Un demi-siècle des RdPs
z Un demi-siècle du travail définissant un point de repère dans la théorie des systèmes dynamiques à événements discrets (SDEDs) mérite d'être souligné. Manuel Silva z On ne prétend que faire une modeste commémoration de sa vie et vision, et présenter des traces sur la Instituto de Investigación en discipline créé, sachant que la science et la technique Ingeniería de Aragón (I3A), sont des constructions sociales. Universidad de Zaragoza z Manuel Silva: “Half a century after Carl Adam Petri’s Ph.D. thesis: A perspective on the field”, Annual Reviews in Control 37:2, 191-219, 2013. Online publication date: 1-Dec-2013.
Carl Adam Petri: Contents of this talk Half a century after his PhD When in Computer Science the paradigm was local 1. Introduction computations on mathematically intricate problems… 2. About Carl Adam Petri … Carl Adam Petri look for a Systems Theory, far 3. Petri Nets: On Basic Properties beyond what was the problematic in Computer and ways of approach Science, applicable in a broad landscape of research fields: 4. Petri Nets: A modeling paradigm z Automatic Control, Operations Research, Informatics… 5. On five interactions over twenty z Automation of manufacturing, transport and logistic systems, communications protocols, software and five years hardware engineering, (bio)chemistry, epidemiology, law/rights, demography…
Carl Adam Petri: Science and Technology are Half a century after his PhD social constructions
z Nevertheless, in their development some people contribute in Interested in the description of some real-life outstanding ways. situations, as mathematician he essentially z The recognition of their achievements is sometimes partially worked at conceptual level, “opening done by giving his/her name to some measurement unit, windows” (i.e., providing foundations for new universal constant, algorithm, etc. ways of thinking), what may be viewed as a z International System of Units (newton, coulomb, ampere, speciality less frequently exercised with volt, farad…) success than “theorem prover”. z Coriolis, Young, Avogadro, Kirchoff, Minkowsky-Farkas, Donsker, Floyd, Dijikstra, Wiener, Kalman, BCMP… At the personal level he always was a warm- z Naming an entire subfield is much less frequent: Markov hearted person Chains, Hilbert Spaces, Galois Fields, Petri Nets… Continuous (“natural”) vs Continuous (“natural”) vs Discrete (“artificial”) Discrete (“artificial”) z Continuous Systems: 300 years of optimal control: from the brachystochrone to the maximum principle z In Future directions in control theory: a (Sussmann and Willems,1997). mathematical perspective (Fleming, 1988): z From logic/discrete automatisms to DEDS (Ho et al., there exist no formalisms for DEDS 1980) mathematically as compact or computationally z Discrete Event System Specification (DEVS; as tractable as the differential equations are for simulation: Zeigler, 1976) continuous systems. [after: SIMSCRIPT, SIMULA, GPSS…] z “Natural” selection (Darwinism…), but NOT z Like Le Bourgeois Gentilhomme of Molière, users or monism… developers on topics related to automata, Markov chains, z But, ideas organized in modeling paradigms queueing networks, Petri nets, process algebras, max/plus algebra, etc., realized that we were speaking PNs: an option (more than a question of taste, but...) not in prose, but “in DEDS”. -
Carl Adam Petri Contents of this talk (Leipzig,1926-Bonn, 2010)
z His father, professor of mathematics, enjoys friendly relationships 1. Introduction with mathematicians as David Hilbert and Hermann Minkowski. z Obliged to join the German army (1944), he becomes prisoner in 2. About Carl Adam Petri England (till 1946), where later he teaches Latin and chemistry. z Petri returns to Germany in 1949, studying mathematics at the 3. Petri Nets: On Basic Properties Technical University of Hannover (till 1956); z From 1959 until 1962 he works at the University of Bonn; and ways of approach z In 1962 he gets the PhD degree from the Technical 4. Petri Nets: A modeling paradigm University of Darmstadt. z Until 1968 Petri is working at the University of Bonn, becoming head 5. On five interactions over twenty of the computer installations. z In 1968 was founded the GMD. He become later one of the directors five years of the Institute for Methodological Foundations. z He retires from GMD in 1991, but in 1988 was named Honorary Professor at University of Hamburg.
