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композиционных микропрограммных устройств управления при любом сочетании разрядности кодов операторных линейных цепей, их компонент и минимальной разрядности адреса микрокоманды. Научная новизна предлагаемого метода заключается в том, что сохраняется возможность оптимального кодирования псевдоэквивалентных ОЛЦ, что позволяет уменьшать аппаратурные затраты в схеме, формирующей коды ОЛЦ и их компонент. Практическое значение метода заключается в том, что применение преобразователя адресов позволяет уменьшить емкость управляющей памяти из-за произвольной адресации по сравнению с известными методами реализации устройств-аналогов. К недостаткам предложенного метода относится увеличение времени цикла КМУУ за счет введения схемы АТ. Исследования авторов показали, что при выполнении условия (12), оптимальном кодировании операторных линейных цепей и уменьшении емкости управляющей памяти предлагаемый метод позволяет на 12-16% уменьшить аппаратурные затраты по сравнению с КМУУ с общей памятью. Применение метода является перспективным и для КМУУ с элементарными ОЛЦ.
Литература: 1. Баркалов А.А., Палагин А.В. Синтез микропрограммных устройств управления. К.: ИК НАН Украины, 1997. 135 с. 2. Баркалов А.А. Синтез устройств управления на программируемых логических устройствах. Донецк: ДонНТУ, 2002. 262 с. З.Грушвицкий Р.И., Мурсаев А.Х., Угрюмов Е.П. Проектирование систем с использованием микросхем программируемой логики. Петербург: БХВ, 2002. 4. Luba T. Synteza ukladow logicznych.
УДК 519.714.5
ON DECOMPOSITION OF PETRI NET BY MEANS OF COLORING
WEGRZYN AGNIESZKA____________________________
In the paper a method of decomposition of hierarchical Petri nets is presented. Such approach is based on P-invariants and testing of dependence between deadlocks and traps. The presented decomposition method can be used for e.g. generation of Finite State Machine (FSM).
1. Introduction
Hierarchical Petri net is very effective tool for modeling of logic controllers. The model is formally verified on the base of well-known Petri net theory. There are a lot of methods of analysis of Petri nets, but majority ofmethods answer only a question, if Petri net have defect or not. These methods have rather small computational complexity, but in some case it is necessary to show places of defect. The presented below method belongs to more precise procedures, although its complexity is exponential.
The proposed method is based on testing dependencies between deadlocks and traps that occur in Petri nets.
Deadlocks and traps of a Petri nets correspond to the decisions of certain logical equations, which can be represented in conjunctive normal form. Deadlocks and 50
Warszawa: WSIZ, 2001. 238 s. 5. Salcic Z. VHDL and FPLDs in Digital systems Design, prototyping and Customization. Kluwer Academic Publishers, 1998. 412 p. 6. Sasao T. Switching Theory for Logic Synthesis. Kluwer Academic Publishers, 1999. 316 p. 7. Баркалов А.А. Принципы оптимизации логической схемы микропрограммного автоматa Мура. / Кибернетика и системный анализ. 1998, №1. С. 65-72. 8. Соловьев В.В. Проектирование цифровых схем на основе программируемых логических интегральных схем. М.: Горячая линия-ТЕЛЕКОМ, 2001. 636 с.
Поступила в редколлегию 06.12.2005
Рецензент: д-р техн. наук, проф. Кривуля Г.Ф.
Баркалов Александр Александрович, д-р техн. наук, проф. каф. ЭВМ ДонНТУ, проф. Университета Зеленогурского (Польша). Научные интересы: цифровые устройства управления. Хобби: научная работа, спорт. Адрес: Украина, 83122, Донецк, ул. Артема, 204А, кв .105, тел. (+3 8062)301 -07-35.
Мальчева Раиса Викторовна, канд. техн. наук, доцент каф. ЭВМ ДонНТУ. Научные интересы: компьютерная графика. Хобби: научная работа, туризм. Адрес: Украина, 83000, Донецк, ул. Артема, 58, к.36, тел. (+38062)301-07-35.
Красичков Алексей Александрович, канд. техн. наук, доцент каф. ЭВМ ДонНТУ. Научные интересы: цифровые устройства управления на ПЛУ. Хобби: научная работа, спорт, музыка. Адрес: Украина, 83005, Донецк, ул. Куприна, 62, кв.33, тел. (+3 8062)301 -07-23.
Халед Баракат, аспирант каф ЭВМ ДонНТУ. Научные интересы: цифровые устройства управления на ПЛУ. Хобби: научная работа. Адрес: Украина, 83000, Донецк, ул. Артема, 58, к.36.
traps can be represented by a special conjunctive form -Horn formulae [4].
