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A note on transition systems
I N F O R M A T I O N S C I E N C E S 10, 347-362 (1976)
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A Note on Transition Systems* Y. EDMUND LIEN Department of Computer Science, University of Kansas, Lawrence, Kansas 66045 Communicated by L. A. Zadeh
ABSTRACT This paper introduces a mathematical structure called transition systems. The notion of transition systems has been developed as a result of the study of Petri nets and vector addition systems. The intended use of transition systems is to model concurrent and asynchronous events. The concept of information flow in a complex system and communication between parts of a complex system can be formulated in this formal structure. This paper is concerned with the mathematical properties rather than the applications of transition systems. Patterns of activities in complex systems are defined in terms of termination and finiteness properties of transition systems. Concepts of conservation and repetitivity have been introduced. Structural properties of restricted classes of transition systems have been studied.
I. I N T R O D U C T I O N Various models have been proposed in the study of parallel computation, concurrent processing and asynchronous events [6, 9, 10, 14, 15, 16, 18]. A m o n g them, Petri nets [15] and vector addition systems [10] are two important models in which the notion of concurrency and asynchrony can be easily formulated. The concept of Petri nets was first introduced by Petri in his doctoral dissertation [15]. The current version of Petri nets is due to Holt and his colleagues in the Information System Theory Project [6]. The intended use of Petri nets in this project was to represent complicated patterns of activities and events in complex systems. Many results of the study of Petri nets have
*This paper is based on the material in the author's Ph.D. dissertation (University of California, Berkeley, 1972) which was supported in part by NSF grant CK-10656X and NESC-N00039-71C-0255. This work is also supported in part by Kansas General Research Grant 3298-5038. © American Elsevier Publishing Company, Inc., 1976
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appeared in the literature [1,3, 5, 6, 17, 19]. Those who have been enthusiastic about the theory and the application of Petri nets have found the notion to be "an elegant formalism for representation of concurrency in processes and of studying asynchronous systems" [19]. Independent of the work mentioned above, K a r p and Miller proposed a formal model of parallel computation called parallel program schemata [10]. The notion of vector addition systems was also introduced for the purpose of studying the decision problems in parallel program schmata. It has been recognized by many researchers in this area that Petri nets and vectors addition systems are very similar in their mathematical structure. In this paper a formal system called a transition system is introduced. In this work, we generalize the common concepts of Petri nets and vector addition systems and preserve the power of describing the behavior of complex systems. The notion of transition systems can also be viewed as a restricted case of production systems used in formal language theory. The main concern of this paper is on the structural property of transition systems. Graph-theoretic properties of the model will be studied. II. PETRI NETS, V E C T O R A D D I T I O N SYSTEMS A N D T R A N S I T I O N SYSTEMS
1. PETRI NETS We first present the model of Petri nets.
A Petri net is a directed graph with two types of nodes, namely, place nodes and transition nodes. A place node is represented by a circle and a transition node by a line segment. Each arc connects either a place to a transition or a transition to a place. A place is said to be an input place to a transition if there is an arc from the place to the transition. Similarly, a place is an output place of a transition if there is an arc connecting the transition to the place. Each place is capable of holding a number of tokens. A transition is said to be enabled orfireable if every input place of the transition holds at least one token. Wefire a transition by removing one token from each of its input places and adding one token to each of its output places. Thus the total number may increase, decrease, or remain the same after firing an enabled transition. Suppose two transitions have a common input place and both transitions are enabled, then only one of these two transitions is allowed to fire. These two transitions are said to be in conflict. We do not specify which one among a group of transitions in conflict is allowed to fire. The system is nondeterministic in this sense. Therefore, two transitions can fire simultaneously, provided that they are enabled and not in conflict.
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EXAMPLE I. Figure 1 shows a Petri net. In this example, transition 2 and transition 5 are enabled. Suppose we fire transition 5, places x , y , and z will hold one token. Thus transition 4 becomes enabled and transition 2 remains enabled. However, only one of these two transitions can be fired since they are in conflict. If transition 2 is fired then a, b, and z will hold one token. If transition 4 is fired instead, there will be one token in x, b, and c.
a
b
e
Fig. 1. A Petri net with tokens. (A dot stands for one token.) 2. VECTOR A D D I T I O N S Y S T E M S
Next, we consider the definitions of vector addition systems. A vector addition system is an ordered pair ( W, d ) where W is a set of n-dimensional integer vectors and d is an n-dimensional integer vector of nonnegative components. The reachability set of ( W , d ) is the set of all vectors x such that (i) x = d + v l + v 2 + . . . + v , with v j ~ W j = l , 2 . . . . . r and (ii) for all integers i, 1 < i < r, d+vl+v2+...+v~>O
(zerovector)
Here " + " is a componentwise addition and " / > " a componentwise comparison. The reachability set gives all the points reachable from the designated point d with the "steps" in W and never leaving the first orthant of the n-space. EXAMVLE 2. Chain Reaction in X2 + R H ~ R X + H X . The. chemical reaction X 2 + R H ~ R X + H X is usually explained by the following four equations t i, t2, t3, and t 4.
t,:
X2-->2.~',
t2 :
f( + RH---~R + H X ,
t3 :
R + Xz--* R X + f(,
t# :
2.¢~X2.
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A six-dimensional vector addition system can represent the above chemical system. The six components in turn correspond to chemicals X2, X, RH, [~, HX, and RX. Let
I) 1 ~
Ill°2 ,
1)2~
0
11 °1 -1
1 0
t9 3
194
0
--1
I
'
0
0 1
0
Suppose initially the system has a concentration of chemicals represented by
d=
n~ and n2 are integers.
The reachability set of < (vl, v2, v3, v4), d> gives all the possible states of this chemical system in terms of concentrations. The system is termed chain reaction in chemistry since once the reaction t~ takes place, reactions t 2 and 13 c a n be carried out alternately. 3. T R A N S I T I O N S Y S T E M S
The concept of transition systems was first presented in [12]. The basic definitions are given below. Let V = (A ~,A2. . . . . An) be a finite set of symbols. Each symbol is called a place. Let + be a commutative and associative operator on V* where V* is a set defined as follows. (1) F o r 1 < i <<.n, A i is in V*. (2) A special element /k (empty term) is in V*. (3) If x and y are in V*, so is x + y . Since we assume that + is commutative and associative, each element in V* can be identified by the number of occurrences of each symbol A i. Namely, each clement in V* has the form a l A l + c t 2 A 2 + . . . +ctnAn where % for 1 < i < n is a nonnegative integer. Each integer a i is called a coefficient. We
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shall also use the abbreviated form ~ aiA i or ~ aiA i when n is understood. i~l
The special e l e m e n t / ~ is ~ ziA; where z; for 1 < i < n is zero. We shall call each element in V* a term or a state. Each place Ai is associated with a function g,4, (or g;) from V* to {0, 1,2 .... } such that gA,(x) gives the coefficient of A; in state x. Two terms x = ~ aiA i a n d y = ~ fl;A i are said to be x < y if and only if cti < /3; for I < i < n. W h e n x < y, y - x is defined to be ~ (fl; -
ai)Ai. A transition t is an ordered pair of terms x a n d y , denoted by x--~y; x is sometimes referred to as LHS(t) a n d y as RHS(t). The set L ( t ) of places is defined as (AjII]./(LHS(t))v~0 }. Similarly we can define R (t) to be the set of places which appear in RHS(t) with positive coefficients. DEHNmON 1. A transition system is a triple ( V, T, S ) where V is a finite set of places ( A l,A 2. . . . . An }, T C_ V* × V* is a set of transitions (tl, t2. . . . . t,, } and S is a term in V* called the initial state. Transitions can be applied on a state to derive new states. If t is in T a n d S i is a state such that LHS(t) < $1, we say that t is enabled orfireable in the state S I. If t is fired in S l, the new state $2 obtained is defined to be S l LHS(t) + RHS(t). The fact that S t yields $2 by firing t is denoted by Si = t ~ S 2 , or S I = ~ S 2 when t is understood. If there is a sequence of states $1,$2 . . . . . $i+ l in V* such that, for some transitions ~-l,r2 . . . . . ~. in T we have 5",.= z i ~ S ; + l for 1 < i < j , we say that Si = ~'lz2""" ~)~Sj+ l or S 1= . ~ S j + 1, or Sj+ I is reachable from S 1. The sequence of transitions r I, r2,..., ~ is called a firing sequence in the state Si, or simply a firing sequence if there is no ambiguity. The concept of reachability set of a transition system can also be defined. DEFINITION 2. The teachability set of a transition system G = ( V , T, S ) denoted by L ( G ) , is the set { x l S = * ~ x ). The generalization from Petri nets or vector addition systems to transition systems is a simple one. A Petri net can be viewed as a transition system such that for each transition ( / a n d place A;, ~;(LHS((/)) < 1 and fl;(RHS(tj)) < 1. A vector addition system is a transition system with a restriction that each transition t has L(t)VI R ( t ) = ~ . I The graph notation of Petri nets suggest a simple way of representing transition systems. Similarly, places in a transition system are denoted by
qndependently, R. Keller modified the model of vector addition systems and introduced vector replacement systems, which has a similar structure as transition systems. His work was presented in [11].
