Contents 1 Project Overview Topic of the research project Application areas of Petri nets Summary

Transcription

1 Evolving Petri Nets with a Genetic Algorithm Holger Mauch October 10, 2002 Abstract In evolutionary computation many dierent representations (\genomes") have been suggested as the underlying data structures, upon which the genetic operators act. Among the most prominent examples are the evolution of binary strings, real-valued vectors, permutations, nite automata, and parse trees. In this project the use of Petri Nets as the structures that undergo evolution is examined. Petri Nets, structurally, can be considered as specialized bipartite graphs. The goal is to evolve Petri Nets, which in their extended version (adding inhibitor arcs) are as powerful as Turing machines. Evolving Petri Nets is therefore a form of Genetic Programming (GP) and could be compared to traditional GP, where parse trees of computer programs are evolved. 1

2 Contents 1 Project Overview Topic of the research project Application areas of Petri nets Summary of Petri net properties The Idea of Evolving Petri Nets Potential Diculties Example Problems Classication Problem: Play Tennis? Learning a Boolean Function: 11-multiplexer Predicting the Primeness of a Number Project Plan for 8 weeks Desired Results and Future Research Plans Petri Net Overview The Origin of Petri Nets Place/Transition Nets Useful Results from Net Theory Existing Literature and Research in this Area 15 4 Design of the Software System 16 References 18 2

3 List of Figures 1 Place/Transition Nets Summary A conict between two transitions Overview of a PN evolving EA system

4 1 Project Overview For this project, I plan to use Petri nets (PN for short, see section 2) as the data structure that undergoes evolution. PNs (in their extended form) have a computational power equivalent to that of Turing machines. Evolving PNs can thus be considered as a form of GP with a special type of graph genome (see chapter 9 in [2]). It appears to me that there are only very few publications that deal with this kind of approach. There exist some papers that deal with both evolutionary computation and Petri nets. But whenever a paper deals with both EC and PN concepts, such as [7] or [14], then most often EC plays only a minor role in the optimization of a certain special feature of the PN, or PNs are used to model parts of the suggested system, and EC is used to optimize certain parts of that system. In [7], for example, the GA merely inuences the ring rules, but not the structure of the PN. After browsing the abstracts of several of these papers, my impression is that some of the EC and PN concepts have been employed together, but none of these papers would follow the rigorous way of using PNs as the data structured to be evolved. This project overview section starts by dening the topic of the research project. Then properties of PNs and application areas of PNs are listed. After that the basic ideas of evolving PNs are outlined. Finally, a project plan is set up, including desired results. 1.1 Topic of the research project The main interest in this project is to describe how EA can be made to work on individuals, which are PNs. Chapter 9 of [2] as well as Chapters 11 and 19 of [1] gave inspiration to me to that there are many interesting ways of generalizing the recombination to arbitrary graphs, not just trees. Syntactically, PNs are bipartite graphs. Since I imagine that recombination is an important genetic operator when it comes to evolving PNs, my approach follows rather the GA stream within EC, rather than ES, or EP. In particular, I suspect that for some problems, there is a chance that the whole solution can be decomposed into subsections (building blocks) of a problem. 1.2 Application areas of Petri nets [5], p.542, lists (among many others) the following application areas: 4

5 Communication protocols Performance evaluation Discrete event systems Manufacturing/industrial control systems Decision models Data ow computing systems 1.3 Summary of Petri net properties \Petri nets are a promising tool for describing and studying information processing systems that are characterized as being concurrent, asynchronous, distributed, parallel, nondeterministic, and/or stochastic" [5], p.541. Petri nets \oer the ability to portray parallel and concurrent activities, the synchronisation of events, and competition for limited resources" [9], p.541. In addition to being a valuable modeling tool, PNs can also be used for classication problems (see subsection 1.6). Advantages of Petri nets are: High illustrative potential. Relatively easy to understand The underlying graphical syntax is precisely dened Certain properties of a Petri net can be proved formally. There exists an underlying mathematical theory of an enormous size (most Petri net literature deals with that theory). Petri Nets are supported by software. In 1998 a study [12] evaluated existing Petri Net tools and came up with a top ten list out of 91 candidates. Concurrency features [11] mentions the following additional properties of Petri nets: Petri nets emphasize locality of causation and eect. 5

