Complexity with costing and stochastic oracles

Transcription

1 Complexity with costing and stochastic oracles Diogo Miguel Ferreira Poças Thesis to obtain the Master of Science Degree in Mathematics and Applications Examination Committee Chairperson: Prof. Maria Cristina Sales Viana Serôdio Sernadas Supervisor: Prof. José Félix Gomes da Costa Members of the Committee: Prof. André Nuno Carvalho Souto Prof. Luís Filipe Coelho Antunes July 2013

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3 Acknowledgements I would like to express my grattitude to my advisor, Félix Costa, for his invaluable dedication and guidance during the ellaboration of this thesis. I also want to thank him for providing me with important lessons and tips for a young mathematician doing research for the first time, and for helping me to get in touch with the scientific community on several occasions. I also want to thank Professor Amílcar Sernadas for his meta-guidance during the entirety of my MSc, and for presenting me a very realistic perspective of research and the future as an investigator. I like to thank all people in the Department of Mathematics of IST, and in the Computer Science Section, for their support and assistance provided when needed. Thanks also to Edwin Beggs, John Case and John Tucker, for their useful suggestions and appreciations during the course of this work. I want to express my grattitude to all my classmates, former classmates and colleagues that provided me with unmeasurable support and close friendship. I am also grateful to my old and recent friends in Scouting, for helping me to mantain my health of mind, body and spirit, and for their true friendship. In particular, I want to thank my friend Rafael Ribeiro also for his insights on some sections of the thesis dealing with Physics. Finally, I wish to give my very special thanks to my family, for their presence, support, interest and unconditional love. iii

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5 Resumo Pretende-se estudar nesta dissertação o poder computacional que se obtêm combinando um dispositivo digital, como a máquina de Turing, com um dispositivo analógico, como uma experiência física que serve de oráculo para a máquina. Definem-se os principais componentes destes sistemas híbridos, a que chamamos máquinas de Turing analógicas-digitais, e analiza-se o seu poder computational sujeito a recursos polinomiais. Investigam-se três tipos de oráculos motivados por experiências físicas, nomeadamente oráculos bilaterais, oráculos de limiar e oráculos de desvanecimento. O oráculo fornece informação qualitativa que auxilia a computação da máquina de Turing. Também se consideram variantes no protocolo entre a máquina de Turing e o oráculo, de modo que as questões ao oráculo sejam traduzidas para números reais com precisão infinita, precisão ilimitada ou precisão finita. Para cada caso são provadas minorações e majorações nas classes de complexidade computadas por estes sistemas. Palavras-chave: Complexidade computacional. Oráculo físico. Sistema híbrido. Teoria da medição. Complexidade não-uniforme. v

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7 Abstract The purpose of this dissertation is to study the computational power obtained when combining a digital device, such as the Turing machine, with an analog device, such as a physical experiment seen as an oracle to the machine. We define the components of such hybrid systems, which we call analog-digital Turing machine, and analyze their computational power subject to polynomial resources. We investigate three types of oracles motivated by experiments in the physical world, namely two-sided oracles, threshold oracles and vanishing oracles. The oracle provides qualitative information that helps the computation of the Turing machine. We also consider variations of the protocol between the Turing machine and the oracle, such that queries can be translated into real numbers with infinite precision, unbounded precision or finite precision. For each case we state and prove lower and upper bounds on the complexity classes computed by such systems. Keywords: Computational complexity. Physical oracle. Hybrid system. Theory of measurement. Non-uniform complexity. vii

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9 Contents Acknowledgements iii Resumo v Abstract vii List of Figures xi List of Abbreviations xiii Glossary xv Introduction 1 1 Two-sided Oracles Introdutory concepts On Turing machines On physical experiments On the interface Examples of two-sided experiments The Sharp Scatter Machine Experiment The Smooth Scatter Machine Experiment The Balance Scale Experiment Characterization of protocols in terms of precision Description and decidability of the AD machine The role of measuring algorithms Feasibility of the Physical Experiments Lower bounds on the AD machines for the SME The AD machine as a means to measure real numbers The AD machine as a means to produce fair coin tosses Some non-uniform complexity classes Encoding advice functions as wedge positions Main results Upper bounds on the AD machines for the SME The boundary numbers Probabilistic trees Main results Threshold Oracles Examples of threshold experiments The Spinal Neuron Experiment The Rutherford Scattering Experiment The Photoelectric Effect Experiment ix