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circles, transitions by line segments, and tokens by dots. The only addition is that each arc in the directed graph is associated with a positive integer, which represents the appropriate coefficient. This graph-theoretic description of a transition system will be referred to as the graph representation of a transition system. The vector notation of vector addition systems also suggests an important representation of a transition system. Let G = (V, T, S ) be a transition system where V = {A i,A2 . . . . . An} and T = {h, t2. . . . . tm}. The characteristic matrix of G is defined to be an n x m matrix F = (~-~) such that ~'0= - fi~(LHS(tj)) + gI(RHS((/)) for 1 < i < n and I < j ~< m. In this paper, we will consider the structural properties of a transition system. Essentially, we will study the relationship between graph representation and the characteristic matrix of a transition system. III. T E R M I N A T I O N A N D F I N I T E N E S S P R O P E R T I E S O F T R A N S I T I O N SYSTEMS F o r the convenience of discussion, we shall assume, in this section, a transition system G = ( V, T, S ) is given where V = {A i,A2 . . . . . A,} and T = {h, t2,..., tm ). Several questions concerning the properties of the transition system arise naturally. Can a certain transition be fired? Is it possible for a certain place to accumulate an infinite number of tokens? Is the reachability set L ( G ) finite? Given a state x, is x reachable from the initial state S? The last problem, usually referred to as the reachability problem, turns out to be extremely difficult and remains open. Before looking at these properties, we need to define some terms. We shall consider the properties related to transitions and places separately. 1. PROPERTIES RELATED TO TRANSITIONS In a state x, a transition t may never become enabled. If there is a state y such that x = . ~ y and t is enabled i n y , we say that t is live in state x. A transition is said to be dead in state x if it is not live in x. A finer classification of live transitions is necessary. We shall use the notation t~j(o) to denote the number of occurrences of (/in the sequence o.
DEFINITION 3. In a state x, (1) a transition (/is terminating if there exists an integer k such that for every firing sequence o in x, hi(o) ~< k; (2) a transition tj is nonterminating if it is not terminating. Thus, a dead transition in state x is terminating in x. A live transition can be either terminating or nonterminating.
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DEFINmON 4. A state x is terminating (or dead) if all transitions in the transition system are terminating (or dead) in x. A state x is nonterminating if it is not terminating. It is obvious that every successor state of a terminating state must be terminating. Note also that a state x is nonterminating if and only if there is a transition which is nonterminating in state x. In fact, a slightly stronger statement can be made. THEOREM 1. A state x is nonterminating if and only if there exist a transition tj and a firing sequence tr such that for any k, ~j(o) > k. Proof. If a transition tj appears in a firing sequence for arbitrarily many times, then (/must be nonterminating in x and hence x is nonterminating. Suppose that for every transition tj and firing sequence o, there is an integer k with [~j(o) < k. It is easy to see that every firing sequence must be finite. We shall prove that x is terminating. A rooted tree with the root labeled x is constructed as follows. ICy is a node label a n d y = ( / ~ z then connect an immediate successor node with label z to the node labeled y. The tree so constructed satisfies two conditions: (1) every node in the tree has no more than IT[, a finite number, immediate successors; (2) there is no infinite path directed away from the root since such a path would correspond to an infinite firing sequence. By the K6nig Infinity Lemma, the tree must have a finite number of nodes. Consequently, the number of firing sequences in x is bounded. F o r any transition tj we can find a constant k such that ~j(o) < k for any firing sequence o. Hence x is terminating. Q.E.D.
COROLLARY. A state x is nonterminating if and only if there are two states y and z such that x = . ~ y , y = o ~ z for a nonempty sequence o a n d y < z. This corollary suggests a way to decide if a state is indeed nonterminating. A special type of nonterminating states is of interest. A state x is said to be strongly nonterminating if there is an infinite firing sequence o in x such that for every transition tj and any integer k, [gj(a) > k. Hence every transition can fire arbitrarily many times in a strongly nonterminating state. Obviously, not every nonterminating state is strongly nonterminating. In the later part of this paper, we shall concentrate on the structural properties of transition systems with a strongly nonterminating state. 2. PR OPER TIES RELA TED TO PLACES
These properties are concerned with the token counts in the places of transition systems. Essentially, we consider the property of whether the token count in a certain place is bounded.
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DEFINITION 5. A transition system G = (V, T, S ) is bounded with respect to place A i ~ V, if there exists an integer k such that for every state x in L(G), g;(x) -<
IV. S T R U C T U R A L PROPERTIES OF T R A N S I T I O N SYSTEMS Let G - - ( V, T, S ) be a transition system. Its termination and finiteness properties depend not only on the structure of transitions, which is reflected in ( V, T), but also on the initial state S. It is possible that G is bounded no matter which state x is substituted for S as the initial state. On the other hand, it is also possible that G is bounded and there is a state x such that ( V, T , x ) becomes not bounded. A similar situation may occur in the study of termination properties. To separate the consideration of the structural properties and the initial state seems to be the first step toward a full understanding of termination and finiteness properties of transition systems. We shall call the ordered pair ( V, T ) a semi transition system. We shall use the term transition system interchangeably with semi transition system when the initial state is irrelevant. This section is concerned with the properties of semi transition systems. DEFI~TION 6. A semi transition system ( V, T) is said to be nonterminating (or strongly nonterminating) if there exists a state S such that S is nonterminating (or strongly nonterminating) in the transition system ( V , T , S ) . DEFrNITION 7. A semi transition system ( V, T) is said to be bounded if for any state S, (V, T, S ) is bounded. If a semi transition system is both strongly nonterminating and bounded, we say that it is well-behaved. Well-behaved transition systems can be defined
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similarly. A well-behaved semi transition system can be assigned an initial state in which every transition has the potential to fire for arbitrarily many times and the token count in every place is bounded. There is a good reason for studying well-behaved transition systems. In [9], Karp and Miller proposed the computation graphs as a model for decision-free parallel computation. A computation graph is essentially a restricted form of a transition system. In their model transitions correspond to a computation step and places correspond to queues holding data to be processed. It is reasonable to assume that only the bounded transition system is of interest. Furthermore, if a transition system is strongly nonterminating, the corresponding computation process is not in danger of blocking a certain computation step. We shall introduce two important concepts and present a sufficient and necessary condition for a semi transition system to be well-behaved. 1. CONSERVATIVE TRANSITION SYSTEMS
Let f be a function from the set V of places to the set N ÷ of positive integers. The value f(A) is referrred to as the weight of a token in place A. The function f can be extended to V* in a natural way. DEFISmON 8. A semi transition system G = ( V, T ) is conservative if there exists a function f: V ~ N ÷ such that for every transition t in T, f(LHS(t)) = f(RHS(t)). The function f if called a weight function of G. Suppose F is the characteristic matrix of a conservative semi transition system of G. It is clear that f. I ' - - 0 where f is considered a row vector. We found the well-known Farkas' Lemma [8] in linear algebra very useful in studying conservation properties. We shall first introduce some terminologies. Let x be a vector with components x~,x 2. . . . . x,. We shall say that x is nonnegative or x >/0 if and only if xi is nonnegative for 1 ,.< i < n. We shall say that x is semipositive or x ~ 0 if and only if x is nonnegative and different from zero vector 0. We shall say that x is positive or x > 0 if x i is positive for l
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system G ' which has the following structure. If t is a transition in G, then both t and RHS(t)---~LHS(t) are transitions in G'. The implication of Theorem 4 is that G is conservative if and only if for any state u in G', u cannot reach a state v such that u ~< v and u is different from v. Hence we have the following observation, which is intuitively clear. The proof follows from Theorem 3. THEOREM 5. I f a semi transition system H is conservative, then H is bounded.