6 Petri nets are neither state-based nor action-based. Both states and actions have a particular status on their own. Petri nets explicitly represent fundamental issues of distributed systems, such as atomicity, synchronization, mutual independence of actions, messages, and shared memory. Disadvantages of Petri Nets are : Petri nets don't exhibit compositionality properties; i.e. merging two nets N 1 and N 2 that are correct for two dierent subproblems to form a net N that is correct for both subproblems is a nontrivial task (see [6], p.39). This could become the major obstacle for using genetic recombination operators in the course of evolutionary computation. Petri nets suer from a lack of abstraction potential. E.g. you cannot just leave out a certain part of an arbitrary design that you want to treat as a \black box", unless you adopt nonstandard conventions. Only under special conditions it is possible to have macro nodes that represent a subnet. Many tests procedures (e.g. matrix based methods) that check a certain Petri net property are static in nature. They ignore the execution order in which the rules/transitions re. For some problems that might just be what is desired. But for other problems this execution order might be of particular interest. This shows that the representation chosen for the individuals of the population will depend on the problem to be solved. Complexity problem: Petri net based models tend to become too large for analysis even for a modest size system ([5], p.542). 1.4 The Idea of Evolving Petri Nets Why might it be promising to evolve Petri nets? In this subsection I list several thoughts about this fundamental question. PNs are a nice formalism that allow the modeling of certain real word problems. Their strength is their \built in" concurrency mechanism, which allows a formal description of systems containing parallel processes. This concurrency feature (transitions can re in parallel) makes the PN a good model for problems, in which the order of certain actions does not matter. Are there interesting problems, whose solutions can be represented as Petri nets, or not? The literature mentions numerous application areas 6

7 for PNs. In particular, the design of industrial production systems can be represented with PNs. Optimizing such a system corresponds to optimizing the design of the associated PN. Existing PN theory and algorithms are able to answer many questions about formal characteristics of a PN, such as reachability orliveness. These theoretical results could be built into the tness function. PNs with poor characteristics would score a low tness value. Also, to encourage parsimony, if 2 PNs show the same behavior, smaller PNs could score higher tness than larger ones. From an operational perspective, there exist numerous software packages (see [12]) that can simulate the dynamic behavior of PNs (\running the PN"). 1.5 Potential Diculties By observing how a PN behaves, a tness value could be assigned to an individual PN. However there are many details that need to be addressed here, and there are various ways how a tness value could be assigned to a PN. A PN does not directly return an output. The dynamical behavior is characterized by the possible ring sequences. The dynamical behavior, which also depends on the static structure (the topology of the PN), determines how \good" a PN is for a given task. The remedy used in this project is to declare certain places P Out P of the PN as output places, and evaluate the PN according to the induced submarking M Out after ring a certain number of transitions, or according to the time trajectory of M Out. The consequence of having designated output places for tness evaluation is that the PNs undergoing evolution are restricted in the sense that only PNs having the appropriate number of output places are viable ([13], p.414). For certain problems, such as those suggested in subsection 1.6, we can x the output places P Out and prevent that they are ever modied by genetic operators. An alternative way of determining when a PN is ready for tness evaluation is demonstrated by Reid in [9], p. 88, where a \qualier function" q : (P! N)! f0; 1g is introduced, which decides whether a possible marking is acceptable for tness evaluation, or not. Another complication is that, given a Petri net and a marking, its dynamic behavior is not deterministic. The nondeterminism has two sources. 7

