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Informally, for any combination of scanned tape symbol and machine con-
figuration, we allow a set of possible instructions.
Formally this is easiest to formulate in terms of the transition function.
Remember the transition function
δ :Q Γ (Q Q ) Γ M,
h
× → ∪ × ×
which maps combinations of scanned symbol and machine configuration into
the set of combinations of configurations, symbols and moves.
Considerthesetofallsubsetsofthe set(Q Q ) Σ M,thisis thepower
h
∪ × ×
set ((Q Q ) Σ M). Allowing several different instructions having the
h
P ∪ × ×
same tape symbol and configurationcan be formulated in terms of a transition
function that maps into this set of subsets
δ :Q Γ ((Q Q ) Γ M).
h
× →P ∪ × ×
How does a non-deterministic Turing machine compute? Suppose that dur-
ing the computation the machine finds a matching pair of scanned symbol and
machine state for which there are several instructions. The computation then
branches off into parallel computations, one for each possible way to proceed.
48
Thestategraphforanon-deterministiccomputationisthereforeadirectedtree,
whereas for a deterministic computation it is a list.
Inordertoactuallycarryoutsuchacomputationinparallel,onewouldhave
to assignnew computationalresourcesateachbranchinthe graph,in the form
of new processors or new Turing machines. In practice, this is not possible in
the general case where the maximal number of processors are limited.
The alternative would be to traverse the tree, breadth-first, using just one
processor.17 Exponential resources are needed in the generic case, in the form
of increasing time and space requirements.
It is quite easy to argue that the set of computable functions are the same.
Suppose a partial function is computable by a non-deterministic Turing ma-
chine. Thismeansthatthefunctionvaluesarefoundattheendsofterminating
branches of the computation graph. Performing the computation breadth-first
on a deterministic Turing machine, we are guaranteed to eventually reach the
halting states, possible after consuming an exponential amount of time and
space. The computation might take exponential time to systematically work
throughtheeverincreasingnumberofbranches,anduseanexponentialamount
oftapetorecordinformationaboutthestateatnotyetprocessedbranchpoints.
Tape can be reclaimed but not time. Still, the function is computable on a de-
terministic Turing machine.
2.6.1 A note on classical parallelism
Non-determinism offers a kind of parallelism. Parallel computation and non-
deterministic computation are overlappingconcepts but they are not the same.
Parallel computation does not involve an unbounded number of parallel pro-
cesses,asthereisalwaysamaximumnumberprocessorsavailableinanyrealma-
chine. On the other hand, parallel processes can communicate, by shared data
orbypassingdata(messages),andthatneednotthe casefornon-deterministic
algorithms. Parallelalgorithms and parallel computation is a huge subject and
there are several different models for parallel computation but no generally
agreed on paradigm.
One might wonder if classical parallelism is a threat to the Church-Turing
thesis? Thatis,isitpossibletocomputenon-computablefunctionsusingparal-
lelcomputation? Theanswerisno,andtheargumentissimilartotheargument
in the case of non-determinism.
2.7 Probabilistic Turing machines
There is a close connection between probabilistic Turing machines and non-
deterministic Turing machines. In non-deterministic machines, the computa-
tioncanbranchofintodifferentsub-computations,inprincipleateverynodein
the computation. Consequently, the computation graph becomes tree. Now, if
17Depth-first traversal runs the risk of going down a non-terminating branch, so breadth-
firstisthebestoptioninanactual simulation.
49
the computational graph edges leading out from a node allowing branching are
assigned probabilities, and a probabilistic choice as to which edge to follow is
made, we get a probabilistic Turing machine. In this case,the computation be-
comesadirectedpaththroughthecomputationgraph. Ofcourse,differentruns
of the same machine, with the same input, will give different paths depending
on the random choices made at each branch node.
Given a perfect random number generator rnd, a probabilistic machine can
be easily simulated on a deterministic machine by performing calls to the rnd
at each step allowing for probabilistic choices.
Formalizing this concept will yield a first step towards an understanding
of quantum Turing machines as well a providing a background for discussing
where quantum computation departs from classical computation. Quantum
Turing machines are treated in chapter 5.
The point of departure is the transition function ∆ of section 2.3.2. There,
thevalues0and1ofthefunctiondeterminedwhetherthetransitionwaspresent
in the programor not. Thinking of the numbers 0 and 1 as probabilities one is
leadtoextendtherangeof∆tonumberspintheinterval[0,1],andinterpreting
p the probability for the transition, i.e. defining
∆:Q Γ (Q Q ) Γ M [0,1].
h
× × ∪ × × →
Referringbacktothecomputationaltreeofanon-deterministicmachine,we
canturnthisintoacomputationtreeforaprobabilisticmachinebymarkingup
eachbranchnode withprobabilities. The probabilityto reacha certainnode in
the tree is the total calculated probability to reach that node from the initial
starting node of the computation.
For theoretical purposes we could then consider the probabilistic machine
as being, at each computational step, in a (classical) superposition of all the
reachable states (from the start). Denoting the states q with s),18 we can
i
L R |
formally write this superposition as a sum p s) where the summation runs
s