Computational complexity with experiments as oracles

25 February 2009 first published online, doi: 10.1098/rspa.2008.0412465 2009 Proc. R. Soc. A

 Edwin Beggs, José Félix Costa, Bruno Loff and J.V. Tucker oracles. II. Upper boundsComputational complexity with experiments as  

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Computational complexity with experimentsas oracles. II. Upper bounds

BY EDWIN BEGGS1,*, JOSE FELIX COSTA

2,3, BRUNO LOFF2,3

AND J. V. TUCKER1

1School of Physical Sciences, Swansea University, Swansea SA2 8PP, UK2Instituto Superior Tecnico, Technical University of Lisbon,

1049-001 Lisbon, Portugal3Centro de Matematica e Aplicacoes Fundamentals, University of Lisbon,

1649-003 Lisbon, Portugal

Earlier, to explore the idea of combining physical experiments with algorithms, weintroduced a new form of analogue–digital (AD) Turing machine. We examined indetail a case study where an experimental procedure, based on Newtonian kinematics,is used as an oracle with classes of Turing machines. The physical cost of oracle callswas counted and three forms of AD queries were studied, in which physicalparameters can be set exactly and approximately. Here, in this sequel, we completethe classification of the computational power of these AD Turing machines anddetermine precisely what they can compute, using non-uniform complexity classesand probabilities.

Keywords: algorithmic procedure; experimental procedure;Turing machines with oracles; analogue–digital computation;

non-uniform complexity; physical oracles

*A

RecAcc

1. Introduction

In Beggs et al. (2008a), we introduced the idea of coupling a Turing machine toan abstract physical experiment as a new form of oracle. We focused on thedesign and operation of these computational models and the question: What canbe computed by Turing machines with such physical oracles? In Beggs et al.(2008a), we chose a particular experiment and studied the hybrid machines madeby combining Turing machines with the experiment.

Specifically, the experiment we chose was the scatter machine experiment(SME), which we had studied from a computational point of view in Beggs &Tucker (2007b). The SME is an experiment in Newtonian kinematics thatmeasures to arbitrary precision the position x of a vertex of a wedge, by firingparticles and observing their behaviour. If the vertex position x2[0,1], then theSME can measure or compute rational approximations x2[pn,qn] to any accuracyjpnKqnj!2Kn, for nZ1, 2, .. Thus, SME is a physical procedure that can

Proc. R. Soc. A (2009) 465, 1453–1465

doi:10.1098/rspa.2008.0412

Published online 25 February 2009

uthor for correspondence ([email protected]).

eived 17 October 2008epted 15 January 2009 1453 This journal is q 2009 The Royal Society

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compute rational approximations of any real number in [0,1], which is a task farbeyond the computational power of a Turing machine. Since the scatter machinecan compute more than the Turing machine, it was an interesting experiment tochoose as an oracle.

In Beggs et al. (2008a), the hybrid machines made by combining the SME andTuring machines were termed analogue–digital (AD) scatter machines. The casestudies revealed some concepts and mathematical tools useful for a generaltheory of Turing machines with experimental oracles.

We introduced the concept of AD protocol for calls to the oracle to cover thecomplex process of

(i) setting parameters to the equipment,(ii) performing the experiment, and(iii) returning the results.

In particular, the AD protocol consumes resources, most obviously time: the costof calling the oracle depends upon the size of the query and contributes to thecomplexity of computation. We use polynomial and exponential timeAD protocols.

Furthermore, since the oracle is a physical process, we introduced three kindsof oracle query:

(i) exact, where a finite binary representation of numerical data z from theTuring machine sets a parameter of the experiment to an exact realnumber value rZz,

(ii) error-prone with unlimited precision, where numerical data z from theTuring machine sets a parameter of the experiment to an approximatereal number value r2[zKe, zCe] for any requested eO0, and

(iii) error-prone with fixed precision, where numerical data z from the Turingmachine sets a parameter of the experiment to an approximate realnumber value r2[zKe, zCe] for an eO0 fixed for the machine.

With these concepts, we were able to prove theorems which partially answeredthe question about the computational power of the machines. Using the non-uniform complexity class P/poly, we characterized the class of sets decidable bythe polynomial-time computable exact AD scatter machines. Specifically, weproved the following for the error-free machines (Beggs et al. 2008a, theorems 7.2and 7.4, respectively):

Theorem 1.1. The class of sets which are decidable in polynomial time byerror-free deterministic AD scatter machines with a polynomial AD protocol isexactly P/poly.