Petri nets: When? Were? How? Robin Milner (Turing Award, 1991)
z In the 1970s, Robin Milner focusses his view of z Against what it is frequently quoted, in his PhD dissertation the graphical notation on PNs does not appear!!! concurrent systems on the “interactions of smaller components”, being one of the more z The well-known bipartite graphs with conditions/places and events/transitions will come some three years later. celebrated steps in the definition of process z But he claims to invent algebras. it in a playful mood in August 1939, and practiced it z In other words, Milner pays central attention to the intensively for the purpose of memorizing chemical process of construction of the model: processes, using circles for substances and squares [Petri] pioneered the scientific modeling of for reactions. discrete concurrent systems. Petri’s work has a z A. Holt (MIT, Project MAC) give his name to the bipartite nets secure place at the root of concurrency z Replacing temporal order by causal order, he looked theory…What I always wanted to advance, to concurrency as mutually causal independent occurrences (synchronization as rendez-vous) complement Petri net theory, is the synthetic or compositional view of systems which is familiar z Causal orders and preservation laws were inspired to Petri by the laws of Physics from programming. (1993) PNs & international meeetings Among other important recognitions to Carl Adam Petri z Since 1980 it exists an annual conference devoted to PNs, the z The Konrad-Zuse-Medal (1993); Int. Conf. on Application and Theory of Petri Nets and z The Society for Design and Process Science Concurrency (ICATPN; in fact, from 1980 until 1988 it was established the Carl A. Petri Distinguished Technical known as International Workshop, IWATPN). Achievement Award (1997); z From 1985 till 2003, ten editions of the IEEE/ACM Int. Symp. z The Werner-von-Siemens-Ring (1997), one of the on Petri Nets and Performance Models (PNPM) took place. highest ranked award for technical sciences in Germany, delivered every three years; This meeting merged in 2004 with others, as that on Process Algebra and Performance Models (PAPM), giving light to the z The Doctorate Honoris Causa by the Universidad de Zaragoza, in 1999. annual Int. Conf. on Quantitative Evaluation of Systems (QEST). z The Order of the Dutch Lion, as Commander (2003); z IEEE Computer Pioneer (2008): z But what is more impressive is the number of meetings in which For establishing Petri net theory in 1962, which not only was cited by PNs is a topic explicitly mentioned, frequently devoting special hundreds of thousands of scientific publications but also sessions to them. For example, in “The Petri Nets: Meetings significantly advanced the fields of parallel and distributed computing. and Events” we can count more that 150 during 2005-2011, so Most of the recognitions to Carl Adam Petri, more than twenty per year. (What is truly a lower bound!!!) a mathematician, came from engineering.
Petri Nets are operational formalisms Contents of this talk (how the system works, vs what it does)
Among the qualities that they enjoy are: 1. Introduction z Minimality in the number of primitives 2. About Carl Adam Petri (trade-off between the engineering and scientific perspectives). 3. Petri Nets: On Basic Properties z Locality and structuration (parallelism, and ways of approach synchronization, conflicts, etc.) >>> z Cooperation & Competition relationships. 4. Petri Nets: A modeling paradigm z Top-down & Bottom-up model building. 5. On five interactions over twenty z Non-determinism (humble position---time five years unpredictability). z Temporal realism and occurrence nets.