10 2.1.4 The Broken Balance Experiment Characterization of protocols Lower bounds on the AD machine for the BBE The AD machine as a means to measure real numbers The AD machine as a means to produce fair coin tosses Main results Upper bounds Boundary numbers revisited Probabilistic trees revisited Upper bounds using boundary numbers as advice Upper bounds assuming explicit time Final remarks Vanishing Oracles Examples of vanishing experiments The Brewster Angle Experiment The Vanishing Balance Experiment Characterization of protocols Lower bounds on the AD machine for the VBE Measuring with the AD machine with infinite precision Measuring with the AD machine with unbounded precision Measuring with the AD machine with fixed precision Tossing coins with the AD machine Main results Upper bounds on the AD machine for the VBE Upper bounds in the first implementation Upper bounds in the second implementation Conclusion 77 Bibliography 81 A Additional proofs 84 A.1 Experimental time for the SmSME A.2 Experimental time for the BBE A.3 Experimental time for the BAE A.4 Generation of coin tosses A.5 BPP/log versus BPP//log A.6 On the size of logarithmic advices B Changing the Schedule 92 C Table of Bounds 95 Index 97 x

11 List of Figures 1.1 The sharp scatter machine experiment The smooth scatter machine experiment The balance scale experiment Protocol Shoot(IP ) Protocol Shoot(UP ) Protocol Shoot(F P, ɛ) Algorithm BinarySearch(IP ) Algorithm BinarySearch(UP ) Algorithm FreqCount(F P, ɛ, h) Behaviour of the oracle Turing machine M The boundary numbers Example of a probabilistic tree Induction step in Proposition The spinal neuron experiment The Rutherford scattering experiment The photoelectric effect experiment The broken balance experiment Protocol Mass(IP ) Protocol Mass(UP ) Protocol Mass(F P, ɛ) Algorithm BinarySearch(IP ) Algorithm BinarySearch(UP ) Algorithm FreqCount(F P, ɛ, h) The boundary numbers Induction step in Proposition The Brewster angle experiment The vanishing balance experiment Protocol Compare(1, IP ) Protocol Compare(2, IP, g) Algorithm BinarySearch(1, IP ) Algorithm BinarySearch(2, IP, g) Algorithm BinarySearch(1, UP ) Algorithm BinarySearch(2, UP, g) Algorithm FreqCount(P, h) Algorithm Simulate(1, IP ) Algorithm Simulate(1, UP, s) xi

12 3.12 Regions of a particular experiment Algorithm Simulate(1, F P, ɛ, σ, h) Algorithm Random(h) Algorithm Simulate(2, IP, g, h) Algorithm Simulate(2, UP, g, s, h) Regions of a particular experiment Algorithm Simulate(2, F P, ɛ, g, σ, h) A.1 The smooth scatter machine experiment A.2 The broken balance experiment A.3 The Brewster angle experiment xii

13 List of Abbreviations δ, γ, p, q probabilities, that is, real numbers in [0, 1] E ɛ I family of experiments fixed precision set of initial conditions E, V expected value and variance R τ T set of outcomes physical constant or experimental constant probabilistic tree ξ, e natural constants, usually in defining advice functions a, y unknown mass or unknown value b, c, d, k natural constants, usually as exponents BP P class of sets decidable by a probabilistic Turing machine in polynomial time C, D, E, F natural constants, usually multiplicative or additive C 3 the set of Cantor numbers f, f, g, g advice functions M M E,T,P Turing machine or oracle Turing machine analog-digital Turing machine n, l the size of a word in {0, 1} P T class of sets decidable by a deterministic Turing machine in polynomial time time schedule T exp experimental time function (arity 2) t exp experimental time function (arity 1) z, z 1, z 2, m, x 0, x 1, x 2, x 3, x 4 dyadic rationals z n cropping or padding of a dyadic rational z k, l k, r k boundary numbers, which are reals in (0, 1) xiii

14 AD machine analog-digital Turing machine BAE BBE BSE FP(ɛ) IP PEE RSE Brewster angle experiment broken balance experiment balance scale experiment fixed precision ɛ infinite precision photoelectric effect experiment Rutherford scattering experiment ShSME sharp scatter machine experiment SME scatter machine experiment SmSME smooth scatter machine experiment SNE UP VBE spinal neuron experiment unbounded precision vanishing balance experiment xiv

