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# Sequential Object Petri Nets and the Modeling of Multithreading Object-Oriented Programming Systems

Written By

Ivo Martiník

Submitted: November 30th, 2011 Published: August 29th, 2012

DOI: 10.5772/48470

From the Edited Volume

## Petri Nets

Edited by Pawel Pawlewski

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## 1. Introduction

Sequential object Petri nets are the newly introduced class of Petri nets, whose definition is the main topics of this article; they feature certain original concepts and can be successfully used at a design, modeling and verification of multithreading object-oriented programming systems executing in highly-parallel or distributed environment. In this article basic characteristics of sequential object Petri nets are very briefly presented including possibilities in their definition of newly introduced tokens as non-empty finite recursive sequences over the set of non-negative integer numbers, functionalities of multiarcs and the mechanism of the firing of transitions. These properties significantly increase modeling capabilities of this class of Petri nets at the modeling of multithreading object-oriented programming systems. Sequential object Petri nets can be used also in the area of recursive algorithms modeling and they are also the initial step to explicitly represent paradigms of functional programming. The fusion of object-oriented and functional programming enables to express new kinds of programming patterns and component abstractions.

The theory of sequential object Petri nets proceeds from the theories of various types of Petri nets, starting with Place/Transition nets (Diaz, 2009) and their sub-classes, followed by High-Level nets(Jensen & Rozenberg, 1991), (Reisig, 2009) such as Predicate-Transition nets and Coloured nets(Jensen & Kristensen, 2009), enabling to model apart from the management structure of the system even data processing, and in connection with modeling of object-oriented programming systems it is Object nets (Agha et. al., 2001), (Köhler & Rölke, 2007), which are being studied lately. But practical usability of Petri nets (in their original form) in the role of the parallel programming language is mainly impeded by the static nature of their structure. They are missing standard mechanisms for description of methods alone, programming modules, classes, data types, hierarchical structures, etc. Positive characteristics of Petri nets demonstrate only in not too much large-scale modules at high abstraction level. That is why Petri nets are often understood as the theoretical abstract module only, whose applicability for design, analysis and verification of extensive programming systems is limited. Therefore, this article briefly describes a special class of sequential object Petri nets and its possibilities of multithreading object-oriented programming systems modeling, which eliminates the stated shortcomings required for design, analysis and verification of these systems in several directions.

This chapter is arranged into the following sections: in the section 2 is described the term of the sequence over the finite set and its properties which denominates the class of the sequential object Petri nets; section 3 explains the base term of this chapter, ie. sequential object Petri net and its properties; section 4 explains in details implementation of the mechanism of firing of transitions in a sequential object Petri net; section 5 then discusses the area of object-oriented programming systems and their representation by the sequential object Petri nets; section 6 explains the example of simple class hierarchy represented by the sequential object Petri net and it is inspired by the several base classes of the Java programming language class hierarchy. Finally, the section 7 gives the conclusions of the research to conclude the chapter.

## 2. Sequences and their properties

Prior to the formal introduction of the term of sequential object Petri net, we present the definition of sequence over the finite set from which the denomination of this class of Petri nets has been derived. N denotes the set of all natural numbers, N0the set of all non-negative integer numbers, P(A) denotes the family of all the subsets of given set A.

Let A be a non-empty set. By the (non-empty finite) sequenceσ over the set A we understand a mapping σ:{1, 2, …, n} A, where n N. Mapping ε: A is called the empty sequence over the set A. We usually represent the sequence σ:{1, 2, …, n} A by the notation σ = <a1, a2, …, an> of the elements of the set A, where ai = σ(i) for 1≤ i n. We also consider any element of the set A as the sequence over the set A, ie. mapping σ:{1} A. Empty sequence ε: A over the set A we usually represent by the notation ε = <>. We denote the set of all finite non-empty sequences over the set A by the notation ASQ, the set of all finite (and possible empty) sequences over the set A by the notationAESQ.

Note also, that the set A can be any non-empty set, which means that it can be also the non-empty set of sequences over some non-empty set B, ie. A = BESQ. Thus member of the sequence over the set BESQcan be then another sequence over the set B. This fact thus also similarly allows sequences over the sets (BESQ)ESQ, ((BESQ)ESQ)ESQ, etc. The term of sequence over some non-empty set has thus recursive character and every such sequence can consists from subsequences consisting from subsequences etc. We denote the set of all finite non-empty sequences without the empty subsequences over the union of sets A, ASQ, (ASQ)SQ, ((ASQ)SQ)SQ, … by the notation ARQ, the set of all finite (and possible empty or with empty subsequences) sequences over the union of sets A, AESQ, (AESQ)ESQ, ((AESQ)ESQ)ESQ, … by the notationAERQ.

The length of the sequenceσ = <a1, a2,…, an>, where σ AERQ, n N, is equal to the natural number n, the length of the empty sequence ε is equal to the number 0. The length of the sequence σ we represent by the notation length(σ), or @σ, the set of all the elements of the sequence σ we represent by the notation elem(σ), ie. elem(σ) = {ai| ai = σ(i) for 1≤ i n}, elem(ε) = . The subsequences of the sequenceσ is the mapping subsq: AERQP(AERQ), such that for i,1≤ i n: ((σ subsq(σ)) (((ai == ε) (aiA)) aisubsq(σ)) (((ai != ε) (aiA)) ((aisubsq(σ)) (subsq(ai)subsq(σ))). The members of the sequenceσ is the mapping memb: AERQP(A), so that memb(σ) = {a | (a subsq(σ)) (a A)}, memb(ε) = .

If σ = <a1, a2,…, an>and τ = <b1, b2, …, bm>are the finite sequences, where σ AERQ, τ AERQ, n N, m N, then by the concatenation of the sequencesσ and τ, denoted by στ,we understand the finite sequence στ = <a1, a2, …, an, b1, b2, …, bm> and its length is equal to n + m. We say, that the sequences σ and τ are equal, denoted by σ == τ, if the following is simultaneously true: (n = m) (i,1≤ i n: (((ai = ε) (bi = ε)) ((aiA) (biA) (ai=bi)) ((aiε) (biε) (aiA) (biA) (ai==bi)))).

If, for instance, τ AERQ, τ = <<a, <a, b>>, <a, <c>, b>, <>>, then length(τ) = 3, elem(τ) = {<a, <a, b>>, <a, <c>, b>, <>}, subsq(τ) = {<<a, <a, b>>, <a, <c>, b>, <>>, <a, <a, b>>, <a, <c>, b>, <a, b>, <a>, <b>, <c>, <>} and memb(τ) = {a, b, c}.

When operating with sequences, notation in the form of n*(σ) can be utilized, where σ AERQ, n N. Informally, that notation expresses sequence consisting of n concatenations of the sequence σ. If, for example A = N0, σ = <1, 2>, then notation 3*(σ) represents the sequence 3*(<1, 2>) == <1, 2><1, 2><1, 2> == <1, 2, 1, 2, 1, 2>.

MultisetM over a non-empty set S is a function m: S N0. By the non-negative number m(a)N0, a S, we denote the number of occurrences of the element a in the multiset m. We usually represent the multiset m by the formal sumaSm(a)a. By SMSwe denote the set of all non-empty multisets over the set S, by SEMSwe denote the set of all multisets over the set S.

