Table Of ContentAssociative Digital Network Theory
Nico F. Benschop
Associative Digital Network
Theory
An Associative Algebra Approach
to Logic, Arithmetic and State Machines
Dr.NicoF.Benschop
AmspadeResearch
Drossaardstraat71
5663GJGeldrop
Netherlands
E-mail:[email protected]
ISBN 978-1-4020-9828-4 e-ISBN 978-1-4020-9829-1
DOI 10.1007/978-1-4020-9865-9
LibraryofCongressControlNumber:2008944293
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———————————————————————————–
Tomyparents,forcaringandpersistingthroughhardtimes.
Andtomywife,forherpatienceandunderstanding.
————————————o0O0o————————————
OldDutchsaying:
—“Wiehetkleinenieteert,ishetgrotenietweerd.”—
(Whodoesnothonorthesmall,doesnotdeservethelarge)
Math=TheArtofseparatingNecessityfromCoincidence.
Life = Makingthebestof Necessity, usingCoincidence.
“Itjusthappenedthatno-oneelsewasfamiliarwithbothfields
atthesametime.” (ClaudeShannononinter-disciplinarywork,1948)
“ItalwaysworkedoutthatwhenIunderstoodsomething,itturnedouttobe
simple.Taketheconnectionbetweenthequantumstuffandelectrodynamicsinmy
book.Ittookmethirtyyearstofigureout,andintheenditwasalmosttrivial.It’s
sosimplethatanyfreshmancouldreadandunderstandit.Butitwashardformeto
getthere,withallthishistoricaljunkintheway.”
(CarverMead:interviewAmericanSpectatorv34n7p68Sep.2001)
“Mathematiciansopenedthegatetoanextensivedomain,
but they did not enter this domain themselves.
Bytheirnaturetheyaremoreinterestedintheway
thegateisopened,thaninthegardenbehindit.”
(MauritsEscherin“RegularDivisionofthePlane”,1958)
— IftheWholeismorethanthesumofitsparts, —
— thenthecouplingofpartsisthedifference. —
H Logic aa = a
/ \
< + ^ * > Quint Essence C U Arithmetic ab = ba
\ /
L R Memory a(bc)=(ab)c=abc
Preface
This book is intended for researchers at industrial laboratories, teachers and stu-
dents at technical universities, in electrical engineering, computer science and ap-
pliedmathematicsdepartments,interestedinnewdevelopmentsofmodelingandde-
signingdigitalnetworks(DN:statemachines,sequentialandcombinationallogic)
in general, as a combined math/engineering discipline. As background an under-
graduatelevelofmodernappliedalgebra1 willsuffice.Essentialconceptsandtheir
engineeringinterpretationareintroducedinapracticalfashionwithexamples.The
motivationinessenceis:theimportanceoftheunifyingassociativealgebraoffunc-
tion composition (viz. semigroup theory) for the practical characterization of the
threemainfunctionsincomputers,namelysequentiallogic(state-machines),arith-
meticandcombinational(Boolean)logic.
Knownprinciplesofdiscretemathematics,especiallyfinitesemigroups,residue
arithmeticandbooleanlogic(lattices)areinterpretedintermsofpracticalDN de-
sign issues. The main three levels of state machine synthesis form a natural ‘top
down’hierarchyofassociativealgebras:
Application Algebratype Syntax Objects Operations
sequentiallogic associative (ab)c=a(bc) functions sequencing
arithmetic commutative ab=ba numbers (+) (.)
combinat’llogic idempotent aa=a sets ∪ ∩
Historically,non-commutativeandidempotentalgebrasdivergedfromarithmeticin
the nineteenth century. Our aim is to emphasize again their arithmetic nature, for
practicalengineeringpurposessuch as efficientsynthesis of binarylogicandstate
machines.The‘static’(combinational,idempotentx2≡x)and‘iterative’(commu-
tative,xi+1=xix=xxi)aspectscanbemodeledbyfiniteresiduearithmetic.Apart
from the two non-commutative components of memory type (branch- and reset-
machines, shown to be each others dual), non-commutative aspects of sequential
behaviorcanberepresentedbycouplingfunctionsbetweencomponents.
• In the first of three parts, on state machines (Chaps. 1–4), an introductory
chapter recalls basic principles in theory and practice. The five basic components
ofsequentialbehavior(withindecomposablesemigroup)arederived,withwaysto
couple them efficiently—only required in the non-commutative case. They define
thefivebasictypesofstatemachinesfornetworkcomposition.