8 1. Concurrency. That is, two or more transitions are enabled and can re at the same time, but dierent ring orders do not result in dierent behavior. This is a feature of PNs and poses no problem. 2. Conicts. A situation as in gure 2 leaves it open which transition to re next in a given state. Therefore there are many dierent ring sequences possible. This nondeterministic behavior might result in dierent output values even if we execute the PN on the same training case. Therefore, there is some randomness involved when assigning a tness value to a PN. To solve the problem of the nondeterminism due to conicts, a tness measure could be dened by taking the mean value of a sample of simulation runs. An easier way out is to restrict ourselves to a subclass of PNs called \marked graphs", which due to restrictions on their structure do not allow conicts. However, the expressional power of marked graphs is lower than that of PNs. Another solution to this problem is suggested by Reid in [9], p. 87. He incorporates a ring order relation into the denition of a PN. This relation is transitive, reexive, and antisymmetric and forms a priority ordering on the set of all transitions. (I do not understand why Reid only denes a partial ordering on the transitions. I think a total ordering is needed to resolve conicts completely). Conicts are resolved by ring the transition having the highest priority rst. This modied PN denition also requires genetic operators on this priority ordering. The ideas for genetic operators on permutations from [1] can be used, if I assume a total ordering of all transitions. Yet another conict resolution mechanism would be to use stochastic PNs as dened in [5], p Example Problems The following examples can be used to test whether the approachofevolving PNs to solve very simple, clear cut problems, is promising. The common theme among all examples is that we run t training cases (often called \tness cases" in the GP literature, see [2], p.10) on each PN; i.e. we simulate each individual PN with t dierent initial markings. 8

9 1.6.1 Classication Problem: Play Tennis? This problem is taken from [4], p.59. It is used by Mitchell as an introductory level problem to decision tree learning. A set of t = 14 training examples is provided, and the task is to classify examples into one of a discrete set of possible categories. The task here is to determine whether to play tennis or not (i.e. there are 2 categories for the target attribute PlayTennis), given a value of sunny, overcast, or rain for the attribute outlook, a value of hot, mild, or cold for the attribute temperature, a value of high, or normal for the attribute humidity, and a value of weak, or strong for the attribute wind. The goal is to evolve a PN that has designated input places to store the values of the attributes of an example, and that has a designated output place that holds the answer to whether to play tennis, or not. After running the PN for every training example, we can assign a score to the PN, which represents its tness - e.g. a score of 10 means that 10 out of the 14 training examples are classied correctly by the given PN. It would be interesting to compare the results of the PN found by EA with the results of decision tree learning algorithms like ID3 or C Learning a Boolean Function: 11-multiplexer This problem is taken from [3], p.169. Koza uses GP to discover \a composition of Boolean functions, that can return the correct value of a Boolean function after seeing a certain number of examples consisting of the correct value of the function associated with a particular combination of arguments". This is very similar to the Play Tennis problem, but here we can be sure that the training examples do not contain any incorrect classications. If the Boolean function to be learned is the 11-multiplexer function f0; 1g 11! f0; 1g as described in [3], p.171, then it seems appropriate to designate 11 special input places and 1 special output place to all the PNs in the population that undergoes evolution. Evaluating the tness of a PN could be done as in [3], where all t =2 11 = 2048 possible input permutations serve as test cases - i.e. a PN could score between 0 and 2048 matches. The results of my approach could be compared to the results Koza achieved with GP Predicting the Primeness of a Number The idea I had for this problem is very similar to the problem of learning a boolean function. The goal is to feed the input place(s) of a PN with a natural number n, and the output place of the PN shall make a prediction 9

10 whether n is prime, or not. The tness value of the PN could be determined by the percentage of correct predictions. If the input value is supplied to the PN in binary representation, the problem is identical to that of learning a Boolean function (assuming a nite set of binary input places). However, in this problem it also seems possible to represent the input n by putting n tokens into the one and only input place. 1.7 Project Plan for 8 weeks Ihave already done some work on how I plan to mutate and recombine PNs in a previous report. These ideas are ready to implement. The remaining project plan is: 1. Literature review of articles that deal with the evolution of rather complex data structures such as graphs. A good starting point might be [2], chapter Set up a PN core system that is able to evaluate a training case, i.e. make sure that we can\execute" PNs. 3. Determine the tness of a particular PN by evaluating a training set of cases. 4. Finally, put everything together: implement genetic operators acting on PNs and evolve them. This is the 8-week milestone for this project. Since software development sometimes takes more time than expected, I might implement simplied versions of the genetic operators to save some time. 5. If time permits, analyze the results and compare them to other GP techniques. This 8 week project should result in a conclusion that says either Yes, the idea of evolving Petri nets is a promising approach. The genetic operators designed are reasonable, and the evolved PNs show satisfying performance. No, evolving Petri does not look promising. This is because [... enter reasons...]. (And this is probably why there are no publications in the literature that deal with this idea.) 10

