Table Of ContentBernd Steinbach
Christian Posthoff
Logic Functions
and Equations
Fundamentals and Applications using
the XBOOLE-Monitor
Third Edition
Logic Functions and Equations
Bernd Steinbach • Christian Posthoff
Logic Functions and
Equations
Fundamentals and Applications
using the XBOOLE-Monitor
Third Edition
BerndSteinbach ChristianPosthoff
ComputerScience ComputingandInformationTechnology
TUBergakademieFreiberg(retired) UniversityoftheWestIndies(retired)
Chemnitz,Sachsen,Germany Chemnitz,Sachsen,Germany
ISBN978-3-030-88944-9 ISBN978-3-030-88945-6 (eBook)
https://doi.org/10.1007/978-3-030-88945-6
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Foreword
Logic functions and equations are foundational to many areas of Math-
ematics, Computer Science, and Computer Engineering. A thorough un-
derstanding of the fundamentals and how they are applied is critical to
solving problems and developing applications in Information Technology.
This textbook provides students and practitioners with a well-structured
coverageofthetheoryandapplicationoflogicfunctionsandequations.
PartIintroducesthefundamentalsunderpinningthepracticalapplication
of the programming system XBOOLE and the XBOOLE-monitor XBM 2,
whichisapowerfulinteractivetooldevelopedandmadefreelyavailableby
the authors. Provision of this software will be of tremendous assistance to
readers in exploring the concepts covered in this book and in undertaking
theirsubsequentworkinthearea.
The conceptual background of XBOOLE is of importance in its own
rightandknowledgeofthatbackgroundwillprovidethereaderwithavery
practical understanding of an effective approach to computation involving
logic functions and equations. For this book, XBOOLE and the XBOOLE-
monitor XBM 2 are of particular importance, as they are used in solving
exercises emphasizing the ideas introduced in each chapter. Indeed, it is a
majorstrengthofthistextbookthattheauthorshaveprovidedexerciseswith
sample solutions and challenging supplementary exercises throughout the
book.
Part II provides a comprehensive progressive introduction of the funda-
mentals of logic functions and equations. The order of topics is very well
planned.
Basicalgebraicstructuresareintroduced,leadingtothedetailedintroduc-
tion of logic functions and logic equations. The coverage of these areas is
verywelldoneandtogetherwiththeexercisesprovidedwillgivethereader
afirmfoundationforthetopicstofollow.
The more advanced fundamentals: Boolean differential calculus, sets,
lattices, and classes of logic functions are also well covered. A good un-
derstandingofthematerialinPartIIwillservethereaderwellinconsidering
boththeoreticalandpracticalapplicationsoflogicfunctionsandequations.
Part III covers a variety of application areas based on the fundamentals
introduced in Part II using XBOOLE and the XBOOLE-monitor XBM 2
introduced in Part I. The breadth of application areas introduced is a
particularstrengthofthistextbook.
v
vi Foreword
Part III begins with discussions of logic and arithmetic applications that
aretraditionallycoveredinbooksonlogicfunctionsandequations.However,
herewealsofindcoverageofspecialfunctionsand,inparticular,bentfunc-
tions that are fundamental to cryptography applications and cybersecurity,
twoparticularlycriticalareasinmodernIT.Thesolvedexercisesinthisarea
areparticularlyinformativeandwouldbeverydifficulttoundertakewithout
thesupportofXBOOLE.
ThechaptersonSAT-problemsandextremelycomplexproblemsarevery
significantinthattheyillustratethecomplexitiesofvariousproblemsaswell
asmethodsforaddressingthosecomplexitiesemployinglogicfunctionsand
equations. Study of these chapters will provide the reader with new ways
to visualize, formulate, and solve complex problems they may encounter in
future work. The presentation of this material is again greatly facilitated by
theuseofXBOOLE.