Minimality-understanding (theory) vs rich ontology-use (engineering) … ways of approach to PNs
if all roads lead to Rome … more than “probably”, Rome was/is important! … to “the town” of Petri Nets (with several nuances) we may arrive using different roads …: z An axiomatic view (the C.A. Petri), +++ z Continuous System Theory, from state transition (…like in VAS) z Theory of Regions: a kind of field-coding in labelled graphs (synthesis problem / “mon cher Philippe Darondeau”…) z Non-monotonic logic: Linear logic, Girard / R. Valette Salient features: P/T systems constitute a From 1950’s • Bipartite --- dual objects ! “frontier formalism” to 1960’s • Locality --- compositionality ! • Timed vs untimed ? z P/T systems are able to model infinite state • Discrete vs fluid views ? (i.e., unbounded) systems. z Constitute a “frontier formalism”: Reservoirs Activities z Are very expressive and most of classical properties are decidable, Queues/ Stations/ 1957… QNs z But “small” extensions transform it into a formalism Clients Servers able to simulate Turing machines, thus Deposits/ Valves/ excellent expressivity, but plenty of undecidabilities. 1961… FDs z Among the evoked extensions, just the addition of: Levels Flows z inhibitor (or zero test) arcs, Places/ Transitions/ z priority levels to the firing of transitions; or 1962… PNs Conditions Events z certain time-interpretations.
One “children” of ordinary PNs: The Grafcet z GRaphe de l’AFCET (Association Française de Cybernétique Économique et Technique) z GRAphe (Fonctionnel) de Commande Étape-Transition z “Like” an ordinary P/T net system in which a flip- flop approach is used for implementing places, even if they are not 1-bounded; also problems with unresolved conflicts! (be careful because this destroy the tokens conservation laws!) Sequential Function Chart (SFC) Norms IEC 848 (1988) + IEC 61131-3 (1993)
24 as the Chevaliers de la Table Ronde ISO norm (mainly) in the framework Maturity & engineering disciplines of software engineering. The necessity of formal methods is beyond question in mature engineering disciplines. z International Standard Organization (ISO) Maturity of an engineering discipline requires: z International Electrotechnical Commission z Formalisms (“secure reasoning”) (IEC) z Methodological support (easiness of use, reusability…) z The ISO/IEC 15909, publicized into two parts: z Existence of expertise (e.g., in terms of catalogs of models (1) Systems and software engineering - and methods for specific domains) High-level Petri nets – Part 1: Concepts, z Analysis techniques z Synthesis procedures definitions and graphical notation, 2004; z (Software) Tools (2) Systems and software engineering - z Standards & international norms (ISO, IEC…) High-level Petri nets – Part 2: Transfer z Implementation with different technologies (+ fault- format, 2011 tolerance, etc.)
Contents of this talk
1. Introduction 2. About Carl Adam Petri 3. Petri Nets: On Basic Properties and ways of approach 4. Petri Nets: A modeling paradigm 5. On five interactions over twenty five years … même si tous du petit village! (société ...) -
From a “formalism” vs PNs constitute a modelling paradigm a “modeling paradigm”
z Long life-cycle of many systems >>> family of related formalisms rather than a collection of • Coherence “unrelated” formalisms. • Economy z A paradigm is (Kuhn, 1962): [...] the total pattern of perceiving, conceptualizing, • Synergy acting, validating, and valuing associated with a particular image of reality that prevails in a science or a branch of science. + relaxions: z A modeling paradigm is a conceptual framework •Hybrid that allows to obtain formalisms from some common • Continuous concepts and principles. z But multidisciplinarity!!!… also multiformalisms…
An elementary example: Basic The basic manufacturing manufacturing cell & life-cycle cell & life-cycle
z Performance vs buffer capacity: z Two processing machines (M1 and M2) monotonicity (?) z An assembly machine (M3) z Frequential view: z A robot (for transportation purposes, R) filtering effect due to the buffer z A buffer (inventory, deposit,...) to uncouple z Gantt diagrams: two scheduling strategies upstream and downstream of M1 z Marking diagram: control • Cooperation relationships (3 machines) z Error detection & correction: • Machine-buffer-machine: decoupling z Duplication vs observation • Competition (mutex-robot) z Observable view Stochastically Timed model
Performance vs Capacity. Frequential view: Monotonicty (not always) “filtering effect” due to the buffer
Gantt diagrams: two Performance optimization: Scheduling // Control scheduling strategies Deterministically Timed model
¾ Pb.: how to order the firing of a,b,c,d? z Cyclic: 3!=6 cases BUT: db ĺ 2! = 2 cases! Performance optimization: Performance optimization: Scheduling-Control Scheduling-Control ¾ Pb.: how to ¾ Pb.: how to order the time order the firing of a,b,c,d? of a,b,c,d? z repetitive: 3!=6 z repetitive: 3!=6 cases cases With: ac ĺ 1! = 9 db ĺ 2! = 1 case !!! 2 cases ! R becomes implicit (so 9 ac ĺ 1! = places {db,ac,cd,ba} is 1 case !!! an optimal schedule- control): remove it, a marked graph appears!!!