11 1.8 Desired Results and Future Research Plans The desired result would be a conclusion saying that the idea of evolving Petri nets is worth pursuing further. There are many ways to evaluate the tness of a PN, and hopefully the suggested way with input and output places along with a random ring order will do well. Another goal is to list, in an organized way, papers and published articles that deal with both Petri nets and evolutionary computation, such that any ideas that somehow resemble that of evolving Petri nets can be extracted and discussed for future research plans. Further ahead in the future would be plans to evaluate the tness of Petri nets in a time ecient way. How can existing Petri net simulators improve the eciency of my proposed system? How can parallel implementations improve the eciency of my proposed system? If the results of the evolution process are not satisfying, which parameters should be changed to improve the process? Which practical application problems, apart from rather trivial academic problems, have solutions that can be expressed as Petri nets? 2 Petri Net Overview This section gives a brief introduction to Petri nets and mentions some fundamental publications in that eld. The denitions given here are limited to those, which seem necessary to focus on the goal of breeding Petri nets - the evolution-directed search of the space of possible PNs for ones, which, when executed, will exhibit high tness. 2.1 The Origin of Petri Nets Petri Nets are named for Prof.Dr. Carl Adam Petri, University of Hamburg, Germany, who showed in his dissertation in 1962 that there is a need for a theory of asynchronously working machines. (The English translation of his dissertation is [8].) At that time the term \process" had not been established yet in the area of computer science, but Petri's schemata allowed the modelling of parallel and independent events in an illustrative manner. His method turned out to be very useful when designing distributed systems and processes and by 1968 it had become increasingly popular all over the world. As a consequence his formalism was referred to as \Petri Nets". This also sparked the emergence of a new research area within the branch of theoretical computer science known as general net theory, which is concerned with the laws of information ow. 11

12 2.2 Place/Transition Nets Good introductory references to Petri nets are [10] and [5]. The following descriptions use their notation. Denition 2.1 (Net) A triple N =(P; T; F) is called a net i 1. P and T are disjoint sets and 2. F (P T ) [ (T P ) is a binary relation called the ow relation of N: In graph theoretical terms a net is a directed bipartite graph. The following denition characterizes a rather simple class of low level Petri nets. Very often when the term \Petri net" is used, it refers to place/transition nets, which are dened as follows. Denition 2.2 (Place/Transition Net) A 6-tuple N =(P; T; F; K; M 0 ;W) is called aplace/transition net (elements of P are called places, elements of T are called transitions) i 1. (P; F; T) is a nite net, 2. K : P! N [ f1g gives a (possibly unlimited) capacity for each place, 3. W : F! N,f0g attaches a weight to each arc of the net, and 4. M 0 : P! N [ f1g is the initial marking where all the capacities are respected, i.e. M 0 (p) K(p) for all p 2 P: Graphically, places will be represented by circles; transitions will be represented by squares. A marking assigns to each place p 2 P a nonnegative integer or 1, thus describing the state of the place/transition net. If, for example, p is assigned the value k, i.e. p contains k tokens, this is depicted by writing the number k inside p's circle. More formally, a marking M of a place/transition net is dened (in the same way as the initial marking of the net) as M : P! N [ f1g, such that M(p) K(p) for all p 2 P (respect capacities). This only denes the syntactic structure of a place/transition net, but not yet its semantic properties. See gure 1 for a graphical overview of place/transition nets. The following denition helps to dene the semantic properties of place/transition nets. 12

