Modelling the Formation of Virtual Buying Cooperatives with

Grammars of Regulated Rewriting

Suna Bensch

, Sigrid Ewert

and Mpho Raborife

Department of Computing Science, Ume

a University, Ume

a, Sweden

School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

Keywords:

Coalition Formation, Virtual Buying Cooperative, Regulated Rewriting, Random Permitting Context Gram-

mar, Random Context Grammar.

Abstract:

In this paper we model virtual buying cooperatives (VBC) with grammars of regulated rewriting and show

that, if VBC relevant information is distributed over several successive VBC processes and must, in a later

stage, be synchronised and co-ordinated, the formal grammar needs to be very powerful with respect to mode

of derivation and thus generative capacity. In particular, we show how to model the supplier phase, invitation

phase, and declaration phase of a VBC with random permitting context grammars and the VBC reservation

phase with random context grammars under a special kind of leftmost derivation. If we use random permit-

ting context grammars for all processes, we can only model a VBC formation during which information is

introduced and processed locally and successively rather than being spread over different VBC processes.

1 INTRODUCTION

Cooperatives are generally goal-directed and have a

short lifespan; they are formed with a purpose and

dissolve when that purpose no longer exists and the

agents leave the cooperative (Horling and Lesser,

2005). They are most useful in situations where a

single organisation or business cannot perform a par-

ticular task or the efﬁciency of the task is increased

if more than one organisation or business performs it.

We consider cooperatives formed by physically dis-

tributed enterprises with the purpose of purchasing

items from a supplier as a single entity in a virtual

marketplace. The authors in (Ngassam and Raborife,

2013) term such a cooperative a virtual buying coop-

erative (VBC). Since a VBC acts as a single entity and

can buy a larger amount than an individual enterprise,

the VBC can negotiate favourable pricing.

In this paper we divide the VBC formation process

into four phases, supplier phase, invitation phase,

declaration phase and reservation phase and propose

that the reservation phase is guided by a purchasing

strategy that either promotes close business associ-

ations or a wide association distribution. The VBC

formation is basically the process during which enter-

prises invite their associates to participate in a VBC.

The VBC itself is the ﬁnally formed cooperative, that

is, the enterprises that purchase items from a supplier.

We model the formation of a VBC and the VBC it-

self with grammars and strings that these grammars

generate. Grammars or rewriting systems are genera-

tive devices that generate inﬁnitely many strings in a

so-called language but are themselves ﬁnite represen-

tations. Grammars are a suitable tool for modelling

VBCs because they

• describe the formation of a VBC as a generative

process reﬂecting the joint effort of all involved

enterprises,

• generate inﬁnite possibilities of VBC structures

(i.e. VBCs),

• can easily be implemented, and

• shed light on the structural conditions of a VBC

and thus can guide the implementation and re-

duce the costs for the actual technological devel-

opment.

An implementation of our grammar model should

enable the enterprises participating in a VBC, to

• form the VBC cooperatively and according to

their needs and networks,

• decide on a purchasing strategy for the VBC,

• be able to see the current state of the VBC forma-

tion (e.g. the participating enterprises, the quan-

tity provided by the supplier), and

Bensch, S., Ewert, S. and Raborife, M.

Modelling the Formation of Virtual Buying Cooperatives with Grammars of Regulated Rewriting.

DOI: 10.5220/0006403700450055

In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017), pages 45-55

ISBN: 978-989-758-265-3

• have proof of how many items an enterprise will

obtain once the overall quantity is purchased from

the supplier.

A technological realisation should take into ac-

count that in rural areas Internet is not always acces-

sible, but a text message based implementation would

let the isolated small enterprises communicate and

form the VBC via text messages. The content of such

text messages should contain information about the

current state of the VBC formation which is reﬂected

in our model as sentential forms during a derivation

process (explained in more detail later). We believe

that an implementation of our VBC model would en-

able such small businesses that operate in isolation to

access markets, to exploit current market opportuni-

ties in a virtual marketplace whilst maintaining their

autonomous business operations in the real world.

We model VBCs with grammars of regulated

rewriting and show that, if VBC relevant information

is introduced in different VBC formation phases and

must during the reservation phase be co-ordinated, the

grammars need to be very powerful with respect to

derivation mode and thus generative capacity. In par-

ticular, we show that the reservation phase requires

random context grammars under a special kind of

leftmost derivation, whereas supplier phase, invita-

tion phase, and declaration phase of a VBC can be

modelled with random permitting context grammars

(without leftmost restriction). Finally we show that, if

we only use random permitting context grammars to

model all four VBC phases, the VBC model will be

restricted, since information has to be processed “lo-

cally” and successively and cannot be spread across

the four phases.

