=Paper=
{{Paper
|id=Vol-2951/paper18
|storemode=property
|title=Interactive Granular Computing Connecting Abstract and Physical Worlds: An Example
|pdfUrl=https://ceur-ws.org/Vol-2951/paper18.pdf
|volume=Vol-2951
|authors=Soma Dutta,Andrzej Skowron
|dblpUrl=https://dblp.org/rec/conf/csp/DuttaS21
}}
==Interactive Granular Computing Connecting Abstract and Physical Worlds: An Example==
https://ceur-ws.org/Vol-2951/paper18.pdf
Interactive Granular Computing Connecting
Abstract and Physical Worlds: An Example
Soma Dutta1 , Andrzej Skowron2,3
1
University of Warmia and Mazury in Olsztyn, Słoneczna 54, 10-710 Olsztyn, Poland
2
Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw, Poland
3
Digital Science and Technology Centre, UKSW, Wóycickiego 1/3; b. 21, 01-938 Warsaw, Poland
Abstract
This short paper is an attempt to clarify the role of Interactive Granular Computing (IGrC) as a com-
putation model which respects that a real cognition about a real physical complex phenomenon and
making decisions based on that cannot be formalized only being in the language of mathematics. In this
regard, the paper focuses on presenting a real life example of computation where in order to move for-
ward, without stumbling over the obstacles, a blind person needs to explore and learn the surrounding
environment through interactions with the environment. The paper simply describes different compo-
nents and features of IGrC model in the light of the concerned example and explains how this computing
model has the potential to handle the grounding problem by bridging a connection between the abstract
mathematical modeling and the real physical semantics.
Keywords
interactions, granular computing, perception, knowledge specification, implementational language, com-
plex granule, informational granule, grounding problem, dynamic transition relation
1. Introduction
In a few of our previous papers [8, 9, 19] we already put forward our arguments in favour of a
need to develop a model for computing and reasoning which is not purely mathematical and
isolated from its real physical semantics, and which has the possibility to learn from the real
physical environment through real physical interactions. That such an endeavour is necessary
for building an intelligent system, dealing with complex phenomenon, is supported by several
opinions of different researchers from different fields of research [1, 2, 3, 4, 6, 7, 10, 11, 12, 13,
16, 18]. Without repeating many such inspiring thoughts of different researchers let us start
with citing one from [18].
[. . .] the often implicit stand one takes with regard to the question of the bridge
between physical and symbolic descriptions determines in a fundamental way how
one views the problems of cognition. A primary question here is, Exactly what kind
of function is carried out by that part of the organism which is variously called the
29th International Workshop on Concurrency, Specification and Programming (CS&P’21)
" soma.dutta@matman.uwm.edu.pl (S. Dutta); skowron@mimuw.edu.pl (A. Skowron)
0000-0002-7670-3154 (S. Dutta); 0000-0002-5271-6559 (A. Skowron)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)
sensor, the transducer, or in the case of Gibson, the “information pickup”? One’s an-
swer to this question determines the shape of any theory of perception and cognition
one subsequently constructs. As I shall argue, unlike the symbolic processes that char-
acterize cognition, the function performed by the transducer cannot be described as
a computation, that is, by a purely symbol-manipulation function. Like all primitive
operations of the functional architecture, the transducer fundamentally is a physical
process; that is, its behavior is explainable (its regularities can be captured) in terms
of its intrinsic properties - physical, chemical, biological, and so on. Thus, typically
it does not come under study by cognitive psychologists who simply presuppose that
an organism is equipped with certain transducer functions. The task of discovering
these mechanisms is left for others, for example, “sensory encoding neurophysiologists,
biophysicists, and engineers.
Let us also cite the opinion concerning the need of a new computing model in Cyber-Physical
Systems (CPS) [5].
The inherent cross-disciplinary nature of CPS requires distinct modelling techniques
to be employed, thus prompting for a common background formalism that enables
communication between all specialities. However, to this date, no such single super-
formalism exists to support the multiple dimensions of the design of a CPS. Indeed, to
effectively design a CPS, engineers (in the role of modellers) either need to be versed
in multiple formalisms, or a fundamentally new modelling approach has to
emerge.