Theorem 1.2. The class of sets which are decidable in polynomial time by error-free deterministic AD scatter machines with a strictly exponential AD protocol isexactly P/log�.

However, for the error-prone machines, we proved only lower bounds in Beggset al. (2008a). In this paper, we answer questions left open in Beggs et al. (2008a)by proving upper bounds, which enable us to complete the classification of the

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power of the three kinds of scatter machines. Combining upper and lowerbounds, we formulate our new results as equivalence theorems. For the error-prone machines, we prove the following in §2.

Theorem 1.3. The class of sets which are decidable in polynomial time by error-prone arbitrary precision deterministic AD scatter machines with a polynomialAD protocol is exactly P/poly.

Thus, there is no difference in computational power between using exactqueries and approximate queries having arbitrary precision.

In the proof of this equivalence, we need to be rather precise about justwhat we mean by ‘arbitrary precision’, specifically about how the variable sizee of the error is communicated from the Turing machine to the experiment.A scatter machine with arbitrary precision operates as follows. The word z onthe query tape represents the binary expansion of a dyadic rational and aparticle is fired by a cannon that is placed somewhere in the accuracy interval[zK2KjzjK1, zC2KjzjK1]. This means that for different words representing thesame dyadic rational, the longest word will give the highest precision. Toincrease the accuracy, it is only necessary to add more zeros to the binaryrepresentation of z. With this simple mechanism, we proved the lower boundin Beggs et al. (2008a). However, to prove the upper bound, we shall supposethat a uniform probability distribution governs the cannon position in thisinterval [zK2KjzjK1, zC2KjzjK1].

The method of analysis and proof uses probabilistic Turing machines and soleads to the complexity class of BPP//poly (which equals BPP/poly; see

Balcazar & Hermo 1998). Now, this is known to be equal to P/poly.1 Previousappearances of P/poly in models based on real numbers in the literature werediscussed in Beggs et al. (2008a). However, in §3, we prove that the case of fixedprecision is different.

Theorem 1.4. Assume an independent uniformly distributed cannon position inthe accuracy interval. The class of sets which are decidable in polynomial time byerror-prone fixed precision deterministic AD scatter machines is exactlyBPP//log� independently of the complexity of the protocol.

Thus, using approximate queries having fixed finite precision results in a lossof computational power.

By fixed precision, we mean that the error is 2KN for fixed N and the accuracyinterval is [zK2KN, zC2KN ]. We assumed an independent uniformly distributedcannon position in the accuracy interval to prove the lower bound in Beggset al. (2008a).

We could, of course, consider other fixed values, but we then have toconsider the fact that the error chosen does affect the probabilities. To allowfor the case when the error is not a dyadic rational, we could code the errorinto the advice and deal with a system with two physical constants (theerror and the wedge position). But this sort of generality, at this stage, would

1The reasons why BPP/polyZP/poly are as follows: on the one hand, for any two classes A and B,if A4B, then A/poly4B/poly; thus, since P4BPP, we have that P/poly4BPP/poly; on theother hand, it is well known that BPP has polynomial circuits, i.e. BPP4P/poly, and that poly isidempotent; we deduce that BPP/poly4P/poly/polyZP/poly. See Balcazar et al. (1995).

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only serve to obscure what is happening: our message is that we can computewith the system even if there is an absolute lower bound on the accuracyof queries.

The equivalence theorems here suggest that non-uniform complexity classesmay have an important role to play in the exploration of the computationalpower of physical systems, a point we discuss in §4.

This paper is a sequel to Beggs et al. (2008a). We assume the reader is familiarwith our earlier publication Beggs et al. (2008a) and so dispense withpreliminaries that have been already summarized there. The essential topicsare as follows: the definition and operation of the scatter machine (see §§3 and 5in Beggs et al. 2008a), and the theory of oracle Turing machines, probabilisticTuring machines and non-uniform complexity classes (see Balcazar et al. 1995;§4 of Beggs et al. 2008a).

To appreciate fully the results in this paper, the reader will need to know themotivating problems, and the methodology that we have been developing tosolve them, both of which are efficiently explained in Beggs & Tucker (2007b)and Beggs et al. (2008a). Fuller treatments of this background material can befound in Beggs & Tucker (2006, 2007a, 2008) and Beggs et al. (2008c).

