1

A Sketch of

Multiresolutional Decision Support Systems Theory

A. Meystel

Department of Electrical and Computer Engineering, Drexel University

Philadelphia, PA 19104

Abstract

Multiresolutional Decision Support Systems gain better

performance and higher accuracy by the virtue of building

highly efficient multiresolutional representation and

employing multiscale Behavior Generation Subsystem

(Planning and Control ). The latter are equipped by devices for

unsupervised learning that adjust their functioning to the

results of self-identification. We show planning and learning to

be joint processes.

Keyswords: behavior generation, control, decision support,

generalization, knowledge, learning, instantiation,

multiresolutional, multiscale, planning, randomized,

representation, resolution, search

1. Introduction

Multiresolutional Representation (MR)

of the World should be considered one of the tools

in the arsenal of Knowledge Management [1]. It is

the tool that is widely applied but is scarcely

noticed, probably because of its overwhelming

omnipresence. The concept of MR can be

illustrated by the series of pictures shown in

Figure 1 (a - f). The enhanced set of these pictures

with much more details can be seen in [2]. The

resolution of each subsequent image is increased

by an order of magnitude while the are of

observation is simultaneously reduced by two

orders of magnitude. It is not difficult to deduce

that as far as the underlying knowledge is

concerned, the objects in the image f are contained

in the image e [(skin texture)⊃(hand)], the objects

of the image e are contained in the image d

[(hand)⊃(sleeping person)], the objects of the

image d are contained in the image c [(sleeping

person)⊃(picnic)], the objects of the image c are

contained in the image b [(picnic)⊃(green lawn)],

and the objects of the image b are contained in the

image a [(green lawn)⊃(part of the city)].

This MR nestedness of sub-processes

and sub-systems of the overall processes and

system in not obvious in a standard cursory

analysis, it can be discovered only as a result of

special observations (computer vision equipment)

and investigative analysis. More importantly, it is

not obvious that this nestedness of entities and

their properties is important (if necessary at all)

for supporting the decision making activities at

each level of resolution. Yet, all images in Figure

1 are tightly linked by the prior cognitive

activities that are unified by identical processes of

generalization performed upon higher resolution

images to obtain a lower resolution image. Similar

processes of instantiation allow for receiving each

higher resolution image from the lower

Resolution. Actually, not the process of sensing or

the process of image edges detection and

segmentation (and others) determine further

image understanding and interpretation but rather

the joint processes of generalization and

instantiation that are executed upon these images

top-down and bottom-up.

As the signals measuring and

processing is conducted, at each particular level of

resolution they contain a different package of

frequencies (Figure 2). It demonstrates that the

granularity of representation is correlated with the

bandwidth of the signals at a level.

Why do we encounter this phenomenon:

multiresolutional knowledge representation? Why

the mechanisms emerged of generalization and

instantiation? The reduction of complexity via

reduction of “multiplicity” could only be done by

the virtue of grouping and representing the group

by a single symbol. This semiotic principle

emerged because of the need to reduce

computational burden. Computational benefits for

a particular example of knowledge representation

associated with planning is given in [8, 9].

The system of representation based upon

recursive grouping/decomposition incorporates

and uses the algorithms of generalization and

instantiation in different incarnations that depend

on circumstantial factors as for example, the

information we are dealing with, or the

subsystem of the world where the results are

applied. Thus, the learning system must employ

the same tools: labeling the entities in order to

deal with concise notations (symbols), grouping

the entities, decomposing them if information

details are required. Learning systems use the

same computational mechanisms.
2

a b

c e

e f

 Figure 1. Consecutive increasing the resolution of representation

The decision support system (DSS)

treats knowledge as an entity suggested in [1]: it

employs the awareness of familiarity gained by

experience for storing experiences as well as for

constructing decisions (including plans and

controls) that ensure functioning of a goaloriented system with increased performance

index. MR gives an opportunity to minimize the

value of computational complexity in a subset of

DSS that organizes knowledge by joint processes

of generalization and instantiation and use nested

MR-search for converging to a recommended

solution. This concept was introduced for

planning and control purposes in 1986 [2] and

explored in depth in subsequent works [3-10].
3

 Figure 2. Multiresolutional representation (alow resolution, b-mid-resolution+low

resolution, c-the sum of signals of all levels)

All these processes dwell on the processes of

learning employed by MR-DSS.