15 Glossary C n Given an open set U R and a function f : U R and a natural n, f is said to be of class C n if all derivatives f,..., f (n) exist and are continuous. O For a given function g over the natural numbers, we define O(g) as the class of functions f over the natural numbers such that there exists positive constants M and n 0 such that f(n) Mg(n) for all n > n 0. Ω For a given function g over the natural numbers, we define Ω(g) as the class of functions f over the natural numbers such that there exists positive constants M and n 0 such that f(n) Mg(n) for all n > n 0. ω For a given function g over the natural numbers, we define ω(g) as the class of functions f over the natural numbers such that for any positive real number ɛ there exists a positive constant n 0 such that f(n) ɛg(n) for all n > n 0. o For a given function g over the natural numbers, we define o(g) as the class of functions f over the natural numbers such that for any positive real number ɛ there exists a positive constant n 0 such that f(n) ɛg(n) for all n > n 0. dyadic rational A dyadic rational is a number of the form n 2 k where n is an integer and k a natural number. Any dyadic rational in [0, 1) has a unique binary expansion of the form 0.a 1 a 2 a k, a i {0, 1}. xv

16 tally set time constructible A tally set or unary set is a subset of 0, that is, a set whose words have the form 0 k. A function f : N N is said to be time constructible if there is a deterministic Turing machine M and a natural n 0 such that for any input word z of size z = n > n 0 the machine M halts after exactly f(n) steps. xvi

17 Introduction The main focus of computability and complexity theories is to study the limits of what can be computed or done in a feasible way. The main tool for studying these objects is the algorithm a sequence of instructions, executable in an unambiguous way by a machine, allowing to decide, for example, membership to a given set. A property of these algorithms is that they operate on a discrete set of data, using unbounded (yet finite) resources. Physical experiments, on the other hand, operate on a real world, which can be seen as continuous. However, the physical experiment can also be seen as a sequence of instructions, departing from an initial set of data and materials, and reaching an experimental result. The idea that physical experimentation can be seen as algorithmic is nicely presented in [GH86], where the notion of measurable number is proposed in parallel with the notion of computable number. In this dissertation we intend to bring together computability theory and physical experiments. We may imagine a model of computation based on a Turing machine with the possibility of performing physical experiments during a computation. We call this model the analog-digital Turing machine; the digital component is given by the Turing machine; the analog component is given by the physical experiment. The thesis of analog computation preceded digital computation. The first analog computers were used to solve differential equations and provided a great help in gunfire control during World War II. However, digital computation emerged and improved greatly in the last half century, surpassing analog computation in present days. Despite this, analog computation is regaining interest. Several examples of recent results include the optical computer for factorization ([Bla07]), the optical computer of Woods, Naughton ([WN05]), the dynamic and hybrid systems of Koiran, Cosnard, Garzon ([KCG94]) and Bournez ([BC96]) and also the neural networks of Siegelmann and Sontag ([Sie99, Sie95, SS95]). Most notably, in the work of Siegelmann and Sontag an analogy to the Church-Turing thesis was proposed, stating that analog devices do not extend polynomial-time computation beyond the non-uniform limit of P/poly. We found that a good way to articulate the interaction between the Turing machine and the physical experiment is the theory of relative computation, i.e, computation with oracles. We can make an analogy between this theory and the activity of a scientist in a laboratory - the Turing machine is the scientist, following a sequence of instructions; the oracle is the physical experiment, aiding the scientist. In modelling the physical experiment as an oracle, we are in fact implying that this dissertation could be seen as a study of the classes of sets decidable by some particular families of oracles. We could in fact make this study abstracting completely from the concept of physical experiment. However, the physical experiment is useful, not only as a motivating example of the theory, but also as a help in the understanding of the proof and techniques used. In addition, most of the results obtained are valid for a general family of oracles, including but not limited to the physical experiment that we use as example. The oracles used for representing physical experiments possess certain features: they model certain binary relations; they possess a computational consultation cost; and they may be stochastic. The two latter features represent a shift from the usual definition of oracles. In this dissertation all the information 1

