PRINCIPALES UTILIZACIONES CULINARIAS  CAZA DE PLUMA

of Figure4.4 demonstrates the operation of a colour-sensitive inhibitor arc1. The transition t4 can fire in the white mode but not in the black one. This is due to

the fact that a black token is in place p3 and blocks the execution of the transition

for that particular colour of token. We could interpret this situation by saying that the ‘white’ user has obtained a response from the system (see place p0) and

can proceed to the next step, while the ‘black’ user still awaits for the system’s advice. Contrary to the colour-sensitive inhibitor arc, the ordinary inhibitor arc blocks the firing of a transition for any token that resides in its pre-places. This happens in case that at least one token of any type is in the inhibitor place of the transition. Generally, the ordinary inhibitor arcs lack this extra flexibility that is not needed in the P/T nets since there is only one type of tokens.

Figure 4.4: A colour-sensitive inhibitor arc.

4.2

Defining Building Block

In this section, we define the basic step net, which captures the intuitive concept of a unidirectional step with control system. The definition of that net follows 1_{For the sake of simplicity the control system of the second unidirectional step is omitted.}

4.2 Defining Building Block

from the above discussion about the building block and its components: the step system and the control system. Regarding the functioning of the basic step nets, it is determined by the initial marking of the APN net that may be constructed out of several basic step nets. The marking of the APN net usually requires only the database places of the basic step nets to be marked. All the other places should be unmarked except for those step places of the net that belong to the initial marking. Finally, the initial state of an ambient system will be provided by a root net, which is described at the end of this section.

4.2.1

Basic Step Nets

At this point, the basic step net, which captures the structure of the introduced building block, is defined. Thereafter, the general structure of a basic step net is presented in Figure 4.5.

Definition 6 (basic step net). A basic step net is an APN net NS = (PS, TS, PreS, PostS, IS, Cl, CS, KS, M0S, GS)

consisting of two parts: step system and control system satisfying the following structural conditions:

• PS = PST ] PCS, where PST is a set of places of the step system and PCS

is a set of places of the control system. More specifically, PST = {ps, pf}

contains the starting (ps) and finishing (pf) place of the unique step tran-

sition ts, while PCS = Pr] Pd] {pc} is built out of three subsets such that

(below n ≥ 1):

1. Pr = {p1r, . . . , pnr} is a set of response places.

2. Pd= {p1d, . . . , pnd} is a set of database places.

3. pc is a unique control place of the finishing place pf associated with

transition ts. We will denote this by pc= cpts(pf).

• TS = TST]TCS, where TST is a set of transitions of the step system and TCS

is a set of transitions of the control system. More specifically, TST = {ts}

contains a unique transition, called a step transition. TCS = Tr ] Tem is

built out of two subsets:

1. Tr = {t1r, . . . , tnr} is a set of retrieve transitions.

4.2 Defining Building Block

• PreS, PostS : TS → µPS are such that:

PreS(tiem) = {pc, pid} for i = 1, . . . , n. PostS(tiem) = {pir} for i = 1, . . . , n. PreS(tir) = {pir} for i = 1, . . . , n. PostS(tir) = {pid} for i = 1, . . . , n. PreS(ts) = {ps}. PostS(ts) = {pf, pc}. • IS = ∅.

• All the places except for the database places are unmarked. The database places are marked by at least one token of some allowed colour.

MS

0(p, c) = 0 , for every p ∈ PST ∪ Pr∪ {pc} and c ∈ CS(p).

MS

0(p, c) 6= 0 , for every p ∈ Pd and some c ∈ CS(p).

• GS = {ps}.

A similar structure to that of the basic step net has been previously defined in [62] representing a circuit in a tracker-bouncer architecture. The structural difference between the tracker-bouncer and the basic step net lies on the number of places and transitions used in these nets. The structure of tracker consists of two step systems and the bouncer only of two places and two transitions that are linked with each other in a cyclic way resembling in some sense the structure of a control system without a control place, but with only one response place, one database place, one retrieve transition and one emptying transition.

4.2.2

Root Net

Apart from the basic step net that is defined above, the root net that is important for the development of the APN models of the ambient systems is defined. The root net is significant for the composition of the APNs since it indicates the ‘start- ing’ state of the APN model of the examined ambient system. The composition of the APNs is described in section 4.3.

Definition 7 (root net). A root net is defined as an APN net that has two places: a place p that acts as both starting and finishing place of a ‘collapsed’ step and

4.2 Defining Building Block

Figure 4.5: General structure of an unmarked basic step net.

its control place pc = cp(p), where  denotes the ‘collapsed’ transition of the

collapsed step.1 The root net will be denoted by:

NRt = (PRt, TRt, PreRt, PostRt, IRt, Cl, CRt, KRt, M0Rt, GRt),

where

• PRt= {p, pc},

• TRt= ∅,

• PreRt, PostRt are empty functions,

• IRt = ∅,

• GRt = {p}.

A root net is used to provide the initial information of the system by marking each of its places with some chosen set of tokens. The place p will be marked with the set of initial users of the system. For example, in Figure 4.6(a, b), we have two users represented by black and white tokens. The control place, pc, is

provided to define the initial restrictions. The root net of Figure 4.6(a) states no 1 _{We can extend our notation to: p ∈}•_{ and p ∈ }•_.