13 Places look like: K=7 The capacity of this place is 7 tokens. Transitions look like: An arrow from p to t with weight 4 indicates that t needs 4 tokens from place p to be able to fire. p 4 t An arrow from t to p with weight 3 indicates that p gets 3 additional tokens after t fires. t 3 p A place that contains 5 tokens looks like: 5 Figure 1: Place/Transition Nets Summary Denition 2.3 Let N =(P; T; F) be a net. For x 2 P [ T x = fyj(y; x) 2 F g is called the preset of x, x = fyj(x; y) 2 F g is called the postset of x: Now the semantics of place/transition nets can be dened. Denition 2.4 Let N =(P; T; F; K; M 0 ;W) be a place/transition net. A transition t 2 T is activated (also called enabled) if 1. for every place p in the preset t the number of tokens at p is greater or equal to the weight W (p; t) of the arrow from p to t and 2. for every place p in the postset t the number of tokens at p plus the weight W (t; p) of the arrow from t to p does not exceed the capacity K(p) of p. Denition 2.5 Let N =(P; T; F; K; M 0 ;W) be a place/transition net. An activated transition t 2 T may re by 13

14 1. decreasing the number of tokens for all p 2t by W (p; t) and 2. increasing the number of tokens for all p 2 t by W (p; t): See [5], p.543, for illustrations of the ring rule. Firing a transition t changes the marking (i.e. the state) of the net from M to M 0 (note that M = M 0 is possible) and is denoted by M =) t M 0. A ring sequence is denoted by t = 1 t M 0 =) 2 M1 =) tn M2 =) M n : If we do not impose restrictions on the class of place/transition nets, then for a given a marking M it is possible that two (or more) transitions t 1 and t 2 are enabled, but ring one of the two disables the other one. This situation is referred to as conict, decision, or choice, see gure 2. 1 Figure 2: A conict between two transitions Petri net structures exhibiting such conicts lead to an inherent nondeterminism within the net, since ring t 1 might lead to a behavior that is dierent from the behavior after ring t 2 instead. The following subclass of place/transition nets does not allow for structures exhibiting conicts. Denition 2.6 (Marked Graph) A marked graph is a place/transition net, where all arc weights are 1's (i.e. W (f) =1for all f 2 F ) and each place p has exactly one input transition and exactly one output transition, i.e., jpj = jp j=1 for all p 2 P: \Marked graphs basically model decision-free (or deterministic) concurrent systems" ([5], p.560). 14

15 2.3 Useful Results from Net Theory The goodness of a PN can be judged to a certain extent by applying algorithms that have been developed in net theory. These algorithms answer questions about behavioral properties of PNs. There are algorithms for the following problems: Given a PN, and an initial marking, is a marking reachable? (a transition cannot re due to capacity con- Are there contacts? straints) Are there conicts? (an enabled transition becomes disabled because another transition res) Is the net bounded (at most k tokens per place) or safe (at most 1 token per place)? Is deadlock possible or is the net alive (requires for each marking, reached from the initial marking, the possibility that every transition might become enabled and might re in the future)? By using those algorithms we can verify whether a given PN fullls certain properties, or not. 3 Existing Literature and Research in this Area A literature search for documents containing any of the following pairs of keywords such as (EA,PN), (EC,PN), or (GA,PN) did not result in any papers using the idea of evolving a population Petri nets. I summarize the content of some of the few references that employ the concepts of EC and PN together now. Wong in [13] uses an approach that is similar to the one suggested here. He (indirectly) evolves fuzzy Petri nets (FPN), a special PN class that allows to deal with imprecise information - but the dierence between FPN and PN is not of major importance. The main dierence is that [13] introduces the following indirection: instead of evolving FPN data structures directly, Wong evolves specications using GP; these specications are then used to generate FPNs from \embryonic FPNs". It is claimed that this approach has also been successfully used to evolve generators for neural networks. The paper does not address how to deal with conicts (and thus with the inherent nondeterminism) in the FPNs. All the example problems are learning problems. 15