Applicationsrelatedtocombinationalandsequentialcircuitsarecovered
inmanybooks.Thestrengthhereisthatthematerialprecedingthosetopics
setsthediscussionofcircuitissuesinaclearcomputationalandcomplexity
context. The reader has been given both the in-depth fundamentals and
a strong understanding of approaches to complex problems to thoroughly
understandtheproblemsandpotentialsolutions.
I strongly recommend this textbook to students and others seeking a
comprehensive knowledge of the theory and practical application of logic
functions and equations. The authors have addressed a broad spectrum of
issuesinaclearandconsistentmanner.
Victoria,Canada, D.MichaelMiller
January2021
Preface
Thistextbookprovidescomprehensiveknowledgeaboutlogicfunctionsand
equations. The basics of this knowledge are used, at least intuitively, by
everyone.Thesebasicsaresuccessivelyextendedbydetailsandapproaches
that belong to the state of the art of applications of logic functions and
equationsandreach topicsbelongingtorecentareas ofresearch. Hence,no
previous knowledge is needed to read this book, and the study of this book
increases the level of knowledge to get degrees from universities and even
beyond.
The common ground of logic functions and equations are variables that
can store only two different values; hence, these variables are the simplest
variables of all. Depending on the application, the two different values
can arbitrarily be chosen. The values True and False are used for logic
application,the values On and Off can be used to determine the position of
aswitch,butveryoftenthedigits1and0areusedasvaluesofsuchalogic
variable,synonymouslyalsodenotedbyBooleanorabinaryvariable.
Due to the use of these simplest variables, K. Zuse successfully built the
first computer, and the strong progress in the recent age of digitization is
still based on these variables. The drawback of these simplest variables is
thatmanyofthemareusuallyneededtoexpresstherequireddifferentstates
and the number of different binary vectors of logic variables exponentially
increaseswiththenumberoflogicvaluesinsuchavector.
Themanipulationofalargenumberoflogicvaluesisbothtimeconsuming
and error prone for human beings; computer solves such elementary tasks
much faster and provides the correct results. However, the computer must
knowin which manner the logicvalues must be combined.Logicfunctions
and solution-sets of logic equations are convenient objects in algorithms
thatsolvetasksforawiderangeoflogicproblems.However,programming
languagesprovideonlyoperationstocomputesinglelogicvaluesorvectors
of such values, but due to the already mentioned complexity problem, such
languagesdonotdirectlyprovideoperationsforlogicfunctionsorsolution-
sets of logic equations. To close this gap, we developed the programming
systemXBOOLE.
Learning a comprehensive and ready-to-use body of knowledge requires
not only the study of the theory but also the application and consolidation
of the insights. Therefore, we extended this third edition of this textbook
byexercises.Providedsolutionsforanumberoftheseexercisessupportthe
vii
viii Preface
readerswhenverifyingtheirownsolutions.Allexercisescanbesolvedusing
the XBOOLE-monitor XBM 2 that provides easy access to the functions
of the XBOOLE-library and can be used by everyone free of charge. The
benefitoftheXBOOLE-monitorXBM2isthatthelogictasksaresolvedvery
efficiently,skillsaboutprogramminglanguagesarenotneeded,andthehuge
overheadofelementarymanipulationsofbitswillbeavoided.Wenoticedthat
theexplorationofproblemsandthecreationofXBOOLE-problem-programs
fortheirsolutionsignificantlyincreasedthelevelofeducationofourstudents.
Wehopethatthistextbookcontributestospreadingthiswelcomeobservation
toallreaders.
The two chapters of Part I introduce in a compact manner the concepts
ofXBOOLEandtheuseoftheXBOOLE-monitorXBM2sothatexercises
based on this monitorcan be solved after reading these two chapters. Part I
canlaterbeusedasahelpfulreference.Moredetailsaboutthefundamentals
ofXBOOLEareprovidedinPartIIofthistextbook.
The five chapters of Part II explain the fundamentals needed to solve
problems that can be specified using the already mentioned logic variables.