According to the view of Edmund Husserl, the founder of phenomenology [7], non-standard
models of computation such as natural computing, reaction systems, Harel’s algorithmics and
Gurevich’s abstract state machines, or neural network computing are closed in the abstract
space, in the mathematical manifold.
Husserl was frustrated by the idea that science and mathematics were increasingly
conducted on an abstract plane [treating nature itself as a “mathematical mani-
fold"] that was disconnected from human experience and human understanding,
independently of questions of truth and applicability. He felt that the sciences
increasingly dealt with idealized entities and internal abstractions a world apart
from the concrete phenomena of daily life.
Till now, in different works (see, e.g., [8, 9, 14, 19, 20]), we tried to introduce what do we mean
by Interactive Granular Computing (IGrC) and how it is different from other existing theories
from the perspective of modeling computations in a complex system. In a very brief description,
Interactive symbolizes interaction between the abstract world and the real physical world, and
Granular Computing symbolizes computation over imperfect, partial, granulated information
abstracted about the real physical world. Here, once again we take the opportunity to explain
briefly the notion of complex granule (c-granule), the basic building block of IGrC.
A c-granule is composed of three parts, known as soft_suit, link_suit and hard_suit. These
three parts correspond to three sets of physical objects, together which determines the scope of
that particular c-granule1 . The soft_suit represents the objects from the physical reality which
are directly accessible or about which already some information is obtained. The hard_suit
corresponds to those objects which are in the scope of the c-granule but not yet accessed or
are not in the direct reach at that point of time of the c-granule. The link_suit represents a
chain of objects that creates a communication channel between the soft_suit and the hard_suit.
A c-granule, when associated with an information layer with it, is known as informational
c-granule, in short ic-granule. The information layer of an ic-granule contains different forms
of information, such as specification of the already perceived properties of the objects from its
soft_suit, specifications of the windows describing where and how some specific part from the
scope of the ic-granule can be accessed or reached etc. Based on the purposes and types of the
information specifications of an ic-granule there can be different types of ic-granules, such as
ic-granule representing perception of the objects from the scope (perception based ic-granule), ic-
granule representing domain knowledge (knowledge based ic-granule), ic-granule representing
plan of actions (planner ic-granule), ic-granule representing plan into an implementational
level language (implementational ic-granule) etc. Here to be emphasized that the ic-granule,
responsible for implementation of a plan of actions, serves the task of connecting between the
abstract and the real worlds.
A computation process over a c-granule is described by a network of ic-granules lying
within the scope of the concerned c-granule. The informational layer of all these ic-granules
constitutes the domain knowledge of the control of the c-granule, may be also called control of
the computation process; this informational content is endowed with a reasoning mechanism.
The information content and the reasoning mechanism together designs the control mechanism
of a computation process. More specifically, the whole information layer is clustered based
on the information relevant to different sub-scopes of the whole scope of the c-granule. For
example, in the ic-granules representing the domain knowledge, there can be different sub-
clusters in the informational layer corresponding to different aspects of the domain knowledge.
That is, an ic-granule may contain several other ic-granules inside its scope. On the other
hand the reasoning mechanism of the control of a c-granule is responsible for aggregating,
deleting, or generating information from the existing clustered of information layers. Thus new
information layers are generated over time based on (i) initial (partial) perception and domain
knowledge of the concerned fragment of the environment (ii) initiation of interactions through
already accessible objects to access the information about the not directly reachable objects
(iii) perception of the physical world after interactions and (iv) verification of the perceived
properties of the newly obtained configuration with the expected specifications of the target
environment.
Having this much of preliminary relevant details about a computation process over a c-
granule, in this paper our target is to present a real life example through which different aspects
of computation over a c-granule can be visualised.
In this regard, Section 2 presents an example of computing along with an explanation of how
such a real life computation can be modeled in the framework of IGrC. Section 3 presents the
concluding remarks explaining how different components and features of IGrC incorporate a
1
The scope of a c-granule at a given moment 𝑡 of the local time of the c-granule is the part of the physical space
corresponding to the formal specifications of all spatio-temporal windows active at 𝑡.
possibility of building a mathematical model which is not a simplified static image of its real
physical semantics; rather it is grounded in the real physical world.