IDENT denotes the set of all identifiers and it is understood to be a set of non-empty finite sequences over the set of all letters of the selected national alphabet and the set of all decadic digits that starts with a letter. Identifiers are recorded in a way usual for standard programming languages. Examples of correctly formed identifiers for example involve the thread, var22, , etc. On the contrary, for example sequences 2main, first goal, _input, are not identifiers. Moreover is it true, that if ID1, ID2, …, IDnIDENT, where nN, n>1, then we call the sequence in the form ID1.ID2. ….IDncompound identifier (i.e. for example the sequence Main.Thread.Variable1 is a compound identifier).#IDENT set is understood to be the set of all non-empty finite sequences in the form #A, where AIDENT. Then, elements of #IDENT set for example include sequences #thread, #var22, #, etc.

The set (N0)RQ we will denote by the symbolTokens. The set ArcSeq (arc sequences) is defined by the following:

1. if x (IDENT#IDENT), then <x>ArcSeq,

2. if xTokens, then xArcSeq,

3. if xArcSeq, then <x>ArcSeq and also <length(x)>ArcSeq,

4. if xArcSeq and yArcSeq, then xyArcSeq,

5. if n (IDENTN)and xArcSeq, then n*(x) ArcSeq.

The elements of Tokens set for example involve sequences <1>, <22, <0, 0>>, <<3, 2>, <4, <7, 8>>>, etc.). The set of arc sequences ArcSeq is, informally said, the set of non-empty final recursive sequences over the set (IDENT#IDENTTokens) which do not contain empty subsequences and which can contain as their members even selected operations over those recursive sequences, between which is the determination of the recursive sequence length and concatenation of recursive sequences. So examples of elements of the set ArcSeq can be sequences <a, b, 1>, <#s, @(s), <1, <2, 3>>>, <a, thread, a*(<thread>), <1, 0>>, etc.

Let ASArcSeq, AS = <a1, a2,…, an>, where nN. Then the mapping variables: ArcSeqP(IDENT#IDENTN0) is defined so that for ASArcSeqi, 1 ≤ in:

1. if ai (IDENT#IDENT), then aivariables(AS),

2. if aiTokens, then memb(ai) variables(AS),

3. if ai = <x>, where xArcSeq, then variables(x) variables(AS),

4. if ai = length(x), where xArcSeq, then variables(x) variables(AS),

5. if ai = xy, where xArcSeqand yArcSeq, then (variables(x) variables(AS)) (variables(y) variables(AS)),

6. if ai = n*(x), where n (IDENTN)and xArcSeq, then (nvariables(AS)) (variables(x) variables(AS)).

Thus mapping variables assigns to each arc sequence ASArcSeq, AS = <a1, a2,…, an>, where nN, the set of members from the sets IDENT,#IDENT a N0 contained in it. The set of variables(AS) associated with a particular arc AS will be identified in the text by the term variables of the arc sequence AS. So if for example AS = <a, thread, a*(<thread>), <1, 0>>, then variables(AS) = {a, thread, 1, 0}.

## 3. Sequential object Petri nets and their properties

Sequential Object Petri Net is an ordered pair SOPN = (, PN), where

1. is a finite non-empty set of pages,

2. PN is a page number function, PN:N, that is injective.

By elements of the finite non-empty set of pages we routinely mark identifiers from the set IDENT. Injective function PN of numbering of pages of the net assigns to each page of sequential object Petri net SOPN the unique natural number within the net.

Let SOPN = (, PN)is a sequential object Petri net. Page of the sequential object Petri net SOPN is an ordered touplePG = (P, IP, OP, T, A, MA, IOPN, AF, MAF, TP, IPF, OPF, SP, IF), PG , where:

1. P is a finite set ofplaces,

2. IP is a finite set of input places, P IP = ,

3. OP is a finite set of output places, P OP = ,

4. T is a finite set of transitions, (P IP OP) T = ,

5. A is a finite set of arcs, A ((P IP) T) (T (P OP)),

6. MA is a finite set of multiarcs, MA ((P IP) T) (T (P OP)), A MA = ,

7. IOPN is a function of input and output place numbers, IOPN:(IP OP)N, that is injective,

8. AF is an arc function, AF:(A MA) ArcSeq,

9. MAF is a multiarc function, MAF: MAArcSeq,

10. TP is a function of transition priorities, TP: TN,

11. IPF is an input place function of multiarcs, IPF:(T (P OP))AIP, where (T (P OP)) MA, AIP = {p | γ : p IP γ},

12. OPF is an output place function of multiarcs, OPF:((P IP) T) AOP, where ((P IP) T) MA, AOP = {p | γ : p OP γ},

13. SP is a finite set of subpages, SP ,

14. IF is an initialization function, IF:(P IP OP) TokensEMS.

The finite set of places P is used for expressing of conditions of a modeled programming system and in the net layout we notate them with circles. IP is a finite set of input places of the net page representing its input interface. Additionally, no input place of the net page can be identical with any of its places. Input places are represented in the page layout with circles of highlighted upper semicircle. Then, OP is a finite set of output places of the net page representing its output interface. Additionally, no output place of the net page can be identical with any of its places. The definition admits even such possibility that the selected input place is identical with any of output places of the given net page. Output places are represented in the net page layout with circles of highlighted lower semicircle.

Likewise the finite set of transitions T is used for describing events in the programming system and we notate them with rectangles. That set is disjoint with the set of places P of the given net page. A is the finite set of arcs being principally oriented while connecting the place with transition or transition with place and in the layout of net we represent them by oriented arrows drawn in full line. It is worth considering that none of output arcs of any transition can be associated with any input place of the net page, and none of input arcs of any transition can be associated with any output place of the net page. MA is finite set of multiarcs, newly introduced type of arc in sequential object Petri nets. Functionalities of multiarc are used for the modeling of synchronous and asynchronous calling of methods in the given programming system and they follow the principles of the multiarcs in the bi-relational P/T Petri nets (Martiník, 2011). Multiarcs are represented in layouts of the net with oriented arrows drawn with dash line. The set of arcs of the given page is disjoint with the set of its multiarcs, hence it is not allowed the existence of the ordered pair (place, transition) or (transition, place) connected by both types of oriented arcs.

IOPN function of the identification of input and output places of the net page assigns to each input and output place of the particular net page unique natural number which is used at the implementation of mechanism of execution of transitions associated with multiarcs of the net page. With each arc or multiarc of the net page is associated the value of its arc function AF, which assigns to each arc or multiarc (one) arc sequence, i.e. the element of ArcSeq set. With each multiarc of the net page is additionally associated the value of its multiarc function MAF, which assigns to each such multiarc (one) arc sequence, i.e. the element of ArcSeq set. The layout of the net page shows values of AF and MAF functions associated with particular multiarc in the form AF | MAF. With all transitions of the net page are associated values of their functions of transition priorities TP assigning each transition with (the only) value of such transition priority, which is the value of a certain natural number. If the value of function of transition priorities is not explicitly indicated in the net layout with the particular transition, we assign it to the value of natural number 1.

The input place function of multiarcs IPF assigns each multiarc of the net page connecting ordered pair (transition, place) a certain input place of the selected net page. The definition admits even the possibility of assigning the selected multiarc of the particular net page with some of the input places of the same net page (ie. it is allowed to model recursive methods). The particular input place p of the selected net page γ (γ ) is in the layout of network identified by ordered pair of natural numbers (PN(γ), IOPN(γ.p)), where the first member of the pair indicates the value of page number function PN and the second member of the pair identifies the selected input place p on the net page γ with the particular value of function IOPN. We present this ordered pair in layouts of net pages usually in the form PN(γ).IOPN(γ.p). The output place function of multiarcs OPF assigns each multiarc of the net page connecting the ordered pair (place, transition) with a certain output place of the selected net page. The definition again admits the possibility of assigning to the selected multiarc of the given net page some of the output places of the same net page. The particular output place p of the selected net page γ is marked in a similar way as in case of the function IPF.