• In the second part, on combinational (Boolean) logic (Chaps. 5, 6) the con-
ceptofspectrumasacharacteristicsequenceofnumbers,isborrowedfromFourier
analysis for order-independent (symmetric) synthesis of Boolean functions (BFs).
Ausefularithmeticcompositionalruleholds:thespectrumofaproductoffunctions
(ofdisjointinputs)istheproductofthecomponentspectra.InfactBoole(1854)in-
troduced his algebra—a calculus of binary properties—as an idempotent form of
1Birkhoff-Bartee1970—ModernAppliedAlgebra
Hartmanis-Stearns1970—AlgebraicStructureofSequentialMachines.
vii
viii Preface
arithmetic. This allows convolution-like composition rules (as in linear filters), to
bedeveloped.
SymmetricBFsareimplementedasacrossing-freeandcompactorthogonalgrid
networkofMOStransistorsinthesiliconplane,toobtainaregularlystructuredVLSI
implementation.SimplyremovingtransistorsfromsuchgridyieldsplanarBFswith
thedesiredcrossing-freeproperty,coveringamajorityofBooleanfunctions.Using
thisrepresentation,thecomplexityofBFsgrowspolynomial,andnotexponential,
withthenumberofinputs.Itappearsthatbypermutingand/orinvertingtheninputs,
eachBF ofatmostfourinputsisplanar.
n
AfastO(n2)algorithmforsymmetriclogicsynthesisisdeveloped,andapplied
tooptimizefault-tolerantlogicusingHamming-orproduct-codesforerrorcorrec-
tion,withsynthesizedgatecountascostcriterium.
•Thethirdandlastpart,onarithmetic(Chaps.7–11),analysesresiduearithmetic
withtwoextremaltypesofprimerelatedmoduli:pk andm =p p ···p typical
k 1 2 k
for ‘sequential’resp. ‘parallel’ arithmetic.By expandingr mod m residues witha
‘carry’ c as multiple of modulus m: n=cm+r, integer arithmetic obtains a dual
focus on closure- and generative properties of residues and carry, as independent
resp.dependentnetworkcomponents.
This balanced approach to arithmetic provides new insights into old and well
known problems in finite additive number theory (Fermat, Goldbach, Waring:
Chaps. 8, 9, 10) with practical engineering results. For instance each odd residue
mod2k isauniquesignedpowerof3,allowingefficientlog-arithmeticoverbases
2 and 3 [patent US-5923888]. Moreover, a binary log-arithmetic microprocessor
(32bits,in0.18μCMOStechnology)isdescribed,designedaspartofaEuropean
Esprit project,2 comparing favorably with the known floating point arithmetic de-
vices.
NicoF.Benschop ♠AmspadeResearch, Geldrop,TheNetherlands, Oct.2008.
Acknowledgements
TheauthorisgratefultoPhilipsResearch(Eindhoven,NL)forallowingtopublish
this material, developed during his 32 years there. Also gratefully acknowledged
arethecontributions,toChaps.6and11,ofcolleaguesRichardKleihorst,Renévan
der Vleuten (Philips Research), G. Muurling, prof. J. Simonis (TU Delft), and of
Esprit project co-workers Chris Softley, dr. Nick Coleman (Univ. Newcastle UK)
andRudolfMatousek,prof.JiriKadlec(UTIA,Prague).