2. STRONGLY REPETITIVE TRANSITION SYSTEMS In a transition system G, a state x is repetitive if there is a nonvoid firing sequence a such that x = o ~ x . If every transition in G appears in a, we say that x is strongly repetitive. DEFINITION 9. A semi transition system (V, T ) is said to be strongly repetitive if there exists a state S which is strongly repetitive in ( V , T , S ) . Similar to the property of conservation, a strongly repetitive semi transition system must have a positive vector g such that F . g = 0. In the following theorems we link the concepts of conservation and strong repetitivity. THEOREM 6. If a semi transition system is strongly repetitive and bounded, then it is conservative.
Proof. Let G be a semi transition system with characteristic matrix F. If G is strongly repetitive, then there is a positive solution to F . y = 0. When G is also bounded, then both F.y _~0 and F.y ~< 0 have no nonnegative solutions. Together, they simply that F-y _~0 has no solution. By Theorem 4, we conclude that G is conservative. Q.E.D. Finally, we give the following result to characterize well-behaved semi transition systems. THEOREM 7. A semi transition system is well-behaved if and only if it is both conservative and strongly repetitive.
Proof. The "if" part is trivial. By Theorem 6, it suffices to show that a strongly nonterminating and bounded semi transition system must be strongly repetitive. If G is strongly nonterminating, there exist an infinite firing sequence a = a~a2" " and an infinite sequence S of states $1,$2 .... such that S I -~ tll=~S2, S2 -~ a2==~S3, etc. Furthermore the number of occurrences of any transition tj in a, or ~)j(a) is not bounded. Hence for every tj, one can find an infinite subsequence Si of S such that for every state S; in $i, a~ = (/. Since G is bounded, we have a finite set L ( ( V, T,S,)). Consequently, there must be two identical states S~ and Sk in the sequence Sj. Let the firing sequence
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oio;+~" "" ok-, be called zj. Letyj be a vector such that its lth component is bt('~). For every transition (/we can find suchyj. Definey to be ~.. Yi" It is jml
clear that F.y = 0 and y is positive. Therefore, G is strongly repetitive.
Q.E.D.
3. STRONGLY CONNECTED TRANSITION SYSTEMS
The results in the previous two subsections were based on the matrix representation of semi transition systems. In this section, the graph representation is considered. We shall consider only the connectivity of a graph representation and ignore the coefficient lables associated with arcs. A graph representation will be viewed as a bipartite directed graph. A semi transition system G is said to be connected if the underlying undirected graph of its graph representation is connected. G is said to be strongly connected if its graph representation is strongly connected. In a directed graph D, one can define an equivalence relation as follows. Two nodes n and n' are said to be n ~ n ' if there exists a directed path from n to n' and a directed path from n' to n. Hence, the set of nodes in D can be uniquely partitioned into equivalence classes under the relation ~ . A subgraph consisting of the nodes of an equivalence class and the arcs of D between these nodes is usually called a strong component of D. It is a well-known fact that strong components of a directed graph are the maximal strongly connected subgraphs of the original graph. Let D I , D 2.... , D r be the strong components of D. We say that D~ covers Dj if there exist nodes a in D i and b in Dy such that a is connected to b by an arc. A partial ordering /> on the set of strong components is defined as follows. a i/> Dj if i = j or there exist D~,,Dk2 . . . . . Dk,+a such that Dk, = Di, Dy= Dk,+~, and Dk, covers Dk,÷, for 1 < s < l. With respect to this partial ordering, we will say that D~ is a minimal element if there is no other Dj(iC=j) such that D~> Dy. THEOREM 9. I f G is a connected and well-behaved semi transition system then G is strongly connected. Proof. Assume to the contrary that G is not strongly connected. Let D be the graph representation and let Dj, D2 ..... D r be the strong components (f ~ 2). Suppose D i is a minimal element. Let tl,t2 . . . . . tr, A i , A 2. . . . . A s be the nodes which are not in D,. but connected directly to nodes Aj~,Aj2. . . . . Aj, tk~, tk~..... tk, in D~, respectively. Here t with a subscript is used to represent a transition node and A with a subscript represents a place node. If G is conservative and strongly repetitive, D~ should have at least one place and at least one transition. Let [ be a weight function of G. Since G is strongly
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repetitive, there exist states $1 and $2 and a firing sequence 0 such that $1 = 0 ~ $ 2 with S~ = $2 and o contains every transition in G. Let Ao, be the set of places in Dr It is obvious that gA(S0-- ~A(S2) for all A in Ao,. Hence g A ( S 0 - f ( A ) = ~ gA(S2).f(A). That is, the total weight in D~ should AEA~
AEA~
be equal in both S~ and $2. However, after firing transitions tq, 1 <~q <~r, a number of tokens are added to A~, and hence the total weight in Di is increased. Similarly, after firing transitions t~, 1 < q < s, the total weight in D i is also increased. Since D~ is a minimal element, it has no transitions to fire to reduce its total weight, thus ~ flA(Sl)'f(A)< ~. gA(S2)'f(A). A EAo,
A contradiction.
A EAo,
Q.E.D.
It is easy to see that the converse of the previous theorem is not true.
V. C O N F L I C T - F R E E A N D C O N C U R R E N T - F R E E T R A N S I T I O N SYSTEMS Two special classes of transition systems, namely, conflict-free transition systems and concurrent-free transition systems, are considered in this section. Both conflict-free and concurrent-free Petri nets have been studied in great detail [5]. Conflict-free Petri nets, which are also called marked graphs, have been studied intensively [1,3, 5]. Computation graphs in [9] are essentially conflict-free transition systems. The following definitions can be extended to semi transition systems. DEFINITION 10. A transition system G = (V, T, S ) is said to be conflict-free if for every pair of distinct transitions t 1 and t 2 in T, L ( t j ) N L ( I 2 ) = R (tl) f"l R (t2)-- 0 . DEHNmON 11. A transition system G( V, T, S ) is said to be concurrent-free if for every transition IL(t)l = IR (t)l-- 1. In other words, every place node in the graph representation of a conflictfree transition system must have its outdegree and indegree bounded by 1. Every transition node in the graph representation of a concurrent free transition system has outdegree and indegree less than or equal to one. These two classes of transitions systems are closely related in an interesting way. Suppose G is a semi transition system. If we replace each transition node (a line segment) by a place node (a circle), replace each place node by a transition node and reverse all the directed arcs, we obtain a new semi transition system. We call it G t, the transpose of G. Note that the characteristic matrix F' of G t is indeed the transpose of the characteristic matrix F of G.
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We have the following straightforward results.