2. Example of a computation over c-granule
The example described in [15] goes well with the idea of how a computation process, based on
perception and interactions, should look like according to IGrC model.
perceiving is a way of acting. [. . .] Think of a blind person tap-tapping his or her
way around a cluttered space, perceiving that space by touch, not all at once, but
through time, by skillful probing and movement. This is or ought to be, our paradigm
of what perceiving is.
Let us consider the above cited example as an example of a computation over a c-granule,
where the c-granule has in its scope a blind person2 and its surrounding. More precisely, the
person and the top part of the stick are directly accessible part of this environment, and hence
belongs to the soft_suit. The part of the stick, which is distant from the direct touch of the
person, belongs to the link_suit as a partial information about the end of the stick can be derived
based on the part belonging to the soft_suit, and it creates a link to the not directly accessible
objects, such as holes or stones lying in the surrounding environment, that is to the hard_suit.
The goal of the computation is to have a successful forward movement of the blind person by
deriving information about the unseen objects based on the already available knowledge and
the perceived information about the directly accessible objects. The whole computation process,
leading to the goal, is conducted by the control of the mentioned c-granule. The behaviour of
the control is based on transformations of collections (finite families) of the actual ic-granules,
or more exactly the actual networks of ic-granules, into the new ones.
All the information layers corresponding to different of ic-granules involved in a computation
is clustered in the control of the computation; using this information the reasoning mechanism
of the control makes the computation process to happen and dynamically move from one layer
to another layer. Below a step-by-step process of the computation is described in the context of
this example.
Layer:0
1. We assume 𝑔𝑠 to be an ic-granule having in its scope a blind person or robot with a
stick and the objects lying in the surrounding. At 𝑡0 , the beginning of the control’s cycle,
𝑔𝑠 is labelled with the perceived information of the directly accessible objects from its
soft_suit. Here the directly accessible objects can be the blind person himself and the
part of the stick directly in contact with the person (see Figure 1). To be remembered that
here 𝑔𝑠 represents a perception based ic-granule. The informational layer of 𝑔𝑠 contains,
in particular information concerning perception of the current perceived situation; in
more complex situations it may also contain link to the domain knowledge related to
general perception of the environment, formal specifications of the transformations of
2
One can even consider a robot instead of a human being.
informational physical
objects objects
informational
label
Part of real
S physical
A blind person or robot
objects and
with directly accessible
their dynamics
stick in the soft_suit (S)
are captured
in ic-granule gs.
A link created in the link_suit (L) The information
through the stick interacting layer at the top
with other objects connecting consists of the
L
the hole. perceived
properties in the
A hole in the hard_suit (H) H scope of gs.
gs
Figure 1: ic-granule operating on a particular scope of the physical world.
the ic-granules within its scope, database with rules for selection of transformation of ic-
granules for realisation, etc. In our illustrative example we discuss only a very simplified
version of 𝑔𝑠 .
2. Let the description of the general goal of the computation be attached as the information
layer of a planner ic-granule 𝑔0 . So, here 𝑔0 corresponds to those particular cells of a
human brain where the goal description is set. So, for 𝑔0 the soft_suit, link_suit and the
hard_suit can be like different layers of those brain cells where the reachability to more
deeper layer in the hard_suit happens through the directly reachable layer in the soft_suit
and reactions of the brain cells propagating from the soft_suit to the hard_suit.
3. The role of the knowledge base is represented by another ic-granule 𝑔𝑘𝑏 . Here, 𝑔𝑘𝑏 can be
considered as the brain parts related to the memory locations. The information layer of
𝑔𝑘𝑏 is labelled with the addresses of different relevant properties of different fragments
of the c-granule. The soft_suit of 𝑔𝑘𝑏 consists of the objects which form the outer box
of the memory location whose address is attached to the information layer; in order
to access the detailed information about some fragments some more inner boxes, lying
in the hard_suit, are to be opened. Such ic-granules representing knowledge base may
be called information granules. Their physical parts create local memories for storing
and transmitting information. One may also look on 𝑔𝑘𝑏 as on a compound ic-granule
representing a network of ic-granules determining the structure of 𝑔𝑘𝑏 .