In the following section theoretical basis related to Petri nets, especially deadlocks and traps, is presented. In section 3, method of computation of all deadlocks and traps in Petri net is described. In this section, the method of decreasing time consumption of proposed algorithm is presented too. In the next section, an example of using proposed method is presented.
2. Definitions, theoretical basis
In this section definition and the theoretical basis of Petri nets and theirs properties is presented.
2.1. Petri net
Petri net is a graphical and mathematical modeling tool describing system characterized as being concurrent, distributed, and parallel. As a graphical tool, Petri nets can be used for simulation of logic controllers. As a mathematical tool, Petri nets can be used for analysis and synthesis of concurrent controllers [5,11].
A Petri net [10,11] is a quadruple PN = (P, T, F, M0) where: P - is a set of places; T - is a set of transitions; M0 - initial marking; P n T = 0',
F c (P x T) u (T x P).
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Set of input and set of output transitions into place p are defined, as follow respectively:
• p = {t є T : (t,p) є F}, p» = {t є T : (p,t) є F}.
Set of input and set of output places into transition t are defined, as follow respectively:
• t = {p є P:(p,t) є F}, f = {p є P: (t,p) є F} .
An ordinary Petri net is said to be a state machine if for all transition ti є T,I* t| = 1 and It *1 = 1 [11].
For Petri net from Fig. 1:
- the set of input transitions into place p1 is • p1 = (t4);
- the set of output transitions into place p1 is p1* = (t1);
- the set of input places into transition t1 is • t1 = (p1);
- the set ofinput places into transition t1 is t1« = (p3,p4).
2.2. Colored Petri net
A colored Petri net (similar to Petri net described in section 2.1) is represented as a graph with two kinds of nodes - places and transitions. A place is represented by a circle, a transition by a bar and coloured bullets represent explicitly coloured tokens. Directed implicitly coloured arcs connect the explicitly coloured places and the implicitly coloured transitions. Transitions are allowed to or prevented from occurring with respect to a particular colour if the attached coloured Boolean expression is respectively true or false [6,10].
By means of using colored Petri net (CPN), each color can represent a sequential process, i.e. the particular color can be related only to one process. The total number of the colored tokens indicates the number of concurrent processes being active at any particular global state. Such colored Petri net can be decomposed on several State Machine Petri Nets (SM-PNs).
If Petri net is colored, it means that the net is covered by a set of subnets, which all can be either state machine or marked graph.
2.3. Hierarchical Petri net
Most practical systems have a very large number of states and transitions. A large Petri net model is unclear and ineffective. A solution to such problem is hierarchical Petri net.
A net can be constructed as the multiple modules that can be modified independently of each other. V arious modules can be linked together as needed to create a net. The same module can be used repeatedly at different places in a net, so that redundant logic is not needed to handle the same situation in different contexts [2,11].
Hierarchical Petri nets consist of macroplaces and macrotransitions. There are different approaches to constructing hierarchical Petri nets. In one of them, macromodules can be created for P-net and T-net. At the beginning, it is being checked, whether Petri net contains a P-net and a T -net, then it is exchanged into macromodule. The next step is searching for parallel branches that have the same input and output transition. Such branches are exchanged into macromodules. These steps are executed as long as all parallel branches will be exchanged [2,11].
2.4. Deadlock and trap
A deadlock is a subset of places that, once unmarked, will never be marked again. Otherwise, a deadlock in a Petri net is a transition or a set of transitions that cannot be fired
[11].
A nonempty subset of places S in an ordinary net N is called a deadlock if • S c S •, i.e., every transition having an output place in S has an input place in S .
A nonempty subset of place Q is an ordinary net N is called a trap if Q» c »Q , i.e., every transition having an input place in Q has an output place in Q .
Union of two deadlocks (traps) is again deadlock (trap). A deadlock (trap) is called basic deadlock (trap) if it cannot be represented as a union of other deadlock (trap). All deadlocks (traps) in Petri net can be generated by the union of some basis deadlock (trap). A deadlock (trap) is said to be minimal if it does not contain any other deadlock (trap). Minimal deadlocks (traps) are basis deadlocks (traps), but not all basis deadlocks (traps) are minimal [11].
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2.5. Horn clause and Petri net
A Horn formula is a conjunction of basic Horn formulae. The basic Horn formula (Horn clause) is a disjunction of literals, with at most one positive literal. A literal is either a propositional letterP (a positive literal) or the negation/P of a propositional letter P (a negative literal) [4,9].