2 RELATED WORK

The authors in (Ngassam and Raborife, 2013) con-

sider VBCs, that is, cooperatives formed by geo-

graphically distributed enterprises in a virtual market-

place. These enterprises meet at a virtual marketplace

and form cooperatives as and when needed based on

the items they are interested in. The enterprises pool

their buying power and negotiate a favourable pricing

based on the number of items that they will purchase.

Once they have made the purchase, the cooperative is

disbanded and another cooperative can be formed.

A VBC is especially beneﬁcial to very small enter-

prises (VSEs) where the owners usually work in iso-

lation and are not connected to economically strong

regions and markets. Due to their small sizes, VSEs

do not buy signiﬁcantly large amounts of goods and

this hinders their ability to fully aggregate demand

and negotiate discounted prices from their suppli-

ers. In emerging economies such as the Republic

of South Africa, VSEs are essential in driving eco-

nomic growth and creating employment (Ngassam

and Raborife, 2013). These businesses are crucial to

emerging economies but are usually operated in infor-

mal environments which are typically characterised

by poor infrastructure, poor inventory management,

bad working habits, and lack of direct access to mar-

kets. This leads to the exploitation of such enterprises

by their suppliers (Merz et al., 2007) (Merz, 2010).

The authors in (Mashkov et al., 2015) model coali-

tion formation as unselﬁsh agents using Petri Nets.

We model the formation of a VBC and the VBC it-

self with random context grammars and one of their

variants. Various regulated rewriting grammars and

their languages and the motivations thereof are ex-

tensively described in (Dassow and P

aun, 1989). We

choose random context grammars to model the VBC

formation since many actions during the formation

are context-dependent and the grammar formalism is

a relatively simple extension of the easy-to-handle

context-free grammars. Random context grammars

(rcgs) (van der Walt, 1972) belong to the class of

context-free grammars with regulated rewriting (Das-

sow and P

aun, 1989), i.e., the productions of a gram-

mar are context-free, but are applied in a non-context-

free manner.

In rcgs, the application of a production at any step

in a derivation depends on the set of symbols that ap-

pear in the sentential form of the derivation at that

step. As opposed to context-sensitive grammars, the

context may be distributed in a random manner in the

sentential form. Context is classiﬁed as either per-

mitting or forbidding: permitting context enables the

application of a production, while forbidding context

inhibits it. When a grammar uses permitting context

only or forbidding context only, it is called a random

permitting context grammar (rPcg) or random forbid-

ding context grammar (rFcg), respectively. The corre-

sponding generated languages are called random per-

mitting context languages (rPcls) and random forbid-

ding context languages (rFcls).

The authors in (Dassow and P

aun, 1989) showed

that rcgs without erasing productions lie strictly be-

tween the context-free and context-sensitive gram-

mars. When erasing productions are allowed, rcgs are

as powerful as the recursively-enumerable grammars.

It is not known whether rFcgs without erasing produc-

tion rules have an erasing equivalent. However, every

rPcg with erasing production rules has a non-erasing

equivalent (Zetzsche, 2010).

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

supplier phase

invitation phase

declaration phase

reservation phase

invitation tree

purchasing strategy

Figure 1: An illustration of the VBC formation process

divided into supplier phase, invitation phase, declaration

phase and reservation phase. The invitation phase is reg-

ulated by an invitation tree and the reservation phase is reg-

ulated by a chosen purchasing strategy.

3 VBC FORMATION

The formation of a VBC involves an enterprise

(termed the initiator enterprise) approaching a sup-

plier with the intent to purchase items (Ngassam and

Raborife, 2013). The supplier in turn replies with the

overall available quantity of the requested items. The

initiator enterprise then purchases items, and invites

selected associates, who in turn invite their associates,

etc., to join the VBC in order to purchase the items

from the supplier. In our model of a VBC formation

we deviate from (Ngassam and Raborife, 2013) as fol-

lows. The initiator enterprise approaches a supplier

that replies with the overall quantity of purchasable

items as in (Ngassam and Raborife, 2013) but the ini-

tiator enterprise does not necessarily get to purchase

before all other enterprises. Our model of VBC for-

mation, which is illustrated in Figure 1 consists of the

four following successive phases.

The ﬁrst phase is the supplier phase in which the

supplier provides the quantity of goods that can be

purchased to a favourable pricing. The supplier phase

is followed by the invitation phase in which all en-

terprises that belong to a network of associates and

want to participate in the VBC are invited. In the

subsequent declaration phase all enterprises declare

how many items they would want to purchase. The

last phase is the reservation phase where enterprises

reserve the quantity of goods they will obtain after

the overall quantity is purchased from the supplier.