THEOREM 9. Let G be a semi transition system. (1) G is conservative if and only if G t is strongly repetitive. (2) G is conflict-free if and only if G' is concurrent-free. Therefore, we can determine if a semi transition system is well behaved by looking at its transpose. Since a conflict-free semi transition system G has a concurrent-free transpose a t, G is well behaved if and only if G ' is well behaved. A sufficient and necessary condition for a conflict-free transition system to be well behaved can be obtained from Theorem 8 in [9]. Here we shall present a new proof, based on concurrent-free semi transition systems, which is believed to be simpler than the one in [9]. Similar results in marked graphs have been reported in [1,3, 5]. DEFINITION 12. A semi transition system G = ( V, T ) is said to be simple if V = (A l,A2 ..... An ) and the transition t,. in T is aiAj~biA~+ l for 1 ~< i < n where ai and b,. are positive integers. Note that An÷ t is understood to be A ~. The graph representation of a simple semi transition system is a simple directed circuit. Properties o f a simple semi transition system G can be characterized by a special rational number qc, gain of G, which is defined to be the ratio bib 2. . • b n / a l a 2 . • • a n [9]. It is easy to see that a simple semi transition system G is well behaved if and only if qo = I. Next, we shall consider concurrent-free semi transition systems. Let G be a semi transition system ( V, T). A subsystem Gt of G is defined to be ( Vi, Ti) such that T~ c_ T and V 1 contains the places in V which appear in transitions in Ti. Clearly, if G is concurrent-free, so is any subsystem G t.
THEOREM 10. Let G be a strongly connected and concurrent-free semi transition system. If every simple subsystem g of G has qg = 1, then G is strongly repetitive. Proof. Every transition in G must be in some simple subsystem of G. If qs = I, g is strongly repetitive. By induction, a firing sequence containing every transition can then be found. Q.E.D. Note that the converse is not true. The semi transition system (A---)B, B ~ 2 A , 2A ~ C, C ~ A } is strongly repetitive, but gain of every simple subsystem is not 1. THEOREM 11. Let G be a strongly connected and concurrent-free semi transition system. I f every simple subsystem g of G has qs = 1, then G is conservative. Proof. Suppose Go = (V0, To) and Gi = (Vj, Tl) are two concurrent-free semi transition systems. We define Got_3G~ to be the semi transition system
360
G2= (VoU VI, T0U Tl).
Y. E D M U N D L I E N Since G is strongly connected and concurrent-free, k
we know from graph theory that G = U gi for some finite k, where gi, i~l
1 ~< i ~< k, is a simple subsystem of G with qg, = 1. The proof proceeds by induction on k. When k = 1, G is conservative if and only if q~ = I. Assume k
that G = GoU G I where G O= U gi -~ ( Vo, To) is conservative and GI = ( Vj, TI) i=l
is a simple subsystem of G. Since G~ has qc, = 1, there exists a weight function f~ of G~. Let fo be a weight function of Go. We claim that for any pair of places Bl and B2 in VoA VI, fo(BO/ft(BO=fo(B2)/fl(B2). For if this is not the case, one can find a directed path from Bt to B2 in G Oand a directed path from B2 to B~ in G~, and this circuit must contain a simple directed circuit with gain not equal to 1. A contradiction. Suppose fo(Bl)/fj(Bi) = do/d t for integers d o and d I. Define a function f such that
f(A)=
dlfo(A) dofl(A )
ifAEV 0 ifAEgt.
Clearly f is a weight function of G. Hence G is conservative. Note that the case IVon Vt[.<< 1 is trivial. Q.E.D. The converse of Theorem 11 is trivially true. Hence a concurrent-free semi transition system G is well behaved if and only if every simple subsystem g in G has q8 = 1. A necessary and sufficient condition for the conflict-free case can be similarly established. VI. POSSIBLE E X T E N S I O N S One possible direction is to investigate classes of transition systems or semi transition systems larger than conflict-free and concurrent-free cases. Free choice Petri nets and simple Petri nets have been identified [9], parallel work on transition systems would be desirable. The relation between transition systems and formal languages is also an area of interest. One may link a special class of formal languages with a class of transition systems. One such instance is Parikh's Theorem [13], which is useful in context-free languages. Another direction of extension is to study the problem of scheduling and resource allocation in the context of transition systems. F o r example, one may associate with each transition a time slice and firing of a transition is assumed to take the amount of time specified by the corresponding time slice [17]. Such a model can be interpreted as a production system in which firing of a transition should be initiated by some units of resources specified by its
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input places. If e n o u g h resources are available, then c o n c u r r e n t firings of e n a b l e d transitions are possible. Otherwise, to schedule the firing sequence so that the n u m b e r of firings within a certain period is m a x i m u m w o u l d be an i m p o r t a n t p r o b l e m in operations research. D e a d l o c k p r o b l e m o c c u r r e n c e in resource allocation is yet a n o t h e r a p p e a l i n g topic related to the notion of Petri nets and transition systems [2,4,7].
The author wishes to express his gratitude to Professor Lotfi A. Zadeh for his thoughtful suggestions and invaluable criticisms. REFERENCES I. F. Commoner, A. W. Holt, S. Even, and A. Pnueli. Marked directed graphs, J. Computer Syst. Sci. 5, 511-523 (1971). 2. E. W. Dijkstra, Co-operating sequential processes, in Programming Languages, F. Genuys, Ed., Chapter 2, Academic, New York, 1968. 3. H. J. Genrich, Einfache Nicht-Sequentielle Prozesse, Gesellschaft fiir Mathematick und Datenverarbeitung, Birlinghoven, West Germany, 1970. 4. P. G. Hebalkar, Deadlock-free sharing of resources in asynchronous systems, Report MAC TR-75, Project MAC, M.I.T., 1970. 5. A. W. Holt and F. Commoner, Events and Conditions (in three parts), Applied Data Research, New York, 1970. (Chapters I, II, IV and VI appear in Record of the Project MAC Conference on Concurrent Systems and Parallel Computation, ACM, New York, 1970, pp. 3-52.) 6. A. W. Holt, Final Report of the Information System Theory Project. Technical Report RACD-TR-68-305, Rome Air Development Center, Griffiss Air Force Base, New York, 1968. 7. R. C. Holt, Some deadlock properties of computer systems, Computing Survey, 4, No. 3, 179-196 (1972). 8. T. C. Hu, lnteger Programming and Network Flows, Addison-Wesley, Reading, Mass., 1969. 9. R. M. Karp and R. E. Miller, Properties of a model for parallel computations: Determinacy, termination, queueing, SlAM J. Appl. Math., 14, No. 6, 1390-1411 (1966). 10. R. M. Karp and R. E. Miller, Parallel program schemata, J. Computer Syst. Sci. 3,.No. 2, 147-195 (1969). 11. R. M. Keller, Vector replacement systems: A formalism for modeling asynchronous systems. Technical Report 117, Department of Electrical Engineering, Princeton University, Princeton, N.J., December 1972. 12. Y. E. Lien, Study of theoretical and practical aspects of transition systems. Ph.D. dissertation, University of California, Berkeley, August 1972. 13. R. J. Parikh, On context-free languages, J. ACM 13, No. 4, 570-581 (1966). 14. S. S. Patil, Coordination of asynchronous events, Report MAC-TR-72, Project MAC, M.I.T., June 1970. 15. C. A. Petri, Communication with automata. Supplement I to Technical Report RADC-TR-65-377, Vol. 1, Griffiss Air Force Base, New York, 1966. [Originally published in German: Kommunikation mit Automaten, University of Bonn, 1962.] 16. J. E. Rodriguez, A graph model for parallel computation, Report MAC-TR-64, Project MAC, M.I.T., 1969.
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17. R. M. Shapiro and H. Saint, A new approach to optimization of sequencing decisions, Annual Review in Automatic Programming, Pergamon Press, Oxford, England, 1970, Vol. 6, Part 5. 18. D. R. Slut.z, The flow-graph schemata model of parallel computation, Report MAC-TR53, Project MAC, M.I.T., 1968. 19. Project MAC Progress Report Vol. VIII, July 1970-July 1971. Project MAC, M.I.T., pp. 13-51. Received August, 1975.