18 relevant to one particular oracle may be condensed in a real number; in this way, we are basically working with a family of analog-digital devices that is parametrized by one real value. We now present a summary of the development of the theory of analog-digital Turing machines. In [BT07], Beggs and Tucker present the Scatter Machine Experiment, an experiment in Newtonian kinematics that can measure a real number encoding the position of a triangular wedge. Later in [BCLT08, BCLT09], Costa suggested a new idea and introduced the concept of analog-digital Turing machine, coupling the Scatter Machine Experiment with a Turing machine. It was shown that these machine could decide sets in P/poly or BPP//log, depending on the precision considered. For the case of infinite precision they have thus obtained the same class as Siegelmann and Sontag did with their neural networks. In [BCT10b] a new type of experiment was studied, the Collider Machine Experiment. This experiment diverged from the previous experiment in a fundamental way. The time taken for an experiment is not constant, and increases with the number of desired digits of precision. Thus, the concept of time schedule was introduced to deal with the growing costs of the experiment. As a consequence, the lower bounds of decidable sets dropped from P/poly to P/log and BPP//log, depending on the precision considered. In further papers [BCT12b] new experiences were studied with the same results. The above experiments could be characterized as two-sided, in the sense that we could approach the unknown value either by above or below. For this reason the techniques used were essentialy the same. However, in [BCT10d] two new kinds of experiments came across, namely threshold experiments (exemplified by the Rutherford Experiment) and vanishing experiments (exemplified by the Brewster Angle Experiment). It was hinted that these experiments could boost the computational power of a Turing machine to P/log and BPP//log, but for these experiments the previous techniques were not applicable. The main contributions from this work are as follows. First, a general form of the techniques for proving lower bounds is presented. The general formalism can be applied to every type of experiment that was studied so far. Second, more robust upper bounds are proved for the threshold experiments, using a new technique based on a different choice of advice. The new technique generalizes for twosided experiment, for which the former techniques were only applicable for polynomial cost experiments (such as the Sharp Scatter Machine) and for a very specific case of exponential cost experiments (seen in [BCT12a]). Third, the case of vanishing experiments is analised and the respective protocols are defined with rigour, with an additional degree of (im)precision, the time tolerance. For the case of upper bounds yet another technique is used that differs from the previous types. This dissertation is organized as follows. In Chapter 1 we study two-sided oracles and present the main concepts that are used throughout the thesis. We consider the Smooth Scatter Machine as our canonical model of two-sided experiments and provide lower and upper bounds. In Chapter 2 we focus on threshold oracles. We introduce the Broken Balance Experiment as our canonical model and we also present the new technique for proving upper bounds. Chapter 3 is devoted to vanishing oracles. In this chapter the canonical model is the Vanishing Balance Experiment. We present time precision and study its implications in the algorithms of measurement. We also provide lower and upper bounds for this class of experiments. The work done in this dissertation is the answer to two open problems of the research project of my advisor José Félix Costa, namely to determine the upper bounds of computational power of Turing machines coupled with oracles arising from deterministic and stochastic measurement of physical quantities of (a) a threshold type and (b) a vanishing type. 2

19 Chapter 1 Two-sided Oracles Since our objective is to analyze the power gained using certain oracles, we should begin by studying the most natural type of these oracles. As we will see, this corresponds to two-sided experiments. In this chapter we will also provide the introduction to this theory of coupling Turing machines with physical experiments. The chapter will follow recent paper on the subject such as [BCLT08, BCT10b, BCT12a, BCT12b]. 1.1 Introdutory concepts Our main object of study will be what we can define as an analog-digital Turing machine, or AD machine. So we must attempt to characterize such types of machines, and then characterize the complexity classes decided by such machines. According to [BCT12a], there are three important components of an analog-digital Turing machine: analog-digital Turing machine = Turing machine + physical experiment + interface. Thus we should describe each component individually On Turing machines Here we will just state the properties that our Turing machines will satisfy, mostly following [BCT10b]. The Turing machines that we consider are equipped with several tapes: one input tape for input and several work tapes for performing calculations. The control unit of our machine also has at least three special states to begin and end the computation: these are called the initial state, the accepting state and the rejecting state, respectively. The above means that we will be using our Turing machines mostly for the purpose of deciding sets. Ocasionally, very rarely, we will use a Turing machine to compute some value; in those sporadic cases we will replace the two last states by an halting state and we will equip the machine with an extra output tape for returning the result of the computations. Our Turing machines will be either deterministic, that is, with only one transition function, or probabilistic with two transition functions. The Turing machines that we will use in AD machines have additional properties: they will have one query tape for the purpose of instructing the physical experiment, and a finite amount of states that are used for interacting with the physical experiment: one of these is called the query state, and the others refer to each possible outcome of the experiment. For the time being we consider three outcomes, and thus three additional states: the yes state, the no state and the timeout state. We shall call oracle 3

