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would then have gone beyond the boundaries of a masters project. For this
reason, these topics will be left for a part II.
Acknowledgment
This work was done with support from the Swedish Knowledge Foundation
under the Promote IT program.
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I wouldlike to thank professorBengtNordstro¨mfor supervising the project
and for valuable discussions on computing science in general and the theory of
computability in particular.
IalsothankIngemarBengtssonforreadingandcommentingonthemanuscript.
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Chapter 1
Introduction
Computer science, and in particular the theory of computation, can be studied
without explicit regard to physics. The whole area of research into classical
computabilityisphrasedwithoutanyreferencetophysicsorevenrealcomputing
machines. Therelatedareasofsyntaxandsemanticsofprogramminglanguages
make no reference to anything more real than symbol shuffling by abstract
machines.
Classical computation is a discrete process. Whether viewed in terms of
Turingmachines,RAM-machinesoroperationalsemanticsofprogramminglan-
guages in terms of abstract stack machines1, it really just amounts to string
processing or symbol manipulation. The number of symbols is finite and the
number of basic operations is finite. A program is a finite set of instructions
in terms of the operations acting on the set of strings built out of the symbols.
Seen in this way, computation seems to be detached from physical reality, and
any ’system’ that ’understands’ the rules can perform the computation.
From a practical point of view, the software/hardwaredivision also stresses
this apparent independence of physics. The software in the form of computer
programs written in any of the many hundreds of invented programming lan-
guagesareagainjuststringsofsymbols. Theyseemtohavenomoreconnection
to physics than the ink with which they are recordedon paper. When they are
compiledandstoredelectronically,the link with physicsis somewhatmorepro-
nounced but still weak. It is upon actually running the program,which always
entails the motion of some physical system, that the physical nature of compu-
tation comes into focus. This is obvious if the algorithmis carriedout by hand
or using some mechanical computing device.
So there is a link, howeverweak, to some physical substratum, and it is not
possible to severe this link completely. On the other hand, it is a fundamental
propertyofrealitythatitis possibleworkandsolvecomputationalproblemsat
abstractlevelswithouthavingtocheckphysicalrealizabilityateverystep. This
isanalogoustothe processofabstractionwhichissocharacteristicofcomputer
1Anabstractstack machineisanotational system forgivingstep-by-stepmeaningtothe
primitivesofaprogramminglanguage.
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science. By abstraction, ever more powerful and complicated computational
tools can be invented, which, once it has been ascertained that they can be
implemented in terms of more primitive structures, can be used to solve more
difficult computationalproblemswithout checking the implementation atevery
step.
Butifweworktheotherway,fromabstractlevelsofprogrammingstructures
tomoreconcreteprimitives,theneventuallywewillarriveatsomephysicalsys-
tem, a computing machine, that actually performs the physical motion needed
in order to carry out the computations. In digital computers this is switching
voltage levels in transistors, which in its turn involves the collective motion of
large numbers of electrons.
Thus,aswaspointedoutandstudiedbyLandauer[5],informationisalways
carriedbysomephysicalmedium,andlikewisecomputationisaphysicalprocess
constituted by some well defined motion of a physical system.
1.1 The theory of computation
The theory of computation arosein the nineteen thirties as a response to prob-
lemsinthefoundationsofmathematicsandlogic[6],inparticularinconnection
toDavidHilbert’sEntscheidungsproblem. TheEntscheidungsproblemisaprob-
lemwithintheformaloraxiomaticapproachtomathematics. Hilbert’sprogram
was to formalize mathematical theories into a set of axioms defining relations
between the undefined primitive notions of the theory, and a set of rules of de-
duction. In this way one should be able secure the foundations of mathematics
aswellasmechanizethe processoftheoremproving. Goodproperties,likecon-
sistencyandcompleteness,shouldbe possibleto ascertainwithin the axiomatic
system.
The axiomatic approach itself has a long history dating back to antiquity.
AftertheinventionofthecalculusbyNewtonandLeibnizinthemidseventeenth
century,therewasaveryrapidprogressinthefieldsofappliedmathematicsand
physics. The new mathematics was phrased in an axiomatic language but the
underlyingconceptswereintuitiveandoftenvague. Inthebackgroundhistoryof
Hilbert’s approachwe find attempts to secure the foundations of such concepts
as infinitesimals, limits, real numbers, functions and derivatives to name a few.
Asanasideitisinterestingtonotetheverycloseinterplaybetweenmathematics
and physics during this period. Apart from being a theoretical subject of its
own,mathematicsisalsothelanguageofthephysicalsciencesandoftechnology.
Hilbert’s formalistic approach to mathematics made a distinction between
thesyntacticaspectsofmathematics,i.e. theaxiomsandtherulesofdeduction,
and the semantic aspects, i.e. what the mathematical concepts and theorems
actually mean.
Physicist, engineers and applied mathematicians are normally interested in
the meaning of mathematics. Phenomena in the real world, and whole areas of
science,aremodeledusingmathematics. Onthe otherhand,oncethemodeling
isdone, the actualcalculationscanbe performedwithoutconsideringthe inter-
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pretation. Inpracticethereisalwaysanintricateinteractionbetweenmodeling,
calculationandinterpretation. Butthepointisclear,thestrengthofmathemat-
icsderivesfromthisdivisionintosyntacticrulesofcalculationanditssemantic,
or intuitive, interpretation in terms of objects in the physical world.
The same interplay between syntax and semantics is, of course, present in
computerscienceitself. Wewritecomputerprogramsinordertosolvescientific,
engineering, economic, administrative, everyday and entertainment problems.
Buttheprogramsrunoncomputersthatperformpurelysyntacticsymbolshuf-
fling. Inthetheoryofprogramminglanguagesthereisalsothisdivisionbetween
thesyntacticandthesemanticaspectsofprogrammingandprogramexecution.
1.2 The input/output model of physics and com-