2Esprit33544HSLA,1999–2002,maincontractorUniv.Newcastle(dpt.ECE)UK.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Part1 SecuentialLogic:FiniteAssociative . . . . . . . . . . . . . . . . . 3
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 SequentialandCombinationalLogic . . . . . . . . . . . . . . . . 3
1.2 FiveBasicStateMachines,asNetworkComponents . . . . . . . . 5
1.3 Subset/Partition,Local/Global,Additive/Mult’ve . . . . . . . . . . 8
1.3.1 AssociativeClosure:SemigroupandSub-Semigroup . . . . 9
1.3.2 PreservedPartition:CongruenceandImage. . . . . . . . . 10
1.4 IntegerArithmetic:ResidueswithCarry . . . . . . . . . . . . . . 11
2 SimpleSemigroupsandtheFiveBasicMachines . . . . . . . . . . . 15
2.1 StateMachine:SequentialClosureandRank . . . . . . . . . . . . 15
2.2 BasicMachinesandSimpleSemigroups . . . . . . . . . . . . . . 16
2.2.1 Iterations:Monotone,Periodic,Idempotent . . . . . . . . . 17
2.2.2 OrderedIdempotentsH forCombinationalLogic . . . . . 18
2.2.3 TheFiveMinimalSemigroupsandBasicMachines . . . . 19
2.3 EquivalentIdempotents:MemoryComponentsL,R . . . . . . . . 20
2.4 MaximalSubgroups:PeriodicG . . . . . . . . . . . . . . . . . . 24
2.5 ConstantRankMachines,andSimpleSemigroups . . . . . . . . . 25
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 CouplingStateMachines. . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 NoCoupling:SemigroupZ(.)modm,Compositem . . . . . . . . 30
3.3 MachineDecomposition:RightCongruenceSuffices . . . . . . . . 34
3.4 CascadeComposition:FullGroupsFG andFG . . . . . . . . . 36
3 4
3.5 DecomposingtheFull-andAlternatingGroupoverFourStates . . 41
3.6 DecomposingSimpleGroupsAG ⊂FG forn>4 . . . . . . . . 44
n n
3.7 LoopCompositionSuperfluous . . . . . . . . . . . . . . . . . . . 48
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
ix
x Contents
4 GeneralNetworkDecompositionofStateMachines . . . . . . . . . . 51
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 ImplementingM=(Q,A)byItsAlphabetA . . . . . . . . . . . 52
4.2.1 DecompositionbyLocalInputClosures . . . . . . . . . . 52
4.3 Bottom-UpRankDrivenDecompositionofS=A∗/Q . . . . . . 53
4.4 PartialDirectProducts,UnusedCodes,Efficiency . . . . . . . . . 53
4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5.1 Top-DownDecompositionbyLocalInputClosures . . . . 62
4.5.2 GlobalDecompositionbyMaximalIterativeComponents . 63
4.6 Invariants:OrderedCommutingIdempotents . . . . . . . . . . . . 65
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Part2 CombinationalLogic:CommutingIdempotents . . . . . . . . . . 69
5 SymmetricandPlanarBooleanLogicSynthesis . . . . . . . . . . . . 69
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 LogicSynthesisIndependentofInputOrdering . . . . . . . . . . . 70
5.2.1 OrthogridPlotandRankSpectrum . . . . . . . . . . . . . 70
5.2.2 FactoringPathsbyaPlanarNode . . . . . . . . . . . . . . 71
5.3 SymmetricandThresholdBF’s . . . . . . . . . . . . . . . . . . . 72
5.3.1 SymmetricFunctions‘Count’ . . . . . . . . . . . . . . . . 73
5.3.2 T-CellLibrary,ThresholdLogicCells . . . . . . . . . . . 74
5.4 PlanarCutandFactoring . . . . . . . . . . . . . . . . . . . . . . 74
5.5 FastSymmetricSynthesis:Quadraticinnr.Inputs . . . . . . . . . 75
5.6 ExperimentsandConclusion . . . . . . . . . . . . . . . . . . . . 75
5.7 PlanarBooleanLogicSynthesis . . . . . . . . . . . . . . . . . . . 76
5.7.1 AllBF ArePlanarupton=4Inputs . . . . . . . . . . . . 77
n
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 FaultTolerantLogicwithErrorCorrectingCodes . . . . . . . . . . 83
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 FaultTolerantICDesignEnvironment . . . . . . . . . . . . . . . 84
6.2.1 ImplementationatRegisterTransferLevel . . . . . . . . . 85
6.2.2 ProtectingRegistersandConnections . . . . . . . . . . . . 86
6.3 ThreeErrorCorrectionMethodsforLogicCircuits . . . . . . . . . 86
6.3.1 MajorityVoting . . . . . . . . . . . . . . . . . . . . . . . 87
6.3.2 HammingCodes(BlockCodes) . . . . . . . . . . . . . . . 87
6.3.3 ProductCodes(ArrayCodes) . . . . . . . . . . . . . . . . 87
6.4 DemonstrationofExperimentalCircuit . . . . . . . . . . . . . . . 88
6.5 ResultsforTypicalDesigns . . . . . . . . . . . . . . . . . . . . . 93