20 Turing machines to these machines, but often we may refer to them as simply Turing machines. Observe that there is a difference between what we call oracle Turing machine and the usual definition; in the usual definition an oracle Turing machine has only two additional states, the yes state and the no state, which represent the possible answers of a query to the oracle (which is a set). However, in our definition the number of additional states is arbitrary (but finite) and equals the number of possible outcomes On physical experiments In this work we will take a look at a lot of different physical experiments. There are many features that they have in common, namely: some initial conditions that can be tuned to some specific values; a physical process, depending on the initial conditions, which takes a (possibly infinite) amount of time; a finite set of possible results or outcomes. Thus we will say that a physical experiment is completely characterized by its set of initial conditions I, its set of outcomes R, its experimental time function t exp : I R and its outcome function r : I R. On a later section we will see examples of physical experiments On the interface The interface between the Turing machine (digital) and the physical experiment (analog) has two main components. First, we must describe the protocol, that is, the sequence of instructions that defines a physical experiment. The protocol should begin by reading a query word from the query tape, and it should end by resuming the computation of the Turing machine in a particular state. Second, we must state a time schedule, first mentioned in [BCT10b]. In most cases we will want some way to interrupt a physical experiment that has been going on for too much time. We introduce the concept of a time schedule, that is used to specify the time allowed for the experiment. Definition 1.1. A time schedule T is a time-constructible function T : N N. This condition can be weakened, that is, we could require only that T is computable and that T (n) n. The reason is that any computable function f can be majorized by some time-constructible function f ; see [BDG95]. 1.2 Examples of two-sided experiments In this section we will present our first model of physical experiment, the Scatter Machine Experiment (or SME for short), in its two variants (sharp and smooth). This should be useful to clarify the components introduced in the previous section, and to get an insight on the rest of the work. We will also present another example of a physical experiment, the Balance Scale Experiment The Sharp Scatter Machine Experiment This experiment is already explained in detail in [BT07] and [BCLT08]. The physical apparatus consists of a cannon for shooting particles, a barrier in the shape of a wedge and two collecting boxes, as in Figure 1.1. We can place the wedge at any position in the interval [0, 1], but we will assume that it is fixed for the duration of the experiment. The cannon can be used to find the position of the wedge up to some accuracy. The wedge has a triangular shape, and the sides are at 45 to the line of the cannon, so that the particle shot by the cannon is deflected perpendicular to its original direction. 4

21 RIGHT COLLECTING BOX x 1 cannon 1 5 m y 0 w 10 m/s z 0 limit of traverse of point of wedge limit of traverse of cannon cannon aims at dyadic z [0, 1] LEFT COLLECTING BOX 5 m Figure 1.1: Schematic depiction of the sharp scatter machine. To perform the experiment, we set the cannon to some position in [0, 1] and then shoot a particle with initial velocity 10ms 1. A different velocity will correspond to a different waiting time. Depending on the position of the wedge, y, and the position of the cannon, z, we have three possible behaviours: if z < y, then the particle hits the left side of the wedge and enters the left collecting box; if z > y, then the particle hits the right side of the wedge and enters the right collecting box; if z = y, then we assume that the particle can be deflected to any of the boxes or it can be deflected elsewhere and not enter either box. Other assumptions could be made about the behaviour, depending on how we would prefer to model the experiment, but they would be largely irrelevant to the complexity classes. In the cases that z y, we can see that the particle enters each box after at most 1s. Thus the experimental time is bounded by a constant. Let us identify the event particle is detected in the left collecting box with the outcome LEFT and the event particle is detected in the right collecting box with the outcome RIGHT. Definition 1.2 (Beggs et al., [BCLT08]). For y [0, 1], we denote by ShSM E(y) the sharp scatter machine experiment with wedge at position y, which is defined by the following properties: initial conditions I = [0, 1] and set of outcomes R = {LEFT, RIGHT}; experimental time function t exp defined on I {y} such that t exp (z) < 1 for all z I {y}; outcome function r defined on I {y} such that { r(z) = LEFT if z < y RIGHT if z > y. We also denote by ShSME the family of experiments ShSME = {ShSME(y) : y [0, 1]} The Smooth Scatter Machine Experiment The sharp SME made its first appearance in [BT07] as a machine that could be used as a form of hypercomputation. Later on, it was considered as a physical oracle in [BCLT08]. It was seen that 5