A part of each net page can be the finite set SP of its subpages, which are by themselves the net pages (i.e. elements of the set ). Initialization function IF assigns each place including input and output places of the net page with a multiset of tokens. That function is also identified in literature as M0.We routinely mark identifiers from the set IDENT by elements of the set of places, input places, output places and transitions.

Figure 1 shows the sequential object Petri net SOPN = (, PN), where = {Main, Sub}, PN = {(Main, 1), (Sub, 2)}. Netpage of this sequential object Petri net Main = (P, IP, OP, T, A, MA, IOPN, AF, MAF, AP, IPF, OPF, SP, IF), where P = {P1, P2}, IP = {in}, OP = {In}, T = {T1, T2}, A = {(in,T1), (P1,T1), (T2,In)}, MA = {(T1,P2), (P2,T2)}, IOPN = {(in, 1), (In,2)}, AF = {((in,T1), <a>), ((P1,T1), <b, 1>), ((T1,P2), <a>), ((P2,T2), <a>), ((T2,In), <b>)}, MAF = {((P2,T2), <b>), ((T1,P2), <b,1>)}, TP = {(T1, 1), (T2, 1)}, IPF = {((T1,P2), (2.1))}, OPF = {((P2,T2), (2.2))}, SP = , IF = {(in, 1<1>), (P1, ), (P2, ), (In, )}. Net page Sub = (P, IP, OP, T, A, MA, IOPN, AF, MAF, AP, IPF, OPF, SP, IF), where P = , IP = {start}, OP = {Start}, T = {T1}, A = {(start,T1), (T1,Start)}, MA = , IOPN = {(start,1), (Start,2)}, AF = {((start,T1), <c, 1>), ((T1,Start), <c>)}, MAF = , TP = {(T1, 1)}, IPF = , OPF = , SP = , IF = {(start, ), (Start, 1<3>)}.

If page_identifier is the identifier of the selected net page and element_identifier is the identifier of a place, input place, output place or transition of the net page, we call the compound identifier in the form page_identifier.element_identifier so calleddistinguished identifier of the element of net page, which uniquely identifies it within the given sequential object Petri net. Designs of distinguished identifiers of subpages of the net and of its elements can be also executed for cases of sub subpages of the net pages, etc.

For the sake of better transparency we will not indicate in layouts of nets explicit values of page number function PN, and explicit values of function of input and output place numbers IOPN of identification of input and output places of individual net pages any more. Moreover, we will not indicate values of functions IPF and OPF of particular multiarcs in the form of a.b pair of natural numbers, but in the form of the pair of identifiers page.ioplace, where page is the net page and ioplaceIP page, perhaps ioplaceOP page is particular input, or output place of the net page while it holds that a = PN(page) and b = IOPN(page.ioplace).

Layouts of sequential object Petri nets are usually further adjusted in the sense of notations of declarations of headings of methods and their calling within the text of the program, similarly as shown in Figure 2. Here, identifiers of input and output places of the net pages are complemented by (informative) notation of the shape of tokens, which are accepted by those input and output places (see the notation of the input places Main.in<a>,Sub.start<c, 1> and of the output places Main.In<b>,Sub.Start<c>). We will not record values of functions AF and MAF in the form of the ordered pair separated by | line any more. The value of the arc function AF is indicated separately and the value of the multiarc function MAF is indicated behind the value of the input place function IPF on the net page, perhaps with a value of the output place function OPF of particular multiarc (see notation Sub.start<b, 1> and Sub.Start<b> of the net page Main) in the sense of declaration of calling of methods with the entry of values of input parameters or output values of these methods.

LetSOPN = (, PN) is a sequential object Petri net,PG is its net page. By the marking M of the net page PG we understand the mapping M:(P IP OP) TokensEMS, whereP PG, IP PG, OP PG. By the marking of the net SOPN we understand the marking of all its net pages.

LetSOPN = (, PN) is a sequential object Petri net,PG is its page. Then:

1. by InputArcs(x)we denote the set of all input arcs of selected place, output place or transition x, ie. x (P OP T) PG: InputArcs(x) = {a A | y (P IP T): a = (y, x)}.

2. by InputMultiArcs(x)we denote the set of all input multiarcs of selected place, output place or transition x, ie. x (P OP T) PG: InputMultiArcs(x) = {a MA | y (P IP T): a = (y, x)}.

3. by InputNodes(x)we denote the set of all input nodes of selected place, output place or transition x, ie. x (P OP T) PG: InputNodes(x) = {y (P IP T) | a (A MA): a = (y, x)}. We denote the set InputNodes(x)by x.

4. by OutputNodes(x)we denote the set of all output nodes of selected place, input place or transition x, ie. x (P IP T) PG: OutputNodes(x) = {y (P OP T) | a (A MA): a = (x, y)}. We denote the set OutputNodes(x)by x.

5. by TransitionInputVariables(x) we denote the set of all variables included in the values of arc functions AF (resp. multiarc functions MAF) of all the input arcs and multiarcs of the transition x, ie. x T PG: TransitionInputVariables(x) = {v |((v variables(AF(y)) (v variables(MAF(y))) ((yInputArcs(x)) (yInputMultiArcs(x)))}.

## 4. Firing of transitions in sequential object Petri nets

An important term used at the implementation of the mechanism of firing of transitions in a sequential object Petri net is the term of binding of the arc sequence contained in the value of AF, or MAF, function of the particular arc, or multiarc, to the token found at a certain marking of the net page in the place associated with this arc (for short, we will refer to that in the text also as binding of token).

Let T Tokens, T = <t1, t2,, tn>, where nN, AS ArcSeq. We denote, that there exists input binding of the arc sequenceAS to the token T, if there exists mapping InputBind: AST that satifies the following:

1. If AS = <a1, a2, …, an>, where a1, a2, …, an (IDENT#IDENTN0), then:

2. if ai (IDENT #IDENT) for 1 ≤ in, then InputBind(ai) = ti,

3. if ai N0 for 1 ≤ in, then (InputBind(ai) = ti) (ai = ti),

4. u variables(AS)vvariables(AS): ((u = v) (InputBind(u) == InputBind(v))).

5. If AS = <a1, a2,…, ak-1, ak, ak+1,…, am>, where m <n, 1 ≤ km, ak #IDENT, a1,…, ak-1, ak+1,…, am (IDENT N0), mN, then:

6. InputBind(ak) = <tk, tk+1, …, tk+n-m>,

7. if ai IDENT for 1 ≤ ik-1, then InputBind(ai) = ti,

8. if aiIDENT for k+1 ≤ im, then InputBind(ai) = tn-m+i,

9. if ai N0 for 1 ≤ ik-1, then (InputBind(ai) = ti) (ai = ti),

10. if ai N0 for k+1 ≤ im, then (InputBind(ai) = tn-m+i) (ai = tn-m+i),

11. uvariables(AS) v variables(AS): ((u = v) (InputBind(u) == InputBind(v))).