347
A Note on Transition Systems* Y. EDMUND LIEN Department of Computer Science, University of Kansas, Lawrence, Kansas 66045 Communicated by L. A. Zadeh
ABSTRACT This paper introduces a mathematical structure called transition systems. The notion of transition systems has been developed as a result of the study of Petri nets and vector addition systems. The intended use of transition systems is to model concurrent and asynchronous events. The concept of information flow in a complex system and communication between parts of a complex system can be formulated in this formal structure. This paper is concerned with the mathematical properties rather than the applications of transition systems. Patterns of activities in complex systems are defined in terms of termination and finiteness properties of transition systems. Concepts of conservation and repetitivity have been introduced. Structural properties of restricted classes of transition systems have been studied.
I. I N T R O D U C T I O N Various models have been proposed in the study of parallel computation, concurrent processing and asynchronous events [6, 9, 10, 14, 15, 16, 18]. A m o n g them, Petri nets [15] and vector addition systems [10] are two important models in which the notion of concurrency and asynchrony can be easily formulated. The concept of Petri nets was first introduced by Petri in his doctoral dissertation [15]. The current version of Petri nets is due to Holt and his colleagues in the Information System Theory Project [6]. The intended use of Petri nets in this project was to represent complicated patterns of activities and events in complex systems. Many results of the study of Petri nets have
*This paper is based on the material in the author's Ph.D. dissertation (University of California, Berkeley, 1972) which was supported in part by NSF grant CK-10656X and NESC-N00039-71C-0255. This work is also supported in part by Kansas General Research Grant 3298-5038. © American Elsevier Publishing Company, Inc., 1976
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appeared in the literature [1,3, 5, 6, 17, 19]. Those who have been enthusiastic about the theory and the application of Petri nets have found the notion to be "an elegant formalism for representation of concurrency in processes and of studying asynchronous systems" [19]. Independent of the work mentioned above, K a r p and Miller proposed a formal model of parallel computation called parallel program schemata [10]. The notion of vector addition systems was also introduced for the purpose of studying the decision problems in parallel program schmata. It has been recognized by many researchers in this area that Petri nets and vectors addition systems are very similar in their mathematical structure. In this paper a formal system called a transition system is introduced. In this work, we generalize the common concepts of Petri nets and vector addition systems and preserve the power of describing the behavior of complex systems. The notion of transition systems can also be viewed as a restricted case of production systems used in formal language theory. The main concern of this paper is on the structural property of transition systems. Graph-theoretic properties of the model will be studied. II. PETRI NETS, V E C T O R A D D I T I O N SYSTEMS A N D T R A N S I T I O N SYSTEMS
1. PETRI NETS We first present the model of Petri nets.
A Petri net is a directed graph with two types of nodes, namely, place nodes and transition nodes. A place node is represented by a circle and a transition node by a line segment. Each arc connects either a place to a transition or a transition to a place. A place is said to be an input place to a transition if there is an arc from the place to the transition. Similarly, a place is an output place of a transition if there is an arc connecting the transition to the place. Each place is capable of holding a number of tokens. A transition is said to be enabled orfireable if every input place of the transition holds at least one token. Wefire a transition by removing one token from each of its input places and adding one token to each of its output places. Thus the total number may increase, decrease, or remain the same after firing an enabled transition. Suppose two transitions have a common input place and both transitions are enabled, then only one of these two transitions is allowed to fire. These two transitions are said to be in conflict. We do not specify which one among a group of transitions in conflict is allowed to fire. The system is nondeterministic in this sense. Therefore, two transitions can fire simultaneously, provided that they are enabled and not in conflict.
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EXAMPLE I. Figure 1 shows a Petri net. In this example, transition 2 and transition 5 are enabled. Suppose we fire transition 5, places x , y , and z will hold one token. Thus transition 4 becomes enabled and transition 2 remains enabled. However, only one of these two transitions can be fired since they are in conflict. If transition 2 is fired then a, b, and z will hold one token. If transition 4 is fired instead, there will be one token in x, b, and c.
a
b
e
Fig. 1. A Petri net with tokens. (A dot stands for one token.) 2. VECTOR A D D I T I O N S Y S T E M S
Next, we consider the definitions of vector addition systems. A vector addition system is an ordered pair ( W, d ) where W is a set of n-dimensional integer vectors and d is an n-dimensional integer vector of nonnegative components. The reachability set of ( W , d ) is the set of all vectors x such that (i) x = d + v l + v 2 + . . . + v , with v j ~ W j = l , 2 . . . . . r and (ii) for all integers i, 1 < i < r, d+vl+v2+...+v~>O
(zerovector)
Here " + " is a componentwise addition and " / > " a componentwise comparison. The reachability set gives all the points reachable from the designated point d with the "steps" in W and never leaving the first orthant of the n-space. EXAMVLE 2. Chain Reaction in X2 + R H ~ R X + H X . The. chemical reaction X 2 + R H ~ R X + H X is usually explained by the following four equations t i, t2, t3, and t 4.
t,:
X2-->2.~',
t2 :
f( + RH---~R + H X ,
t3 :
R + Xz--* R X + f(,
t# :
2.¢~X2.
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Y. E D M U N D L I E N
A six-dimensional vector addition system can represent the above chemical system. The six components in turn correspond to chemicals X2, X, RH, [~, HX, and RX. Let
I) 1 ~
Ill°2 ,
1)2~
0
11 °1 -1
1 0
t9 3
194
0
--1
I
'
0
0 1
0
Suppose initially the system has a concentration of chemicals represented by
d=
n~ and n2 are integers.
The reachability set of < (vl, v2, v3, v4), d> gives all the possible states of this chemical system in terms of concentrations. The system is termed chain reaction in chemistry since once the reaction t~ takes place, reactions t 2 and 13 c a n be carried out alternately. 3. T R A N S I T I O N S Y S T E M S
The concept of transition systems was first presented in [12]. The basic definitions are given below. Let V = (A ~,A2. . . . . An) be a finite set of symbols. Each symbol is called a place. Let + be a commutative and associative operator on V* where V* is a set defined as follows. (1) F o r 1 < i <<.n, A i is in V*. (2) A special element /k (empty term) is in V*. (3) If x and y are in V*, so is x + y . Since we assume that + is commutative and associative, each element in V* can be identified by the number of occurrences of each symbol A i. Namely, each clement in V* has the form a l A l + c t 2 A 2 + . . . +ctnAn where % for 1 < i < n is a nonnegative integer. Each integer a i is called a coefficient. We
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shall also use the abbreviated form ~ aiA i or ~ aiA i when n is understood. i~l
The special e l e m e n t / ~ is ~ ziA; where z; for 1 < i < n is zero. We shall call each element in V* a term or a state. Each place Ai is associated with a function g,4, (or g;) from V* to {0, 1,2 .... } such that gA,(x) gives the coefficient of A; in state x. Two terms x = ~ aiA i a n d y = ~ fl;A i are said to be x < y if and only if cti < /3; for I < i < n. W h e n x < y, y - x is defined to be ~ (fl; -
ai)Ai. A transition t is an ordered pair of terms x a n d y , denoted by x--~y; x is sometimes referred to as LHS(t) a n d y as RHS(t). The set L ( t ) of places is defined as (AjII]./(LHS(t))v~0 }. Similarly we can define R (t) to be the set of places which appear in RHS(t) with positive coefficients. DEHNmON 1. A transition system is a triple ( V, T, S ) where V is a finite set of places ( A l,A 2. . . . . An }, T C_ V* × V* is a set of transitions (tl, t2. . . . . t,, } and S is a term in V* called the initial state. Transitions can be applied on a state to derive new states. If t is in T a n d S i is a state such that LHS(t) < $1, we say that t is enabled orfireable in the state S I. If t is fired in S l, the new state $2 obtained is defined to be S l LHS(t) + RHS(t). The fact that S t yields $2 by firing t is denoted by Si = t ~ S 2 , or S I = ~ S 2 when t is understood. If there is a sequence of states $1,$2 . . . . . $i+ l in V* such that, for some transitions ~-l,r2 . . . . . ~. in T we have 5",.= z i ~ S ; + l for 1 < i < j , we say that Si = ~'lz2""" ~)~Sj+ l or S 1= . ~ S j + 1, or Sj+ I is reachable from S 1. The sequence of transitions r I, r2,..., ~ is called a firing sequence in the state Si, or simply a firing sequence if there is no ambiguity. The concept of reachability set of a transition system can also be defined. DEFINITION 2. The teachability set of a transition system G = ( V , T, S ) denoted by L ( G ) , is the set { x l S = * ~ x ). The generalization from Petri nets or vector addition systems to transition systems is a simple one. A Petri net can be viewed as a transition system such that for each transition ( / a n d place A;, ~;(LHS((/)) < 1 and fl;(RHS(tj)) < 1. A vector addition system is a transition system with a restriction that each transition t has L(t)VI R ( t ) = ~ . I The graph notation of Petri nets suggest a simple way of representing transition systems. Similarly, places in a transition system are denoted by
qndependently, R. Keller modified the model of vector addition systems and introduced vector replacement systems, which has a similar structure as transition systems. His work was presented in [11].