22 Turing machines combined with the sharp SME as oracle computed in polynomial time the class P/poly. However, following case studies have given rise to different complexity classes. One of these, the smooth SME (first seen in [BCT12b]), can be seen as a modification of the previous experiment, but its corresponding complexity class is quite different. The unique difference from the sharp SME to the smooth SME is that the wedge does not have a triangular shape. Instead, the sharp wedge is replaced by a smooth curve with vertex at the same point. See Figure 1.2 for more details. RIGHT COLLECTING BOX φ x 1 φ cannon z 1 5 m y 0 w V 10 m/s sample trajectory 0 limit of traverse of point of wedge limit of traverse of cannon cannon aims at dyadic z [0, 1] LEFT COLLECTING BOX 5 m Figure 1.2: Schematic depiction of the smooth scatter machine. We assume that the shape of the wedge is given by a map w : R R with the following properties: w is C n in ( 1, 1), for some n 2; w has a unique maximum at x = 0, and w(0) = 1; w(x) = 0 for all x ( 1, 1); and w (n) (0) 0. The wedge is shifted so that the position of the maximum is located at some unknown value y [0, 1] where 0 is the zero line of the cannon. To perform the experiment, the cannon is set to some position z [0, 1] and shoots a particle. The particle will collide with the wedge at the point with x coordinate x = z y, with the following behaviour: if x < 0, then the particle will be reflected and enters the left collecting box; if x > 0, then the particle will be reflected and enters the right collecting box; if x = 0, then the particle will be reflected in the direction it came from and thus will not enter any box. Observe that, as opposed to the sharp scatter, the time taken for the particle to enter some box, when z y, will increase to infinity as z approaches y. A majorization of this time can be found in the Appendix. For the moment, let us identify the event particle is detected in the left collecting box with the outcome LEFT and the event particle is detected in the right collecting box with the outcome RIGHT. Then we can proceed with the formalism of the smooth SME. Definition 1.3 (Beggs et al., [BCT12b]). For y [0, 1], we denote by SmSME(y) the smooth scatter machine experiment with wedge at position y, which is defined by the following properties, for some constants C, D and n: initial conditions I = [0, 1] and set of outcomes R = {LEFT, RIGHT}; experimental time function t exp defined on I {y} such that D z y n 1 < t C exp(z) < for all z I {y}; z y n 1 6

23 outcome function r defined on I {y} such that { r(z) = LEFT if z < y RIGHT if z > y. We also denote by SmSME the family of experiments SmSME = {SmSME(y) : y [0, 1]} The Balance Scale Experiment The balance scale experiment was mentioned for the first time in [BCT09]. It consists of a balance scale with two pans (see Figure 1.3). In the right pan we have some body with an unknown mass a. R h z O a Figure 1.3: Schematic depiction of the balance scale experiment. To measure a we place test masses z on the left pan of the balance: if z < a, then the right pan of the scale will move down; if z > a, then the left pan of the scale will move down; if z = a, then we assume that the scale will not move since it is in equilibrium. The balance experiments will be thorougly discussed in Chapters 2 and 3. For now we will just present the definition of the BSE, alerting the reader that a derivation of the experimental time (for z > a) can be found in the Appendix. Definition 1.4. For a [0, 1], we denote by BSE(a) the balance scale experiment with unknown mass a, which is defined by the following properties, for some constant τ: initial conditions I = [0, 1] and set of outcomes R = {LEFT, RIGHT}; experimental time function t exp defined on I {a} such that t exp (z) = τ z+a z a for all z I {a}; outcome function r defined on I {a} such that r(z) = { RIGHT if z < a LEFT if z > a. We also denote by BSE the family of experiments BSE = {BSE(a) : a [0, 1]}. 1.3 Characterization of protocols in terms of precision In this section we dwell further into the concept of protocol, and see that different assumptions on the type of precision can be described as different protocols. 7

24 From this section on we focus on the scatter machine experiment as our canonical model of twosided experiments. To use that experiment in a computation, we would like to set the position of the cannon, then shoot the cannon and find out which box (if any) did the particle went through. In the classical theory of computation with oracles, we have the concept of queries, which are possible words that can be asked to the oracle. In our case, the query word will correspond to a dyadic rational in [0, 1] that represents a possible initial condition of the experiment. Thus all protocols will attempt to perform the experiment with test position z, where z is the dyadic rational corresponding to the query word. However, we may allow some imprecision in the initial conditions; for instance, we could attempt to perform the experiment with initial condition z but a different z is used instead. This is our way to incorporate imprecision in the system. As such, there are three types of precision that we will study in these experiments: infinite precision: when z is read in the query tape, the cannon is set to position z; unbounded precision: when z is read in the query tape, the cannon is set to position z, where z is randomly and uniformly sampled in ( z 2 z, z + 2 z ) ; fixed precision ɛ, for some real valued ɛ > 0: when z is read in the query tape, the cannon is set to position z, where z is randomly and uniformly sampled in (z ɛ, z + ɛ); Let us look at the case of infinite precision. The experiment will basically consist of setting the cannon, shooting a particle, waiting some amount of time, and examine the outcome of the experiment. And so the experiment is abstracted by means of the procedure Shoot(IP ), which behaves as in Figure 1.4. PROTOCOL SHOOT(IP ) 1. Input dyadic rational z (possibly padded with 0s); 2. Set the cannon to the position z; 3. Shoot a particle; 4. Wait T ( z ) units of time; 5. If the particle was detected in the left box, Then Return LEFT, Else If the particle was detected in the right box, Then Return RIGHT, Else Return TIMEOUT. Figure 1.4: Procedure that describes the SME with infinite precision, for some unknown position of the wedge y and some time schedule T. PROTOCOL SHOOT(UP ) 1. Input dyadic rational z (possibly padded with 0s); ) 2. Set the cannon to the position z, where z (z 2 z, z + 2 z ; 3. Shoot a particle; 4. Wait T ( z ) units of time; 5. If the particle was detected in the left box, Then Return LEFT, Else If the particle was detected in the right box, Then Return RIGHT, Else Return TIMEOUT. Figure 1.5: Procedure that describes the SME with unbounded precision, for some unknown position of the wedge y and some time schedule T. The other types of precision can then be seen as just different protocols for the same experiment, which we will call Shoot(UP ) and Shoot(F P, ɛ) for the unbounded and fixed precision. Notice also that, 8