Performance optimization: Autonomous Scheduling-Control Distributed State Diagram: Marking
Controlled Diagram
Duplication vs Observation Checking points: Observable view
Possibility: to increase the
Hamming distance
+ Fault detection & diagnostic… (consistent markings…) multiformalisms… The important fact: the existence of multi-paradigmatic bridges inter-paradigmatic transformations z even if we believe that PNs constitute an adequate (bridges) conceptual framework or paradigm for the operational description of DEDS: “we do not believe that it is always possible to select a single formalism, or family of them, to deal in a reasonable way with every aspect of every DEDS. The complexity and variety of systems suggest instead the interest of having multi-paradigm environments where the existence of sound and efficient bridges between different paradigms becomes a major issue” (Silva and Teruel, 1996). z In a different, but close context: “I reject the idea that there can be a unique conceptual model, or one preferred formalism, for all aspects of something as large as concurrent computation.” (Milner, 1993)
5. On five interactions Contents of this talk over twenty five years
1. Introduction In this part I will consider five points of contact with Carl Adam, over more than 25 years. 2. About Carl Adam Petri The first recall our fully disjoint set of 3. Petri Nets: On Basic Properties interests at the given time (1976) and ways of approach Synchronic relations and fluidization are 4. Petri Nets: A modeling paradigm two topics I sporadically discuss with him, 5. On five interactions over twenty Carl Adam showing big interest. Even five years sometimes we arrive to agree on “doing something together”, but…
Just five points… over more I. I was a PhD student in Grenoble, than a quarter of century LAG (+ENSIMAG), 1975-1978
1. A student in Grenoble, 1976 z The first time I meet C.A. Petri was at the ENSIMAG (Mme. Saucier, J. Sifakis…), 1976 2. At the Bocconi University & Oxford, 1986 z My interest was important because months before I 3. At he European Workshop on Petri Nets, abruptly change the topic of my PhD from: Zaragoza, 1987 z interconnected automata & modular hardware synthesis (CUSAs / “hazards under control”) to 4. Carl Adam Petri, Doctor Honoris Causa z Petri nets & programmable logic controllers (PLCs) by the University of Zaragoza, 1999 z The topic of his talk? --Synchronic distance related…? 5. The International Conference on Petri z But at that period I deal with problems of quite different Nets and Related Models, Miami, 2005 nature (software implementation issues, performan- ce evaluation…/ he does not like explicit timing…) Just five points… over more II. At the Bocconi University than a quarter of century & EWPN-Oxford, 1986
z The context (Bocconi Univ.: A. Pagnoni): 1. A student in Grenoble, 1976 (¿1977?) z I was interested on transitions firing dependences in 2. At the Bocconi University & Oxford, 1986 “generalized” (weigthed) PNs (P/T) z Petri (1975/76): Synchronic Distance, S-completion… 3. The European Workshop on Petri Nets, Zaragoza, 1987 z Other previous works by: J. Sifakis, U. Golz, I. Suzuki, T. Kasami, T. Murata… 4. Carl Adam Petri, Doctor Honoris Causa z My cooperation with T. Murata (USA-Spain Joint Com. by the University of Zaragoza, 1999 for Scientific & Technological Research, 1985/88) 5. The International Conference on Petri z The fact: existence on “Non-linear Synch. Relations” Nets and Related Models, Miami, 2005 z I did show Carl Adam one (the) “smallest” ordinary, pure, live & bounded net system…