12. In other case the mapping InputBind is not defined.

Then, input binding of the arc sequence AS to the token T via mapping InputBind: AST can be successfully realized in the following two cases:

1. Arc sequence is in the form AS = <a1, a2,…, an>, where nN, i.e. it holds that length(AS) = length(T) = n, while generally more than one of elements a1, a2,…, anof that sequence can be the element of the set #IDENT. Then, at the input binding of the arc sequence AS to the token T we execute, informally said, binding of mutually corresponding elements of sequences according to their order. If an element ai of the arc sequence AS is a nonnegative integer, then such element must be bound to single-element ti of token T, where 1 ≤ in, which is also nonnegative integer and the value of both those numbers must be identical. If u and v are two identical variables of the arc sequence AS, then the values of elements of the token T bound to them must be identical. Figure 3 shows a very simple example of input binding of the arc sequence AS = <1, a, a, #c, 3> to the token T = <1, 10, 10, 2, 3>:

1. Arc sequence is in the form AS = <a1, a2,…, ak-1, ak, ak+1,…, am>, where mN, holds that length(AS) <length(T), i.e. m<n,and at the same time just a single element ak #IDENT, where 1 ≤ km. Then, that only element ak is bound to the sequence<tk, tk+1,, tk+n-m>of elements of the token T. In binding of other elements of the sequence AS the same rules hold as it was in the case of (i). An example of that type of binding of the arc sequence AS = <x, #y, 5, z> (and thus element a2 #IDENT) to the token T = <4, 8, 10, 2, 5, 19> is shown in Figure 4.

Next examples of the arc sequences binding to tokens in a sequential object Petri net involve:

• arc sequence <a, a, 1> can be successfully bound to token <2, 2, 1>, where InputBind(a) = 2, InputBind(1) = 1,

• arc sequence <a, a, 1> cannot be successfully bound to token <1, 2, 3> (it would hold that InputBind(a) = 1 and InputBind(a) = 2),

• arc sequence <#x> can be successfully bound to token <1, 2, 3>, where InputBind(#x) = <1, 2, 3>,

• arc sequence <x, #y> can be successfully bound to token <1, 2, 3>, where InputBind(x) = 1, InputBind(#y) = <2, 3>,

• arc sequence <x, y> can be successfully bound to token <<1, 2>, <3, 3>>, where InputBind(x) = <1, 2>, InputBind(y) = <3, 3>,

• arc sequence <x, #y> can be successfully bound to token <<1, 2>, <3, 3>, 4>, where InputBind(x) = <1, 2>, InputBind(#y) = <<3, 3>, 4>.

Let SOPN = (, PN)is a sequential object Petri net, PGis a net page, tT is a transition of the net page PG, pt (P IP) is a place or input place of the net page PG, qt (P OP) is a place or output place of the net page PG, M is a marking of the net SOPN. Transition t is enabled in the marking M of the net SOPN, if:

1. (p, t) (InputArcs(t) InputMultiArcs(t)) InputBind: AF(p, t) e, where eM(p),

2. (p, t) InputMultiArcs(t) InputBind: MAF(p, t) e, where eM(OPF(p, t)),

3. uTransitionInputVariables(t) vTransitionInputVariables(t):

((u = v) (InputBind(u) == InputBind(v))).

If transition t is enabled in the marking M of the net SOPN, we record that fact symbolically in the form of tenM.

Let AS = <a1, a2,, an>ArcSeq, nN. If transition t is enabled in the marking M of the net SOPN, then we say, that there exists partial mapping OutputBind:ArcSeqTokens, if

OutputBind(AS) = OB(a1) OB(a2) … OB(an)

where OB: ArcSeqTokens and i, 1 ≤ in:

1. OB(ai) =<InputBind(ai)>, if aiIDENT,

2. OB(ai) =InputBind(ai), if ai#IDENT,

3. OB(ai) =ai, if ai Tokens,

4. OB(ai) =<OB(x)>, if ai = <x>, where xArcSeq,

5. OB(ai) =<@(OB(x))>, if ai = @(x), where xArcSeq,

6. OB(ai) =OB(b1) OB(b2) … OB(bk), if ai = b1 b2bk, where b1,b2,…, bk ArcSeq, kN,

7. OB(ai) =OB(b)*(OB(x)), if ai = b*(x), where b (IDENT N0), xArcSeq.

Thus transition t on the net page PG of the net SOPN is enabled, if the following is satisfied:

1. for all the input arcs (p, t), resp. input multiarcs (p, t), of the transition t there exists input binding of the value of the arc function AF(p, t) to some token e in the place p of the marking M,

2. for all the input multiarcs (p, t) of the transition t there exists input binding of the value of the multiarc function MAF(p, t) to some token e in the output place of the net page that is given by the value of the output place function OPF of the multiarc (p, t) in the net marking M,

3. if u and v are two equal variables of the set TransitionInputVariables(t), then the values of elements (resp. subsequences) bound by them in the frame of mapping InputBind must be equal.

Figure 5 shows the fragment of sequential object Petri net in its marking M and the construction of the mapping InputBind: AF(P1, T1) <2, 0>, where <2, 0>M(P1) and InputBind: AF(P2, T1) <1, 1, 1>, where <1, 1, 1>M(P2). It is easily to find that transition T1 is enabled.

Partial mapping OutputBind:ArcSeqTokens is for the given transition t of the net page realized only in case that the transition t is enabled in the marking M of the net. Hence, partial mapping OutputBind assigns the selected arc sequence AS the token, being the element of the set of all Tokens (i.e. that token is not generally located in any of places of the net SOPN in its current marking M). The definition assumes that arc sequence AS is generally in the form AS = <a1, a2,…, an>, nN. The value OutputBind(<a1, a2,…, an>), which is generally the element of the set Tokens, is given by concatenation of sequences in the form OB(a1) OB(a2) … OB(an), while individual values OB(ai) are for 1 ≤ in determined according to specified rules.

Regarding recursive nature of the partial mapping OutputBind we will include several examples of binding of the arc sequences to elements of the set Tokens. Let us assume in all cases that a certain sequential object Petri net SOPN is given containing the transition T, whose set of input variables TransitionInputVariables(T) = {a, b, c, x, #x} and in certain marking M of the net SOPN there exists binding of those input variables given as follows: InputBind(a) = 10, InputBind(b) = 2, InputBind(c) = <<1, 1>, 3>, InputBind(x) = <1, 2, 3>, InputBind(#x) = <1, 2, 3>. In the following examples we will investigate values of partial mapping OutputBind applied to various values of the arc sequence AS.

• if AS = <a>, then OutputBind(<a>) = OB(a) = <InputBind(a)> = <10>.

• if AS = <c, 1>, then OutputBind(<c, 1>) = OB(c) OB(1) = <InputBind(c)><1> = <<<1, 1>, 3>><1> = <<<1, 1>, 3>, 1>.

• if AS = <x, a, 5>, then OutputBind(<x, a, 5>) = OB(x) OB(a) OB(5) =

<InputBind(x)><InputBind(a)><5> = <<1, 2, 3>><10><5> = <<1, 2, 3>, 10, 5>.

• if AS = <#x, a, 5>, then OutputBind(<#x, a, 5>) = OB(#x) OB(a) OB(5) =

InputBind(#x)<InputBind(a)><5> = <1, 2, 3><10><5> = <1, 2, 3, 10, 5>.

• if AS = <b*(#x)>, then OutputBind(<b*(#x)>) = OB(b*(#x)) = OB(b)*(OB(#x)) = InputBind(b)*(InputBind(#x)) = 2*(<1, 2, 3>) = <1, 2, 3><1, 2, 3> =

<1, 2, 3, 1, 2, 3>.