2. Decision Support of Behavior Generation

The structure of power station control

system shown in Figure 3 was successfully

tested at Delmarwa Power Station, DE, USA

[11]. is required in all faculties of a system

shown in Figure 3. Three levels of resolution are

demonstrated Low (“Task Level”), Middle

(“Component Level”) and High (“Actuation

Level”). Each level forms a loop closed through

connection 1. Each of these loops is a loop of

“closure” [9] and is equivalent to the Elementary

Loop of Functioning (ELF) described in [6, 7, 8].

The vertical subsystem 5 (Plant) from

Figure 3 is equivalent to subsystem Sensors,

World and Actuators from the Elementary Loop

of Functioning that is described in [8]..

Figure 3. Architecture of the Power Station Multiresolutional Decision Support System for energy efficient

Planning/Control

1

2 4 3 6 5

7

8

9

1

1

1

1

1

LEARNING

DISPLAY

PLANNING

(SEARCH)

ARBITRATION PLANT PREPROCESSIN REPRESENTATION

STABILIZATION/

SAFETY OVERRIDE

PREDICTION

ASSIGNMENT

ASSIGNMENT

TASK

LEVEL

COMPONENT

LEVEL

ACTUATION

LEVEL

TASK

LEVEL

COMPONENT

LEVEL

ACTUATION

LEVEL

TASK

LEVEL

COMPONENT

LEVEL

ACTUATION

LEVEL

TASK

LEVEL

COMPONENT

LEVEL

ACTUATION

LEVEL

TASK

LEVEL

COMPONENT

LEVEL

ACTUATION

LEVEL

TASK

LEVEL

COMPONENT

LEVEL

ACTUATION

LEVEL

1
4

Figure 3 contains several subsystems

that should be added to the standard ELF if there

is a need to equip the system by

Multiresolutional Decision Support: these are the

systems of Learning and Prediction. Learning

enhances Representation, while Prediction,

together with Stabilization arrives at the

subsystem of Arbitration (6). Standard ELF is

the core structure demonstrating the important

property of any functioning system, including

Intelligent Systems: it shows the property of

closure. The meaning of closure is in the fact

that the proper functioning requires the loop of

information flows to be closed.

What happens in the subsystem of

Behavior Generation? The latter has two

mechanisms: 1) maintenance of the ontology that

organizes the inventory of the known symbols

and their definitions and keeps relationships of

nestedness with ontology subsystems of other

resolution levels, 2) combinatorial search engine

that performs planning i. e. creates alternatives

of imaginary (possible and desirable) worlds,

and 3) simulator engine that explores expected

behaviors of the alternatives of the imaginary

worlds and/or monitors the execution processes.

The goal of functioning to be achieved

by the system arrives at the subsystem of

Behavior Generation [9] that is equipped by

mechanisms of planning and execution. At the

present time, these mechanisms cannot be

considered as thoroughly studied, and the general

theory of planning can hardly be attempted. We

will discuss a subset of problems in which the

goal is defined as the attainment of a particular

state or a particular string of states. Other types

of problems can also be imagined: in chess the

goal is clear -to win but this goal demand

achieving a special configuration (matesituation) but it cannot be achieved by arriving at

a particular pre-determined position in a space

(even in a descriptive space.) Most of the

problems related to the theory of games and

linked with pursuit and evasion are characterized

by a similar predicament and are not discussed

here.

Planning is understood as searching for

appropriate future trajectories of motion leading

to the goal. Searching is performed within the

system of representation (simulation) that gives a

tremendous advantage in comparison with

searching via trying.

3. Planning in a Representation Space with a

Given Goal

The world is assumed to be judged upon

by using its Space of Representation, or its State

Space which is interpreted as a time tagged

vector space with a number of properties. Any

activity in the World (State Space or

Representation Space) is called motion. It can be

characterized by a trajectory of motion with the

“working point” or “present state” (PS) that is

traversing the space from one point (initial, or

state, IS) to one or many other states (goal states,

GS.) The goal states are given initially from the

external source as a “goal region”, or a “goal

vicinity” in which the goal state may not be

completely defined in a general case. This vision

of the problem of Behavior Generation was

dominating in the area of Control Systems.

