A Generic Model for Advanced Networks Handling Imprecise Information

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A Generic Model for Advanced Networks Handling Imprecise Information

Dr. Gábor Németh

¹

and Gábor Árpád Németh

²

1Communication Department, Budapest University of Technology and Economics, Hungary, e-mail: nemeth@hit.bme.hu

2Department of Telecommunications and Media Informatics, Budapest University of Technology and Economics, Hungary, e-mail: gabor.nemeth@tmit.bme.hu

Abstract: The previously separate fields of computing and com- munications are converging rapidly. Besides, the intelligence aspects necessitate the integration of formerly separate various processing modes. The emerging intelligent network aims to provide advanced services, extending well beyond the traditional voice, data transfer and computing services, handling intelli- gence aspects of possibly imprecise and inconsistent information as well.

These services are offered on dynamically varying network structures and processing nodes for arbitrary number of users in a transparent way. The widely used design models are based on separate problem classes, thus their extension to these problems is very difficult, if possible at all. A generic model for the clari- fication of the issues involved and determining the theoretical limitations is investigated, thus forming a starting point for various design models with precisely known limitations.

1. INTRODUCTION

The previously separate fields of computing and com- munications are converging rapidly. The emerging network consists of various resources (different both in technology applied and their underlining information processing and transfer models) and these resources should be used in a distributed way, transparently for the user. There are two major driving forces behind this approach. First, within a given technology applying parallel/distributed processing can theoretically increase the processing speed.

Second, in a large network the probability of failures and reintroduction of repaired modules is not negligible. More- over, the structure and capabilities of the network are changing very rapidly with the number and types of resources con- tinuously introduced because of new developments. An im- portant question for service providers is how many resources of a particular type should be offered? In distributed envi- ronment the optimisation problem is intractable in general because of the incoherent observability [1]. What is even worse, the suboptimal use of a particular, but only inconsistently observable set of resources requires the rearrangement of user procedures [2]. In case of dynamic task assignment the problem of assigning the various resources as observed to be available to the user tasks is the generalisation of the classical multiprocessor routing and scheduling problem, which is intractable even in its classical form [3]. The classical design models are not sufficient to tackle most of these problems.

Moreover even the goals of processing are changing. With the extension of the problem classes handled, the traditional information processing, transfer and retrieval services must be

modified to handle possibly imprecise and inconsistent information as well. For this, besides intelligence aspects, general topological considerations must be considered.

Consider, for example, the new concept of general service providing intelligent network. In this case a user enters its processing request at any node of the network. In the network the currently independent parts of the same algorithm(s) are concurrently executed by co-operating resources in such a way that all observable resources in the network are theoretically available for the user task [4]. However, in practice a number of problems must be solved. The concurrently executable parts form dynamically variable subsets of the logically concurrent parts. Available resource types depend upon the dynamically changing system structure and loading conditions as observed (incoherently) by the system. The various resources may be based on different information processing models (e.g. one node may be of classical control driven, and its co-operating partner of information driven neural network type).

The existing design models are based on classical mathe- matics and logic. However, both are timeless and lack causal- ity, containing expressions implying all elements exist forever with their relevant set of (possibly implied) values. But a computer system consisting of physical elements evaluates the expressions always in non zero time and - because of the limited resources available - in successive phases. A new model is needed to handle this situation.

2. RELATIONALSTRUCTURESANDOPERATIONS The generic processing element of the formal model proposed is the knowledge base system (KBS). A KBS is a da- tabase system with logical, temporal and topological structures together with operations on these structures.

The necessary mathematical concepts for modelling such a system are parameterised hierarchical relations, hierarchies of variables with their hierarchical control operators, and neighbourhood/similarity structures. These concepts are then applied to define a model of a knowledge module. By composition of knowledge modules the KBS is obtained.