Each basic Horn formula is equivalent to a clause of one of three types:
- Q , a propositional letter;
- /Pi v ... v/Pq where q > 1 and Pp.Pq are distinct propositional letters;
- /Pi v... v/Pq v Q where q > 1 and Pp.Pq are distinct propositional letters and Q is a propositional letter.
All the deadlocks and traps of Petri net can be specified by the roots of the logical equation (1) and (2).
П teT П et • (yi pj e»t/yj), (1)
ПteTne.t(yi +Zpj et• /yj), (2)
where t,T - (respectively) transition, set of transitions;
• t,t • - input and output places of transition t; pi,pj -places; yi - variable represents place pi i.e. pi ^ yj = o.
For Petri net from Fig. 1 Horn formula for deadlocks is following:
HF = (/p1 v p3) л (/p1 v p4) л (/p3 v /p4 v p2) л (/p2 v p3) л (/p2 v p4) л (/p3 v /p4 v p1)
For Petri net from Fig. 1 Horn formula for traps is following:
HF = (p1 v/p3 v/p4) л (p3 v/p2) л (p4 v /p2) л (p2 v /p3 v /p4) л (p3 v / p1) л (p4 v / p1)
Deadlocks and traps of a Petri net can be calculated by solving the logical equations respectively (1) and (2).
Theorem: Deadlocks in Petri net [9]
All the y vectors satisfying (1) are in 1-1 correspondence with the deadlocks.
Theorem: Traps in Petri net [9]
All the y vectors satisfying (2) are in 1-1 correspondence with the traps.
2.6. Boundedness
A Petri net is said to be k-bounded or simply bounded if the number of tokens in each place does not exceed a finite number k for any marking reachable from initial marking. The Petri net for vending machine is 1 -bounded. A 1-bounded Petri net is also safe [11].
Since no more than one token with a particular color can ever be in any place, the net should be 1 -bounded (safe) in respect to any colour. The interpretation is extended to: coloured tokens, coloured external inputs and outputs. A description of CCIPN is more intuitive, than other similar coloured Petri net models.
Boundedness, similar to liveness is the properties which depends on occurrence deadlocks and traps in Petri nets.
3. Method of decomposition of Petri net
During decomposition a Petri net is divided into a set of subnets. These subnets have to satisfy some restriction, e.g. a subnet must include only places which are sequential to each other or cannot contain multi-input or multioutput transitions [6].
Decomposition of Petri net can be based on coloring of Petri net.
In this section decomposition method of Petri net based on coloring is presented. The method is using algorithm for finding deadlocks and traps in Petri net (presented in section 3.1 and widely described in [13]).
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3.1. Algorithm for finding deadlocks and traps in Petri nets
Algorithm for finding all deadlocks and traps in Petri nets can be used for checking liveness and boundedness. For checking such properties it is necessary testing dependencies between deadlocks and traps. For FC and EFC class of Petri nets [10], for liveness analysis, Commoner properties can be used [10]. For boundedness analysis method described in [4] is used.
First step of presented method is generation Horn formulae for analyzed Petri net. Such formula is generated twice: once for deadlocks and once for traps. Then for both equations Thelen trees are generated.
Thelen has proposed an efficient algorithm for converting a conjunctive form into the sum of all prime implicants
[12]. It is based on building a search tree, such that every level of it corresponds to a clause of the CNF, and the outgoing arcs from a node correspond to the literals of the clause. Conjunction of all the literals corresponding to the arcs on the path from the root of the tree to a node is associated with the node. The tree is searched in DFS order, and several pruning rules are used to minimize it. The rules are listed below.
R1: An arc is pruned, if its predecessor node-conjunction contains the complement of the arc-literal.
R2: An arc is pruned, if another non-expanded arc on a higher level still exists which has the same arc-literal.
R3: A disjunction is discarded, if it contains a literal which appears also in the predecessor node-conjunction.
The next rule R4 was added by Mathony [8]. This rule leads application to a significant reduction of the search tree.
R4: An arc j is pruned, if another already expanded arc k with the same arc-literal exists on a higher level i and if rule R2 was not applied in the subtree of arc k with respect to arc p on level i which leads to arc j.
This method has an exponential time-complexity and a linear space-complexity [12]. For decreasing in time calculation of all prime implicants, firstly, heuristic methods have been used. A size of the tree can be reduced by sorting clauses and literals in the input formula. There have been proposed three methods [14]:
H1 - Heuristic 1 (Sort by Length): Choose disjunction D j with the smallest number of literals.
H2 - Heuristic 2 (Sort by Variables): Choose disjunction D j with the smallest number of literals that do not appear in the disjunctions chosen before.