• if AS = <@(<x, a, 5>)>, then OutputBind(<@(<x, a, 5>)>) = <OB(@(<x, a, 5>))> =

<@(OB(x) OB(a) OB(5))> = <@(<InputBind(x)><InputBind(a)>OB(5))> =

<@(<<1, 2, 3>><10><5>)> = <@(<<1, 2, 3>, 10, 5>)> = <3>.

Figure 6 shows the net pages Prim and Sub of a certain sequential object Petri net in their marking M and we are interested, if there exists input binding of transition variables associated with the transitions Prim.T1 and Prim.T2. With the transition Prim.T1 is associated one input arc (Prim.in, Prim.T1) whose value of the arc function AF(Prim.in, Prim.T1) = <a>, and thus the set TransitionInputVariables(Prim.T1) = {a}. We can easily determine that for the input arc (Prim.in, Prim.T1) there exists mapping InputBind: <a><2>, and thus holds that InputBind(a) = 2. With the transition Prim.T2 is associated one input multiarc (Prim.P2, Prim.T2) whose value of arc function AF(Prim.P2, Prim.T2) = <a>, the value of multiarc function MAF(Prim.P2, Prim.T2) = <y> and thus the set TransitionInputVariables(Prim.T2) = {a, y}. And again, we can easily determine that for the input multiarc (Prim.P2, Prim.T2) there exists mapping InputBind: <a><1>, and thus holds that InputBind(a) = 1. With that input multiarc it is also necessary to determine the mapping InputBind: MAF(Prim.P2, Prim.T2) e, where eM(OPF(Prim.P2, Prim.T2)). The value of the output place function OPF(Prim.P2, Prim.T2) = Sub.First, whose markingM(Sub.First) = 1<3>. So we investigate, if there exists mappingInputBind: <y><3>. We can easily determine that the mapping exists and it holds that InputBind(y) = 3. Generally, for the transition Prim.T2 holds that InputBind(a) = 1 and InputBind(y) = 3.

So it can be stated that for both transitions Prim.T1 a Prim.T2 exist particular input bindings of all the transition input variables associated with their input arcs and thus, both transitions are enabled. We are further interested, if there exists the mapping OutputBind of the values of functions AF and MAF associated with output arcs (or multiarcs) of both transitions. With transition Prim.T1 is associated the only output multiarc (Prim.T1, Prim.P2) whose value of the arc function AF(Prim.T1, Prim.P2) = <a> and the value of the multiarc function MAF(Prim.T1, Prim.P2) = <a>. So we can easily find out that OutputBind(<a>) = <2>. With the transition Prim.T2 is associated the only output arc (Prim.T2, Prim.In) whose value of the arc function AF(Prim.T2, Prim.In) = <a, y>. So again, we can easily find out that OutputBind(<a, y>) = <1, 3>.

Let SOPN = (, PN)is a sequential object Petri net, PGis a net page, tT is a transition of the net page PG, M is a marking of the net SOPN.

1. If the transition t is enabledin the marking M, then we obtain by its firing marking M’ of the net SOPN, defined as follows:

M’(p) = M(p)\ InputBind(AF(p, t)), if (p t) ((p, t) (A MA))

(InputBind: AF(p, t) e, eM(p)),

M’(p) = M(p) OutputBind(AF(t, p)), if (p t) ((t, p) (A MA)),

M’(q) = M(q)\ InputBind(MAF(p, t)), if (p t) ((p, t) MA) (q = OPF(p, t))

(InputBind: MAF(p, t) e, eM(OPF(p, t))),

M’(q) = M(q) OutputBind(MAF(t, p)), if (p t) ((t, p) MA) (q = IPF(t, p)).

M’(p) = M(p), otherwise.

1. Firing of transition t T, which will change the marking M of the sequential object Petri net SOPN into the marking M’, is symbolically denoted as M [ t M’.

2. Stepis understood as firing of non-empty subset from the set of enabled transitions in the given marking M of the sequential object Petri net SOPN. Step Y which will the marking M into the marking M’ is symbolically denoted as M [ Y M’.

3. Let step Y be enabledat the marking M of the net SOPN. If t1, t2 Y and t1 t2,we say then that transitions t1a t2are concurrently enabled and that fact is symbolically denoted in the form of {t1, t2}en M.

Firing of transition will result in the new marking of given sequential object Petri net, which we will obtain as follows:

• from each input place p of the fired transition t we will remove the (unique) token in the marking M, which is bound to the value of the arc function AF(p, t),

• to each output place p of the fired transition t we will add up the (unique) token which is the value of partial function OutputBind(AF(t, p)),

• from each output place of page q, being the value of function OPF of the input multiarc (p, t) of the fired transition t, we will remove the (unique) token in the marking M, bound to the value of the multiarc function MAF(p, t),

• to each input place of page q, being the value of function IPF of the output multiarc (t, p) of fired transition t, we will add up the (unique) token being the value of partial function OutputBind(MAF(t, p)),

• in all the remaining places of the net we will leave their original marking.

Figure 6 shows the net pages Prim and Sub of a certain sequential object Petri net in its marking M. From previous text we know that transitions Prim.T1 and Prim.T2 are concurrently enabled. Hence, firing of transition Prim.T1 consists in:

• removing token <2> from the input place Prim.in,

• adding token <2> to the place Prim.P2,

• adding token <2> to the input place Sub.first.

Hence, firing of transition Prim.T2 consists in:

• removing token <1> from the place Prim.P2,

• removing token <3> from the output place Sub.First,

• adding token <1, 3> to the output place Prim.In.

Marking M’ of the net after concurrent firing of transitions Prim.T1 and Prim.T2 is shown in Figure 7.

Relatively complicated mechanism of firing of transitions in sequential object Petri nets can be better explained by notional substituting of all the multiarcs of the net by standard arcs, which can be realized as follows:

• if p is place and t transition of the given net page and (p, t) its multiarc whose value of arc function equals to AF(p, t), the value of output place function equals to OPF(p, t) and the value of multiarc function equals to MAF(p, t), we substitute this multiarc by notional pair of the following standard arcs:

• by the arc (p, t) with the value of the arc function equal to AF(p, t),

• by the arc (OPF(p, t), t) with the value of the arc function equal to MAF(p, t).

• if p is place and t transition of the given net page and (t, p) its multiarc whose value of arc function equals to AF(t, p), the value of input place function equals to IPF(t, p) and the value of multiarc function equals to MAF(t, p), we substitute that multiarc with notional pair of the following standard arcs:

• by the arc (t, p) with the value of the arc function equal to AF(t, p),

• by the arc (t, IPF(t, p)) with the value of the arc function equal to MAF(t, p).

That notional substitution of multiarcs in the previous net is shown in Figure 8 where:

• multiarc (Prim.T1, Prim.P2) was substituted by the following pair of arcs:

• the arc (Prim.T1, Prim.P2) with the value of the arc function AF equal to <a>,

• the arc (Prim.T1, Sub.first) with the value of the arc function AF equal to <a>,

• multiarc (Prim.P2, Prim.T2) was substituted by the following pair of arcs:

• the arc (Prim.P2, Prim.T2) with the value of the arc function AF equal to <a>,

• the arc (Sub.First, Prim.T2) with the value of the arc function AF equal to <y>.