We suggest that the reservation phase (i.e. the actual

purchasing) in the VBC is regulated by a purchasing

strategy that is chosen after the declaration phase. We

propose two different strategies:

1. (Early strategy or E-strategy) Enterprises that

were invited at an earlier time step can purchase

items before enterprises that were invited at a

later time step. This reﬂects a purchasing strat-

egy where close associations or ﬁrst-degree asso-

ciations are promoted.

2. (Maximal strategy or M-strategy) Maximize the

number of enterprises that purchase. This strategy

reﬂects a network-wide distribution of purchasing

opportunities to as many enterprises as possible.

One can think of other possible purchasing strate-

gies, including a mixture between E- and M-strategy.

In this paper we assume that a purchasing strategy

is chosen outside our model. Regardless of the pur-

chasing strategy, the difference between the number

of items provided by the supplier and the number of

items purchased by the VBC should be as small as

possible. This is in order to guarantee that the VBC

can negotiate a favourable pricing. Moreover, the to-

tal number of items purchased by members of the

VBC cannot be more than the quantity made avail-

able to them by the supplier. In our model the four

phases of a VBC and the results thereof are explicitly

modelled as generative processes. We will construct

grammar fragments for all four phases. The supplier

phase is part of our model and represents the supplier

providing the available overall quantity of goods. For

the subsequent invitation phase we assume the follow-

ing:

• The VBC formation starts with a given network

graph. A network graph represents a network of

associates and is represented as a graph. A net-

work graph is ﬁnite and predetermined at the start

of a VBC formation. Network graphs can be ex-

tended by adding new enterprises to the network

(but not during a VBC formation).

• Enterprises can invite as many associates from

their network as desired, regardless of whether the

invited associates are interested in purchasing or

not, or whether they invite other associates or not.

• An enterprise may only be invited once.

During the invitation phase an invited enterprise

can either invite other enterprises from its network or

not invite any further enterprises. During the declara-

tion phase an invited enterprise can either declare the

amount it wants to purchase or opt out of the VBC by

not declaring items. During the reservation phase and

after a purchasing strategy has been chosen, we model

the binding allocation of goods to the enterprises that

have chosen to purchase. The reservation phase is the

part of the coalition formation that requires grammars

with high generative capacity to model, since the co-

ordination of information must be modelled with de-

vices that have the capacity to keep track of informa-

tion distributed over several phases.

Modelling the Formation of Virtual Buying Cooperatives with Grammars of Regulated Rewriting

4 DEFINITIONS AND

PRELIMINARIES

Let N denote the integers, and N

= {1, 2,.. .}. An

alphabet is a ﬁnite set of symbols. A word over an

alphabet Σ is a ﬁnite ordered list of symbols chosen

from the set Σ. A language L is a set of words over

some alphabet (Xavier, 2005).

Deﬁnition 1. (Ewert and Van der Walt, 2002)

A random context grammar (rcg) is a quadruple

G = (V

,P, S), where

1. V

is a ﬁnite set of non-terminals,

2. V

is a ﬁnite set of terminals,

3. P is a ﬁnite set of productions of the form

A → x (P ; F ),

where A ∈ V

, x ∈ (V

∪V

)

∗

and P , F ⊆ V

, and

4. S ∈ V

is the start symbol.

Let V denote V

∪V

. For two strings y

∈ V

∗

and a production A → x (P ;F ) in P, we may write

=⇒ y

if every B ∈ P is in the string y

and no B ∈ F is in the string y

. We refer to

=⇒ y

as a derivation step and to the strings

, y

as sentential forms. The reﬂexive and

transitive closure of =⇒ is denoted by

∗

=⇒. The lan-

guage generated by a grammar G is deﬁned as

L(G) = {w | w ∈ V

∗

and S

∗

=⇒ w}.

A random permitting context grammar (rPcg) is

a random context grammar G = (V

,P, S), where

for each production A → x (P ; F ) ∈ P, F =

0. A ran-

dom forbidding context grammar (rFcg) is a random

context grammar G = (V

,P, S), where for each

production A → x (P ; F ) ∈ P, P =

0. A context-

free grammar (cfg) is a random context grammar

G = (V

,P, S), where P = F =

0 for each produc-

tion A → x (P ; F ) ∈ P. A language is context-free,

random permitting context, random forbidding con-

text, or random context if it is generated by a context-

free, random permitting context, random forbidding

context, or random context grammar respectively. In

the remainder of this paper, we write A → x instead

of A → x(

0), if the permitting and forbidding con-

text are empty sets. If a context consists only of one

element, we write A instead of {A}.

Example 1. Let G = ({S,A, B,C},{a, b, c},P, S),

where P is given by

{S → ABC,

A → aA

(B;

0),

B → bB

(C;

0),

C → cC

;

0),

→ A(B

;

0),

→ B(C

;

0),

→ C(A;

0),

A → a(B;

0),

B → b(C;

0),

C → c}.