LogicfunctionswillbedefinedbasedonthesetsBandBn.BooleanAlgebras
facilitateusefuldifferentviewsoflogicfunctionsandequations.Thestrong
interrelation between logic functions and logic equations becomes obvious
withthepossibilitytouseternaryvectorlists(TVLs),themaindatastructure
of XBOOLE, to represent different forms of logic functions as well as
solution-sets of logic equations. We explain this data structure in detail and
showthatasingleternaryvectorthatcontainsd dash(−)elementsexpresses
2d binary vectors; hence, TVLs are able to reduce the required memory as
wellastherequiredcomputationtimeexponentially.
Digital systems are specified not only by the values of certain logic
variables but also by the changes of these values. The Boolean Differential
Calculus expands the static concepts of logic functions and equations by
operationsthatspecifyseveraltypesofchanges.Averywelcomepropertyof
asubsetofderivativeoperationsisthattheirresultsdependonfewervariables
than the evaluated logic functions. Efficient algorithms that compute the
results of derivative operations of the Boolean Differential Calculus have
beenimplementedintheXBOOLE-libraryandcanbeusedintheXBOOLE-
monitorXBM2.
Thereareapplicationsinwhichnotonlyasinglelogicfunctionbutaset
of such functionsoccur.In many cases, a set oflogic functionssatisfies the
conditionsofalattice.Weshowthatnotallfunctionsofsuchalatticemust
becomputedseparately,thattwomark-functionssufficetodescribealllogic
functions of the lattice, and that the computation can be restricted to these
mark-functions. Even all derivative operations of the Boolean Differential
calculus can be computed for lattices of logic functions using both mark-
functions;generalizedlatticesresultcanbeusedinsubsequentcomputations.
The solution of logic equations with regard to variables is a very fun-
damental concept. Several beneficial applications of this concept will be
demonstrated in the third part of this textbook. Derivative operations of the
Booleandifferentialcalculusareusedtocheckwhetheragivenlogicequation
canbesolvedwithregardtoacertainsubsetofvariables,todecidewhether
Preface ix
the solution is a single logic function or a lattice of logic functions, and to
computethesinglesolution-functionorthemark-functionsofthelattice.
Boolean differential equations extend the field of applications. Different
toalogicequationthathasasetofbinaryvectorsasitssolution,aBoolean
differential equation has a set of logic functions as its solution. For certain
Boolean differential equations, the solution consists of a set of equivalence
classesoflogicfunctions.WeshowhowBooleandifferentialequationscan
besolvedusingthefunctionsoftheXBOOLE-monitorXBM2.
The five chapters of Part III demonstrate the use of logic functions and
equationswithinanimportantsubsetoffieldsofapplications.Weencourage
the readers to utilize the provided fundamentals of Part II and computation
poweroftheXBOOLE-monitorXBM2alsofortasksinotherfields.
Logicfunctionsinfluenceoureverydaylife:weusethepropositionallogic
forourdecisions,arithmeticoperationsareexecutedinsideofthecomputers
usingthebinaryarithmetic,orcodinghelpstoavoidtheerrorsoftransmitted
dataorincreasethesecurityofsecretinformation.Thespecificnormalforms
ormostcomplexfunctionsextendourknowledgeaboutlogicfunctions,and
bentfunctionsareusedinthefieldofcryptography.
Manypracticalproblemscanbemodeledassatisfiability(SAT)-problem
thatisarestrictivelogicequation,wherethelogicfunctionontheleft-hand
sidehasaconjunctiveform.SAT-problemsbelongtothesetofmostcomplex
problems. We demonstrate how ternary vector lists can be used to solve
suchproblemsanddescribeapproachestoshortentherequiredsolutiontime.
Severalexamplesareusedtoshowhowagivenproblemcanbemodeledand
solvedasaSAT-problemusingtheXBOOLE-monitorXBM2.