When enabling individual steps of the sequential object Petri net, so called conflicts can originate in certain markings of the net (or conflict transitions). At the enabling of transitions t1 and t2 of the given net and its marking M the conflict occurs, if both transitions t1 and t2 have at least one input place, each of t1 and t2 transitions is individually enabled in the marking M of the net, but t1 and t2 transitions are not in that marking M concurrently enabled, i.e. enabling of one of them will prevent enabling the other. The term of conflict transitions can be obviously easily generalized for the case of finite set t1, t2, …, tn (n N)of transitions of the given net.

A typical example of the conflict at enabling transitions in the particular marking of the net is shown in Figure 9, where transitions T1 and T2 of the net have a common input place P1, both are enabled (particular binding of tokens can be easily found), but not concurrently enabled, i.e. enabling of the transition T1 will disable enabling of the transition T2 and vice versa. When solving conflicts at enabling of transitions in sequential Petri nets we will therefore follow the rule which determines, informally said, that from the set of conflict transitions at the given binding of tokens the one will be enabled, whose value of transition priority function TP is the highest. If such transition from the set of conflict transitions does not exist, the given conflict would have to be solved by other means. In our studied example will be then on the basis of that rule the transition T2 enabled (because TP(T1) = 1 and TP(T2) = 2).

## 5. Object-oriented programming systems and their representation by the sequential object Petri nets

This section deals with main principles applied at modeling of multithreading object-oriented programming systems with the sequential object Petri nets. All program listings are developed in the Java programming language (Goetz et. al, 2006), (Lea, 1999), (Subramaniam, 2011).

Each declared class of the object–oriented programming system is in the sequential object Petri net represented by a net page containing declared data items and methods. Their declarations are made by using elements of the net page. Individual input places of the net page then represent input points of static and non-static methods as a part of the class declaration, and output places of the net page then represent their output points. Input and output places of the net page are associated with identifiers of the particular method whose input and output point they represent while each method has one input and output place. In so doing, we abide to the convention whereby input place identifier of the particular method starts with a small letter and identifier of the output place of the same method with a capital letter. In order to differentiate graphically in layouts of the net declaration of static methods from non-static methods on a given net page, we demarcate identifiers of input and output places of the net page representing input and output points of static methods with the square brackets. Moreover, it is possible on the net page via a position of input and output places in its layout represent visibility of public, protected and private type of individual declared methods thus implementing the characteristic of encapsulation of the object-oriented programming.

Figure 10 illustrates the net page representing the following declaration of the class Sys:

 public class Sys { public static void compute() { … } protected void run() { … } private void init() { … }}

Static and non-static data items are on the net pages represented in the form of tokens in the given net marking, i.e. by elements of the set Tokens. When representing values of static and non-static data items being the elements of particular tokens of the net, it is for example possible to proceed as follows:

• If the data type of the particular data item is a non-negative integer (int), its actual value equals to that non-negative integer.

• If the data type of the particular data item is boolean, we represent the truth value false most frequently with constant 0 and the truth value true with constant 1, while in layouts of the net we are also using symbolic values false and true.

• If the data type of the particular data item is char, whose value is the element of the set contained in the code table given by the Unicode standard, we represent its value corresponding to the value of the symbol code given in the Unicode Standard, i.e. the letter 'A' can be represented by the value 65. In order to make again layouts of nets more readable, the element of token can be entered as the particular symbol bounded by apostrophes, i.e. instead of token <65> we will indicate in layouts of nets token <’A’>.

• If the data type of the particular data item is string, whose value is text string, the values can be represented by sequences of codes of symbols according to the Unicode standard. Again, for better transparency of notations, it is possible to use instead of codes of symbols directly the sequence of symbols bounded by apostrophes, i.e. in layouts of nets represent data items of the data type string with the tokens of <’H’, ’e’, ’l’, ’l’, ’o’> shape, or to bound sequence of symbols of the string with quotation marks, i.e. to note the tokens in the form of <”Hello”>.

• Data items whose data type are numbers with floating decimal point (float, double), or other numerical data type being the superset of the set of non-negative integer numbers, it is not possible to declare it directly on the basis of the stated definition of the sequential object Petri net. If a need emerges to operate with those numerical sets during simulation of multithreading object-oriented programming system, it is always easy to extend in this sense the definition of the token, arc sequence and other particular definitions, in order to ensure the support of those numerical sets (e.g. it is possible to define the token of sequential object Petri net as the finite non-empty sequence over the set of real numbers R, etc.).

• In the case of data items whose data type is the pointer to instance of class, we represent their values with natural numbers. Each instance of the class has at its creation allocated unique natural number expressing its address within the programming heap. Number 0 than represents the value of pointer null.

Static data items are usually represented in layouts of nets with the tokens containing only their actual values. Non-static data items are usually represented by tokens containing both the value of pointer to a given instance of the class in which the particular data item is declared, and its actual value according to particular data type. So if within the declaration of the class First for example the following data items are declared:

 public class First { private static boolean indicator = true; private char status = ‘a’;}

then, the static data item indicator can be represented by the token <true> and the non-static data item status by the token <11, ‘a’>, where numerical value 11 represents the value of the pointer to the particular instance of class.

The dynamic creation of the instance of the class is not in the sequential object Petri net realized by the creation of a new instance of the particular net page representing declaration of the given class, but by creation of all the tokens representing non-static data items of the declared class. At the same time, each such token contains (usually as its first element) the value of the pointer to the newly created class instance. The fact, that during dynamic creation of instances of classes it is not necessary to create instances of net pages, dramatically simplifies its analysis.

All methods represented by elements of the net pages are executed by the programming threads. Each such programming thread is represented by the particular instance of class (usually by the class Thread). Thus, the token accepted by the input place of the net page must contain the pointer for particular programming thread realizing execution of the given method, while that pointer is within arc sequences of the net represented by default by some of the identifiers T, U, V, etc. By the element of the token accepted by the input place of the page representing the input point of some of the non-static methods must be then the pointer of the particular instance of class whose non-static method is (i.e. pointer this). That pointer is then within the arc sequences of the net standardly represented by the identifier O (while indeed, within representation of static methods that pointer cannot be used). When entering identifiers of static and non-static data items, parameters and local variables of methods, we use by default in layouts of the net so called Hungarian notations, where the first letter (or the first part) of identifier expresses its data type. For identification of standard data types we use the acronym i even for data type int, b for boolean, c for char, s for string and p for pointer. Hence for example the identifier sName within the arc sequence represents the variable of the data type string with the identifier name.

Figure 11 shows the net page representing the declaration of the following class Obj in its marking M:

 public class Obj { private char val = ‘a’; public synchronized char getVal() { return value; } public synchronized void setVal(char value) { this.val = value; }}

The current notation of that net page suggests, that within the program execution modeled by that net two instances of the class Obj have been already created, as in the place Obj.val appear tokens <4, ‘a’> a <6, ‘c’>, where the first element of each of those tokens is the pointer to the instance of the class Obj and second the actual value of the non-static data item val of the data type char. The non-static method getVal will be executed above the instance of class with the pointer 4 by the programming thread with the pointer equal to the value 1, the non-static method setVal will be executed above the instance of the class with the pointer 6 by the programming thread with the pointer equal to the value 2, while the value of method parameter value equals to character ‘b’. The shape of the net demonstrates, that over the selected instance of the class can be concurrently executed with any programming thread maximum one of getVal or setVal methods.