2. Consulting the oracle with unlimited precision

In this section, we investigate the class of sets decidable in polynomial time by anerror-prone AD scatter machine with arbitrary but finite precision and using apolynomial protocol. We will conclude that the computational power of thesemachines is not altered by considering such a small error, since they decideexactly BPP//polyZBPP/polyZP/poly in polynomial time.

As in our former paper (Beggs et al. 2008a), we now show that only a numberof digits of the vertex position, linear in the length of the computation, influencethe decision of an error-prone AD scatter machine. Note that the behaviour ofthe error-prone AD scatter machines is probabilistic, because, if the machineshoots close enough to the vertex position, and with a large enough error 3, theparticle can go both left or right with a non-zero probability. Thus, we cannotensure that, when we truncate the vertex position to O(t(n)) digits, the state ofthe machine will be exactly the same after t(n) steps. Instead, we will show that ifa machine decides in time t(n), then by truncating the vertex position to O(t(n))digits, the machine will decide the same set for inputs up to size n. In thefollowing lemmas, remember that if a set is decided by an error-prone AD scattermachine with error probability bounded by g, we may assume without loss ofgenerality that g!1/4.

The whole computation can be summarized as follows. The Turing machinesets up an initial cannon position on the query tape. The experiment takes placeand the result is either left or right. Owing to the error involved in settingup the actual physical position from the data on the query tape, the result ofthe experiment is given as a probability. Given the result of the experiment, theTuring machine sets a second cannon position and the process repeats. Giventhe bound on the time taken by the experiment, there can only be a certainnumber of experiments. Without loss of generality, it will be convenient tosuppose that the number of experiments is always the same (we will justify this

Proc. R. Soc. A (2009)

initial cannon positionscatter to the rightscatter to the left

A

0 1 1—2

1—2

1—2

1—5

4—5

1—2

2—3

1—3

2—3

2—3

1—3

1—3

A R A A R R R

Figure 1. A binary probabilistic tree of height 3.

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later). In figure 1, we show a tree describing the situation where there are threeexperiments. It is a binary tree as every experiment has two possible results, eitherleft or right. At the end of the process, the Turing machine must either accept orreject, and this is shown by an ‘A’ or ‘R’ on each leaf of the tree. To obtain theprobability of acceptance in figure 1, we take each descending path from the initialvertex, multiply the probabilities along its edges and add the results for eachpath ending in acceptance. Thus, the probability of acceptance in figure 1 is

1

3!

1

2!0C

1

3!

1

2!1C

1

3!

1

2!

1

2C

2

3!

1

5!

2

3:

The reader should note that this tree, with its accept and reject states, iseffectively ‘coded’ into, or specified by, the Turing machine, with the results of theexperiments merely dictating which branch of the tree is followed at any stage.Each node or vertex of the tree could be labelled with the cannon position codedon the query tape if and when that experiment is performed. However, theprobabilities attached to the branches are determined by the actual wedgeposition in the scatter machine. We shall study how the probabilities vary if weslightly change the wedge position. To do this, we shall develop some technicalconcepts and a useful proposition.

A probabilistic subtree will be a tree with probabilities attached to the edges asbefore, but now the probabilities may sum to less than 1. Thus, a subtree offigure 1 including the top node (the root) would be an example. Having the sumpotentially less than 1 will be useful in a later proof by induction, but it alsoallows us to talk about (for example) a subtree ending in accepting states.

For a rooted tree T, we denote by P(T ) the set of assignments of probabilities tothe edges of T, making T into a probabilistic subtree. Such an assignment is a mapof the form D:E/[0,1], where E is the set of edges of the tree T. Then, forD2P(T ), the total probability P(T, D) denotes the sum of the probabilities of eachpath from the root to the leaves of the tree T. (The probability of a path is just theproduct of the probabilities in each of its edges.) In the case of figure 1, this is 1.

We denote the set of rooted trees of height %m, having %2 outgoing edges atevery note, by T m.

Next, we define a distance between probabilistic subtrees on P(T ) by settingd(D, D 0) to be the maximum of the absolute values of the differences of theprobabilities over the edges. The number d(D, D 0) is the largest deviation of theprobabilities associated by D and D 0 to any edge.