The generated language is L(G) = {a

| n ≥ 1}.

The derivation starts with rewriting S with the ﬁrst

rule and we obtain the sentential form ABC. Now the

second and third are applicable, each of which intro-

duces an A

or B

into the sentential form. After, for

example, applying the second and third rule we ob-

tain the sentential form aA

C. Then the fourth rule

is applied (since there is an A

in the sentential form)

and we obtain the sentential form aA

. Note

that at this point of the derivation process the ﬁfth

rule as well as the sixth rule can be applied. If we,

at this point, would apply the sixth rule we would ob-

tain the sentential form aA

bBcC

and the ﬁfth rule

would not be applicable since the symbol B

does not

occur in the sentential form. In fact, such a deriva-

tion would block and never complete since no rule

could be applied. So, after obtaining the sentential

form aA

as described above, we apply the ﬁfth,

sixth, seventh rule and obtain aA

∗

⇒ aAbBcC.

After this we start the ﬁrst cycle again (applying sec-

ond, third, fourth rule again) or terminate the deriva-

tion by using the last three rules. We can generate all

strings consisting of an equal number of occurrences

of a, b, and c.

5 RANDOM PERMITTING

CONTEXT GRAMMARS

In this section we construct successively fragments of

an rPcg for the supplier phase, invitation phase, decla-

ration phase and reservation phase and show that the

reservation phase cannot be described by an rPcg. We

illustrate the modelling with the Examples 2, 3, 4, and

Let q ≥ 1 be the number of enterprises participat-

ing in the VBC. The language that we want to gener-

ate is L

VBC

and reﬂects the outcomes of the supplier

phase, invitation phase, declaration phase, and reser-

vation phase. We divide a string in L

VBC

by the des-

ignated symbol $, where to the left of the $ symbol

invitation and declaration are represented and to the

right of $ the reservation and supplier phase are rep-

resented. L

VBC

is deﬁned as:

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

D E

Figure 2: The graph represents a network of associates.

Each node label corresponds to a name of an enterprise and

an edge connecting a node n

with a node n

represents that

the enterprises at n

and n

are associates.

VBC

= {a

.. . q

for1 ≤ i ≤ q, m

= n

or 0,n

≥ 1, k ≥ 0}.

The non-context-free language L

VBC

reﬂects the

VBC formation and the VBC itself (that is, the enter-

prises that will purchase goods and were selected by

a purchasing strategy). The letters a, b,. . ., q repre-

sent different enterprises and the number of their oc-

currences (n

occurrences of a, n

occurrences of b,

etc.) represent the quantity of goods they declared

(i.e. would want to buy). To the right of the

$ symbol

we represent the enterprises that will purchase goods,

possibly followed by some letters x representing items

provided by the supplier that the VBC will not buy.

In particular, a string in the language represents the

result of the invitation phase and declaration phase to

the left of the symbol

$ and the reservation to the right

of the symbol

$. In what follows, we describe how a

random permitting context grammar G is constructed

to describe the phases up to the reservation phase.

We assume there exists a network of associates

represented by a graph. Let the network graph in Fig-

ure 2 represent such a network of associates, where

A is the name of the initiator enterprise. Each node

label corresponds to a name of an enterprise and an

edge connecting a node n

and a node n

represents

that the enterprises at nodes n

and n

are associates.

In Figure 2, for example, enterprise A is linked with

enterprises B, C, D, E and G and enterprise B is linked

with its associates A, C and E, etc.

Assume the network graph N in Figure 2 as

given. We start with modelling the initiator enter-

prise approaching the supplier and the supplier pro-

viding the overall quantity that the VBC can purchase.

To this end, we construct a fragment of our rPcg

G = (V

,P, S) as follows.

Example 2. We construct rules of the following

form: S → A

inv

, X

→ X

X, X

→ $, where

inv

,X ,$ ∈ V

. The ﬁrst rule rewrites the start

symbol and is applied once, representing the initiator

approaching the supplier. The second rule is applied

as many times as the total number of purchasable

items the supplier is providing and the third rule is ap-

plied once, indicating that the supplier provided a to-

tal number of items and that the supplier phase ended.

After applying these three initial rules and assuming

that the supplier provides ten items, we have a sen-

tential form of the following form (representing the

result of the supplier phase):

inv

$XXXX XXXXX X.

Now the invitation phase begins where the initia-

tor enterprise begins inviting its associates. Note that

a network graph represents the entire network of en-

terprises, not necessarily the enterprises that are in-

vited to join a VBC. Moreover, note that from a net-

work graph one cannot see when and which enter-

prises invited which enterprises. Therefore, given a

network graph N, we construct an invitation tree θ

according to the sequence of invitations and then con-

struct production rules for G. Since the initiator en-

terprise and all other invited enterprises must not be

invited again (by other enterprises at a later time), the

set of node labels (i.e. enterprises) in an invitation

tree θ are distinct. We construct the invitation tree

such that a node labelled with a nonterminal symbol

is never introduced more than once. The Algorithm 1

constructs an invitation tree θ from a given network

graph N according to the sequence of invitations.