Planning was unified with Control only recently

when it became clear that both Planning and

Control are involved into anticipation of the

preferable motion (off-line) with some

appropriate correcting activities (on-line).

One of the stages of planning (often the

initial one) serves for defining where exactly is

the GS within the “goal region.” In this paper,

we will focus upon a subset of planning

problems where one or many GS remain

unchanged through all period of their

achievement. Traversing from IS to GS is

associated with consuming time, or another

commodity (performance index, or cost.).

Planning Problems in Behavior

Generation is frequently associated with the

domain of robotics or automated control systems

although it is absolutely equivalent to planning

in all other domains. Robotics became the

integrated research domain that provided for

blending the goals and testing the means of

achieving them, i.e. a domain with a direct need

for planning. In 1983, T. Lozano-Perez has

introduced the idea of search in "configurations

space". From the experience of using this search,

it became clear that the exhaustive search would

be computationally prohibitive if the

configuration space is tessellated with the final

accuracy required for motion control. The theory

of configuration space made one important thing

obvious: planning is searching for admissible

alternatives. This development helped to realize

that planning should combine the exhaustive

(often meaningfully complex) search off-line,

and an efficient algorithm of an off-line control.
5

At this period of time the engineering

community stopped talking about control of

actions and introduced a more balanced term of

Behavior Generation

The overview of the situation in the area

of planning and control can be found in [9]. The

recommended algorithm should be aligned with

the following suggestions. Consider the Ω state

space in which the start and final points SP and

FP are given. The minimum cost path from SP to

FP is to be found with the final accuracy ρ. Let

us consider particular cases Ω=Ω1 and ρ=ρm. To

declare the final accuracy is equivalent to

applying some mechanism od space tessellation.

One of mechanisms of tessellation is distributing

discrete points in the state space. We will

distribute them in a random fashion and then,

will determine the minimum cost path while

considering these points as vertices of an

imaginary graph. The condition of constructing

random tessellation reflects uncertainties of the

system that should be available for evaluation

from the existing representation. In Figure 4,a

the random points are distributed in the state

space with obstacles. An example of the result of

running a minimum-cost algorithm in the

tessellated state space with obstacles is shown in

Figure. 4,b.

Since the graph is randomized, the

trajectory is a random one, too. If one runs the

search algorithm a number of times, we receive

the results of searching as a “stripe of solution”

as shown in Figure 5,a. Then, we get a privilege

to continue with constructing tessellations of

higher resolution only within this stripe as shown

in Figure 5,b. Then, when we run the minimum

cost search-algorithm only within this “finely”

tessellated stripe with high resolution density of

tessellata, we receive a high resolution plan. This

process can be repeated recursively within as

many levels as necessary.

We will introduce three operators that

describe the above computations.

I. Operator of Representation (Â)

R:(Ω,ρ)→M, or M=R(Ω,ρ)

, (1)

where M- is the map representing the statespace Ω, ρ is the level of resolution of this map

determined by the density of the search-graph that we

intend to run at this particular level of resolution

(determined by the accuracy ρ). This is a non-trivial

operator because it presumes discovery of entities,

putting them into relationships with each other,

generalization, instantiation and measuring

relationships (including costs).

a

b

Figure 4 Randomized tessellation of the state space (a)

and a single running of the minimum cost trajectory

algorithm (a rectangle and a trapeze – obstacles)

II. Operator of state space search (S3)

S

3: (M, SP, FP, J, ρ)→P , (2)

or P=S3(M),where P- is the optimum path connecting

the start point SP and the finish point FP with

tessellation constructed for the accuracy ρ. J- is the cost

of operation which should be minimized as a result of

search S3. This operator should be based upon one of

the minimum-cost algorithms (e.g. Dijkstra) and

tailored for specifics of the problem.

III. Operator of space contraction (C)

which determines the width of stripe and the new final

goal for the algorithm of search.