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circles, transitions by line segments, and tokens by dots. The only addition is that each arc in the directed graph is associated with a positive integer, which represents the appropriate coefficient. This graph-theoretic description of a transition system will be referred to as the graph representation of a transition system. The vector notation of vector addition systems also suggests an important representation of a transition system. Let G = (V, T, S ) be a transition system where V = {A i,A2 . . . . . An} and T = {h, t2. . . . . tm}. The characteristic matrix of G is defined to be an n x m matrix F = (~-~) such that ~'0= - fi~(LHS(tj)) + gI(RHS((/)) for 1 < i < n and I < j ~< m. In this paper, we will consider the structural properties of a transition system. Essentially, we will study the relationship between graph representation and the characteristic matrix of a transition system. III. T E R M I N A T I O N A N D F I N I T E N E S S P R O P E R T I E S O F T R A N S I T I O N SYSTEMS F o r the convenience of discussion, we shall assume, in this section, a transition system G = ( V, T, S ) is given where V = {A i,A2 . . . . . A,} and T = {h, t2,..., tm ). Several questions concerning the properties of the transition system arise naturally. Can a certain transition be fired? Is it possible for a certain place to accumulate an infinite number of tokens? Is the reachability set L ( G ) finite? Given a state x, is x reachable from the initial state S? The last problem, usually referred to as the reachability problem, turns out to be extremely difficult and remains open. Before looking at these properties, we need to define some terms. We shall consider the properties related to transitions and places separately. 1. PROPERTIES RELATED TO TRANSITIONS In a state x, a transition t may never become enabled. If there is a state y such that x = . ~ y and t is enabled i n y , we say that t is live in state x. A transition is said to be dead in state x if it is not live in x. A finer classification of live transitions is necessary. We shall use the notation t~j(o) to denote the number of occurrences of (/in the sequence o.
DEFINITION 3. In a state x, (1) a transition (/is terminating if there exists an integer k such that for every firing sequence o in x, hi(o) ~< k; (2) a transition tj is nonterminating if it is not terminating. Thus, a dead transition in state x is terminating in x. A live transition can be either terminating or nonterminating.
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DEFINmON 4. A state x is terminating (or dead) if all transitions in the transition system are terminating (or dead) in x. A state x is nonterminating if it is not terminating. It is obvious that every successor state of a terminating state must be terminating. Note also that a state x is nonterminating if and only if there is a transition which is nonterminating in state x. In fact, a slightly stronger statement can be made. THEOREM 1. A state x is nonterminating if and only if there exist a transition tj and a firing sequence tr such that for any k, ~j(o) > k. Proof. If a transition tj appears in a firing sequence for arbitrarily many times, then (/must be nonterminating in x and hence x is nonterminating. Suppose that for every transition tj and firing sequence o, there is an integer k with [~j(o) < k. It is easy to see that every firing sequence must be finite. We shall prove that x is terminating. A rooted tree with the root labeled x is constructed as follows. ICy is a node label a n d y = ( / ~ z then connect an immediate successor node with label z to the node labeled y. The tree so constructed satisfies two conditions: (1) every node in the tree has no more than IT[, a finite number, immediate successors; (2) there is no infinite path directed away from the root since such a path would correspond to an infinite firing sequence. By the K6nig Infinity Lemma, the tree must have a finite number of nodes. Consequently, the number of firing sequences in x is bounded. F o r any transition tj we can find a constant k such that ~j(o) < k for any firing sequence o. Hence x is terminating. Q.E.D.
COROLLARY. A state x is nonterminating if and only if there are two states y and z such that x = . ~ y , y = o ~ z for a nonempty sequence o a n d y < z. This corollary suggests a way to decide if a state is indeed nonterminating. A special type of nonterminating states is of interest. A state x is said to be strongly nonterminating if there is an infinite firing sequence o in x such that for every transition tj and any integer k, [gj(a) > k. Hence every transition can fire arbitrarily many times in a strongly nonterminating state. Obviously, not every nonterminating state is strongly nonterminating. In the later part of this paper, we shall concentrate on the structural properties of transition systems with a strongly nonterminating state. 2. PR OPER TIES RELA TED TO PLACES
These properties are concerned with the token counts in the places of transition systems. Essentially, we consider the property of whether the token count in a certain place is bounded.
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DEFINITION 5. A transition system G = (V, T, S ) is bounded with respect to place A i ~ V, if there exists an integer k such that for every state x in L(G), g;(x) -<
IV. S T R U C T U R A L PROPERTIES OF T R A N S I T I O N SYSTEMS Let G - - ( V, T, S ) be a transition system. Its termination and finiteness properties depend not only on the structure of transitions, which is reflected in ( V, T), but also on the initial state S. It is possible that G is bounded no matter which state x is substituted for S as the initial state. On the other hand, it is also possible that G is bounded and there is a state x such that ( V, T , x ) becomes not bounded. A similar situation may occur in the study of termination properties. To separate the consideration of the structural properties and the initial state seems to be the first step toward a full understanding of termination and finiteness properties of transition systems. We shall call the ordered pair ( V, T ) a semi transition system. We shall use the term transition system interchangeably with semi transition system when the initial state is irrelevant. This section is concerned with the properties of semi transition systems. DEFI~TION 6. A semi transition system ( V, T) is said to be nonterminating (or strongly nonterminating) if there exists a state S such that S is nonterminating (or strongly nonterminating) in the transition system ( V , T , S ) . DEFrNITION 7. A semi transition system ( V, T) is said to be bounded if for any state S, (V, T, S ) is bounded. If a semi transition system is both strongly nonterminating and bounded, we say that it is well-behaved. Well-behaved transition systems can be defined
A N O T E ON T R A N S I T I O N SYSTEMS
355
similarly. A well-behaved semi transition system can be assigned an initial state in which every transition has the potential to fire for arbitrarily many times and the token count in every place is bounded. There is a good reason for studying well-behaved transition systems. In [9], Karp and Miller proposed the computation graphs as a model for decision-free parallel computation. A computation graph is essentially a restricted form of a transition system. In their model transitions correspond to a computation step and places correspond to queues holding data to be processed. It is reasonable to assume that only the bounded transition system is of interest. Furthermore, if a transition system is strongly nonterminating, the corresponding computation process is not in danger of blocking a certain computation step. We shall introduce two important concepts and present a sufficient and necessary condition for a semi transition system to be well-behaved. 1. CONSERVATIVE TRANSITION SYSTEMS
Let f be a function from the set V of places to the set N ÷ of positive integers. The value f(A) is referrred to as the weight of a token in place A. The function f can be extended to V* in a natural way. DEFISmON 8. A semi transition system G = ( V, T ) is conservative if there exists a function f: V ~ N ÷ such that for every transition t in T, f(LHS(t)) = f(RHS(t)). The function f if called a weight function of G. Suppose F is the characteristic matrix of a conservative semi transition system of G. It is clear that f. I ' - - 0 where f is considered a row vector. We found the well-known Farkas' Lemma [8] in linear algebra very useful in studying conservation properties. We shall first introduce some terminologies. Let x be a vector with components x~,x 2. . . . . x,. We shall say that x is nonnegative or x >/0 if and only if xi is nonnegative for 1 ,.< i < n. We shall say that x is semipositive or x ~ 0 if and only if x is nonnegative and different from zero vector 0. We shall say that x is positive or x > 0 if x i is positive for l
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Y. E D M U N D L I E N
system G ' which has the following structure. If t is a transition in G, then both t and RHS(t)---~LHS(t) are transitions in G'. The implication of Theorem 4 is that G is conservative if and only if for any state u in G', u cannot reach a state v such that u ~< v and u is different from v. Hence we have the following observation, which is intuitively clear. The proof follows from Theorem 3. THEOREM 5. I f a semi transition system H is conservative, then H is bounded.