H3 - Heuristic 3 (Reordering Literals): Split the set of literals of every clause D j into two parts. One part (C) is formed from literals that appear in any of the clauses Di+1„ .D k (where k is the number of clauses) and the
second part (NC) contains remaining literals. Optimized disjunction contains literals in order
Di = {ft v...vP v Pn+1 v ... vPm}
' NC ' ' C '
The literals in (C) are sorted growing frequency in Di+1...Dk order.
The experiments show that the best way is sorting disjunctions according to Heuristic 2, and literals in the disjunctions according to Heuristic 3. But time reduction is not enough, especially for bigger model of system. Result of computer experiments was presented in [14].
For minimization of time computation parallel algorithm described in [15] is used.
As a result, ternary vectors are received. These vectors represent sets of deadlocks and traps. Testing of dependences between sets of deadlocks and traps give information about such properties of Petri nets, like: liveness, boudedness, safeness. For decomposition of Petri nets, dependencies between deadlock and trap are analyzed too.
3.2. Coloring of Petri net
Decomposition of concurrent controller into sequential automata is used to simplify synthesis process of digital circuits [1,7]. If such circuits are modeled by Petri nets, decomposition resolves into separate automata subnets, i.e. subnet contain one token, only.
For explanation decomposition method of Petri net into automata subnet, basic information is presented [3]. Two places pi and p j are concurrent, if exist such marking of net, in which these two places pi and pj are marking simultaneously.
The decomposition method described in [3] is based on coloring of Petri net. Such method attribute to each place of net, minimum number of color, that two concurrent nodes should have different colors.
Discussed method of Petri net coloring can be described using following rules [3,6]:
- each transition and place should have at least one color;
- if place has color, each input and output transition must have the same color;
- each input place of transition should have different color;
- each output place of transition should have different color;
- sets of input and output colors of transitions are equal;
- do not exist two (or more) initial marked places which share exactly the same set of colors;
- each color should have yours marking, i.e. number of color for marking is equal number of color.
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Colored Petri net carry information about belonging to sequential process. Each automata component is represented by P-invariants [13].
[CM1]
u_
c^>[C1]
[C3]
3.3. Decomposition of Petri net
Basing on algorithm for finding all deadlocks and traps in Petri net (section 3.1) and checking dependencies between sets of deadlocks and traps, it is possible to answer the question if Petri net is bounded and live. Such algorithm can be used for decomposition of Petri nets.
For decomposition and checking boundedness P-invariants are used. Subsets of places satisfying both Horn formulae for deadlocks and traps at the same time can be P-invariants. Each such vector is verified by incidence matrix. Each P-invariant is nonnegative integer vector satisfying the matrix equation у • C = 0, where C is the incidence matrix of the net [4].
For Petri net from Fig. 1 following sets of deadlocks are generated:
(p1 л p2 л p4) v (p1 л p2 л p3)
and traps:
(p1 л p2 л p4) v (p1 л p2 л p3).
These sets are representing by vectors: [1,1,0,1] and [1,1,1,0] for deadlocks, and [1,1,0,1] and [1,1,1,0] for traps.
Incident matrix C for Petri net is following:
1 0 -1 -1
0 -111 C =
0 1 -1 -1 -10 11
Sets of deadlocks and traps are equal, i.e. each vector can be P-invariant. Results of multiplication of vector and incident matrix are 0, i.e. both vectors are P-invariants.
Such P-invariant corresponds to one automat and to one color in Petri net. The number of colors depends on number of P-invariants for flat Petri net. In hierarchical Petri nets it is not so clear.
If it is possible to color Petri net, i.e. that Petri net could be decomposed into automata. P-invariants correspond to automata subnets.
In case, coloring of hierarchical Petri net, it is not provide number of parallel branches of net, which was reduced in net preparing process. In connection with, it is not possible to define number of color of flat Petri net. The number of color of net depends on number of parallel branches and it does not depend on the number of P-invariants.
For example, color [CM1] of macroplace is represented by three colors [C1 C2 C3] for explication of this macroplace (Fig. 4).
[CM1] ^ [C1] [C2] [C3]
Fig. 4. Explication of color 3.4. Example of decomposition
As an example Petri net from Fig. 5 is considered. For such Petri net, two Horn formulae were prepared (one for deadlocks and one for traps).