C:(P, w)→Ω, or Ω=C(P), (3)
6

where w- is the width of the “stripe” obtained

after several runs of the search algorithm. Instead of

“stripe” one can use the term “envelope”, (e.g. the

“width” is the parameter of the envelope).

a

b

Figure 5 Multiple running of the minimum cost

trajectory algorithm (a) and the uncertainty stripe

obtained as a result of the multiple running (b)

The hierarchical control algorithm can be

described as follows:

for k=1, ..., m do the following string of procedures:

a) Ω k=C(Pk-1), or at k=1 assume Ω k=Ω,

b) Mk= R(Ω κ, ρk),

c) Pk= S3(Mk).

The algorithm of control can be represented as a

diagram

w ρk SP, FP, J

 ↓ ↓ ↓

Pk-1 →Χ → Ρ → Σ3 →Pk(4)

 

↑ Ω k ↑ Mk

or a recursive expression

Pk=S3(R (C (Pk-1, w), ρk ) SP, FP, J) (5)

The algorithm (5) has proven to be good

for off-line search in the state space. In the class

of on-line problems the process of control is to

be described by the trajectory of “working point”

moving in the state space.

4. Learnable Representations

All Representation Spaces are acquired

from the external reality by the processes of

Learning. Many types of learning are mentioned

in the literature (supervised, unsupervised,

reinforcement, dynamic, PAC, etc.). We will

focus primarily on processes of unsupervised

learning [12]. Before classifying the needs for a

particular method of learning and deciding how

to learn, we would like to figure out what should

we learn. Now, it is not clear whether the

process of learning can be separated

algorithmically into two different learning

processes: a) of objects representation, and b) of

the rules of action representation, or are these

two kinds of learning just two sides of the same

core learning process. In both cases, learning is

storing and generalizing information of

experiences with their values associated with

achieving particular goals.

The following knowledge should be

contained in the Representation Space. If no GS

is given, any pair of state representations should

contain implicitly the good rule of moving from

one state to another. In this case, we consider any

second state as a provisional GS.

 We will call “proper” representation a

representation similar to the mathematical

function and/or field description: at any point of

the space, the derivative is available together

with the value of the function. The derivative can

be considered an action required to produce the

change in the value of the function.

We will call “goal oriented”

representation a representation in which at each

point a value of the action is given required for

describing not the best way of achieving an

adjacent point but the best way of achieving the
7

final goal. Both “proper” and “goal oriented”

representations can be transformed in each other.

Neither is mandatory for functioning: valued

memories of experiences (ME) are sufficient.

5. The Artifacts of Representation Space:

The Phenomenon of “Sea Weeds”

Representation as sets of valued ME is

characterized by the following artifacts:

 • existence of states with its boundaries

determined by the resolution of the space

each state is presented as a tessellatum [9], or

an elementary unit of representation, the

smallest discernible unit of attention)

 • characteristics of the tessellatum which is

defined as an indistinguishability zone; we

consider that resolution of the space shows

how far the “adjacent” tessellata (states) are

located from the “present state” (PS)

• lists of coordinate values at a particular

tessellatum in space and time

• lists of actions to be applied at a particular

tessellatum in space and time order to achieve

a selected adjacent tessellatum in space and

time

• existence of strings of states intermingled

with the strings of actions to receive next

consecutive tessellata of these strings of

states

• boundaries (the largest possible bounds of

the space with similar properties, i. e. the

obstacles

• costs of traversing from a state to a state and

through strings of states.

Figure 6. The sea weeds

When ME are clustered into classes of

similarity (e.g. of adjacency) they remind

visually masses of “sea weeds”. In many cases,

the states contain information pertaining to the

part of the world beyond our ability to control it,

and this part is called “environment.” The part of

the world to be controlled is the system for

which we plan often referred to as “self.” Thus,

the representation is a part of “self” including

knowledge about actions that “self” should

undertake in order to traverse the environment.

Plans are formed as strings of preferable “sea

weeds” combined together.

6. Planning in Redundant Systems

Non-redundant systems have a unique

trajectory of motion from one state to another.

Redundant systems are defined as systems with

more than one “the best” trajectory of motion

from initial (IS) to final states (FS)

These systems contain a multiplicity of

alternatives of space traversal. Redundancy

grows when the system is considered to be a

stochastic one. The number of available

alternatives grows even higher when we consider

also a multiplicity of goal tessellata at a

particular level of resolution. This happens when

the goal is being assigning at a lower resolution

level which is the fact in multiresolutional

systems (such as NIST-RCS [8, 9])

In non-redundant systems, there is no

problem of planning. The problem is to find the

unique trajectory and to provide tracking of it by

an appropriate classical control system.