4. Now based on the information gathered from the informational layers of 𝑔𝑠 , 𝑔0 and
𝑔𝑘𝑏 , the control with its reasoning mechanism aims to better understand the perceived
situation necessary for decomposition; this leads to construction of a more detailed plan
of actions. This detailed plan is represented in informational layer of the ic-granule 𝑔1 at
the next time point 𝑡1 . For a visual representation the readers are referred to Figure 2.
In our simplified illustrative example, we mention only one mechanism for enriching the
I: informational P: physical
objects objects
Layer:0
S
L
label H gkb g0
gs
of ic-granule
info translation info expansion
Layer:1
gi1 gs gkb g1
Figure 2: Computation over ic-granules passing from layer-0 to layer-1.
(i) 𝑔𝑠 : Current perception of some objects of P at time 𝑡0 , indicating the scope. S contains directly
perceivable objects, L contains objects creating communication channel to the objects in H where
some actions are to be performed.
(ii) 𝑔𝑘𝑏 : Relevant information about general laws related to objects in 𝑔𝑠 and specifications of
where this information is stored. Here S contains directly accessible part of the storage memory
and L contains the objects linking to the directory at H.
(iii) 𝑔0 : Relevant information regarding a goal that to be implemented in the H part of 𝑔𝑠 . This
specification of goal is stored in some object in the H of 𝑔0 , and is accessible by some object lying
in the S of 𝑔0 .
(iv) info expansion denotes decomposition of the plan available at 𝑔0 to a more detailed plan.
informational layer of 𝑔𝑠 . In a more realistic example, one should consider other mechanisms, e.g.,
measuring different parameters by sensors (e.g., stick in our example), recording the perceived
results of measurements in the corresponding informational layers of ic-granules etc. In a more
general case, a sequence of steps of reasoning is realised, which are based on transformations
of ic-granules, leading to better understanding of the currently perceived situation. It should
be also noted that in general decomposition problems are challenging and they are related, in
particular to the idea of information granulations and computing with words [23, 24, 25, 26].
Layer:1
1. From the perspective of the example, at 𝑡0 the information attached to 𝑔0 encodes the
general goal of the blind person that primarily gets registered in his brain, i.e., in the
soft_suit of 𝑔0 . At time 𝑡0 the hard_suit of 𝑔0 , such as more deep analytical brain cells,
remains still unaccessed. Based on the collected information of 𝑔𝑠 and 𝑔𝑘𝑏 , at time 𝑡1 the
person ponders more analytically; this in a sense activates interaction with the previously
unaccessed part of 𝑔0 . This gradually gives access to the hard_suit of 𝑔0 , and thus at time
𝑡1 the hard_suit of 𝑔0 becomes the soft_suit of 𝑔1 , labelled with a more detailed plan for
the person.
2. Now in order to implement the abstract description of the plan available at 𝑔1 through
real physical actions, the plan needs to be transformed from the abstract level to an
implementational level language. From the perspective of our example, this can be a
translation of the plan from the person’s analytical brain cells to a language of actuators,
like hands, legs, and the stick of the person. So, a new ic-granule is manifested at this layer.
We call it as 𝑔𝑖1 , an implementational ic-granule. To be noted that 𝑔𝑖1 does not concern
about the actual actuators; rather it is like another hard-drive in the brain of the person
where the action plan can be stored in the language of actuators. The information layer of
𝑔𝑖1 also contains the specification of the conditions for initiating the implementation plan
through a real actuator. Figure 3 presents the computation process described in layer-1.
Layer:0
gs gkb g0
info translation info expansion
Layer:1
gi1 gs gkb g1
info embedding
gi2 gs Layer:2
g2
Figure 3: Computation over ic-granules passing from layer-1 to layer-2.
(i) 𝑔1 : at time 𝑡1 specification of the plan of 𝑔0 is expanded. Detailed specification is generated
based on 𝑔𝑠 , 𝑔𝑘𝑏 and 𝑔0 of Layer:0. In particular the H part of 𝑔0 can be now the S of 𝑔1 which is
gradually reached through the L part of 𝑔0 .