The generic system model KBS consists of the following components [8]:

S a set of primitive objects,

A(S) a hierarchy of relations over S, all parameterised by indices of hierarchically structured index sets, F(S) an explicitly given part of A(S), the facts,

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D(S) = def A(S)\F(S) the implicitly given part of A(S), ob- tainable by composite applications of functions of R a set of inference rules, the application of which is

subject to constraints and conditions collected in a set G(R) the grammar of R.

var x variables defined on sets of components on all hierarchical levels.

P control functions; assignments to variables and re- ciprocal, reassignments to substitutable components are performed by a hierarchy of control functions, whereby a control function val: P×{var x}→X is associated with each variable var x, and where X = {x[p] | p∈P} is the variability domain (type) of var x and P is a set of control parameters p. Domains of variables can contain variables of lower hierarchical level and variables can be defined on sets of lower level control parameters of variables.

To operate on the components of KBS a set of operations OP has to be given (e.g. selectors: subset forming, pro-

jections, cuts, selection of substructures by properties, set constructors, concatenation of relations, and transformations of objects and indices, counting cardinalities).

G(OP) grammar for the application of operations of OP may be given.

To express structural properties of KBS, we need PR a set of predicates, e.g. generalised quantors, is part of

property, etc.

Given a partially or linearly ordered logical or physical model time

(T, <) all components of KBS can be indexed by time points and processes (KBSt)t∈U⊆T with varying states KBS[t] at time points t∈U⊆T can be considered.

Temporal properties can be adjoined to P, thus an evolutionary system can be considered, both in its capabilities and active structure [1].

Finally, on each hierarchical level, sets of objects, rules and parameters can be topologised by introducing a topological structure [6]. This is the generalisation of fuzzy sets [7].

As an illustration let us consider knowledge represented by binary relations with valuated elements. If (y, x) is a proposi- tion (object y has property x), it can be valuated by v ∈ V = {T, F}, yielding ((y, x), v). This includes of course composite objects (relations) y and composite properties (relations) x and arbitrary sets V with any structures. Given a valuation v´ to y, a valuation v´´ to x, and a function ϕ: (v´, v´´) |→ v, then to ((y, v´), (x, v´´)) can be assigned ((y, x), v). The valuations are of course arbitrary, thus extending the traditional data base concepts. If the knowledge module contains variables, e.g.

((yj, xi), var vji), they express indeterminacy. This case has to be distinguished from elements not appearing in the module, e.g. index pairs (j´,i´) ∈ (J×I)\U. A query operation in such a data base may yield a valuated approximate result (Fig. 1).

However, in a particular application x can be considered as an object, while y can be one of its attributes (with proper

valuation), while in another application y can be considered as an object and x of one of its attributes (with proper, but in general different valuation, Fig. 2). E.g. this can be utilized to describe the case when one pin of an IC can be both input and output, depending upon some control parameter. In that case composite search operations must be introduced.

Fig. 1. Topologised query operation gives qualified result.

Fig. 2. Symmetrical construction of data base.

In system design starting on top, we consider:

varKBS = (varS, varA(varS), varF(varS), varD(varF), var R(var F), varΓ(varR), varOP, varΓ(varOP), varP, var(T, <).

All variables varx range on given domains X parameterised by P[X] and have control functions val: P[X]→ X with control parameters p[X] ∈ P[X] .

The assignment steps in logical time are:

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varS := S ≠ ∅, selection of the primitive objects; varI := I ≠ ∅, selection of the primitive indices;

varOP := OP, varΓ(varOP) := Γ(OP), selection of admitted structors for S;

varP := P selection of structural predicates;

For bottom up construction of the hierarchy F:

varF⁽⁰⁾ : pow A\∅, selection of varF⁽⁰⁾ := F⁽⁰⁾ ; F(0) = ∅; varN := N, for n = 0,1,2,...N−1: varI⁽ⁿ⁺¹⁾ : pow I\∅, selection of varI⁽ⁿ⁺¹⁾ := I⁽ⁿ⁺¹⁾;

varF⁽ⁿ⁺¹⁾ : pow ( ^{( )})

var ⁽

Fⁿ ^J

J⊆

∪

Iⁿ⁺¹⁾ ^\^∅, selection of varF⁽ⁿ⁺¹⁾ := F⁽ⁿ⁺¹⁾ , varF(n+1) =: F(n) ∪ F⁽ⁿ⁺¹⁾ ;

Selection of admitted rules:

varR(F(N)) := R(F(N)), varΓ(R(F(N))) := R(F(N)).