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Contents xi
Part3 FiniteArithmetic:Associative,Commutative . . . . . . . . . . . . 97
7 Fermat’sSmallTheoremExtendedtorp−1modp3 . . . . . . . . . . 97
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.1.1 Divisorsr|p±1andResidues(p±1)p modp3 . . . . . . 99
7.2 LatticeStructureofSemigroupZ(.)modq . . . . . . . . . . . . . 99
7.2.1 Distinctep−1modp3 forIdempotentse∈Zp−1 . . . . . . 101
7.3 Distinctrp−1 modp3 forDivisorsr|p±1 . . . . . . . . . . . . . 103
7.3.1 IdempotentsofZp+1(.)andDivisorsofp+1 . . . . . . . 104
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8 AdditiveStructureofUnitsGroupmodpk,withCarryExtension
foraProofofFermat’sLastTheorem . . . . . . . . . . . . . . . . . 107
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2 StructureoftheGroupG ofUnits . . . . . . . . . . . . . . . . . 109
k
8.3 CubicRootSolutioninCore,andCoreSymmetries . . . . . . . . 111
8.3.1 AnotherDerivationoftheCubicRootsof1modpk . . . . 111
8.3.2 Core Increment Symmetry mod p2k+1, Asymmetry
modp3k+1 . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.4 SymmetriesasFunctionsYield‘Triplets’ . . . . . . . . . . . . . . 115
8.4.1 ATripletforEachUnitninG . . . . . . . . . . . . . . . 117
k
8.4.2 TheEDSArgumentExtendedtoNon-CoreTriplets . . . . 118
8.5 RelationtoFermat’sSmallandLastTheorem . . . . . . . . . . . 119
8.5.1 ProofoftheFLT Inequality . . . . . . . . . . . . . . . . . 120
8.6 ConclusionsandRemarks . . . . . . . . . . . . . . . . . . . . . . 121
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
9 AdditiveStructureofZ(.)modm (Squarefree)andGoldbach’s
k
Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9.2 LatticeofGroups . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9.2.1 OrderingofCommutingIdempotents . . . . . . . . . . . . 125
9.2.2 LatticeofIdempotents:AddvsJoin . . . . . . . . . . . . . 125
9.3 Primes,CompositesandNeighbors . . . . . . . . . . . . . . . . . 126
9.3.1 EachIdempotent’sSuccessorisinG orG . . . . . . . . 127
1 2
9.4 EuclideanPrimeSieve . . . . . . . . . . . . . . . . . . . . . . . . 128
9.4.1 PairSumsofCarryExtendedUnits . . . . . . . . . . . . . 129
9.4.2 InductionBase:PairSumsofPrimesinG(3) . . . . . . . . 130
9.4.3 ExcludingCompositesin G(k), Baseprimesand1 as
Summands . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.5 ProvingGCviaGR(k)byInductiononk . . . . . . . . . . . . . . 133
9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
(cid:2)
10 Powersums xp Represent Residues mod pk, from Fermat
toWaring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
xii Contents
10.2 CoreIncrementsasCosetGenerators . . . . . . . . . . . . . . . . 138
10.3 CoreExtensions:A toF ,andPairsumsmodpk . . . . . . . . . 140
k k
10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
11 Log-Arithmetic,withSingleandDualBase . . . . . . . . . . . . . . 147
11.1 Log-ArithmeticwithDualBase2and3 . . . . . . . . . . . . . . . 147
11.1.1 ProposedNewBinaryNumberCode . . . . . . . . . . . 148
11.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 149
11.1.3 ApplicationtoMultipliers . . . . . . . . . . . . . . . . 149
11.1.4 SignedMagnitudeBinaryCodeoverBases2and3 . . . 150
11.1.5 AdditioninLogCode:‘Odd’Arithmetic(Base2and3). 151
11.2 EuropeanLogarithmicMicroprocessorELM . . . . . . . . . . . . 152
11.2.1 Introduction:Log-ArithmeticwithSingleBase2 . . . . 153
11.2.2 Log-ArithmeticAlgorithms,anOverview . . . . . . . . 155
11.2.3 DataFormat,RangeandPrecision . . . . . . . . . . . . 156
11.2.4 MeasurementofAccuracy . . . . . . . . . . . . . . . . 157
11.2.5 ConventionalLNSAdditionandSubtraction . . . . . . . 158
11.2.6 NewErrorCorrectionAlgorithm . . . . . . . . . . . . . 160
11.2.7 ErrorCorrectionforSubtraction . . . . . . . . . . . . . 163
11.2.8 Adder/SubtractorDesignandEvaluation . . . . . . . . . 163
11.2.9 ArchitectureandPerformance . . . . . . . . . . . . . . 165
11.2.10 VLSIImplementation . . . . . . . . . . . . . . . . . . . 166
11.2.11 TheELM:SomeMoreArchitecturalDetails . . . . . . . 167
11.2.12 AccuracyComparisonsLNSvs.FLP . . . . . . . . . . . 168
11.2.13 TheTMS-320C6711 . . . . . . . . . . . . . . . . . . . 170
11.2.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 171
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