Fig. 5. The flat Petri net
Below two formulae are shown, first for deadlocks and second for traps. Horn formula for deadlocks is as follow:
HF_D= (/p 1v p2)A (/p2v p3) л л (/p2v p6) л (/p2v p11 )л (/p3 v p4) л (/p4v p5) л л (/p6v p7)A (/p7v p8)A (/p8v p9)A (/p9v p10)A A (/p11v p12)A (/p12v p13)A (/p13v p14)A (/p14v p15)A л (/p5v /p10v /p15v p16)A (/p16v p1)A (/p16v p2).
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Horn formula for traps is as follow:
HF_T = (p1 v /p2) л (p2 v /p3 v /p6 v /p 11) л л (p3 v /p4) л (p4 v /p5) л (p6 v /p7) л л (p7 v /p8) л (p8 v /p9) л (p9 v /p10) л л (p11 v /p12) л (p12 v /p13) л л (p13 v /p14) л (p14 v /p15) л л (p5 v /p16) л (p10 v /p16) л л (p15 v /p16) Л (p16 V /p1) Л л (p16 v /p2).
For such formulae Thelen trees were generated. Results were compared with vectors, which were deadlock and trap at the same time, were multiplied by incident matrix. Three P-invariants were given
([1,1,1,14,0,0,0,0,0,0,0,0,0,0,1], [1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1], [14,0,0,0,0,0,0,0,0444444]).
Each element of vector represents place in Petri net, i.e. first element corresponds to place p1, second element -place p2 , etc.
Fig. 6. Coloring of flat Petri net
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These three P-invariants represent three colors, which cover whole Petri net. It means that Petri net is bounded [10]. Each color presents one automaton. First vector corresponds to color C1 , second vector corresponds to color C2 and the last - to C3 color, i.e. that places p1, p2, p16 and transitions t1, t2, t13, t14, t15 are colored by three colors C1, C2 , and C3 (Fig. 6).
Each color corresponds to one automaton. In presented case color C1 represents automaton on Fig. 7,a, color C2 represents automaton on Fig. 7,b and color C3 represents automaton on Fig. 7,c. Because places p1, p2, p16 are combined for such three automata, in one automaton occur these places, and the remainder automata are added additional place, which closes automata [16].
X1
p1^Y31 t11 ІХ31
SP SP
p12Qy32
0
t12
X32
p1VY
t13
33
X33
p15.
t16
p14Qy34
t14 TX34
г
p5*p10
b
Fig. 7. Decomposed Petri net
t2
а
c
Because transition t16 is fired without any conditions, for each automaton for transition t16 , condition is added, i.e. for automaton on Fig. 7,a, if token is in place p10 and p15 (in automaton on Fig. 7,b and Fig. 7,c respectively), transition t16 can be fired.
55
3.5. Example of decomposition of hierarchical Petri net
On the other hand, if hierarchical Petri net (Fig. 8, e.g. which corresponds to flat Petri net on Fig. 5) is analyzed, one P-invariant is calculated ([1,1,1]). This P-invariant covers whole Petri net, and represents one color (CM1). Notwithstanding CM1 color is three-color [C1], [C2], [C3], because when macrotransition will be explicated, it will be three parallel places.
Fig. 8. Coloring of hierarchical Petri net
On Fig. 9 coloring of Petri net after expansion of macrotransition TM1, is shown. Transition t1 and t13 is colored by three colors C1, C2 and C3, and macroplaces M1, M2 and M3 are colored, respectively, by colors C1 , C2 and C3 .
On the last stage, macroplaces M1 , M2 and M3 are expanded, and the flat Petri net is colored (Fig. 6).
4. Conclusions
A symbolic method of decomposition of a Petri net is discussed in the paper. The algorithm is based on Thelen method of a prime implicants calculation. The method uses a symbolic way of analysis methods. Decomposition
of Petri net rests on retrieve P-invariants and coloring of Petri net. Using such algorithm, hierarchical or flat Petri net can be analyzed.
This method has an exponential time of calculation, but for acceleration of computing, an advantage of parallel approaches can be taken.
Acknowledgment. The research has been financially supported by (Polish) Committee of Scientific Research (KBN) in 2004-2006 (grant № 3 T11C 046 26).
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Поступила в редколлегию 23.01.2006
Рецензент: д-р техн. наук, проф. Хаханов В.И.
Wegrzyn Agnieszka, PhD., lecturer at University of Zielona Gora. Interested in: formal analysis, Petri net, databases, DBMS, Internet applications. Address: Computer Eng. and Electronics Institute, University of Zielona Gora, ul. Podgorna 50, 65-246 Zielona Gora, Poland, E-mail: A.Wegrzyn@iie.uz.zgora.pl Ph.: (+48 68) 3282 484.
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РИ, 2006, № 1