16 Reid in [9], p.89, claims that PNs are not \directly conducive to algorithmic manipulation; the genetic operators require that solutions be represented within a population of vectors or strings." He proceeds by coding single transitions as strings of nonnegative integers, representing the input arc weights and the output arc weights of transitions. Then a population of transitions is evolved with a GA. The population as a whole represents a PN. While this paper addresses many of the problems I encountered (how to evaluate the tness of a PN; how to deal with the intrinsic nondeterminism), the fundamental dierence to my approach is what constitutes an individual in the population. I do not see the necessity that individuals have tobe string- or vector-coded, so I suggest that an individual is a PN. In [7] PNs are used to limit the GA search to only legal individuals in a highly constrained scheduling problem. I found it to be an interesting approach to the ever present question of what to do with the many illegal individuals produced by GA, if the solution space is highly constrained. In essence, however, [7] use the GA concepts merely to optimize the dynamic behavior of the PN, which represents the solution to a highly constrained scheduling problem. There is no population of PNs. I compiled a list of around 15 more references that use both the EC concept and the PN concept. After browsing through the title and/or abstract, they do not seem to breed PNs. 4 Design of the Software System In this section the design of a software systems that evolves PNs is described. Figure 3 gives an overview of such a system. (The overview is itself modeled as a PN.) There is a basic GA cycle of evaluation, selection, recombination, and mutation. To prepare a PN for evaluation, it is \run" on a PN simulation system, which res a certain number of transitions until the PN is ready for evaluation. The PN simulator can be an external system (this allows the easy reuse of an existing simulator), in which case a fast interface to the GA system must be developed. Alternatively, the PN simulator could be integrated into the GA system. The PN simulation system takes as an input a PN N = (P [ P in [ P out ;T;F;K;M 0 ;W) with xed input places P in and xed output places P out in its initial state. These places P in and P out remain xed for every viable PN and must not undergo mutation or recombination. The PN simulator then determines the behavior of the given PN by ring a certain number of transitions. (If the structure of an individual PN is to be simulated 16

17 GA System recombine mutate Population of PNs (each in initial state) PN Simulator Software select evaluate Population of PNs (each in final state) Figure 3: Overview of a PN evolving EA system with many dierent initial markings on P in, as required in the examples of subsection 1.6, multiple simulation runs for a single individual in the population are necessary.) The resulting PN state (and also the trajectory of the red transitions if desired) is returned to the GA system. From this information the evaluation procedure can assign a tness value to the PN. 17

18 References [1] Thomas Back, David B. Fogel, and Zbigniew Michalewicz, editors. Evolutionary Computation 1 - Basic Algorithms and Operators. Institute of Physics Publishing, Bristol, UK, [2] Wolfgang Banzhaf, Peter Nordin, Robert E. Keller, and Frank D. Francone. Genetic Programming - An Introduction: On the Automatic Evolution of Computer Programs and its Applications. Morgan Kaufmann Publishers, Inc., San Francisco, CA, [3] John R. Koza. Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT Press, Cambridge, MA, [4] Tom Michael Mitchell. Machine Learning. WCB/McGraw-Hill, Boston, MA, [5] Tadao Murata. Petri nets: Properties, analysis and applications. Proceedings of the IEEE, 77(4):541{580, April [6] Ernst Rudiger Olderog. Nets, Terms and Formulas: Three Views of Concurrent Processes and their Relationship. Cambridge University Press, Cambridge, New York, [7] M.R. O'Neill, V.H. Allan, N. Flann, and H. Chen. Software pipelining via stochastic search algorithms. Technical report, Department of Computer Science, Utah State University, Cambridge, MA, [8] Carl Adam Petri. Communication with automata. Technical Report RADC TR vol-1-suppl 1, Applied Data Research, Princeton, New Jersey, [9] D. J. Reid. Constructing petri net models using genetic search. Mathematical and Computer Modelling, 27(8):85{103, [10] Wolfgang Reisig. Petri Nets: an Introduction. Springer-Verlag, Berlin, Germany, [11] Wolfgang Reisig. Elements of Distributed Algorithms: Modeling and Analysis with Petri Nets. Springer-Verlag, Berlin, Germany, [12] Harald Storrle. An evaluation of high-end tools for petri nets. Technical Report 9802, Institute for Computer Science, Ludwig-Maximilians- University Munich, Munich, Germany,

19 [13] Man Leung Wong. A exible knowledge discovery system using genetic programming and logic grammars. Decision Support Systems, 31:405{ 428, [14] W. H. R. Yeung and P. R. Moore. Genetic algorithms and exible process planning in the design of fault tolerant cell control for exible assembly systems. International Journal of Computer Integrated Manufacturing, 13(2):157{168,