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1—2

1—2

1—3

1—2

10

(a) (b)

2—3

2—3

1—5

4—5

Figure 2. (a,b) Two probabilistic subtrees of height 2.

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For example, the distance between the two trees in figure 2 is

max1

2K

1

5

��������; 1

2K

4

5

��������; 0K

2

3

��������; 1K

1

3

��������; 1

2K

2

3

��������

� �

and their respective total probabilities are

1

2!0C

1

2!1C

1

2!

1

2and

1

5!

2

3C

1

5!

1

3C

4

5!

2

3:

Now, with these concepts, we define a function f(m, k) for m2N andk2[0,1] by

f ðm; kÞZ supT2T m

supD;D 02PðTÞ:dðD;D 0Þ%k

jPðT ;D 0ÞKPðT ;DÞ j :

The function f(m, k) gives the largest possible difference in total probabilityfor two different assignments of probabilities to a subtree of height %m, wherethe distance between the probabilities is %k. The function provides a measurethat is uniform for trees of height up to m and assignments whose probabilitiesdeviate up to k.

Proposition 2.1. f(m, k)%2mk.

Proof. By induction on m. The result is true for mZ0. Assume that it is truefor m and consider f(mC1,k). A tree T2T mC1 of height R1 can be written asone of the following, where X,Y2T m:

X XY

We shall only deal with the first case, the second (and easier case) is left to thereader. The tree can be assigned probabilities as follows, which we labelD2P(T ) and D 02P(T ), respectively:

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Here, E,E 02P(X ) and F,F 02P(Y ). Now,

PðT ;DÞZ pPðX ;EÞCqPðY ;FÞ; PðT ;D 0ÞZ p0PðX ;E 0ÞCq 0PðY ;F 0Þ:From this, we have

PðT ;D 0ÞKPðT ;DÞZ p0ðPðX ;E 0ÞKPðX ;EÞÞCq 0ðPðY ;F 0ÞKPðY ;FÞÞCðp0KpÞPðX ;EÞCðq 0KqÞPðY ;FÞ;

so we have

jPðT ;D 0ÞKPðT ;DÞ j%p0f ðm; kÞCq 0f ðm; kÞC j p0Kp j C j q 0Kq j%f ðm; kÞC2k: &

Lemma 2.2. Let M be an error-prone AD scatter machine with arbitraryprecision, with the vertex at x, deciding some set A in time t(n) with errorprobability bounded by g%1/4. Let M0 be an identical AD scatter machine, withthe exception that the vertex is placed at x 0. If

j xKx 0 j! 2KtðnÞ

16tðnÞ ;

then for any word of size %n, the probability of M0 making an error whendeciding A is %3/8.

Proof. We set the depth of the tree to t(n), as there may be at most t(n)experiments carried out in time t(n). The change in any particular probability isat most

j xKx 0 j2e

;

where e is the error specified in the protocol. After carrying out t(n) steps or less,the size of the word representing z, which was written in the query tape, isbounded by t(n), and thus the error in aiming the cannon is eZ2Kt(n)K1—this isthe smallest error possible. We then set

k Z 2tðnÞ j xKx 0 j :

For any particular word w of length jwj%n, take the tree T to be the subtree ofall possible experimental results which give the incorrect answer (this does notdepend on the vertex position). Then, let D2P(T ) be the probabilities on theedges given by the vertex at x, and let D 02P(T ) correspond to the vertex at x 0.The probability of an incorrect result for the wedge at x 0 is, by proposition 2.1,

PðT ;D 0Þ%PðT ;DÞC jPðT ;D 0ÞKPðT ;DÞ j%gC2tðnÞk

%gC2tðnÞ2tðnÞ j xKx 0 j ;

so if we choose x0 so that t(n)2t(n)jxKx0j!1/16, then P(T, D 0)!3/8. &

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Lemma 2.3. Let M be an error-prone AD scatter machine with arbitraryprecision, with the vertex at x, deciding some set in time t(n) with errorprobability bounded by g!1/4. Let Mn be an identical AD scatter machine,with the exception that the vertex placed at x ^2t(n)C5.