Itisveryhardtosolveaproblemthatrequiresalargenumbernoflogic
variables for its specification because 2n potential solution patterns must
be considered. We demonstrate how we solved such an extremely complex
problem for n = 648, which means we were faced with the gigantic set of
more than 10195 potential solutions. The key to solving such an extremely
complex problem consists of the detection of all hidden properties of the
givenproblemandtheirutilizationwiththeappropriateapproachesoflogic
functionsandequations.
All digital systems contain combinational circuits. There is a strong
relation between logic functions and combinational circuits. Both the logic
gates of combinationalcircuits and the behaviorof the outputsof combina-
tional circuits can be described by logic functions. New results in analysis,
synthesis, and test of combinational circuits contribute to the technological
progressofdigitalsystemsandviceversa;therequirementsofsuchsystems
forcethedevelopmentofinnovativeapproachesforimprovedcombinational
circuits. In addition to well-known covering methods, we explain power-
ful decomposition methods that utilize the properties of lattices of logic
functionstosynthesizecompletelytestablecombinationalcircuitswithshort
delays,smallareas,andlowpowerconsumption.
Sequential circuits realize finite state machines. These circuits combine
combinational circuits and memory-elements. Sequential circuits must be
distinguished between asynchronous and synchronous circuits. For these
two types of sequential circuits, we explain the methods of analysis and
x Preface
synthesis.Thesolutionoflogicequationswithregardtovariablesfacilitates
thecommonutilizationofpropertiesdeterminedbytherequiredbehaviorand
thelatticesthatspecifythebehavioroftheselectedflip-flops.
In this textbook, we combine the fundamentals of logic functions and
equations that represent the state of the art with the results of our own
research in this field over more than four decades. Some more or less
independentareasexistedwhenwestartedthiscooperativeresearch.Onepart
is very related to Mathematics or can even be considered as a fundamental
toolfortheconstructionofaxiomaticMathematicsaltogether.Thispartgoes
backtoG.Booleandotherfamousmathematicians.
The second field is the use of the binary number system that has its
roots in the papers of G. F. Leibniz and was electrically implemented by
C. E. Shannon for relay circuits. It is nowadays a giant part of Information
Technologyandoneofthefundamentalconceptsofcircuitdesign.
The third intention, which only started at that time, had the aim to
contribute to the development of Artificial Intelligence by using rule-based
systems.Implicationhasplayedamajorroleinthisfield.Wesawonlylater
thatourconceptswillalsocoverthispart.Thedevelopedalgorithmsarevery
efficient;itwaspossibletosolvemanyproblemsinaveryinstructiveway.
Atthebeginningofourcooperativeresearch,onlysomerootsofBoolean
DifferentialCalculuswereknown.Thisencouragedustoextendthiscalculus
andapplyittofindsuccessfulsolutionsforpreviouslyopenproblems.
The most important stimulus for this third edition of this textbook was
thepossibilitytopresenttheresearchworkofabout40yearstootherswho
canbenefitfromtheseresults,theoreticallyaswellaspractically.Thecom-
binationofthepresentationofthetheoreticalbasiswithexercisestoenlarge
the knowledge learned supports the teaching starting from zero to reaching
a level as high as necessary or possible. Logic functions, logic equations,
and the Boolean differential calculus are now an inseparable theoretical
basisofalgorithmstosolveproblemsbelongingtotheBooleandomain.The
XBOOLE-system supplements the theoretical basis with software tools for
practicalsolutions.Weseethistextbookasboththesummitandcompletion
ofoursuccessfulcollaboration.
Many people, colleagues, and students contributed to this book by com-
ments and discussions. We are grateful to all of them. We are also very
grateful to Springer, who gave us the excellent possibility to transform the
research work of many years into a textbook that is available for everyone
andhelpfulforallthereaders.
Chemnitz,Germany BerndSteinbach
Chemnitz,Germany ChristianPosthoff
January2022