Additionally, in layouts of nets can be simply represented the relation of a simple inheritance between two declared classes. The identifier of the net page representing the class declaration being superclass of the given class, we indicate in the layout of the given net page of the class in its left bottom corner. If that identifier of the net page of the superclass is not explicitly indicated in the layout of the net, we consider the given class to be implicitly subclass of the top class in the hierarchy of classes created on the basis of relation of simple inheritance of classes (e.g. within the hierarchy of classes of the Java programming language it is the class java.lang.Object).

With the inheritance relation of classes is organically connected even the term of polymorphism of the object-oriented programming and the possibility of declaration of so called virtual methods. Figure 12 shows two net pages out of which the first represents declaration of the class Object and the second declaration of the class System, being subclass of the class Object. Within the class Object is also declared virtual method with the head public int hashCode(), which is in the declaration of the class System overwritten with the virtual method of identical method head. Input and output places of net pages Object and System, which appertain to input and output points of both virtual methods are indicated in the net layout by default while in the case of declaration of the virtual methods it is necessary that all input places representing the input point of the given virtual method had assigned on all the net pages containing the declaration of this method the identical value of IOPN function and all output places representing the output point of the given virtual method had assigned on all relevant net pages the identical value of IOPN function (see Figure 12).

Next example demonstrates representation of classes containing declaration of virtual and abstract methods. Virt and Add classes are declared as follows:

 public abstract class Virt { public abstract int compute(int a, int b); public int make(int a, int b) { return compute(a, b); }}public class Add extends Virt { public int compute(int a, int b) { return (a + b); }}

The Virt class is an abstract class with the declared abstract method compute. That method is called as a part of the declared method make of the Virt class. The methodcompute is then overwritten at the level of declaration of the class Add, being the subclass of the class Virt. The net page representing the declaration of the class Virt is shown in Figure 13. Abstract method compute is represented only by a pair of the input place Virt.compute and the output place Virt.Compute. Regarding the fact that it is also declaration of the virtual method, the selection of function IOPN values is important (i.e. in our case IOPN(Virt.compute) = 1 and IOPN(Virt.Compute) = 2). The key element of the net page is the manner of calling of the virtual method compute represented by the multiarc (Virt.T1, Virt.P1) and its value of input place function IPF, the multiarc (Virt.P1, Virt.T2) and its value of output place function OPF. The value of input place function IPF of the multiarc (Virt.T1, Virt.P1) is in the form of O.compute (more precisely in the O.1 form), where identifier Orepresents the pointer of particular instance of non-abstract subclass of class Virt, whose non-static method compute is executed. The value of identifierO is in our case non-constant, ie. generalized at the realization of steps, and depends on binding of the specific token to the arc sequence in the form <T, O, iA, iB>, which shares in the given identification of the net execution of transition Virt.T1. According to the numerical value bound to identifier O the particular net page net will be determined whose value of the function PN of numbering net pages is identical with the numerical value bound to the identifier O. Then on this net page the input place with the value of IOPN function equal to number 1 will be selected (representing the input point of the virtual method compute), into which the particular token in the form <T, O, iA, iB> will be placed.

Net page representing the declaration of the class Add, being the subclass of the class Virt, is shown in Figure 14. In the class Add is declared the only non-static virtual method compute, which overwrites the abstract method compute declared in the body of the class Virt. Thus, on the grounds of that fact it is necessary to correctly assign values of IOPN function (in our case IOPN(Add.compute) = 1 and IOPN(Add.Compute) = 2). An interesting detail of this net page is the way of determination of the addition of integral values bound to variables iA and iB of the arc sequence of the arc (Add.compute, Add.T1). The expression in the form @(iA*<0>iB*<0>) represents length of the sequence formed by concatenation of two sequences: first sequence containing in total iA of numbers 0 and the second sequence containing in total iB numbers 0 (in this case, it is worth mentioning, that e.g. via the expression in the form @(iA*(iB*<0>)) product of integral values of variables iA a iB can be determined).

Next generalization of the form of functions IPF and OPF associated with multiarcs is possible within the sequential object Petri nets to implement the explicit support of the mechanism of first order and higher order functions known from functional programming (ie. given method can take another methods as parameters and return the method as the return value). Figure 15 shows the net page of the class Func containing declaration of method call. Arc sequence of the token accepted by the input place of that method contains a variable mO, the value of which is the value of the function PN for the net page with the input place make and output place Make, which are represented by particular integral values of the function IOPN. Values of functions MAF of multiarcs of the net in the form mO.make and mO.Make, whose all components are variables and whose values are not determined until particular arc sequence is bound to token of the net, provide general mechanism for the possibility of declaration of first order and higher order functions.

Example of recursive method represented by the net page can be seen on Figure 16. This static recursive method fact of the class Integer implements calculation of the value of factorial function whose parameter is a certain natural number n. Recursive algorithm for calculation of factorial of the integer value n in the Java programming language is represented by the following listing:

 public class Integer { public static int fact(int n) { if (n == 1) return 1; else return (n * fact(n – 1));}

If the value of the parameter iN equals to number 1, the transition Integer.T4 is enabled and in the output place Integer.Fact the token <T, 1> is stored, whose first element T represents the programming thread and the second element the value of factorial of number 1. If the value of number iN is greater than number 1, by enabling the transition Integer.T2 that number iN is substituted by the sequence consisting of iN numbers 0, which is then used at recursive call modeled by the multiarc (Integer.T3, Integer.P2), whose enabling will always result in elimination of one element of that sequence. The programming stack which is used at realization of recursive procedure of the algorithm is represented by the token in the place Integer.P3 in the form of the sequence <T, …, iN-2, iN-1, iN>. At the moment where the sequence composed from 0 numerals contains after the series of recursive calls of algorithm one element (i.e. it is necessary to determine the value of factorial of number 1 within the stop condition of algorithm), the transition Integer.T4 is enabled and the value of factorial of number 1 is represented in token <T, 1> located in the output place Integer.Fact. Then, by repeatedly enabling the transition Integer.T5 and finally the transition Integer.T6 the return from recursion is implemented with the gradual calculation of the value of factorial function, which is represented in the token of the form <T, @(iM*(iF*0))>. Following completion of the process of reverse return of recursion in the output place Integer.Fact the token in the form <T, iF> is stored, whose second element represents the value of factorial of the natural number iN.

Moreover, in the sequential object Petri nets can be simply represented declarations of inner classes of the selected class via subpages of the net page. So if for example the class Obj which within its declaration contains declaration of its own inner class InnerObj, that declaration can be represented by the subpage InnerObj of the page Obj (see Figure 17)

The representation of declared interfaces, which can contain only declarations of headings of publicly accessible abstract methods, is also easy. Figure 18 shows the representation of interface Runnable (identifiers of interface we indicate in layouts of the net with spaced letters) containing declaration of the method with the head public abstract void run().

## 6. Example of simple class hierarchy represented by sequential object Petri net

In this section we will demonstrate how simple hierarchy of the classes via sequential object Petri nets is built. The model of those examples involve selected classes contained in standard library of classes of the Java programming language. (e.g. classes java.lang.Object, java.lang.Thread, etc.). For better visibility of layouts we usually present individual declared methods in separate figures representing individual parts of the particular net page (while naturally places and transitions of the net which appear in individual parts of the net page and have identical identifiers, always represent the same place or transition of the total page of the net). Thus, top of our hierarchy of classes will be the class Object, whose declaration made in the Java programming language is the following:

 public class Object { private Object monitor; private Thread thread; private Vector = new Vector; private synchronized static int getPointer() { … } public Object() { … } protected void finalize() { … } public void lock() { … } public void unlock() { … } public void wait() { … } public void notify() { … }}

The static method getPointer (see Figure 19) can be executed by the only one programming thread only and it is determined for assigning unique integral values of pointers to individual instances of the classes. At the first execution of the method the integral value 1 is returned in variable P (because InputBind(#p) = 0 a @(#p) = @(<0>) = 1), in the place Object.P1 is then stored the token <0, 0>, and at next execution of the method is in the variable P returned the value 2, etc.