Algorithm 1. Input: A network graph N. Output:

An invitation tree θ with n tree layers for the sequence

of invitations. Method: Construct an invitation tree θ

as follows:

• Let the initiator A label the root of θ and denote it

tree layer L

• For each tree layer L

, 0 ≤ i ≤ n and for each sym-

bol A

,. . ., A

in L

– for A

, 1 ≤ j ≤ k inviting associates

,. . ., B

– for all B

, 1 ≤ l ≤ l

not occurring in any tree

layer L

, 0 ≤ g ≤ i do

– let B

,. . ., B

, 1 ≤ l ≤ l

be the children of

• Denote the resulting layer L

i+1

As an example, assume that enterprise A invites B

and C and that B invites E and C invites G. Then θ is

as illustrated in Figure 3.

Given an invitation tree θ we construct the rules

for the invitation phase. The invitation rules are of the

form A →

.. . G

inv

.. . G

inv

. This rule re-

ﬂects that enterprise A invited enterprises B,C, ... ,G

Modelling the Formation of Virtual Buying Cooperatives with Grammars of Regulated Rewriting

Figure 3: An invitation tree θ representing that enterprise A

invites B and C, that B invites E and that C invites G.

by the subscript A on the nonterminals. The nontermi-

nals with the subscript “inv” give the respective enter-

prises the opportunity to invite their associates. The

Algorithm 2 constructs the invitation rules with per-

mitting context.

Algorithm 2. Input: An invitation tree θ. Output:

Invitation rules with permitting context. Method:

Construct the invitation rules as follows:

• If the root of θ is labelled by A and A

,. . ., A

occur in layer L

as children nodes of A do

– add the rule

inv

→

.. . A

inv

.. . A

inv

($;

to P.

• For all tree layers L

, 1 ≤ i ≤ n and all symbols

, 1 ≤ j ≤ l occurring in L

– for j = 1 add a rule

inv

→

.. . B

inv

.. . B

inv

(

;

to P, where B

, 1 ≤ f ≤ p, are the children of

node A

in L

i+1

and

A is (the rightmost) A

oc-

curring in layer L

i−1

– for j = 2, . .. , l add a rule

inv

→

.. .C

inv

.. .C

inv

(

j−1

;

to P, where C

, 1 ≤ g ≤ r, are the children of

node A

in L

i+1

• If a node A has no children we add the rule

inv

→

to P, where

Y is either (the rightmost) A

occur-

ring in the previous layer L

i−1

j−1

• For the rightmost Z in the last layer L

that has

no children add the rule

inv

→

Z#(

to P, where

Y is either (the rightmost) A

occur-

ring in the previous layer L

i−1

j−1

The nonterminal symbols of the form

A are intro-

duced in order to keep track of the order of the invi-

tations, because no enterprise invited at a later stage

should be able to act prior to an enterprise that was

invited before. Note how the permitting context of

nonterminals is simply carried over to the next layer

in a tree or to a sister node to the right in a layer. The

symbol # indicates that the invitation phase ended.

For our example, the following rules would be

constructed.

Example 3. We construct

inv

→

inv

($;

0),

inv

→

inv

(

0),

inv

→

inv

(

0),

inv

→

0),

inv

→

G#(

0).

Applying the rules in our running example we ob-

tain the sentential form:

G#$XXXX XXXXX X.

In the declaration phase all invited enterprises that

want to buy items declare how much they would want

to buy. For this, we introduce for all nonterminal sym-

bols with a subscript and for A

, simple recursive rules

that are applied as many times as the number of the

items that the respective enterprise wants to purchase.

For A

we introduce A

→ A

A(#;

0) and A

→ A(#;

and for a nonterminal symbol A

we introduce rules

of the form A

→ A

. For an enterprise A

that

does not want to purchase anything, we introduce a

rule of the form A

→ A

the subscript OO standing

for “opt out”. The rules for the declaration phase (ex-

cept the initiator rules) are context-free, since a (left-

most) ordering on the declaration of goods does not

matter.

Now we introduce rules that delete all nontermi-

nals A in the sentential form that are marked with

or the subscript OO. Note that this is a simple design

choice that can be made as desired. We make this

design choice to simplify the formalisation and for

better readability of the paper. In case terminal sym-

bols a are marked with ˆa or the subscript OO, the lan-

guage L

VBC

must be changed accordingly. We intro-

duce for all nonterminals of the form

A and A

the

rules

A → λ(#;

0) and A

→ λ(#;

0), respectively.