Assignments to composite variables can be performed in partial steps. This results in varKBS := KBS.

The deduction steps in logical time are:

varf(varD) : R with Γ(R). Selection of f: varf := f, follows varD := D, varW := f(D). Selection of an argument: vard: D, vard := d, evaluation of varw := w = f(d). Decision on operation: varop(F, w): OP with Γ(OP), varop(F, w) := op(F, w).

Since at system design initially an informal, inconsistent and incomplete problem statement is given (in a natural language), first a formal, consistent and complete system specification is created by a directly executable specification language. The language is interactive (to fill incompleteness), resolve inconsistencies discovered by unconditional executa- bility of formal specifications and is based on existentially quantified logic (to handle unobserved or lost messages and states) [2]. Besides it is extended to handle knowledge structures as well.

Fig. 3. Main procedures of large system design.

This approach is a bottom-up constitution, which cannot guarantee reachability. (A system is reachable, if from any legal state it can be transited with finite number of legal operations to an arbitrary other legal state.) Therefore from this formal system specification the top-down implementation procedure is followed to ensure reachability, however, based on the formalism outlined above (Fig. 3).

For illustration the modification of the classical finite state machine is given. In the classical model M:S × I Æ S (S is the state and I is the input) does not take into account the time and the possible errors, thus a modified model is used:

A = <F, E, S, C, G, M>, where

{ }

fi _i _I

F= ∈

set of functions/services of the system (finite);

{ } ^{( )}

I i i J j j ei I i i E E

∈

= ∈

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎭⎬ ⎫

⎩⎨ ⎧

set of entities implementing the functions (finite), where Ei is the set of the entities implementing the function fi (finite);

( ) ( ) ( )

^j ^J ⁱ

I i j ,i K jk k si i J j

I i j Si S

∈

= ∈

∈

= ∈

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎭⎬ ⎫

⎩⎨ ⎧

⎭⎬ ⎫

⎩⎨ ⎧

set of the states of the entities (finite);

( )

ⁱ

J j

I i j ci

C ∈

= ∈

⎭⎬ ⎫

⎩⎨ ⎧

set of the clocks of the entities (finite), where ₌

⎜ ⎝ ⎛

^j

⎟ ⎠ ⎞

mi j ci j f ti j

ci defines the local time instant of a state transition;

( )

ⁱ

J j

I i j gi

G ∈

= ∈

⎭⎬ ⎫

⎩⎨ ⎧

set of knowledge functions (finite), where jq 1

gi = if the entity j

e senses until the time instant i j t

ci the messages sent by q

e , otherwise 0; i

( )

ⁱ

J j

I i j mi

M ∈

= ∈

⎭⎬ ⎫

⎩⎨ ⎧

set of the state transition functions of the entity j e i (finite), where

( ) ( ) ( )

_j ^j ^J

( )

ⁱ

Si i J j j Si i J j j q , i J q q Si j gi j:

mi ∈

∈ ◊

×

≠ ∈

∗ ∈

⎭⎬ ⎫

⎩⎨ ⎧

⎭⎬ ⎫

⎩⎨ ⎧

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎭⎬ ⎫

⎩⎨ ⎧

Note the term G, taking into consideration that in a real system messages may be lost or may arrive late, thus will be unobserved. The use of relation ◊ (eventually) instead of mapping handles the fact that any physical state transition requires non-zero time.