Within the declaration of the constructor of the object of the class Object (see Figure 20) is by the programming thread firstly obtained the value of the pointer for a newly created object by calling the method getPointer, that value is then stored to the variable O and the value of non-static data item monitor representing the monitor of particular instance of the class is then stored in the form of the token <O> in the place Object.monitor. The destructor of the object represented by the method finalize will then mainly ensure cancellation of the monitor of the object represented by the token <O> in the place Object.monitor.

Entry into the critical section of the object (i.e. execution of the non-static method with the modifier synchronized) is conditioned by getting the object monitor with the particular programming thread. As a part of the execution of the method lock the programming thread T can respectively allocate the object monitor with the value of the pointer O, (i.e. the token <O> in the place Object.monitor) and then enter into the critical section of the object while the programming thread is permitted to enter the critical section of the same object several times in sequence. First entry of the programming thread T into the critical section of the object O is realized by execution of the transition Object.T5. Pointer to the programming thread, which successfully obtained the object monitor, is then stored in the variable thread (i.e. the token <T, O, 1> is located in the place Object.thread). At repeated entry of the programming thread into the critical sections of the object O the transition Object.T6 is executed and particular token in the place Object.thread is added with the next element which is the number 1 (i.e. at second entry of the thread T into the critical section of the object O is in the place Object.thread stored the token <T, O, 1, 1>, at the third entry the token <T, O, 1, 1, 1>, etc). Deal location of the object monitor and initialization of its critical section is then realized by execution of the non-static method unlock, whose functionalities are inverse to functionalities of the method lock (see Figure 21).

Method wait causes the current thread T to wait until another thread invokes the notify method for the object O. The current thread T must own the monitor of the object O. The thread releases ownership of this monitor and waits in the place Object.pool until another thread notifies threads waiting on the object's monitor to wake up through a call to the notify method. The thread T then waits in the place Object.P3 until it can re-obtain ownership of the object monitor and resumes execution (see Figure 22).

Next declared class in our hierarchy will be the class Thread representing the programming thread, which is the subclass of the class Object and it also implements the interface Runnable (see Figure 23). Its declaration in the Java programming language is as follows:

 public class Thread extends Object implements Runnable { private Runnable runnable; public Thread(Runnable run) { … } protected void finalize() { … } public void start() { … } public void run() { … }}

The method Thread is the constructor of the object which during its execution firstly invokes the construction of the instance of the class Object, that is the superclass of the class Thread, which in the variable value U returns the pointer to the newly created programming thread. In the place Thread.runnable (i.e. in the value of the variable runnable) is then stored the token <U, pRun>, where the value of the variable pRun contains the pointer to the class instance implementing the interface Runnable (see Figure 18) whose method run will be then executed by the newly created programming thread. That will be implemented through execution of the method start with programming thread T. The token <U, pRun, 0> is then located into the place Thread.runnable representing the fact that the execution of the method pRun.run was initiated. Its invocation itself is realized by asynchronous method call by the firing of the transition Thread.T3 and by the mechanism of multiarc (Thread.T3, Thread.Start). The class Thread itself has the method run implemented the easiest possible way.

As a part of execution of the destructor of the instance of the class Thread, i.e. the method finalize (see Figure 24), the particular token is removed from the place Thread.runnable and the destructor of the superclass Object is then executed. If in the place Thread.runnable is the token of the form <U, pRun>, the programming thread is not active to obeyed destruction and the transition Thread.T7 can be fired. If, on the contrary, in the place Thread.runnable is the token in the form <U, pRun, 0>, the programming thread is active and the execution of the destructor must be delayed until the completion of the activities of this thread (i.e. returning of the thread U after executing the method pRun.run ensured by the mechanism of the multiarc (Thread.finalize, Thread.T8)).

Next declared class in our hierarchy will be the class Semaphore representing a counting semaphore. Conceptually, a semaphore maintains a set of permits. Method down blocks if necessary until a permit is available, and then takes it. Method up adds a permit, potentially releasing a blocking acquirer. However, no actual permit objects are used and the Semaphore just keeps a count of the number available and acts accordingly. Declaration of the class in the Java programming language is the following:

 public class Semaphore extends Object { private int value = 1; public Semaphore() { … } protected void finalize() { … } public synchronized void down() { … } public synchronized void up() { … }}

The constructor of the object, i.e. the method Semaphore, following execution of the constructor of the instance of the superclass Object will store in the place Semaphore.value the token <O, 1> representing the initial value of the data item value of the class instance O. As a part of the execution of the destructor of the instance of the class Semaphore, i.e. of the method finalize, that token is removed from the place Semaphore.value and then the destructor of the superclass Object is executed (see Figure 25).

Method down acquires a permit from this semaphore, blocking until one is available. Following entry into the critical section of the object instance the programming thread acquires a permit from this semaphore (ie. token in the place Semaphore.value can be bound to the arc sequence <O, 1, #x> after at least one execution of the method up) and will leave the critical section of the object. If no permit is available then the current thread becomes disabled for thread scheduling purposes and waits (i.e. the transition Semaphore.T7 is executed and the method Object.wait is invoked, the programming thread will release the monitor of the object in order to enable execution of the method up by other programming thread) until some other thread invokes the up method for this semaphore and the current thread is next to be assigned a permit (see Figure 26).

Method up releases a permit, increasing the number of available permits by one. If any threads are trying to acquire a permit, then one is selected and given the permit that was just released. That thread is (re)enabled for thread scheduling purposes (see Figure 27).

## 7. Conclusion

Sequential object Petri nets represent an interesting class in the area of object Petri net classes, which can be applied at design, modeling, analysis and verification of generally distributed multithreading object-oriented programming systems. A newly introduced term of token as finite non-empty recursive sequence over the set of non-negative integer numbers, functionalities of multiarcs and the mechanism of the firing of transitions do not increase demands on performance of analysis of characteristics, as seen in other classes of high-level or colored Petri Nets.

Functional programmingis one of the most important paradigms of programming that looks back on a long history. The recent interest in functional programming started as a response to the growing pervasiveness of concurrencyas a way of scaling horizontally. Multithreaded programming is difficult to do well in practice and functional programming offers (in many ways) better strategies than object-oriented programming for writing robust, concurrent software. Functional programming is generally regarded a paradigm of programming that can be applied in many languages - even those that were not originally intended to be used with that paradigm. Like the name implies, it focuses on the application of functions. Functional programmers use functions as building blocks to create new functions and the function is the main construct that architecture is built from. Several programming languages (like Scala, C#, Java, Delphi, etc) are a blend of object-oriented and functional programming concepts in a statically typed language in the present time. The fusion of object-oriented and functional programming makes it possible to express new kinds of programming patterns and component abstractions. It also leads to a legible and concise programming style. Sequential object Petri nets then also fully support design, modeling, analysis and verification of programming systems based on this fusion of object-oriented and functional programming paradigms.

## Acknowledgement

This paper has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme 'Research and Development for Innovations' funded by Structural Funds of the European Union and state budget of the Czech Republic.

## References

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