7. Learning as a Source of Representation:

Storing and Clustering “Sea Weeds”

Learning is defined as knowledge

acquisition via experience of functioning. Thus,

learning is development and enhancement of the

representation space. The latter can be

characterized in the following ways:

• by a set of paths (to one or more goals)

previously traversed

• by a set of paths (to one or more goals)

previously found and traversed

• by a set of paths (to one or more goals)

previously found and not traversed

• by a totality of (all possible) paths

• by a set of paths executed in the space in

a random way.

One can see that this knowledge

contains implicitly both the description of the

environment and the description of the actions

required to traverse a trajectory in this

environment.

All information arrives in the form of

experiences. The “learned” representation as a

set of strings of valued “sea weeds” is equivalent

to the multiplicity of explanations how to

traverse, or how to move.
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8. Types of Problems of Planning

Any problem of planning is associated

with

 • actual existence of the present state

 • actual, or potential existence of the goal state

 • knowledge of the values for all or part of the

strings of executable states as far as some

particular goal is concerned.

Any problem of planning contains two

components: to refine the goal (bring it to the

higher resolution) and to determine the path to

this refined goal. They are performed together, or

separately and can be formulated as follows:

a) given PS, GS and KS find the subset

of KS with a minimum, or a prearranged cost, or

with a cost in a particular interval.

b) given PS, Gs from the lower

resolution level and KS (all paths) find the GS

with a particular value

Finding solutions for these problems is

done by a process of planning. In other words,

planning is a construction of the goal states,

and/or strings of preferable states connecting the

present state with the goal states. There is a

striking similarity and interrelatedness between

planning and learning, actually their

inseparability.

In order to do this, we must learn where

the goal is located by consecutive refinement of

the initial coarse information. In all cases it is

associated with reduction of the

indistinguishability zone and the size of the

tessellatum associated with a particular variable,

i.e. the accuracy of representation grows. We

plan and learn by testing: in the representation,

for planning, and in the reality, for learning.

Learning via testing simulated systems is

becoming more and more wide spread.

The second component is the simulation

of all available alternatives of the motion from

the initial state. Procedurally, this simulation is

performed as a search, i.e. via combinatorial

construction of all possible strings (groups). To

make this combinatorial search for a desirable

group more efficient we reduce the space of

searching by focusing attention.

The need in planning is determined by

the multialternative character of the reality The

process of planning can be made more efficient

by using appropriate heuristics which are

available via processes of learning.

9. The Unified System of Planning and

Learning: A Subsystem of MR DSS

The process of searching for plans is

associated either with collection of additional

information about experiences, or with extracting

from KS the implicit information about the state

and moving from state to state, for the purpose of

learning. In other words, planning is inseparable

from and complementary to learning.

This unified planning/learning process

is always oriented toward improvement of

functioning in engineering systems

(improvement of accuracy in an adaptive

controller, improvement of efficiency in energy

consuming devices) and/or toward increasing of

probability of survival (emergence of the

advanced viruses for the known diseases that can

resist various medications, e.g. antibiotics.)

This joint process can be related to a

system as well as to populations of systems and

determines their evolution.

Figure 7. On the relations between planning and

learning

10. Planning, Learning, and Control: A

Unified Theory

Learning/Planning Automaton. The

joint Planning/Learning process is studied by

using a tool of Learning/planning automata

(LPA) is a tool that allows for jointly exploring

these two fundamental processes of intelligent

systems. Naturally, it becomes a component of

the Multiresolutional DSS.

Elementary Computations. Search (S)

is always performed by constructing feasible

combinations of the states within a subspace

(“feasible” means: satisfying a particular set of

conditions or constraints.) As many as possible

alternatives of feasible motions should be

explored and compared. If search is combined

with formation of alternatives, we call this

procedure combinatorial search (CS).
9

Usually, grouping (G) presumes

exploratory construction of possible

combinations of the elements of space and as one

or many of these combinations satisfy conditions

of “being an entity”, this group generates a new

symbol with subsequent treating it as an object.