Unfortunately, the formalism while sufficiently precise and general, is too complicated for practical use. Its main draw- back is that it is very difficult to interpret it in engineering terms. As a consequence the generic model proposed is mainly for providing theoretically sound foundation for the devel- opment of practically usable limited scope design models. To formal, complete and consistent logical

model constitution

implementation

formal design model (the waterfall procedure ensures the reachability of the system) informal, incomplete

and inconsistent problem definition

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illustrate this difficulty a token ring is modelled with unidi- rectional flow (Fig. 4) [4].

i n link

L[u]

ou t

in po r t

ut po r t o node N[mod (u)+1]

in po r t

ut po r t o node

. . . N[u] . . .

(p) (q) (r) N (s)

a b

f1(p) q◊

f2(q) r◊

f1(r) s◊

Fig. 4. Specification of a ring.

NOTATIONS:

A - messages sent by a node (for the sake of simplicity let us consider instantaneous atomic messages only; the generalisation for finite duration messages is obvious);

B - messages received by a node;

- implication.

SYSTEM DESCRIPTION:

process system is

N: array [identifier] of node;

L: array [identifier] of link;

connection for u in 1..N: ^N

[ ]

^u^.^outport⁼^L

[ ]

^u^.ⁱⁿ^;

^N

[

^mod_N

( )

^u⁺¹

]

^.^inport ⁼^L

^{[ ]}

^u^.^out

end system;

The (sub)process node processes the messages received and sends out the results of the processing as new messages in the same order. The function f1 defines the processing in the node.

process node is inport: input;

outport: output;

constraint

(

^outport^/^a

)

_i^Ö

(

^outport^/^a

)

_i₊₁^;

(

^inport^/^b

)

_i

f1 ◊

(

^outport^/^a

)

_i

end node;

In the link (sub)process an error-free message transfer must be specified. This requirement can be formally described by five orthogonal rules.

process link is in: input;

out: output;

constraint /*R1: the order of messages is preserved*/

( )( )

[

â₁^,â₂^∈Â^,^b₁^,^b₂^∈^B

]

∀ :

[ ( ) ( )

b2

]

a2 b1 a1◊ ∧ ◊

⎩⎨

(

⎧⎢⎣⎡

a1 Ö

) (

b1

a2∧ Ö

) ]

^∨

[ ( ) ( )

⁼ ^∧ ⁼b2

]

^∨

b1 a2 a1 b2

[ (

a2Ö

) (

b2

a1∧ Öb1

) ] }

;

/*R2: no messages are lost during transmis- sion*/

( ) ( )

â^∈Â^∃^b^∈^B^:â^◊^b

∀ ;

/*R3: the link does not generate messages*/

( ) ( )

^b^∈^B^∃â^∈Â^:â^◊^b

∀ ;

/*R4: there is no reflection or echo*/

( ) ( )

[ ] [ ( ) ( ) ] ( )

b2 b1 b2 1 a b a : 2 B b 1, b , A

a∈ ∈ ◊ ∧ ◊ =

∀ ;

/*R5: the message contents are preserved (the output of the link is a function of only the message received at the input of the link, but code conversion is possible)*/

( )( )

[

^a^∈^A ^,^b^∈^B

]

^f:₂

( ) (

ⁱⁿ^/^a _i^◊^out^/^b

)

_i

∀ end link;

port input is

operation receive (a ∨ b);

constraint

[ ( ) ( )

^a^∈^A^∧ ^b^∈^B

]

end input;

port output is

operation send (a ∨ b);

constraint

[ ( ) ( )

^a^∈^A^∧ ^b^∈^B

]

end input.

Since the orthogonal features can be treated separately, it is enough to show how the technique works on a single feature.

For illustration let us prove that a message is not lost while travelling around the ring (see Fig. 4).

PROOF:

Let u be the identifier of an arbitrary processor and the link following it.

1. ∀p ∈ B in N[u].inport ∃q ∈ A in N[u].outport such that p ◊ q (2nd constraint in the node specification: ^f₁

( )

^p ^◊^q^).

2. ∀r ∈ B in

[

mod N

( )

u 1

]

.inport s AinN

[

mod N

( )

^u ¹

]

^.outport

N + ∃ ∈ +

such that r ◊ s (2nd constraint in the node specification:

( )

^r ^s

f1 ◊ ).

