Adapted from the proceedings of the
International Conference on Computing and Information Technologies ICCIT2001,
Montclair State University, Upper Montclair, NJ, pp. 455461
DYNAMICAL COMPUTING, COMMUNICATION, DEVELOPMENT AND HIERARCHICAL INFERENCE
H. M. HUBEY
Montclair State University, Upper Montclair, New Jersey, 07043, USA
Email: hubeyh@mail.montclair.edu
P. B. IVANOV
International Science and Technology Center, 9 Luganskaya
Street, P.O. Box 25, Moscow, 115516, Russia
Email: unism@narod.ru
A model of computing is suggested, combining the approach of
analytical mechanics with the principles of a general
psychological theory of activity. Thus reformulated, the
traditional picture of computation allows generalizations of
interest for distributed and parallel computing, artificial
intelligence, or consciousness studies. The notion of
hierarchical computing is discussed, stressing the
communicative aspect; the directions of increasing the
complexity of both computational universe and the computing
agents are indicated. The idea of computability is
reconsidered in the light of the new approach. The basic
principles of hierarchical logic are presented as a tool for
constructing generic formal systems.

1 Introduction
Using a computer, one has to arrive to useful results starting from
some raw material. The principal question is that of computability.
First computers were relatively simple, and the famous Godel
theorems reformulated for various formal systems [13] indicated the
limits of primitive sequential computing. With the development of
the Internet, the problem of computers talking to each other gained
importance, and the rapid development of parallel computing and
peertopeer technologies requires a different theoretical picture
reflecting the present situation. The inherent insufficiency of the
traditional logical systems in a complex environment has been
demonstrated by Hubey [4].
In studying human behavior, computer analogies are still
popular, which may hinder the inverse process, understanding
computation as a primitive analog of consciousness. A general theory
of activity that has been developed in Russian psychology since
1920s [5,6] could provide a solid framework for analysis of the
communities of computers. The key principle of that theory, the
sociality of development, perfectly reflects the practices of the
World Wide Web and may serve as a source of ideas in designing
efficient computer protocols approaching conscious communication.
Hierarchical structures and systems are necessary for efficient
computation in a developing world [7]. However, the general
principles of hierarchical organization are still poorly explicated
in the literature, and the relation of hierarchy to development is
far from being well understood.
In this paper, we present a summary of a hierarchical approach
to computation. A general model of dynamical computing serves to
translate the traditional static notions into a language more
suitable for description of motion and development. Then we consider
communicating computers and demonstrate how the opposition of the
inner and outer world appears. We also present a formal scheme of a
hierarchy, replacing the traditional idea of inference with
purposedirected construction.
2 Dynamics of computation
Traditionally, theories of computing were developed as formal models
of an isolated computer operating in an essentially static world.
Such an approach complies with the classical paradigm of
mathematical study, but its application to real computation can only
be limited, since the results of one computation serve to shape many
other computation processes. That is why alternative pictures of
computing may be useful.
2.1 The configuration space
Every computation occurs in some universe, so that successive
operations would change the state of that universe. We admit that
its distinct states can be somehow specified, and the collection of
all the possible states forms what physicists usually call a
configuration space X, which may be modeled with some
mathematical structure (e.g. a finite set, a Euclidean space, a
Hilbert space, a functional space, or a manifold). Points x
of the configuration space X represent both the possible
initial data and the possible outcomes of computation.
2.2 The agent
The agent is a device that can perform computation. According to A.
N. Leontiev's theory [6,8], we distinguish the following levels of
any agent's functioning.
2.2.1 Operations
An elementary operation changes the state of the universe, which is
naturally represented as transition from one point of the
configuration space to another. In every particular state (point
x) there is a variety of admissible changes; by analogy to
analytical mechanics [9,10], we will call it the tangent
space to X in point x,
T_{x}; the union of all the
T_{x} is called the tangent space to X and
denoted with T. Configuration space X together with all the tangent
spaces T_{x} forms the phase space of
the system, similar to a stratified manifold. Different agents are
represented by different tangent spaces T.
The points of X that can be connected with a single
operation are considered as adjacent. With thus introduced
notion of relative adjacency, points adjacent for one agent
may be not adjacent for another. For instance, different processors
may emulate each other's functionality on the microprogrammatic
level.
2.2.2 Actions
In this model, a computation process is represented by a trajectory
in the phase space. Formally, there is a mapping d: X
→ T, so that every point x of X
corresponds to a single element dx from
T_{x} . Given the initial and final states,
x_{i} and x_{f} , one can choose an
admissible trajectory to arrive from x_{i} to
x_{f} ; the class of such trajectories is called an
action. That is, contrary to operations that connect only
adjacent points, actions link distant points via a sequence of
operations.
