Table Of ContentLecture Notes in Networks and Systems 45
Ludwik Czaja
Cause-
Effect
Structures
An Algebra of Nets with Examples of
Applications
Lecture Notes in Networks and Systems
Volume 45
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Ludwik Czaja
Cause-Effect Structures
An Algebra of Nets with Examples
of Applications
123
Ludwik Czaja
Vistula University
Warsaw,Poland
Institute of Informatics
University of Warsaw
Warsaw,Poland
ISSN 2367-3370 ISSN 2367-3389 (electronic)
Lecture Notesin Networks andSystems
ISBN978-3-030-20460-0 ISBN978-3-030-20461-7 (eBook)
https://doi.org/10.1007/978-3-030-20461-7
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To Nina and Mateusz
Preface
Cause-effect (c-e) structures are objects of a formal system devised for modeling,
testing, and verifying properties of tasks where parallel execution of actions is the
most characteristic feature. This is an algebraic system called a quasi-semiring,
“quasi” because it differs from the standard semiring by somewhat restricted dis-
tributivity law, that is a(cid:1)ðbþcÞ¼a(cid:1)bþa(cid:1)c fulfilled provided that b6¼h if and
only if c6¼h, where h is a neutral element for both operations: “multiplication”
(denoting simultaneity) and “addition” (denoting nondeterministic choice). Due to
theconditionaldistributivity,this algebraneither reduces totrivial,that is,a single
neutral element, nor makes both operations coincide, but describes objects
behaviorally equivalent to Petri nets, although differing from them in many
respects. The justification of this is in Chaps. 2 and 9. In the graphic presentation,
thec-estructures,likePetrinets,arealsonets,thatis,akindofgraphs,butwithone
type of nodes, thus not bipartite, as in case of Petri nets.
The main motivation to develop this algebra was to integrate structuring and
transformation rules it provides, with graphical and animated presentation of
modeled real-world systems—by a special computerized tool. The algebra is a
formal basisforcombiningc-estructuresofeasytounderstandbehaviorintolarge
system models, some behavioral properties of which can be derived—under
well-determined conditions—from behavior of their smaller parts. Such feature,
knownintheareaofprogramminglanguagesascompositionality,1isacounterpart
ofextensionalityinformallogic.Also,absenceoftransitionsandadjoinedtothem
edgesprovidesmoremonitor(screen)spaceforgraphicpresentationthanincaseof
Petri nets. The appearance on screen is scalable and in many cases uses the
so-called“buslayout”(Chaps.1and2),whichallowstodisplaysomec-estructures
of arbitrary size and to avoid excessive entanglement of edges. Apart from the
standard issues investigated in the Petri net area, like reachability, liveness,
1In some programming languages, breaking the compositionality principle occurs, e.g., in case
ofthe“sideeffects”ofprocedures.Moregeneral,in[5],theoppositeofcompositionalityiscalled
the“emergenteffects”.Itisknownthatinaccuratemathematicalmodelingof“reallife”systems,
thefullcompositionality,desirableforformalanalysisofamodel,isoftendifficulttoobtain.
vii
viii Preface
boundedness, etc. (Chap. 8), such problems like lattice properties (Chap. 6) of c-e
structures, their decomposition (Chap. 7) or generation of process languages with
analysis and synthesis procedures (Chaps. 10–12), are considered.
Cause-effectstructuresdescribedinthebookareanextensionoftheelementary
c-e structures ([2–4]), by the following features: weighted edges, multi-valued
nodes having capacities (counterpart of place/transition Petri nets), inhibitors, and
several models of time (Chaps. 3 and 4). The extensions are accomplished by
modifyingthenotionofstateandsemantics,butleavingunchangedstructureofthe
quasi-semiringexpressions.Therefore,unlikeincaseofPetrinets,allthesefeatures
are included in various definitions of state and semantics, not in the structural
(algebraic) definition of c-e structures. The features are illustrated by respective
examples. An information system called rough c-e structures is exemplified in
Chap. 5.