2 Then, Mn decides the sameset as M, also in time t(n), but with error probability bounded by 3/8.

Proof. This is a corollary of lemma 2.2, using the fact that t(n)%2t(n). &

The reader should recall that the only important bit about the errorprobabilities is that the machine decides the set in polynomial time to an errorstrictly less than 1/2. Thus M and Mn above decide the same set for words oflength %n. However, the important thing about Mn is that its wedge ispositioned at a dyadic rational. Remember that we have also (by construction ofthe machine) a physical cannon position that is specified by a dyadic rationalcentre and width. Then, the probability of left or right is also a dyadic rational,and one which can be calculated given x ^2t(n)C5 and the data on the query tape.This allows us to make a completely digital model of Mn, provided that we haveaccess to a fair coin-tossing machine to generate the dyadic rational probability.Following this line of reasoning gives the following result:

Theorem 2.4. Every set decided by an error-prone AD scatter machine witharbitrary precision in polynomial time is in BPP//poly. This is independent ofthe protocol.

Proof. Let A be a set decided by an error-prone AD scatter machine M inpolynomial time p and with an error probability bounded by 1/4.

Let x be the position of the vertex of M. We use the advice function f2poly,given by

f ðnÞZ ~x Z x 2pðnÞC5; ð2:1Þ

to construct a probabilistic Turing machine ~M with advice which decides A.From lemma 2.3, we know that a machine with vertex at f(n) will make the samedecisions on words of length %n as the one with the original vertex position, butwith error probability bounded by 3/8. Now we will model the behaviour of themachine with dyadic rational vertex at f(n) by using a program with access to afair coin-tossing oracle.

Suppose that M is given a word of length n as input. A call to the AD oracle inM consists of a word z representing the binary expansion of a dyadic rational,and the cannon is placed according to a uniform probability distribution in theinterval [zK2KjzjK1, zC2KjzjK1]. Considering the time taken to write the word tothe query tape, we know that jz j%p(n). If f(n) is outside this interval (this takesa polynomial amount of time to check), we answer left or right with certainty (leftif it is less than the lower bound and right if it is greater than the upper bound). Ifit is inside the interval, we use the fair coin-tossing oracle to simulate the uniformprobability distribution for the cannon position. Using the fact that ~xZ f ðnÞ is arational with denominator 22p(n)C5, we note that we obtain the following integerm and accuracy interval:

m Z 22pðnÞC5ð~xKzC2K j z jK1Þ2 ½0; 22pðnÞC5Kj z j �:

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Now make kZ2p(n)C5Kjzj calls to the fair coin-tossing oracle, interpreting theresults ti (taken in order) as either 0 or 1. Now test if t1t2/tk!m, wheret1t2/tk is taken to be the binary representation of an integer. If the test is true,return right, otherwise return left. The probability of returning right is m/2k, asrequired. The time taken is polynomial in n.

In a run time of p(n), we can call the AD oracle inM at most p(n) times, so thetotal run time of the simulated AD oracle machine ~M is also polynomial. &

In our paper (Beggs et al. 2008a), we proved the statement for lower bounds ofcomputational complexity of the AD scatter machine with unbounded precision.

Theorem 2.5. Every set in BPP//poly can be decided in polynomial timeby an error-prone AD scatter machine with arbitrary precision andpolynomial protocol.

Combining the above theorems, the conclusion for this section is theorem 2.6.

Theorem 2.6. The class of sets decided by error-prone AD scatter machineswith arbitrary precision and polynomial protocol in polynomial time is exactlyBPP//polyZP/poly.

3. Reducing the precision of the cannon

We will now study the class of sets decidable in polynomial time by an error-prone AD scatter machine with fixed precision. We will show that such machinesmay, in polynomial time, make probabilistic guesses of up to a logarithmicnumber of digits of the position of the vertex. We will also prove that thesemachines decide exactly BPP//log�.

The AD scatter machine combines a deterministic Turing machine with aphysical oracle. When the oracle queries involve errors, the coupled machinebehaves probabilistically. Specifically, our analysis used probability to handle thequeries. We have seen that the computation tree of an AD scatter machine isdeterministic except in the nodes that correspond to an oracle consultation. InBeggs et al. (2008a), we gave the lower bound theorem that any set in BPP//log�

can be decided in polynomial time by an error-prone AD scatter machine withfixed precision. Perhaps we left too many details to the reader, so here we takethe opportunity to give a detailed proof before tackling the upper bound result.The reader is reminded that, so that we do not create the unnecessarycomplication of having to deal with two constants in the apparatus, we supposethat the fixed error is of the form eZ2KN.