II. At the Bocconi University & II. At the Bocconi University & EWPN-Oxford…: Synchrony Theory EWPN-Oxford…: Synchrony Theory X1 = (1011) X = (2 1 1) X2 = (2101) Linearly based synchronic properties: Synchronic lead & Distance A new way of measuring: SD is a metric Basic finite delay synchronic properties: B-deviation & B-fairness
Quantitative dependences (value) & synchronic relations (boundedness)
Synchronic relations: behavioural (given m0) Synchrony Theory: a branch of General Net Theory & structural (for any m0) devoted to the study of transitions firing dependences
II. At the Bocconi University & Just five points… over more EWPN-Oxford…: Synchrony Theory than a quarter of century
1. z SD-relation Ź BF-relation, the reverse does not hold A student in Grenoble, 1976 (¿1977?) z Structural BF & SD relations (SBF; SSD): 2. At the Bocconi University & Oxford, 1986 are equivalence relations and 3. The European Workshop on Petri Nets, can be computed in polynomial time. Zaragoza, 1987 z In bounded systems, realizability of min-T-semiflows Ź Ź Behavioural = Structural Synchronic relations 4. Carl Adam Petri, Doctor Honoris Causa by the University of Zaragoza, 1999 DUALITY: 5. The International Conference on Petri T-views (T-semiflows) vs Nets and Related Models, Miami, 2005 P-views (implicit places, eventually on partial nets) III. The European Workshop on III. The European Workshop Petri Nets, Zaragoza, 1987 on Petri Nets, Zaragoza, 1987
z “On the computation of structural synchronic invariants in z Continuous Petri Nets P/T nets”, co-authored with J.M. Colom. z Explicitly (net level): z The idea: systematic use of LPP, weak & strong duality R. David & H. Alla theorems, unboudness theorems & alternatives theorems. z Implicitly (relaxation z Polynomial time computation of bounds ( “like” fluid at the state equation relaxation) of the synchronic values, and of the structural level): Systematic synchronic relations (lead, distance, deviation & B-fairness). use of LPP z Precedents on the use of: z LPP: H.J. Genrich & K. Lautenbach, for marked z A topic received with graphs (but exact values, because the incidence matrix reluctances in the is unimodular) EWPN / ICATPN (like (A small technicality: z Farkas’ lemma / alternatives theorems: G. Memmi & J. stochastic PNs were • consistency & Sifakis sometime before) • no-empty siphon at the initial marking)
III. The European Workshop on Petri Nets, Zaragoza, 1987
z Concurrency & Nets (60th birthday of C. A. Petri), K. Voss, H.J. Genrich & G. Rozenberg, eds. z The book was offered by the PN community to Carl Adam in Zaragoza, in a kind of “birthday party”, at the gala diner of the EWPNs, Linear Systems only June 1987. describe the convex hull, z “Towards a synchrony not the behaviors theory for P/T nets” (a first global view of the topic)
III. EWATPNs, Zaragoza, 1987
EWATPN, Zaragoza 1987 Just five points… over more IV. C.A. Petri, Doctor Honoris Causa than a quarter of century by the University of Zaragoza, 1999
z Foundation of the University of Zaragoza: 1. A student in Grenoble, 1976 (¿1977?) from «Universitas magistrorum» 2. At the Bocconi University & Oxford, 1986 to «Universidad general de todas las ciencias» z Privilege of the Emperor & King Charles V, 1542 3. The European Workshop on Petri Nets, z Papal Bull of Julius III, 1554. Zaragoza, 1987 z … but “no funding”, until some 40 years later…: Pedro 4. Carl Adam Petri, Doctor Honoris Causa Cerbuna, 1582. by the University of Zaragoza, 1999 z Engineering (E.T.S. Ingenieros Industriales: Mechanical & Electrical) & Economics : 1974 5. The International Conference on Petri z As dean of Engineering, 1989: I obtain the transformation of Nets and Related Models, Miami, 2005 the ETSII into Centro Politécnico Superior (+Telecommunication, + Informatics, + Chemical, …)