Example 4. In our example we assume that the en-

terprises A, B and E want to purchase ﬁve items each

and that enterprises C and G opt out. Applying the

rules, that we construct as explained above, we ob-

tain the following sentential forms:

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

G#$XXXX XXXXX X

∗

⇒

G#$XXXX XXXXX X

∗

⇒

#$XXXX XXXXX X.

After the declaration phase a purchasing strategy

is chosen outside our model. The chosen purchasing

strategy determines which enterprises may purchase

items. After the declaration phase (and deleting the

marked nonterminals) we obtain sentential forms of

the form: A . .. AB .. . BC ...C . . .#$X ...X indicating

the quantity each enterprise would want to purchase

to the left of the symbol $ and the quantity provided

by the supplier to the right of the symbol $.

Note that in order to obtain a string in the language

VBC

, we would have to rewrite the leftmost nonter-

minal on the left side of the symbol $ and then we

would have to rewrite the leftmost nonterminal X on

the right side of the symbol $. We basically would

need to walk across the sentential form and back. In

the following we argue that such a mode of derivation

cannot be encoded in the permitting context, but that

we need forbidding context too. This kind of rcg with

a special kind of leftmost restriction is investigated

in (Dassow and P

aun, 1989) (leftmost restriction II,

page 54). In this leftmost restriction at each step of

a derivation the leftmost occurrence of a nonterminal

in a sentential form which can be rewritten has to be

rewritten. It is shown that the generative capacity of

rcg with this kind of leftmost restriction is consider-

ably increased (they generate recursively-enumerable

languages) (see Theorem 1.4.4 in (Dassow and P

aun,

1989)). If no deleting rules are used, rcg with this left-

most restriction generate context-sensitive languages.

A simpler form of leftmost restriction in which at each

step of a derivation the leftmost occurrence of a non-

terminal has to be rewritten, decreases the generative

capacity to context-free.

Lemma 1. Given a sentential form of the form

w = A .. . AB. . .BC . . .C . .. #$X . .. X, there is no rPcg

that alternates between rewriting the leftmost nonter-

minal of the left hand side of $ and the leftmost non-

terminal of the right hand side of $.

Proof idea 1. Assume that there exists such an rPcg.

Then there exists a rule of the form A → α(X;

0),

α ∈ (V

∪V

)

∗

to rewrite the leftmost nonterminal in

w. Then there exists a rule the second nonterminal

A in w from being replaced. One way to do this is a

rule of the form A → α

(Y ;

0), where Y is a nonter-

minal that is introduced by a rule that rewrites X, i.e.

X →

αY (K;

0) for some K ∈ V

. Then we have the

following rules

: A → α(X ;

0) and r

: A → α

(Y ;

0).

Thus, the ﬁrst two A in the sentential form w must

be distinct, otherwise r

is applicable to the second

nonterminal A immediately after the ﬁrst A has been

replaced (i.e. not the derivation we want). Extend-

ing this discussion in a similar fashion, we conclude

that all nonterminals to the left hand side of $ must be

distinct. A similar discussion holds for the nontermi-

nals X to the right of $. Then w consists of distinct

nonterminals and we can build a regular grammar

that rewrites w into a regular string w

∈ L

VBC

, which

shows the contradiction.

6 RANDOM CONTEXT

GRAMMARS

Thus, at this point we know that random permitting

grammars are not powerful enough to model the reser-

vation phase. We can however construct a random

context grammar and impose the leftmost restriction

II described above in order to model the reservation

phase. The Algorithm 3 constructs rcg production

rules describing the reservation phase.

Algorithm 3. Input: An invitation tree θ. Output:

Reservation rules with permitting and forbidding con-

text. Method: Construct the reservation rules as fol-

lows:

• Let A

,. . ., A

be the enterprises that have

been chosen by the E-strategy or M-strategy.

• For every A

, 1 ≤ i ≤ k do

• Construct for A

the following rules:

→ a

(X;{

})

X → a

(

;

)

→ λ(

;

→ λ(

)

where

and

are newly introduced nontermi-

nal symbols in V

• Construct for all other A

, j = 2,. .. , k, the follow-

ing rules

→ a

(X;{A

i−1

})

X → a

(

;

)

→ λ(

;

→ λ(

)

where

and

i−1

are newly introduced nonter-

minal symbols in V

Modelling the Formation of Virtual Buying Cooperatives with Grammars of Regulated Rewriting

• For all other nonterminals B on the left hand side

of $ representing enterprises that were not cho-

sen by a purchasing strategy create the following

rules:

B → b.