Consider the class BPP//log�. A set A is in the class if we can find a suitableadvice f of logarithmic size and a set B in BPP, such that, for any input x, x2A ifand only if hx,f(n)i2B, for all nRjxj. In Beggs et al. (2008a), the outline proof ofthe lower bound theorem concentrated on showing the following.

Lemma 3.1. An error-prone AD scatter machine with fixed precision, giveninput x of size jxjZn, can read O(log(n)) bits of the vertex position in polynomialtime in n, independently of the protocol.

If the advice f is coded as a vertex in the way described in Beggs et al. (2008a),then the lemma implies that the advice can be decoded and read in polynomial

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time, no matter the protocol being polynomial or exponential. The reason for theindependence is that the method uses repeated firings of the cannon from thesame vertex position, so the cost in time is linear in the number of firings.

Next, we have to prove that given the input x and the advice f(jxj), with the sameAD scatter machine, we can show that hx,f(jxj)i2B. Moreover, since this in B isdecided by a probabilistic Turing machineM, we need to show that the AD scattermachine, although deterministic in its digital part, can simulate the probabilisticTuring machine. The proof uses the following, lemma 7.7 from Beggs et al. (2008a).

Lemma 3.2. Take a biased coin (probability of heads q2(d, 1Kd) for some1/2OdO0), and g2(0,1). Then, up to probabilityRg, there is a sequence ofindependent coin tosses of length

n

dð1KdÞ 1C1ffiffiffiffiffiffiffiffiffiffiffi1Kg

p� �

;

which gives a sequence of independent fair coin tosses of length n.

Theorem 3.3. An error-prone AD scatter machine with fixed precision candecide any set in BPP//log� in polynomial time, independently of the protocol.

Proof. Let A be an arbitrary set in BPP//log� and M a probabilistic Turingmachine with advice f2log�, which decides A in polynomial time with errorprobability bounded by g!1/7. Let a,b2N be such that j f ðnÞ jZ ba log nCbc .

We construct an error-prone AD scatter machine with fixed precision,~M , which estimates f(n) with an error probability bounded by 1/16, as in lemma

3.1. (We also make sure that the vertex position is estimated to within e/4,where e is the fixed error, again within the given error probability.) Now we needto simulate independent unbiased coin tosses, and we use lemma 3.2 to do this. Ifwe can set the cannon to a position where the probability q of scattering left is in(d,1Kd) for a known dO0, then we have a known linear bound on the number ofrepeated firings we need to simulate a given number of independent unbiasedcoin tosses (to within a probability of 1/16). If we set the cannon position to ourestimate of the vertex position, we know that we can take dZ1/4, since we haveestimated the vertex position to within e/4, where e is the fixed error. In fact, wedo not need to use the full accuracy of our determination of the vertex position,we need only enough to give the accuracy within e/4, so we need only to set thecannon to a fixed number of binary places. &

Our first result is similar to that of §2. We will show that the decision of anerror-prone AD scatter machine with fixed precision remains the same if we keeponly a number of digits of the vertex position logarithmic in the size of the input.

Lemma 3.4. Let M be an error-prone AD scatter machine with fixed precisioneO0, with the vertex at x, deciding some set in time t(n) with error probabilitybounded by g!1/4. Let M0 be an identical AD scatter machine, with theexception that the vertex is placed at x 0. If

j xKx 0 j! e

8tðnÞ ;

then for any word of size %n, the probability of M0 making an error is %3/8.This result is independent of the protocol.

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Proof. We set the depth of the tree to t(n), as there may be at most t(n)experiments carried out in time t(n). The change in any particular probability isat most

k Zj xKx 0 j

2e;

where e is the fixed error. For any particular word w of length jwj%n, take thetree T to be the subtree of all possible experimental results which givethe incorrect answer (this does not depend on the vertex position). Then,let D2P(T ) be the probabilities on the edges given by the vertex at x, andlet D 02P(T ) correspond to the vertex at x0. The probability of an incorrectresult for the wedge at x0 is, by proposition 2.1,

PðT ;D 0Þ%PðT ;DÞC jPðT ;D 0ÞKPðT ;DÞ j% gC2tðnÞk% gC tðnÞ j xKx 0 j =3:

&

Lemma 3.5. Let M be an error-prone AD scatter machine with fixed precisioneO0, with the vertex at x, deciding some set in time t(n) with error probabilitybounded by g!1/4. Let Mn be an identical AD scatter machine, with theexception that the vertex is placed at x log2tðnÞKlog2eC5. Then, Mn decides the sameset as M, also in time t(n), but with error probability bounded by 3/8. This resultis independent of the protocol.