(ii) 𝑔𝑖1 : Specification of how the abstract plan of 𝑔1 can be implemented in 𝑔𝑠 , that is the specifi-
cation of the plan in a lower level language which can be implemented via physical objects. This
lower level language is also a built-in language of a hardware lying at the H part of 𝑔𝑖1 .
(iii) info translation denotes translation of the plan from the level of abstract description to the
level of implementational language.
(iv) 𝑔2 : Specification in lower level language is embedded to a physical object lying in the S part
of 𝑔2 , which prepares the ground to run the plan via a physical object in H part of 𝑔2 .
(v) info embedding represents embedding the plan of actions on a real physical object.
Layer:2
1. The specification of the plan of implementation of 𝑔𝑖1 is now realized through a physical
object at time 𝑡2 . Let this object belong to the scope of the ic-granule 𝑔2 . In case of the
example, it can be the stick of the blind person on which the abstract implementation
plan is embedded, and 𝑔2 represents the ic-granule containing the stick in its scope. The
physical interaction of the stick with other objects in 𝑔2 is encoded in the information
layer of 𝑔2 . If this information matches to a significant level to the condition for initiating
implementation plan stored at 𝑔𝑖1 then an action compilation signal is passed to the next
implementation granule, may be named as 𝑔𝑖2 .
2. With the action compilation specification of 𝑔𝑖2 the objects lying in its link_suit and
hard_suit propagate actions to realize a desired configuration in the hard_suit of 𝑔𝑠 . In
the context of our example, 𝑔𝑖2 represents the ic-granule which specifies how to move
the stick forward until it touches a stone on its way. This chain of objects between the
stick and a stone creates a communication channel.
3. Through this communication channel the computation process enters into the hard_suit
of 𝑔𝑠 , which was inaccessible at time 𝑡0 . The initiation of the action compilation via 𝑔𝑖2
creates a link to the hard_suit of 𝑔𝑠 . This new interaction gives access to the hard_suit of
𝑔𝑠 which was previously inaccessible.
4. A new cycle starts by perceiving properties of the newly accessible part of 𝑔𝑠 .
Here to be noted, that in the example we only have mentioned about the decomposition of
the plan of actions from the initial stage 𝑔0 to a stage 𝑔1 , from where it gets translated to
the implementational level language. But in practice decomposition of the action plan, say
𝛼 : 𝑔0 ⇒ 𝛽 : 𝑔1 available in the information layer of 𝑔0 specifying the target ic-granule 𝑔1 with
property 𝛽 can have several layers of decomposition in between. For a visual representation
the readers are referred to the Figure 4, which will be discussed in Section 3. One should also
consider that some actions are lunched in the process of perception while the other ones are
related to the main decisions.
The above described idea of computation over a c-granule is in the line of the idea that Luc
Steels has characterised in [21]; complex dynamical systems (complex systems, for short) are
considered as systems consisting of a set of interacting elements
[. . .] where the behavior of the total is an indirect, non-hierarchical consequence of
the behavior of the different parts. Complex systems differ in that sense from strictly
hierarchical systems [...] where the total behavior is a hierarchical composition of the
behavior of the parts. In complex systems, global coherence is reached despite purely
local non-linear interactions. There is no central control source. Typically the system
is open.
3. Beyond pure mathematical modeling: a concluding remark
From the above exemplification of the process of computation over a c-granule, moving from a
configuration of ic-granules to another, it is quite clear that the process deals with a set of hunks
of real physical matter associated with their information layers; the information layers indicate
where, when and how they can be touched, or their properties can be achieved or verified. So,
this already clarifies how in the model of IGrC by a c-granule both abstract world, that is the
information layer, and the real physical semantics, that is the three-layered hunk of objects,
together are referred to. One more point, that needs to be clarified, is how the model designs its
real physical implementation by lifting the static description of a process to the level of actions.