IV. C.A. Petri, Doctor Honoris Causa IV. C.A. Petri, Doctor Honoris Causa by the University of Zaragoza, 1999 by the University of Zaragoza, 1999
z The program was conceptually based on three z The opportunity: basic pillars of technology, and three outstanding z 25 years of the foundation of the Engineering personalities: School: 1974-1999 z Materials: Prof. Steve Tsai, pioneer in Composite Materials, from Stanford University. z 10 years of the transformation in Polytechnic z Energy: Prof. Amable Liñán, pioneer in the fluid Faculty (“High School”): 1989-1999 - mechanics perspective for Combustion z The dean of Engineering: prof. Javier Martínez, Engineering, from the Polytechnic University of the first PhD on Petri Nets at the University of Madrid. Zaragoza z Information: Prof. Carl Adam Petri, as representative of System Theory & Computer z The first three HC on engineering at Zaragoza Science, Univ. of Hamburg (ex-GMD, Bonn).
The day before (14-IV-99): an The entourage: towards the Paraninfo international seminar in his honor
z G. Balbo (Universitá di Torino): On PNs and Performance Evaluation: The GSPN case z J. Billington (University of South-Australia): On the ISO/IEC Petri Net standard (15909) z D. de Frutos (Universidad de Madrid): Decidable properties in Timed Petri nets z M. Koutny (University of Newcastle upon Tyne): Combining Petri Nets and Process Algebras z M. Silva (Universidad de Zaragoza): On continuous Petri Nets Laude UZAR, Paraninfo: the 15 of April at 12h candidatorum
Accipe pileum…: the academic cap The speech
Just five points… over more V. The Int. Conf. on Petri Nets than a quarter of century and Related Models, Miami, 2004
1. A student in Grenoble, 1976 (¿1977?) z Context for fluid models: z System Theory 2. At the Bocconi University & Oxford, 1986 z Automatic Control Theory 3. The European Workshop on Petri Nets, (observability, controlla- Zaragoza, 1987 bility…) z Operations Research 4. Carl Adam Petri, Doctor Honoris Causa z Scalability of models with by the University of Zaragoza, 1999 “large populations” (QNs, Compartmental Systems, 5. The International Conference on Petri SFGs, FDs…) Nets and Related Models, Miami, 2005 z A “last” chapter in the PhD of Laura Recalde, 1998 V. The Int. Conf. on Petri Nets Systems are NOT…: • we always manage models: “views” of systems and Related Models, Miami, 2005 • are we able to move form one to another level? (i.e., when are we able? / how? / …)
z Continuisation of Timed • Discrete: {1,2,3} Petri Nets: From • Continuous? Performance Evaluation • Discrete? (molecules) to Observation and •??? Control (inv. lecture) z We (re)comment on “fluid views” z Molecules vs mols: Predator/prey: Volterra-Lotka equations… z Fractions of moles z He found “not surpri- Manufacturing engineers, sometimes seems to be hydraulic engineers sing” my (first) degree on Chemical- engineering -
On Unfluidisable Models On fluid / continuous PNs
z Deadlock-freeness: discrete vs continuous (?) if there is no dead transition (no siphon unmarked):
p z all the potential repetitive sequences are realizable 3 1 3 p 2 t 2 t t 1 t 1 2 2 1 2 z Behavioural and structural synchronic relations 2 p 2 p 2 2 coincide (in particular bounded= structurally bounded) Dealock as discrete, 3 DF as discrete, but but z Therefore synchronic relations can be computed in DF as continuous Deadlock as continuous polynomial time!!
