cambridge University Press,
United Kingdom, (2003).
[4] A. Borgida, "Towards the systematic development of
description logic reasoners: CLASP reconstructed", In Proc. of
the Third International Conference on Principles of Knowledge
Representation and Reasoning, Cambridge, MA, (1992), 259269
[5] A. Borgida, "On the relative expressiveness of description
logics and predicate logics", Artificial Intelligence Journal,
Elsevier, NewYork , Vol. 82. (1996), 353-367
[6] A. Bouzouane, L.M. Gagnier, C. Dionne, I. Stiharu-Alex,
and D. Gagné, "Towards Virtual Role Playing Based on
Intelligent Agents", Applied Artificial Intelligence Journal,
Vol.13. (1999), 777-794.
[7] R.J. Brachman, D. McGuinness, P.P. Schneider, L.A.
Resnick and A. Borgida, "Living with CLASSIC: When and
How to Use a KL-ONE-like Language",In Principles of
Semantic Networks: Explorations in the Representation of
Knowledge, John F. Sowa (Ed), Morgan Kaufmann, (1991)
[8] R.J. Brachman, and J.G. Schmolze, "An Overview of the
KL-ONE Knowledge Representation System", Cognitive
Science, Vol. 9:2, (1985), 171-216
[9] P.T. Devambu, and D.J. Litman, "Taxonomic Plan
Reasoning", Artificial Intelligence, Vol. 84, (1996), 1-35
[10] R.E. Fikes, and N.J Nilsson, "STRIPS: a new approach to
the application of theorem proving to problem solving",
Artificial Intelligence Journal, Vol.2. (1971), 189-208
[11] J. Heinsohn, D. Kudendo, B. Nebel, and H.J. Profitlich,
"RAT: Representation of Action Using Terminological Logics",
DFKI Technical Report, (1992)
[12] M. Ghallab, A. Howe, C. Knoblock, D. McDermott, A.
Ram, M. Veloso, D. Weld, and D. Wilkins: "PDDL-the planning
domain definition language", In: AIPS-98, Planning Committee.
(1998)
[13] Y. Gill, "Description Logics and Planning", to appear in
AI Magazine, (2005), 1-22
[14] C. Kemke, "A Formal Approach to Describing Action
Concepts in Taxonomical Knowledge Bases", In: N. Zhong,
Z.W. Ras, S. Tsumoto, E. Suzuku (eds.), Foundations of
Intelligent Systems, Springer-Verlag, (2003), 657-662
[15] D. McDermott, "Estimated-Regression Planning for
Interactions with Web Services", In Proc. of the Sixth
International Conference in AI Planning, (2002)
[16] H. Pigot, A. Mayers, S. Giroux, The intelligent habitat and
everyday life activity support, In Proc. of the 5th International
conference on Simulations in Biomedicine, April 2-4, Slovenia,
(2003), 507-516
[17] PowerLoom: Knowledge Representation System, ISI,
University
of
Southern
California,
http://www.isi.edu/isd/LOOM/PowerLoom/index.html, (2002)
[18] R.A. Weida, and D.J. Litman "Terminological Reasoning
with Constraints Networks and an Application to Plan
Recognition", In Proc. of the Third International Conference on
Principles of Knowledge Representation and Reasoning
(KR’92), Cambridge, MA, (1992)