The implementational ic-granules create a specific interface between the abstract and the
physical world. In particular, at some level of the implementational phase the control of the
c-granule launches actions linking the abstract world associated to the informational layer of
the control of the c-granule with the real physical world. These actions are not from abstract
mathematical space and the model of IGrC keep those action functions free from mathematical
formalizations. Their syntactic descriptions can be formalised in the informational layers of the
control of the c-granule. However, their implementation should be realised in the real physical
world and the model only can mathematically formalize their performance quality by perceiving
the changes in the world after initiation of the actions. Of course, the expected properties of
the real physical environment, after the implementation of the actions, also can be formalized
within the description of the action specification.
Thus the IGrC model keeps the possibility of mismatch between expected and real physical
outcome open as we never can a priori formalize all possible outcomes of an action, which is
supposed to be initiated in the physical world based on an abstract description of the action
specification. We can only expect a desired outcome, specified in the information layer. So,
after each implementational phase of a computation over a c-granule the control starts a new
cycle again by perceiving the properties of the objects lying in its scope. Then those observed
properties are verified with the expected properties. Hence, accordingly there is a need to
modify the induced models based on their comparisons with the recorded data.
In Figure 4 we illustrate the idea of decomposition of the description of the action plan in
order to transform an ic-granule with a given property, say 𝛼, into another one with the property
𝛽. After several levels of decomposition it reaches a level at which the relevant actions are
launched by the control so that the expected realisation of the whole chain of actions can lead
to a situation satisfying, to a satisfactory degree, the property 𝛽. Though the conditions of the
chain of actions and the expected properties after the actions are specified by 𝛼’s and 𝛽’s, the
actual actions 𝑎𝑐1 , . . . 𝑎𝑐𝑘 are not in the realm of mathematical formulation. The model of IGrC
keeps this juncture between abstract and physical bridging free from mathematical formulation.
Instead, IGrC proposes to formalize the results of actions by perceiving the properties of outcome
configurations and verifying them with the expected property 𝛽.
Let us outline a scheme of the behaviour of the control of a c-granule.
Often the control aims at achieving its target goal, expressed in the information layer of the
ic-granule 𝑔0 , using complex vague concepts, e.g., related to safeness of the perceived situation,
from a natural language. Below, we assume that for a given specification a set of rules for
selection of the relevant transformations of ic-granules was learned by the control or is given a
priori. The pre-condition of each rule is a condition. The degree of matching of this condition
by the current status of the perceived situation determines selection of the transformation of
ic-granules from the post-condition of the rule.3
The outline of the general procedure of a computation in IGrC, realised by the control
of a c-granule, is based on searching for relevant transformations of ic-granules and their
3
Note that other learning paradigms such as lazy learning can be used in inducing models of complex vague
concepts different from the discussed here.
Transformation specification tr from an
ic-granule with property to an ic-granule with
property available at the planner ic-granule g0
: g0 tr : g
∝: 𝑔𝑜 ⟹𝑡𝑟1 𝛼1 : 𝑔1 𝛼1 : 𝑔1 ⟹𝑡𝑟2 𝛽: 𝑔
...
∝: 𝑔𝑜 ⟹𝑡𝑟1 𝛼1 : 𝑔1 … 𝛼𝑘−1 : 𝑔𝑘−1 ⟹𝑡𝑟𝑘 𝛽: 𝑔
plan : ac1 ... ack
Figure 4: Illustration of a simple case of decomposition of transformation.
implementations; it looks as follows.
perceive accessible parts of the abstract and physical world to understand (up to a
satisfactory degree) the current situation for making decision concerning the selection of
the formal specification of transformation of ic-granules for implementation; the current
status of perception is represented in the informational layer of 𝑔𝑠 ; this is realised by the
control in several steps and one of them is listed below
⇓
consult domain knowledge and required goal of computation (𝑔𝑘𝑏 , 𝑔0 ) to enrich the
information about the status of currently perceived situation
⇓
...
(it may be necessary to perform several steps of reasoning before having a satisfactory
understanding of the perceived situation; it can be achieved by allowing the control to
select the relevant formal specification of transformation of ic-granules for
implementation; some of these steps may be related, e.g., to measurements (by sensors)
of features of the perceived physical objects in the scope of active ic-granules4
...
