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over all states reachable at the given stage iPn the computation.
Thissuperpositionisanentirelytheoreticalconstruct. Physicallythereisno
such superposition for a classical probabilistic Turing machine. Each separate
execution of the machine simply traces out a path in the computation graph.
Theoretically, however, we can speak of the machine as being in a superposed
state. Observing (or measuring) the machine after a certain number of time
steps, we will find the it in a certain state with a certain probability. This
probability is the same as the probability to reach that state from the initial
starting state. The classical computation of a probabilistic machine can be
observed at each state, thus tracing out the particular execution path. This
observation, or measurement is of no consequence for the future execution of
the machine.
Here we have two major differences as compared to quantum machines.
Firstly,forquantummachines,thecorrespondingsuperpositions(somewhatdif-
ferently defined though) do actually exist. Secondly, and as consequence of the
18Thenotation s>willsubsequentlybeusedfortruequantum states.
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realityofthesuperpositionandthenatureofquantumdynamics,themeasuring
a quantum Turing machine will affect the future execution of the machine.
Returning to the classical probabilistic machine, it intuitively appears that
restricting the branching probabilities to 0,1/2,1 , 1/2 corresponding to fair
{ }
coin tossing, ought to be sufficient. This can in fact be proved [??].
2.8 Some Complexity Theory
Computability theory discusses what can be computed in principle by investi-
gating the boundary between computable and uncomputable functions. Com-
plexity theory discusses what can be computed in practice by analyzing the
amount of resources needed in a computation.
The complexity is calculated by analyzing the algorithm, not by running
the computation, so clearly, in order for complexity to make sense, it must be
defined for decidable problems or computable functions.
Complexity is most conveniently discussed in terms of the computational
resources required to decide recursive languages, i.e. in terms of decision prob-
lems. A computational resource can be time, roughly measured as the number
of steps required by an algorithm. Another resource is space, corresponding to
the amount of memory required. It could also be some other physical resource
like energy, but time and space are the measures are the most important from
the point of view of difficulty of algorithmic problems. In the circuit model,
complexityisnaturallymeasuredintermsofthe number ofgatesinthe circuit.
Adistinctionismadebetweentractableproblemsanduntractableproblems.
A problem is tractable if it can be solved on a computer using a reasonable
amountof CPU-time and/ormemory. It is well knownthat there is a dramatic
differenceinthe growthrateofpolynomialfunctionsandexponentialfunctions.
Reasonable amount of resources are those that grows at most as a polynom in
the size of the problem.
Algorithms are not in general intended to solve particular problems, but
rather sets of problems, parameterized in some way. A particular problem in
the set is called an instance. In general the instances are increasing in size
in terms of the parameters. When analyzing a certain algorithm for a certain
problem (for example, insertion sorting for sorting) it is in general the worst-
casebehaviorthatisinteresting. Inthatcaseweareinterestedinupperbounds
on the amount of resources required by the algorithm.
When analyzing classes of algorithms for a certain problem (for example,
the class of sorting algorithms) it is rather lower bounds that are in focus. We
want to know the performance of the best possible algorithm.
We will now make these notions exact. First of all we need a model of
computation. In general, different models of computation can have different
strengths.19 However,thesocalledslowdownbetweendifferentreasonablemod-
els is polynomial, and therefore not important in theoretically. Of, course, in
practical computing, even small increases in speed can be important.
19Thisisincontrasttothesituationasregardscomputability.
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In order to treat the complexity of algorithmic problems in a uniform way,
the problems are formulated in terms of formal language theory. Problems are
codedusingsomealphabetandprobleminstancesthencorrespondstostringsin
the set of all strings Σ . A decision problemthen amounts to deciding whether
∗
agivenstringbelongstothe language(whichisasubsetΣ ,seesectionX.X.X)
∗
or not.
Thissectiononthetheoryofcomplexitywillbeverybriefandjustrecordthe
basicdefinitionsandresultsofthetopicwithoutproofsordetailedexplanations.
A good modern reference is [18], see also [19] and [20].
2.8.1 Measures of complexity
Any’reasonable’modelofcomputationcanbeusedtosetupthetheoryofcom-
plexity. What is needed for measuring the time complexity is some consistent
way of counting computational steps in terms of a unit of time for performing
someelementarystep. Thereisagooddealofarbitrarinessherebothregarding
whatis astepandwhatisanelementaryunit. Thisarbitrarinessis howeverin-
herentto the problem,and in the end does not matter much, as goodmeasures
of complexity only differs polynomially.
It is anyway not the exact number of steps that is important. Rather, we
need a concept of complexity that is robust to incremental improvements in
hardware and software, and yet sensitive to more dramatic developments, of
which quantum computation is an example.20
The Turing machine model will be used here. Time complexity will be
defined in terms of the number of steps taken by the machine during the com-
putation. Space complexity will be defined in terms of the maximal number of
tape squares needed by the computation. These complexity measures will be
functions from the size of the input to the number of computational steps and
the number of tape squares respectively. Input size is defined as the number of
written tape squares.
Let L be a recursive language decided by a Turing machine M. Referring
back to section 2.3.4 we recall that this means that M halts on all inputs x
in either the ’yes’ or the ’no’ configuration depending on whether the string x
belongs to the language or not. Computational resources is measured in terms
of the size of the input x and will be taken as the number of non-blank tape
squares in the start configuration of the machine. This is called the length of