Proof. This is a corollary of lemma 3.4. &

Theorem 3.6. Every set decided by an error-prone AD scatter machine withfixed error (taken to be 2KN) in polynomial time is in BPP//log�.

Proof. The proof is in every way similar to the proof of theorem 2.4, usingthe advice function f ðnÞZx log2tðnÞKlog2eC5. &

Theorem 3.7. The class of sets decided by error-prone AD scatter machines withfixed precision (taken to be 2KN) in polynomial time is exactly BPP//polyZP/poly.

Proof. One-half is theorem 3.3 (see also Beggs et al. 2008a) and the other halfis theorem 3.6. &

4. Conclusion

In our earlier publications (Beggs et al. 2008a–c), we began to explore the idea ofcombining physical experiments with algorithms, by using a physical process asan oracle to a Turing machine. We examined in detail case studies where anexperimental procedure, the SME, based on Newtonian kinematics, is combinedwith a deterministic Turing machine. In Beggs et al. (2008a), we demonstratedthat the notion had both conceptual and technical merits. The physical oracleleads directly to the new concepts of protocol and three forms of AD query, inwhich physical parameters can be set exactly and approximately. In particular,one characterization theorem and two lower bound theorems for computation bythese AD machines were established using non-uniform complexity classes.

Proc. R. Soc. A (2009)

E. Beggs et al.1464

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Here, in this sequel, we have completed the classification of the computationalpower of the two types of error-prone AD Turing machines, determiningprecisely what they can compute. Combining the lower bounds of Beggs et al.(2008a) with the upper bounds here, we have the following computationalcharacterizations: for polynomial-time AD protocols,

AD scatter machines with infinite precision in polynomial time

ZAD scatter machines with finite but unlimited precision in polynomial time

ZBPP==polyZP=poly;

but, using theorem 3.3, there is a reduction in power

AD scatter machines with finite fixed precision in polynomial timeZBPP==log�:

The equivalence theorems tempt us to introduce classes for the sets computedby these new AD machines. Let E be some experiment that is suitable as anoracle. We can define

TMðE0Þð f ðnÞ; gðnÞÞ; TMðEe/0Þð f ðnÞ; gðnÞÞ; TMðEeÞð f ðnÞ; gðnÞÞ;

for Turing machines combined with the exact, error-prone with unlimitedprecision, and error-prone with fixed precision e, experiment E, operating withO(f(n)) computation time and O(g(n)) protocol time, respectively. However, atthis early stage of development, several more experiments need to be studied todetermine an appropriate theoretical framework.

The case EZSME is of general significance. We believe that the scattermachine is an important representative of a class of interesting experimentalprocedures: it has bounded space, linear time and energy and introducesbisection methods into experimental procedures. The equivalence theorems heresuggest that non-uniform complexity classes have an essential role in under-standing the computational capabilities of physical systems. Certainly, giventhat the power of the oracle, the SME, is provably beyond that of the Turingmachine, it is fortunate that the theory of non-uniform classes exists and is ableto analyse and capture this form of hypercomputation. Furthermore, in theproofs, the probabilistic Turing machines play an essential role in relatingexperiments and computers.

We believe that non-uniform complexity theory, and probabilistic algorithms,can classify the computations of a wide variety of combinations of differentexperimental and algorithmic procedures. This is supported by our new resultson the scatter machine and other experiments. Different types of Turing machinecan be coupled to the scatter machine and compared: in particular, the PZNPand similar problems can be relativized to the SME. More simply, the argumentin theorem 3.3 of this paper can be developed to show the following.

Theorem 4.1. Any probabilistic Turing machine working with time t(n) canbe simulated by an error-prone AD scatter machine with fixed precision intime O(t(n)).

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In this paper, precision plays an important role. The role of protocols will bemore prominent when combining Turing machines with a new experimentdesigned to measure mass using Newtonian collisions; this experiment indynamics has an exponential protocol (see Beggs et al. in preparation).

The research of Jose Felix Costa is supported by Fundacao para a Ciencia e Tecnologia,Financiamento Base 2008—ISFL/1/209. E.J.B. and J.V.T. would like to thank EPSRC for theirsupport under grant EP/C525361/1.

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