ZENO For hybrid and discrete net systems, the
z Liveness: yes or not?: lim-reachability (!) above is not necessarily true (usually false)
timed models: example: infinite server semantics-minimum infinite server semantics-minimum Infinite Servers Semantics (variable speed) piecewise m>p@ ½ p4 t4 f (W )>@t Ȝ[t]enab(t,m) Ȝ[t]min® ¾ linear system px t Pre p,t ¯ >@¿ p3 t3
p6
t6 p2 t2 t5 2 Firing rate: proportional to the input “level” p7 4
Analogous to the discrete “markovian” case f (W )>@t1 O >@t1 m(W )>p1 @ 2 p1 t1 p5 2 f (W )>@t2 O >@t2 m(W ) >p2 @ 3 2 x f (W )>@t3 O >@t3 m(W ) >p3 @ °m(W ) Cf (W ) A set of switching ° f (W )>@t4 O >@t4 m(W ) >p4 @ ® differential linear systems m[ p](W ) ½ f (W )>@t5 O >@t5 min^`m(W ) >p5 @4,m(W )>@p6 °f (W )>@t O >@t min x ® ¾ f (W )>@t O >@t m(W ) >p @2 ¯° p t ¯Pre[ p,t]¿ 6 6 7 one of the counterintuitive properties in timed (infinite servers semantics) models R1 R2 13 f (W )>@t O >@t m(W )>p @ 4 m[p5]/4m[p5]/4m[p1] 5 5 5 m[p6]m[p2] z 2.5 m[p6]m[p3] No performance monotonicities with 0.8 m[p4] f (W ) t O t m(W ) p m[p5] respect to firing speeds > 5 @>@5 > 6 @ 2 m[p6] 0.6 m[p7] 1.5 z Discontinuities, bifurcations
0.4 6 1 5 0.2 [100, 0 , 110] 0.5 4 [50, 0, 55] 0 00 2 4 6 0 1 2 3 4 5 6 3
[10, 0, 11] Throughput 2 configurations & regions 1
0 0 0.5 1 1.5 2 2.5 3 3.5 4 O[t1] infinite number of switchings
25 markings 40 decolouring and net m1 35 m2 m3 m4 30 m5 interpretation-product m6 20 25 m7 m8
20 20 15 z Population dynamics: basic predator/prey 10 5 t 25 model 1 0 t 0 2 4 6 8 10 12 14 16 18 20 2 Er
12
+ bounds on decolouring and net rabbits populations 40 interpretation-product 35
k ~r20 ~f k=57 k=47 t1 30 k=37 t2 k=33 Trajectories obtained t1 k=30 2 t2 k=25 with k=20 20 20 f 2 25 k=10 k=15 r D O[t1]=0.75, O[t2]= 20 20 f r O[t2]=20, D=2, 20 t3 D m [r]= m [f]=20, t o o 3 D m [~f]=40, m [~r]=k 15 o o 0 5 10 15 20 25 30 35 40 foxes
P/T models of the predator/prey system, with Dr=0, Df=D, Er=2, Ef=0, |3|=|'|=20. They can be seen either as discrete or as continuous PN. Discrete Continuous x r O r O r f synchronisations (Stochastic) (Deterministic) ° 1 3 ® x (Un)bounded Bounded ° f O (D 1) r f O f leads to products ¯° 3 2 Non-live Live Zaragoza, Part of the family… Thursday, 15 of April 1999 Concluding the selected points … z As a difference with other experiences, usually my conversations with Carl Adam Petri were not really long, I try to keep on very concrete things. z We should remember Carl Adam as: z the founder of a field, that influence others, that combine conceptual elegance Australia & pragmatism; Canada France z a close & warm-hearted person. Germany Italy United Kingdom …and to conclude with an image-souvenir… …& Spain
Zaragoza, Thursday 15 of April 1999