3. ^f₂

( )

^q ^◊^r (R5 constraint in the link specification).

4. p ◊ s (since ◊ transitive).

Fig. 5. Successive phases of candidate token generation.

The introduction of a clock system is one a necessary con- dition of sensing failures. For example, let us consider the case when a control token is lost. The timers of some entities will expire and they will send a candidate token with their identifier around the ring. Because of lack of space the discussion of the distributed election of a new control token is omitted here. But let us look at what the entities sense while their candidates make a full circle. Let us suppose entities i, j and k sense the loss of the control token (in different time instants in

i

k j

i

k

i

k j j

Ti

Tk

Ti Tk

Tj

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general). The situation can be followed on Fig. 5. In this case entity i sees candidates Ti and Tk during a full revolution of its own candidate, while entity k senses Ti, Tj and Tk, i.e. the observations are incomplete, necessitating the introduction of the knowledge function G into the formal model.

3. KNOWLEDGEREPRESENTATION

For the query operations topological structures are used, but the topological structures may also be variable (Fig. 6).

Fig. 6. Variable types (properties) of items.

Fig. 7. Composition of a knowledge base system.

Such modules can be used to compose a KBS by concatenation, i.e. feeding (part of) the answer of one module as (part of a) query to the same or another module (Fig. 7.).

Time may be handled the following way [5]. Let us consider a linear or partial ordered set (T,<) as time, and for simplicity binary modules which are time parameterised i.e. a process (((ytj, xti), vtji)(ji) ∈ U(t))t ∈ T, with U(t) ⊆ J(t) × I(t) . Let us define

( )

^t

I T def t I

= ∈∪ and ^J

( )

^t

T def t J

= ∈∪ , then concatenation of the family of families (( vtji)(ji) ∈ U(t))_{t ∈ T} yields (vtji)_{(tji) ∈ S} with a suitable S ⊂ T×J×I. In other words, history is handled just as another dimension in the hierarchy [9].

4. CONCLUSIONS

The proposed generic model is sufficiently general to describe large systems with nodes of any combinations of the four (control driven, data flow, demand driven and information driven) information processing models. The model can handle various types, various detail levels and various ap- proximation levels of information.

The main goal of the abstract formalism proposed is to gain deep insight into the basic elements and theoretical limitations of large system models, and not to develop engineering design tools. For that latter purpose it is too complicated, however, a proper simplification of the model for a particular problem class is possible to achieve that goal as well.

Since the generic model is theoretically sound, its tailoring to handle particular problem classes preserves its formal properties, thus its limitations are always well defined.

REFERENCES

[1] Németh, G., Formal treatment of time, Trends in Theo- retical Informatics, Oldenbourg, 1996, pp. 145-158.

[2] Németh, G., Parallel Architectures, 3C, 1996.

[3] Németh, G., Scheduling, timing and intractability in massively parallel systems, Computer Systems, vol. 11, no. 4, July 1996, pp. 245-254.

[4] Németh, G., Information Processing Networks, Proc.

ConTEL 97, 1997, pp. 293-302.

[5] Németh, G., Lovrek, I. and Sinkovic, V., Scheduling Problems in Parallel Systems for Telecommunications, Computing, vol. 58, no. 3, 1997, pp. 199-223.

[6] Albrecht, R. F., Systems with Topological Structures, Proc. Int.Conf. on Computing Anticipatory Systems, CASYS97, 1997.

[7] Németh, G., Knowledge Based Generic Model of Intel- ligent Networks and Services, Int. Conf. on Communica- tions and Computer Networks, ISBN CD: 0-88986-630-9, 2006.

[8] Németh, G., Towards a Generic Model of Intelligent Networks and Services, ACST 2007, ISBN CD:

978-0-88986-656-0, 2007.

[9] Németh, G.: Modelling Advanced Services and Net- works Handling Imprecise Information, Information Technology and Applications, 2008, ISBN:

978-0-9803267-2-7, pp.259-263, Australia.

A Generic Model for Advanced Networks Handling Imprecise Information