25 PROTOCOL SHOOT(F P, ɛ) 1. Input dyadic rational z (possibly padded with 0s); 2. Set the cannon to the position z, where z (z ɛ, z + ɛ); 3. Shoot a particle; 4. Wait T ( z ) units of time; 5. If the particle was detected in the left box, Then Return LEFT, Else If the particle was detected in the right box, Then Return RIGHT, Else Return TIMEOUT. Figure 1.6: Procedure that describes the SME with fixed precision ɛ, for some unknown position of the wedge y and some time schedule T. in the two latter cases, the answer to a query may be probabilistic instead of deterministic. Indeed, if z is close enough to the unknown position y, there may be different values of z corresponding to different answers, and so the same experiment may provide more than one possible outcome. There are many examples in the literatures of machines working with imprecision. For example, in [von56] a study is made of finite automata with faulty, probabilistic gates and how to control the errors. Also, [BK68] describes the main components of a hybrid computer, in particular the digital-analog converters that display some similarities with our types of precision. 1.4 Description and decidability of the AD machine The goal of this section is to combine all the concepts previously discussed in order to properly define the behaviour of an AD machine. In the general case, consider some physical experiment E, some oracle Turing machine M with at least as many outcome states as the possible outcomes of the experiment, some protocol P and some time schedule T. Then M E,T,P denotes the corresponding AD machine for that experiment. In the case of the smooth scatter, for example, the physical experiment is identified by the unknown position of the wedge, y, the oracle Turing machine M has outcome states left, right and timeout, and the protocol P is one of IP, UP, or F P (ɛ). Then the corresponding AD machine could be denoted by the expression M SmSME(y),T,P. However, since for the rest of this chapter we shall focus only on AD machines for the smooth SME, we may simplify the subscript in M SmSME(y),T,P and write only M y,t,p for an AD machine coupled with the smooth SME. The behaviour of the above machine is determined by its transition relation, which can be defined as follows. At any step of the computation, if the current state of the machine is different from the query state, the accepting state or the rejecting state, then the next configuration of the machine is determined by its transition function, as in the usual sense of deterministic Turing machines; if the current state is the query state, then the machine executes an instance of the experiment. It follows protocol P, using the query word in the query tape, z, as input and T as the time schedule. After some amount of time T ( z ), it resumes the computation on one of the states left, right or timeout, depending on the answer of the protocol; if the current state is the accepting (rejecting) state, then the computation is terminated and we say that the machine accepts (rejects) its input word. Finally, we can define the notion of decidability of a set by an AD machine coupled with the smooth scatter machine experiment. To do that, we must split the definition for the infinite precision case and the other two types of precision. Definition 1.5 (adapted from Beggs et al., [BCT10a, BCT12b]). Let M y,t,ip be an AD machine with unknown position y, time schedule T and infinite precision. We say that M y,t,ip halts for the input word 9