Chapter 13 contains a number of examples of c-e structures modeling various
tasks taken from the “real world”. Often their graphic shape resembles appearance
of their real-world prototype. In such cases, in the graphic design of a model, the
mentioned “bus layout” and symmetries encountered in the modeled objects are
applied. So, respective c-e structures are drawn as interconnections of similar
(isomorphic) substructures. Adequate examples exhibiting this feature are also in
Chaps. 1 through 5.
All the aforesaid examples have been run and tested by means of a computer
system comprising graphical editor and simulator of c-e structures [1]. Simulation
shows motion of tokens during its run, but sometimes delivers also data about
performance of a task being modeled. For instance, experiments made by simula-
tionconfirmexpectations,howthelikelihoodoftotalblockadeofanervoussystem
decreases with increasing its size, i.e., number of neurons, or provide data about
dependence between a traffic smoothness of a crossroad and street lights change
frequency, given a traffic intensity. A sample of a screen lookout of the editor and
simulator is in Chap. 1.
Anumberofpeoplecontributedtothedevelopmentofelementaryc-estructures
inthelate1980sandinthenextdecade,whenIhavehadnumerousdiscussionson
the issue, resulting in publications listed at the end of successive chapters of this
book and in Related Literature not Referenced in the Chapters. They were:
L. Holenderski, A. Szalas, J. Deminet, K. Grygiel, U. Abraham, D. Wikarski,
H. W. Pohl, M. Raczunas, A. Maggiolo-Schettini, Nguyen Duc Ha, G. Ciobanu,
A. P. Ustimenko, M. Kudlek, who published their contributions, but also others,
mainly authors of M.Sc. or Ph.D. dissertations. Also, the students attending my
course on selected problems in concurrency at the Institute of Informatics,
UniversityofWarsaw,haveoftenbeenhelpfulinrefiningtheideaanddetails.The
research on this concept was greatly stimulated by my visits to the Informatik
Berichte Humboldt Universitat zu Berlin, Computer Science Department of
Ben-Gurion University in Beer-Sheva, Israel, Dipartimento di Informatica
Preface ix
Universita di Pisa, Institute of Informatics Russian Academy of Sciences, Siberian
Branch in Novosibirsk and Fachbereich Informatik Universitat Hamburg. The
extensionsofelementaryc-estructurespresentedinthisbookbeingitsmainsubject
based on a new concept of state and semantics, arose and have been elaborated in
severalrecent months.
Warsaw, Poland Ludwik Czaja
References
1. Chmielewski RE (2003) Symulacja struktur przyczynowo-skutkowych z wykorzystaniem
platformy .NET (Simulation of cause-effect structures using .NET platform), MSc thesis,
InstituteofInformatics,WarsawUniversity
2. CzajaL(1988)Cause-effectstructures,North-Holland,InformationProcessingLetters,vol26,
pp313–319
3. CzajaL(1993)ACalculusofNets,KibernetikaISistemnyjAnalizNo.2March–April1993
(Russianedition).Englishedition:Springer,CyberneticsandSystemAnalysis,vol29,No.2,
pp185–193
4. CzajaL(2002)Elementarycause-effectstructures,WarsawUniversity
5. HedgesJ(2016)Howtomakegametheorycompositional.In:Proceedingsoflogicforsocial
behaviourworkshop,Zurich
Acknowledgements
Many people contributed to elaboration of the primary version of cause-effect
structures, called “elementary” and dating back to the late 1980s. Some of the
contributorsarementionedinthePreface.Thepresentversion,basedonaxiomsof
an algebraic system close to the semiring and encompassing several extensions
oftheelementarycause-effectstructures,hasbeenelaboratedinrecentmonths.Iam
indebtedtoKrzysztofCzajaforreadingthetextandcallingattentiontosomeofits
deficiencies. My special thanks go to Professor Andrzej Skowron for encourage-
menttothebookpreparation,andespeciallyforconvertingitsroughappearance—
the outcome of my editing—into the style appropriate for publication in
Springer-Verlag. I also wish to express my deep thanks to Professor Janusz
Kacprzyk, the editor of the “Lecture Notes in Networks and Systems” series, for
making possible of its publication.
Ludwik Czaja
xi