Different agents may have different classes of admissible
trajectories, and the same action may either be unavailable to some
agents, or be implemented in different ways. The range of possible
actions is intimately related to the nature of the agent, and it can
usually be derived from a few fundamental principles. Thus, in
classical mechanics, the principle of minimum action normally
selects a single trajectory for any fixed x_{i} and
x_{f} . The same holds for quantum mechanics, but the
trajectory in a functional or operator space is considered instead
of the common 3dimesional space.
2.2.3 Activities
In simple configuration spaces, only operations and actions are
possible. In a more complex case, the points of the space X
form a number of classes X_{k}; any trajectory
connecting the points of the same classes X_{1} and
X_{2} belongs to the same action class, which is
called an activity. That is, an activity is like a higherlevel
operation, connecting adjacent classes; on the other hand, any
activity is nonlocal, since it implies actions.
For an example of activity, one can consider an infinite
trajectory in some configuration space X: the points on the
trajectory belong to the same class, and any action that can be
represented by a finite segment of that trajectory connects that
class to itself, hence belonging to the same activity.
Yet another important example: if the initial and final states
are structured, any action transforming a component of the initial
state to some component of the final state will be a representative
of the same activity. Such partial actions (iterations) may fail to
converge; however, such an activity often leads to quite acceptable
results (e.g. using asymptotic series expansions in special function
approximation).
2.3 The computable world
Every agent encounters certain initial (boundary) conditions and
operates following its builtin logic. However, due to the limited
operational capacity, the agent cannot achieve any point of the
world at all. Some points are unachievable because they are not
connected to the initial state by any admissible trajectory; some
other points are only asymptotically approached; there may also be
dynamical singularities that cannot lie on any admissible trajectory
regardless of the initial conditions. That is, any single agent can
only span a subspace of the full configuration space X; this
subspace is called the world W of that particular
agent, in the given dynamical conditions.
A world is an analog of a dynamic flow (a bunch of trajectories)
in analytical mechanics. In the simplest case, the world can be
reduced to a single trajectory.
This individual world may have a structure quite different from
the structure of the configuration space in general, actualizing
only a part of the possibilities available. For instance, in a
Euclidean configuration space, the individual world may form a
sphere, a torus, or a fractal. Also, the worlds spanned by the same
agent with different initial conditions may be quite different. The
agent can never break out of its individual world unless there are
other agents, and hence a hierarchy of agents operating in a
hierarchical world.
3 Hierarchical computing
The very distinction of the levels of operation, action and activity
is already introducing hierarchy in the model, implying a
hierarchical organization of both the configuration space and the
agent. In this section, we consider communication as the source of
hierarchical development, which gives way to numerous implications
of importance in distributed computing and artificial intelligence.
Let there be two agents A_{1} and
A_{2} operating in the same computational universe.
Since a point in the configuration space X is a distinct state of
the universe, and since, in this model, any operation changes the
state of the whole universe, the two agents cannot act
simultaneously, save in the trivial cases, and sequential operation
is the only possibility:
... → x_{1}
→ A_{1} →
x_{2} → A_{2} →
x_{3} → A_{1} → x_{4} → ...
This means that, from the viewpoint of each agent, the state of the
universe between successive operations or actions may
"spontaneously" change, which is impossible in the traditional
approach to computability. That is, the activity of another agent
results in discontinuities of individual trajectories, up to
switching to an entirely different class of trajectories (activity).
Similarly, assuming a universe developing according to some
natural laws, we arrive to the necessity of accounting for the
regularities of such development in the individual computation
processes. However, in this work, we are mostly interested in agent
produces changes in the universe and do not consider naturally
developing worlds.
Agents A_{1} and A_{2} exist in
their individual worlds W_{1} and
W_{2} , in general, spanning different parts of the
whole configuration space X . This leads to a number of
useful notions characterizing the possible relations between the
worlds.
Nonintersecting worlds W_{1} and
W_{2} imply that agents A_{1} and
A_{2} cannot operate together; if one of them works,
the other must be stopped.
An operation of agent A_{1} is
A_{2}compliant iff the resulting state of the
universe belongs to W_{2} . Such operations do not
change the activity of agent A_{2} , rather
influencing the timing of an action; alternatively, they could be
called boosts.
An operation of agent A_{1} is
A_{2}compatible iff it results in a point
x that can lie on some trajectory of A_{2} ,
maybe with different initial conditions. In other words, there are
points of X adjacent to x in A_{1} .
Obviously, all A_{2}compliant operations of
A_{1} are also A_{2}compatible.
Existence of noncompliant operations means that the actual
configuration space of an agent does not coincide with the whole
X and hence is reducible to some subspace of X.
However, as it will be shown below, there are no such domain
limitations in hierarchical agents.