2. STRONGLY REPETITIVE TRANSITION SYSTEMS In a transition system G, a state x is repetitive if there is a nonvoid firing sequence a such that x = o ~ x . If every transition in G appears in a, we say that x is strongly repetitive. DEFINITION 9. A semi transition system (V, T ) is said to be strongly repetitive if there exists a state S which is strongly repetitive in ( V , T , S ) . Similar to the property of conservation, a strongly repetitive semi transition system must have a positive vector g such that F . g = 0. In the following theorems we link the concepts of conservation and strong repetitivity. THEOREM 6. If a semi transition system is strongly repetitive and bounded, then it is conservative.
Proof. Let G be a semi transition system with characteristic matrix F. If G is strongly repetitive, then there is a positive solution to F . y = 0. When G is also bounded, then both F.y _~0 and F.y ~< 0 have no nonnegative solutions. Together, they simply that F-y _~0 has no solution. By Theorem 4, we conclude that G is conservative. Q.E.D. Finally, we give the following result to characterize well-behaved semi transition systems. THEOREM 7. A semi transition system is well-behaved if and only if it is both conservative and strongly repetitive.
Proof. The "if" part is trivial. By Theorem 6, it suffices to show that a strongly nonterminating and bounded semi transition system must be strongly repetitive. If G is strongly nonterminating, there exist an infinite firing sequence a = a~a2" " and an infinite sequence S of states $1,$2 .... such that S I -~ tll=~S2, S2 -~ a2==~S3, etc. Furthermore the number of occurrences of any transition tj in a, or ~)j(a) is not bounded. Hence for every tj, one can find an infinite subsequence Si of S such that for every state S; in $i, a~ = (/. Since G is bounded, we have a finite set L ( ( V, T,S,)). Consequently, there must be two identical states S~ and Sk in the sequence Sj. Let the firing sequence
A N O T E ON T R A N S I T I O N SYSTEMS
357
oio;+~" "" ok-, be called zj. Letyj be a vector such that its lth component is bt('~). For every transition (/we can find suchyj. Definey to be ~.. Yi" It is jml
clear that F.y = 0 and y is positive. Therefore, G is strongly repetitive.
Q.E.D.
3. STRONGLY CONNECTED TRANSITION SYSTEMS
The results in the previous two subsections were based on the matrix representation of semi transition systems. In this section, the graph representation is considered. We shall consider only the connectivity of a graph representation and ignore the coefficient lables associated with arcs. A graph representation will be viewed as a bipartite directed graph. A semi transition system G is said to be connected if the underlying undirected graph of its graph representation is connected. G is said to be strongly connected if its graph representation is strongly connected. In a directed graph D, one can define an equivalence relation as follows. Two nodes n and n' are said to be n ~ n ' if there exists a directed path from n to n' and a directed path from n' to n. Hence, the set of nodes in D can be uniquely partitioned into equivalence classes under the relation ~ . A subgraph consisting of the nodes of an equivalence class and the arcs of D between these nodes is usually called a strong component of D. It is a well-known fact that strong components of a directed graph are the maximal strongly connected subgraphs of the original graph. Let D I , D 2.... , D r be the strong components of D. We say that D~ covers Dj if there exist nodes a in D i and b in Dy such that a is connected to b by an arc. A partial ordering /> on the set of strong components is defined as follows. a i/> Dj if i = j or there exist D~,,Dk2 . . . . . Dk,+a such that Dk, = Di, Dy= Dk,+~, and Dk, covers Dk,÷, for 1 < s < l. With respect to this partial ordering, we will say that D~ is a minimal element if there is no other Dj(iC=j) such that D~> Dy. THEOREM 9. I f G is a connected and well-behaved semi transition system then G is strongly connected. Proof. Assume to the contrary that G is not strongly connected. Let D be the graph representation and let Dj, D2 ..... D r be the strong components (f ~ 2). Suppose D i is a minimal element. Let tl,t2 . . . . . tr, A i , A 2. . . . . A s be the nodes which are not in D,. but connected directly to nodes Aj~,Aj2. . . . . Aj, tk~, tk~..... tk, in D~, respectively. Here t with a subscript is used to represent a transition node and A with a subscript represents a place node. If G is conservative and strongly repetitive, D~ should have at least one place and at least one transition. Let [ be a weight function of G. Since G is strongly
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repetitive, there exist states $1 and $2 and a firing sequence 0 such that $1 = 0 ~ $ 2 with S~ = $2 and o contains every transition in G. Let Ao, be the set of places in Dr It is obvious that gA(S0-- ~A(S2) for all A in Ao,. Hence g A ( S 0 - f ( A ) = ~ gA(S2).f(A). That is, the total weight in D~ should AEA~
AEA~
be equal in both S~ and $2. However, after firing transitions tq, 1 <~q <~r, a number of tokens are added to A~, and hence the total weight in Di is increased. Similarly, after firing transitions t~, 1 < q < s, the total weight in D i is also increased. Since D~ is a minimal element, it has no transitions to fire to reduce its total weight, thus ~ flA(Sl)'f(A)< ~. gA(S2)'f(A). A EAo,
A contradiction.
A EAo,
Q.E.D.
It is easy to see that the converse of the previous theorem is not true.
V. C O N F L I C T - F R E E A N D C O N C U R R E N T - F R E E T R A N S I T I O N SYSTEMS Two special classes of transition systems, namely, conflict-free transition systems and concurrent-free transition systems, are considered in this section. Both conflict-free and concurrent-free Petri nets have been studied in great detail [5]. Conflict-free Petri nets, which are also called marked graphs, have been studied intensively [1,3, 5]. Computation graphs in [9] are essentially conflict-free transition systems. The following definitions can be extended to semi transition systems. DEFINITION 10. A transition system G = (V, T, S ) is said to be conflict-free if for every pair of distinct transitions t 1 and t 2 in T, L ( t j ) N L ( I 2 ) = R (tl) f"l R (t2)-- 0 . DEHNmON 11. A transition system G( V, T, S ) is said to be concurrent-free if for every transition IL(t)l = IR (t)l-- 1. In other words, every place node in the graph representation of a conflictfree transition system must have its outdegree and indegree bounded by 1. Every transition node in the graph representation of a concurrent free transition system has outdegree and indegree less than or equal to one. These two classes of transitions systems are closely related in an interesting way. Suppose G is a semi transition system. If we replace each transition node (a line segment) by a place node (a circle), replace each place node by a transition node and reverse all the directed arcs, we obtain a new semi transition system. We call it G t, the transpose of G. Note that the characteristic matrix F' of G t is indeed the transpose of the characteristic matrix F of G.
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We have the following straightforward results.