• Add X → x(

0;γ) to P to ﬁnally replace possible

nonterminals X, where

γ = {A

,. . ., A

,. . .,

• Add the rules # → λ($;

0) and $ →

0;#) to P,

$ ∈ V

Example 5. Suppose that, in our running example,

the E-strategy determined that the two enterprises A

and B were chosen to purchase items. Then, the fol-

lowing rules would be constructed:

A → a

A(X;{

}),

X → a

(

A → λ(

;

0),

→ λ(

A),

→ b

(X;{A,

}),

X → b

(

;

→ λ(

;

0),

→ λ(

X → x(

0;{A, B,

}),

→ e.

Corollary 1. A rcg modelling a VBC and its four

phases has to work in leftmost restriction II.

Figure 4 illustrates the information that we obtain

after each phase. The information is represented in

the sentential forms that our grammar generates. In

Figure 4 the sentential forms are generalised to the

following two parts: left of the $ symbol and right of

the $ symbol.

7 VBC MODEL CHANGE

If we want to generate L

VBC

with random permitting

context grammars, we have to change the ordering

of the phases in our VBC model and thus the VBC

model itself. We generate the information to the left

of the $ symbol and the information to the right of $

successively with chain rules. In order to be able to

control the rewriting we have to have a constant num-

ber of nonterminals in each sentential form and we

have to build chain rules accordingly. By our previ-

ous discussion we know that the occurrences of the

supplier phase

AAA . . . AAA$XXX . . . XXX

quantity supplied items

invitation phase

AAA . . . AAA$XXX . . . XXX

who invited whom

quantity supplied items

declaration phase

AAA . . . AAA$XXX . . . XXX

quantity supplied itemswho invited whom,

who wants how much

reservation phase

AAA . . . AAA$XXX . . . XXX

quantity supplied items,

who gets how much

who invited whom,

who wants how much

Figure 4: The information that we obtain after each phase

represented as generalised sentential forms. The sentential

forms are divided into left of $ and right of $ and are de-

picted as AAA.. .AAA$XXX ... XXX. Beneath each left and

right part we explain what kind of information is conveyed.

nonterminals X cannot be matched against the occur-

rences of the nonterminals to the left of $ with only

permitting context. Therefore, we cannot initiate a

VBC with letting the supplier provide the quantity of

purchasable items. Instead the phases of the changed

VBC model are the following. Each enterprise partic-

ipating in the VBC (starting with the initiator)

• declares the quantity of goods it wants to purchase

to the left and to the right of $ symbol, and then

• invites its associates from its network.

After the declaration and invitation phases, the pur-

chasable quantity provided by the supplier is inquired

and a purchasing strategy is chosen accordingly out-

side the grammar model. Once the enterprises are

determined by the E-strategy or M-strategy a string

in L

VBC

is generated. In this alternative VBC model

the declaration phase and invitation phase alternate

for each enterprise that is invited to join the VBC.

This process is illustrated in Figure 5 for n enterprises.

Moreover, Figure 5 shows that after all enterprises de-

clared and invited their associates, the supplier pro-

vides purchasable items. Depending on the formed

VBC and chosen purchasing strategy, the process ﬁ-

nally leads to a deal or non-deal.

Algorithm 4 generates the fragments of the rPcg

for the declaration phase and invitation phase. Some

of the design choices can be made differently but the

SIMULTECH 2017 - 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

declaration phase for enterprise A

invitation phase for enterprise A

declaration phase for enterprise A

invitation phase for enterprise A

invitation tree

supplier phase

deal or non-deal

purchasing strategy

Figure 5: An illustration of the individual processes of the

alternative and restricted VBC model.

quintessence is that information cannot be distributed

over several phases and co-ordinated later, but has to

be introduced rather locally.

Algorithm 4. Input: An invitation tree θ. Output:

Production rules for the declaration and invitation

phase for each enterprise participating in the alter-

native VBC model. Method: Construct the rules as

follows:

• Let S → A$X

be a rule.

• For each tree layer L

, 0 ≤ i ≤ n and for each sym-

bol A

,. . ., A

in L

– for A

, 1 ≤ j ≤ k construct declaration rules as

follows

→ A

;

→ X

;

→ A

;

→ X

;

– for A

, 1 ≤ j ≤ k inviting associates

,. . ., B

construct the following rules

→ B

.. . B

→ X

.. . X

({B

.. . B

};

– for the leaf nodes A in θ construct the following

rules

A → λ

Note that the rules have to be applied in the or-

der they are constructed in Algorithm 4, since other-

wise the derivation blocks. After applying rules con-

structed in Algorithm 4 we obtain a sentential form of

the form (here simpliﬁed for better readability):

.. . $X

.. .

Once a purchasing strategy is chosen after con-

sultation with the supplier about the quantity of pur-

chasable goods, context-free rules can be generated as

outlined in Algorithm 5.