The larger the space of search is, the

higher is the computational complexity of search.

This is why a special effort is allocated with

reducing the space of search, i.e. focusing

attention (FA) upon reduced sub-spaces. FA

results in determining two conditions of

searching, namely, its upper and lower

boundaries:

a) the upper boundaries of the space

where the search is to be performed, the scope

b) the resolution of representation (the

lower boundaries, the tesselatum)

Via exploring these experiences in

planning and learning we arrive at a conclusion

that they are always employ these three

procedures: grouping, focusing attention and

combinatorial search (or subsets of them).

The property of Intelligence. Forming

multiple combinations of entities (combinatorial

search, CS) satisfying required conditions of

transforming them into new entities (grouping,

G) within a bounded subspace (focusing

attention, FA) is frequently performed as a

fundamental set of procedures. Since these three

procedures work together we will talk about

them as about a triplet of computational

procedures (the abbreviations GFACS or CFS

are used.) Notice, that in learning it creates lower

resolution levels out of higher resolution levels

(bottom-up) while in planning it progresses from

lower resolution levels to higher resolution levels

(top-down). This algorithmic triplet emerges as a

tool of multiresolutional representation and/or

for the purposes of generating goal-oriented

behaviors.

This triplet of computational procedures

is characteristic for intelligence of living

creatures and constructed systems, and probably

is the elementary computational unit for

The need in GFACS is stimulated by

the property of knowledge representations to

contain a multiplicity of alternatives of space

traversal (i. e. a property of any representations

to be redundant.) Representations reduce the

redundancy of reality. This allows for having

problems that can be solved in a closed form (it

is a form when no combinatorics is possible

and/or necessary).

At each level of resolution, planning is

done as a reaction for the slow changes in

situation which invokes the need in anticipation

and active interference

a) to take advantage of the growing

opportunities, or

b) to take necessary measures before the

negative consequences occur.

The deviations from a plan are

compensated for by the compensatory

mechanism also in a reactive manner. Thus, both

feedforward control (interpreted as planning at

all levels of resolution but the highest one) and

feedback compensation of deviations are reactive

activities. Both can be made active in different

implementation approaches in control theory.

Examples: a) Classical control systems are

systems with no redundancy, they can be solved

in a closed form without searching.

 b) Any stochastic condition, any type of

uncertainty introduced to a control system

creates redundancy and requires either for

elimination of redundancy or performing search.

 c) Optimum control allows for the degree of

redundancy that makes searching feasible.

In Figure 8, the process of

multiresolutional planning via consecutive

search with focusing attention and grouping is

demonstrated for the control problem of finding

a minimum-time motion trajectory. From Figure

8, one can judge the processes that are performed

during the single level S

3

-search in the

randomized tessellated state space. The reader

can identify the processes because a trivial

example is considered: minimum time

functioning of the dynamic system. The

operation won’t change if one is dealing with

higher order and/or non-linear system.

intelligence. Its purpose is transformation of

large volumes of information into a manageable

form that ensures the success of functioning.

This explains the pervasive character of

hierarchical architectures in all domains of

activities including Decision Support Systems.

The space is learned and encoded in

advance by multiple testing, and its

representation is based upon knowing that the

distance, velocity and time are linked by

nonredundant expression. Several methods of

constructing attention envelopes are applied.
10

Figure 8. S3-Search in the state space for the minimum tyme dynamic trajectory

Conclusions

We have demonstrated the advantages

of Multiresolutional Decision Support Systems that

can be listed as follows:

1. Multiresolutional System of Knowledge

Organization allows to reduce complexity and

increase efficiency of representation.

2.Most of the Planning/Control problems are being

solved via S3-search (Search in the State Space). The

latter requires performing randomized state space

tessellation with density of points that reflects the

uncertainty of information. Multiresolutional S

3

-

search allows for stochastic optimization of systems.

3. Representations at each level of resolution are

organized as memories of experiences and do not

require constructing any analytical model: this

system plans and controls with no model required.

4. This representation supports processes of

unsupervised learning that contains self-oriented

information; no special self-identification is required.

5. The MRDS system was tested in applications to

power station energy efficient planning/control

system, for planning/control of an unmanned

autonomous mobile or spray-casting robots.

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