THEOREM 9. Let G be a semi transition system. (1) G is conservative if and only if G t is strongly repetitive. (2) G is conflict-free if and only if G' is concurrent-free. Therefore, we can determine if a semi transition system is well behaved by looking at its transpose. Since a conflict-free semi transition system G has a concurrent-free transpose a t, G is well behaved if and only if G ' is well behaved. A sufficient and necessary condition for a conflict-free transition system to be well behaved can be obtained from Theorem 8 in [9]. Here we shall present a new proof, based on concurrent-free semi transition systems, which is believed to be simpler than the one in [9]. Similar results in marked graphs have been reported in [1,3, 5]. DEFINITION 12. A semi transition system G = ( V, T ) is said to be simple if V = (A l,A2 ..... An ) and the transition t,. in T is aiAj~biA~+ l for 1 ~< i < n where ai and b,. are positive integers. Note that An÷ t is understood to be A ~. The graph representation of a simple semi transition system is a simple directed circuit. Properties o f a simple semi transition system G can be characterized by a special rational number qc, gain of G, which is defined to be the ratio bib 2. . • b n / a l a 2 . • • a n [9]. It is easy to see that a simple semi transition system G is well behaved if and only if qo = I. Next, we shall consider concurrent-free semi transition systems. Let G be a semi transition system ( V, T). A subsystem Gt of G is defined to be ( Vi, Ti) such that T~ c_ T and V 1 contains the places in V which appear in transitions in Ti. Clearly, if G is concurrent-free, so is any subsystem G t.
THEOREM 10. Let G be a strongly connected and concurrent-free semi transition system. If every simple subsystem g of G has qg = 1, then G is strongly repetitive. Proof. Every transition in G must be in some simple subsystem of G. If qs = I, g is strongly repetitive. By induction, a firing sequence containing every transition can then be found. Q.E.D. Note that the converse is not true. The semi transition system (A---)B, B ~ 2 A , 2A ~ C, C ~ A } is strongly repetitive, but gain of every simple subsystem is not 1. THEOREM 11. Let G be a strongly connected and concurrent-free semi transition system. I f every simple subsystem g of G has qs = 1, then G is conservative. Proof. Suppose Go = (V0, To) and Gi = (Vj, Tl) are two concurrent-free semi transition systems. We define Got_3G~ to be the semi transition system
360
G2= (VoU VI, T0U Tl).
Y. E D M U N D L I E N Since G is strongly connected and concurrent-free, k
we know from graph theory that G = U gi for some finite k, where gi, i~l
1 ~< i ~< k, is a simple subsystem of G with qg, = 1. The proof proceeds by induction on k. When k = 1, G is conservative if and only if q~ = I. Assume k
that G = GoU G I where G O= U gi -~ ( Vo, To) is conservative and GI = ( Vj, TI) i=l
is a simple subsystem of G. Since G~ has qc, = 1, there exists a weight function f~ of G~. Let fo be a weight function of Go. We claim that for any pair of places Bl and B2 in VoA VI, fo(BO/ft(BO=fo(B2)/fl(B2). For if this is not the case, one can find a directed path from Bt to B2 in G Oand a directed path from B2 to B~ in G~, and this circuit must contain a simple directed circuit with gain not equal to 1. A contradiction. Suppose fo(Bl)/fj(Bi) = do/d t for integers d o and d I. Define a function f such that
f(A)=
dlfo(A) dofl(A )
ifAEV 0 ifAEgt.
Clearly f is a weight function of G. Hence G is conservative. Note that the case IVon Vt[.<< 1 is trivial. Q.E.D. The converse of Theorem 11 is trivially true. Hence a concurrent-free semi transition system G is well behaved if and only if every simple subsystem g in G has q8 = 1. A necessary and sufficient condition for the conflict-free case can be similarly established. VI. POSSIBLE E X T E N S I O N S One possible direction is to investigate classes of transition systems or semi transition systems larger than conflict-free and concurrent-free cases. Free choice Petri nets and simple Petri nets have been identified [9], parallel work on transition systems would be desirable. The relation between transition systems and formal languages is also an area of interest. One may link a special class of formal languages with a class of transition systems. One such instance is Parikh's Theorem [13], which is useful in context-free languages. Another direction of extension is to study the problem of scheduling and resource allocation in the context of transition systems. F o r example, one may associate with each transition a time slice and firing of a transition is assumed to take the amount of time specified by the corresponding time slice [17]. Such a model can be interpreted as a production system in which firing of a transition should be initiated by some units of resources specified by its
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input places. If e n o u g h resources are available, then c o n c u r r e n t firings of e n a b l e d transitions are possible. Otherwise, to schedule the firing sequence so that the n u m b e r of firings within a certain period is m a x i m u m w o u l d be an i m p o r t a n t p r o b l e m in operations research. D e a d l o c k p r o b l e m o c c u r r e n c e in resource allocation is yet a n o t h e r a p p e a l i n g topic related to the notion of Petri nets and transition systems [2,4,7].
The author wishes to express his gratitude to Professor Lotfi A. Zadeh for his thoughtful suggestions and invaluable criticisms. REFERENCES I. F. Commoner, A. W. Holt, S. Even, and A. Pnueli. Marked directed graphs, J. Computer Syst. Sci. 5, 511-523 (1971). 2. E. W. Dijkstra, Co-operating sequential processes, in Programming Languages, F. Genuys, Ed., Chapter 2, Academic, New York, 1968. 3. H. J. Genrich, Einfache Nicht-Sequentielle Prozesse, Gesellschaft fiir Mathematick und Datenverarbeitung, Birlinghoven, West Germany, 1970. 4. P. G. Hebalkar, Deadlock-free sharing of resources in asynchronous systems, Report MAC TR-75, Project MAC, M.I.T., 1970. 5. A. W. Holt and F. Commoner, Events and Conditions (in three parts), Applied Data Research, New York, 1970. (Chapters I, II, IV and VI appear in Record of the Project MAC Conference on Concurrent Systems and Parallel Computation, ACM, New York, 1970, pp. 3-52.) 6. A. W. Holt, Final Report of the Information System Theory Project. Technical Report RACD-TR-68-305, Rome Air Development Center, Griffiss Air Force Base, New York, 1968. 7. R. C. Holt, Some deadlock properties of computer systems, Computing Survey, 4, No. 3, 179-196 (1972). 8. T. C. Hu, lnteger Programming and Network Flows, Addison-Wesley, Reading, Mass., 1969. 9. R. M. Karp and R. E. Miller, Properties of a model for parallel computations: Determinacy, termination, queueing, SlAM J. Appl. Math., 14, No. 6, 1390-1411 (1966). 10. R. M. Karp and R. E. Miller, Parallel program schemata, J. Computer Syst. Sci. 3,.No. 2, 147-195 (1969). 11. R. M. Keller, Vector replacement systems: A formalism for modeling asynchronous systems. Technical Report 117, Department of Electrical Engineering, Princeton University, Princeton, N.J., December 1972. 12. Y. E. Lien, Study of theoretical and practical aspects of transition systems. Ph.D. dissertation, University of California, Berkeley, August 1972. 13. R. J. Parikh, On context-free languages, J. ACM 13, No. 4, 570-581 (1966). 14. S. S. Patil, Coordination of asynchronous events, Report MAC-TR-72, Project MAC, M.I.T., June 1970. 15. C. A. Petri, Communication with automata. Supplement I to Technical Report RADC-TR-65-377, Vol. 1, Griffiss Air Force Base, New York, 1966. [Originally published in German: Kommunikation mit Automaten, University of Bonn, 1962.] 16. J. E. Rodriguez, A graph model for parallel computation, Report MAC-TR-64, Project MAC, M.I.T., 1969.
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17. R. M. Shapiro and H. Saint, A new approach to optimization of sequencing decisions, Annual Review in Automatic Programming, Pergamon Press, Oxford, England, 1970, Vol. 6, Part 5. 18. D. R. Slut.z, The flow-graph schemata model of parallel computation, Report MAC-TR53, Project MAC, M.I.T., 1968. 19. Project MAC Progress Report Vol. VIII, July 1970-July 1971. Project MAC, M.I.T., pp. 13-51. Received August, 1975.