Algorithm 5. Input: A sentential form w result-

ing from the declaration and invitation phase and

the chosen enterprises by the purchasing strategy.

Output: Production rules for terminating the deriva-

tion. Method: Construct the rules as follows:

• Let w = A

.. . Z

.. . X

• Construct for all nonterminals A

occurring to the

left of $ rules of the form

→ a

• Construct for all chosen enterprises B

rules of

the form

→ b

• Construct for all other nonterminals X occurring

on the right hand side of $ the following rules:

X → λ

8 CONCLUSIONS

We showed how to model the supplier phase, invi-

tation phase, and declaration phase of a VBC with

random permitting context production rules. Then

we argued that to model the VBC reservation phase

we need random context grammars under a special

kind of leftmost derivation. If we interpret the mod-

elling from an information distribution and coopera-

tion view, then we showed that if VBC relevant in-

formation is distributed over supplier phase, invita-

tion phase and declaration phase and must during the

reservation stage be synchronised and co-ordinated,

the formal grammar needs to be very powerful with

respect to mode of derivation and thus generative ca-

pacity. From a perspective of representing the enter-

prises and the supplier as agents, this model is ad-

equate as the VBC formation is a process in which

several agents cooperate and synchronise informa-

tion. In particular, we need rcgs that work under

a special kind of leftmost restriction and generate

Modelling the Formation of Virtual Buying Cooperatives with Grammars of Regulated Rewriting

recursively-enumerable languages (see (Dassow and

aun, 1989)).

If we use random permitting context grammars for

all processes, we can only model a VBC formation

during which information is introduced and processed

locally and successively rather than being spread over

the different VBC processes. Thus, supplier phase,

invitation phase, declaration phase and reservation

phase have to be altered and have to be synchronised

in a more local and successive way. In this alternative

model, the supplier is not part of the model, which

may have its beneﬁts, as the enterprises act instead of

react to a supplier offer. A disadvantage is that VBCs

may be formed without making a purchase in the end

(e.g. if the quantity the VBC wants to purchase does

not coincide with the quantity provided by the sup-

plier). Finally, the alternative VBC model discussed

in Section 7 does not achieve that all enterprises are

able to see what is happening and has happened dur-

ing the VBC formation (e.g. the amount of available

goods, the enterprises participating, etc.), which is an

unfavourable position for the very small enterprises

working in isolation.

We see the two main beneﬁts of our approach,

that is, modelling the VBC and the VBC formation

with grammars of regulated rewriting, as the follow-

ing. One is that grammars model generative processes

and the formation of a VBC is a generative process

that involves several enterprises, which can actively

participate in the formation. A model of such a gen-

erative process can shed light on the structural condi-

tions of a VBC and can thus guide an implementation

(as we have shown in this paper).

The second main beneﬁt is that once an adequate

model is found, one can draw many implications from

the rich mathematical theory behind regulated rewrit-

ing. One example is the observation that we showed

in this paper, namely if we restrict the grammar model

we cannot model the four successive phases of a

VBC formation and if we want an adequate model

we have to have more powerful grammars (in terms

of lifting the rewriting mode to a speciﬁc leftmost

restriction). Another example is parsing. A pars-

ing algorithm decides ﬁrst if an input string w is in

a certain formal language L and then assigns a struc-

tural representation to w (i.e. a derivation tree). A

parsing algorithm can be useful in a VBC context

if several enterprises a

,. . .a

want to form a VBC

(independent of the four successive VBC formation

phases). These enterprises would, for instance, in-

put a string w = a

.. . a

in which the

quantity of the items that each participating enter-

prise wants to purchase is represented and let the pars-

ing algorithm decide whether w ∈ L (representing the

question whether it is possible for those enterprises

to form that speciﬁc VBC). Structural representations

from parsing might give further insights into the VBC

structure. This is a more bottom-up approach differ-

ing from the top-down approach that we described in

this paper. An interesting observation in this context

is that the restricted VBC model, even though it does

not model the four successive phases, might be more

useful due to more efﬁcient parsing algorithms.

Other mathematical properties such as closure

properties for certain formal languages may be use-

ful in situations in which two networks of different

enterprises want to combine into a larger network.

Can we in such a situation use the same grammar for-

malisms? Or does joining two networks and their re-

spective grammars lead beyond the grammar model?

Apart from investigating mathematical properties

relevant for VBCs (e.g. parsing, closure), future work

will also focus on implementations of the two VBC

model approaches presented in this paper and on eval-

uations with respect to efﬁciency of the VBC forma-

tion process, number of closed deals, and user (i.e.

enterprise) satisfaction. In the context of implementa-

tion, one interesting question is how to translate natu-

ral language text into the respective grammar rules or

sentential forms and vice versa.

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