Indeed, one can consider sequential operations performed by
different agents as a higherlevel operation performed by an agent
A consisting of both A_{1} and
A_{2}:
... → x_{1}
→ (A_{1} →
x_{2} →
A_{2}) →
x_{3} → ...
The intermediate state x_{2} (point of X) can be
interpreted as internal for A, and the elementary
operation of A transforming x_{1} into
x_{3} is a composition of the operations of
A_{1} and A_{2} ; the point
x_{2} of X, beside being a specific state of
the universe, represents a particular composition of
operations. Points of X that serve as internal for some agent
A (and hence mediate communication of lowerlevel agents) are
called products. State s of the universe that is
exclusively used to switch activity from one agent to another is
called a symbol.
Alternatively, one can consider a hierarchy of operations. The
original tangent space T now contains only direct
operations, while there also are indirect operations mediated
by other agents. In the above example, x_{3} may be
unachievable for A_{1} with any direct operation, but
it becomes achievable with a secondorder operation involving
another agent.
Hierarchical agents imply hierarchical worlds composed of many
individual worlds, plus the points x achievable via collective
actions. In the above scheme, the points x_{i} and
agents A_{i} become interchangeable:
... → A_{1}
→ (x_{1} →
A_{2} → x_{2})
→ A_{3} → ...
Like the points x may become internal for hierarchical
agents, transformations of the world (operations and agents) can be
considered as occurring in the interior of a higherlevel point of
the hierarchical configuration space. The difference between agents
and the states of the computational universe hence becomes relative.
From the hierarchical viewpoint, one could consider any action
as an operation of a higherlevel agent arising from selfcommunication:
... → x_{1} →
(A → x_{2}
→ A) →
x_{3} → ...
The agent A thus becomes composed of two specialized
components: one of them (the afferent component) transforming an
outer state of the universe x_{1} into an inner state
of the agent x_{2} , and the other (efferent)
component producing an outer state x_{3} from the
inner state.
4 Hierarchical inference
So far, we considered hierarchical computing in a static universe,
so that only its state could be changed. Beside the already
mentioned natural development, this picture can be complicated by
new objects produced by the agents. Once the states of the universe
become represented by some other states (symbols), the operations on
symbols may develop in a very complicated area. After all, agents do
not stop on symbolic computation, and they eventually pass to
material production, which enormously extends the configuration
space and opens new directions of hierarchical development.
Formally, this process could be modeled in a peculiar logic,
containing the following rules of inference.
 (Reflexivity) If there is an object O, there is a link
→ of this object to itself: O
→ O
 (Unfolding links) For any link →
there is an object O' mediating it, so that →
is equivalent to → O'
→ ; the resulting links are different from the
original and denoted with the same arrow merely for brevity.
 (Folding links) The reverse of (2): any mediated link can be
folded in a higherlevel link.
 (Abstraction) For any linked objects O_{1} →
O_{2} , there is an object O representing the link.
 (Unfolding objects) Any object mediating a link →
O' → is a contracted form of a triad of input,
inner state, output: → (S'
→ C' → R')
→ ; this rule might be replaced with an equivalent:
→ O' → implies
→ (S' → R')
→ , and then →
(S' → C' →
R') → by rule (2).
 (Refoldability) → (O_{1}
→ O_{2}) →
is equivalent to → O_{1})
→ (O_{2} → ,
with a proper reinterpretation of links.
The entity obeying these laws is called a hierarchy.
This set of rules is not minimal, and there may be many
equivalent formal systems. Explicitly specifying the levels of
hierarchy for both objects and links, one can construct rather
complex structures, then fold them into simple schemes, and unfold
in a different way. As one can easily see, in a hierarchy as a
whole, the objects do not differ much from the links, while they
will certainly be different in every particular hierarchical
structure derived from that hierarchy.
Obviously, no hierarchy can be complete, since any element can
be unfolded in a complex structure, producing additional elements
and additional types of links. Hierarchical logic is a method of
construction, rather than a description of a completed construction.
However, a hierarchy possesses a kind of absolute integrity, since
every element is related to each other, and the hierarchy can always
be unfolded in a structure, in which these elements are connected
with a direct link.
One could put forward the hypothesis that any static formal
system, as known in modern mathematics, can be obtained via
hierarchical development, as one of the possible unfoldings.
5 Conclusions
We have outlined an alternative approach to computation based on the
hierarchical ideas. This approach conveniently links the traditional
notions of analytical mechanics to the studies of human behavior
within a general psychological theory of activity. Such a synthesis
may be productive enough, to give birth to various nonstandard
theories of computation and inference, efficient methods of
distributed and parallel computing, new forms of artificial
intelligence. Even if not so, it presents one more possible
conceptualization, which is not reducible to any known mathematical
structure, rather being a tool for reconstructing any integrity at
all.
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