⇓
select (from the proper knowledge base) the relevant formal specification of the
transformation of the ic-granules for implementation5
⇓
4
It should be noted that the mentioned steps of reasoning are also realised by transformations of some ic-
granules.
5
This step is especially compound; details will be discussed in an extended version of our paper.
...
the selected formal specification may require several decomposition steps before the
proper level for the direct implementation can be achieved (embedding in the physical
world)
⇓
after reaching the satisfactory level of decomposition generate the formal specification of
the action plan; in the simplest case such a plan is represented by a linear order of
ic-granules which can be implemented by the considered control of the c-granule in the
physical world6
⇓
initiate (in proper order) implementational granules from the plan responsible for
implementation of actions of plan
⇓
during plan realisation, perceive the relevant accessible parts of the physical world
(through the implementational granules) and store the perceived data in informational
layers of the relevant ic-granules
⇓
follow the same loop as above taking, in particular into account the results of matching
the expected results with the perceived ones7 .
The discussed above issues of decomposition and implementation should be treated as an
illustrative example only. In the real-life projects one should take into account many other
issues.
Here to be noted that the transition from one configuration of ic-granules to the other,
as described above in the general plan of computation, by creating and accessing different
ic-granules and their information layers is not also purely mathematical. Usually, the state
transition relation is presented by a given family of sets {𝑋𝑖 }𝑖∈𝐼 where a transition relation is
represented as a relation 𝑡𝑟𝑖 ⊆ 𝑋𝑖 × 𝑋𝑖 . Here, this cannot be purely mathematical as we need
to incorporate the components which can specify (i) how elements of 𝑋𝑖 are perceived in the
real physical environment, and (ii) how the transition relation 𝑡𝑟𝑖 is implemented in the real
physical world. This reasoning mechanism, as described in the Introduction, is conducted by
the control of the c-granule over which the computation process is running.
So, in contrary to a static transition relation, in IGrC the control of a c-granule incorporates
the possibility of dynamic as well as not purely mathematical formulation of a state transition
relation. The structure of a control is designed to have two interacting modules, called the
abstract module (AM) and the physical module (PM) (see Figure 5).
Communication between AM and PM is designed by two mechanisms. The first one provides
a possibility of encoding given information from AM by a relevant state of a set of physical
objects from PM, and the second one provides a possibility of encoding the considered state of
6
Plans may be generated on different levels of decomposition, e.g., for better understanding the perceived
situation helping the control to generate the high quality plans for realisation of the target goals.
7
In a more compound case of the control of a c-granule, strategies for adaptation of the previously used plan
to the new situation are used. Moreover, the decision about the necessity of plan adaptation may be taken often
during of the actual plan.
AM PM
Figure 5: Interacting abstract (AM) and physical (PM) modules in control of c-granules.
a set of physical objects from PM by a relevant information represented in the language of AM.
These two mechanisms can be implemented by atomic actions.
AM module sends to PM a formal specification of a transformation of a ic-granule as well as
formal specifications of some spatio-temporal windows. Some of them are labelled by already
perceived information or properties from the scope of the c-granule and others are labelled by
formal specifications of the required information from PM. In case when the delivered specifica-
tion can be directly embedded by PM in the physical world then PM, by creating a network of
interacting ic-granules, aims to deliver a network of physical pointers matching information,
perceived by AM, with the expected properties, expressed formally in AM. Otherwise, PM sends
to AM a message about the necessity of the specification decomposition.
AM may send to PM messages consisting of formal specification(s) of the required trans-
formation(s) of ic-granules together with the expected results provided by AM. The messages
sent by AM are encoded by the states of some physical objects in PM. In this way, atomic
actions changing states of physical objects to the ones specified by the given information are
implemented. However, it is to be noted that the modeling of the behaviour of AM and commu-
nication of AM with PM can be based on mathematical modeling; whereas PM is composed out
of physical objects which, being from the real world, are outside of the abstract mathematical
modeling. However, partial information about properties of these objects and their interactions
may be communicated to AM through interaction of PM with AM. Thus, IGrC incorporates
both dynamic as well as mathematical modeling grounded in the real physical world.
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