26 x if the respective computation reaches a halting state. We also say that M y,t,ip decides a set A if it halts for all input words and, for any input word w: if w A, then M y,t,ip accepts w; and if w A, then M y,t,ip rejects w. Finally, we say that a set A is decidable by an infinite precision smooth scatter AD machine in polynomial time if there is an oracle Turing machine M, an unknown position y and a time schedule T such that M y,t,ip decides A and runs in polynomial time. Definition 1.6 (adapted from Beggs et al., [BCT10a, BCT12b]). Let M y,t,up (resp. M y,t,f P (ɛ) ) be an AD machine for unknown position y, time schedule T and unbounded precision (resp. fixed precision ɛ). We say that M y,t,up (resp. M y,t,f P (ɛ) ) halts for the input word x if all computations reach a halting state. We also say that M y,t,up (resp. M y,t,f P (ɛ) ) decides a set A if it halts for all input words and there is some 0 < γ < 1/2 such that, for any input word w: if w A, then M a,t,up (resp. M a,t,f P (ɛ) ) accepts w with probability at least 1 γ; if w A, then M a,t,ip (resp. M a,t,f P (ɛ) ) rejects w with probability at least 1 γ. Finally, we say that a set A is decidable by an unbounded precision (resp. fixed precision ɛ) smooth scatter AD machine in polynomial time if there is an oracle Turing machine M, an unknown position y and a time schedule T such that M y,t,up (resp. M y,t,f P (ɛ) ) decides A and runs in polynomial time. 1.5 The role of measuring algorithms In this section we discuss the first and most obvious use of physical experiments in a computation: to measure a real number. As a byproduct we will present an algorithm for our first AD machine. All of the oracles that we consider in this work were first conceived as physical experiments of measurement; this means that the experiment was designed to find the unknown value of a physical quantity, such as the position of a wedge, the mass of a body, the threshold frequency for photoelectric emission of a material, the Brewster angle of a surface, etc. For a discussion on measurable quantities the reader is invited to consult [GH86]; the references [Hem52, Car66, Sup51] are relevant in studying theories of measurement. When examining the SmSME for the first time, the most obvious use of it is to measure the value of the unknown wedge position by means of a binary search procedure; initially send a particle from position 1/2 and, depending on the result, we can deduce that the unknown wedge position is in either [0, 1/2] or [1/2, 1], depending on the answer of RIGHT or LEFT. 1 By repeating the process, we could obtain consecutive approximations to the unknown wedge position y. We can formalize this idea for any family of experiments E. When we write E we are referring to a class of experiments parametrized by some physical quantity, for example E = SmSM E is the class of all experiments SmSME(y), y (0, 1). When we couple a family of experiments E with a protocol P we obtain the pair (E, P ) which we call setup from now on. In the following definition we adopt the terminology of oracle Turing machines with an output tape and one halting state. Definition 1.7. We say that setup (E, P ) permits measurements in time U if there is an oracle Turing machine M and a time schedule T such that, for any real number r (0, 1) there exists E E such that for all size n, M E,T,P halts for input word 1 n and the content of the output tape is a dyadic rational z such that z r < 2 n ; the number of steps of M E,T,P is bounded by a function in O(U). 1 Of course, this is not the full story; one has to deal with the answer of TIMEOUT as well. 10

27 Definition 1.8. We say that setup (E, P ) permits probabilistic measurements in time U if for every real number γ (0, 1/2) there is an oracle Turing machine M and a time schedule T such that, for any real number r (0, 1) there exists E E such that for all size n, every computation of M E,T,P for input word 1 n halts and, with probability of failure at most γ, the content of the output tape is a dyadic rational z such that z r < 2 n ; the number of steps of M E,T,P is bounded by a function in O(U). Thus, the goal of this section is to show that the setup (SmSME, IP ) permits measurements, and discover how much time (most of the cases polynomial or exponential) is required. We will answer this question shortly by presenting a measuring algorithm. We will use the notation z n, where z {0, 1} ω is an infinite word to indicate the prefix of size n of z. Furthermore, when z {0, 1} is a finite word we use z n to denote the prefix of size n of the word z0 ω. That is, for example, = 1010, = , and = 110. We shall use a time schedule T such that T (l) = C2 (n 1)l, where C and n are the constants related to the smooth SME of Definition In this way the procedure in Figure 1.7 can obtain approximations to the unknown mass. This is a modification to an algorithm previously presented in [BCT10b]. ALGORITHM BINARYSEARCH(IP ) 1. Input a natural number l number of places to the right of the left leading 0; 2. x 0 := 0; m := 0; x 1 := 1; 3. While x 1 x 0 > 2 l Do Begin x0 + x m := ; s := Shoot(IP ) (m l ); %Recall that this step takes T (l) units of time 3.3. If s = LEFT, Then x 0 := m; 3.4. If s = RIGHT, Then x 1 := m; 3.5. If s = TIMEOUT, Then x 0 := m; x 1 := m; 4. End While; 5. Output the dyadic rational denoted by x 0. Figure 1.7: Procedure to obtain an approximation of an unknown wedge position y of the scatter machine. We can see that in algorithm BinarySearch(IP ) the protocol is called in instruction 3.2 with position m l. We pad m with zeros to ensure that we wait the same amount of time for all experiments. In this way, we do not get the result of timeout too early. Observe also that the input l does not necessarily correspond to the precision of the output in terms of the unknown wedge position. Running the above algorithm for input l only guarantees that the result of the algorithm is a dyadic rational of size l. We will see that the distance between the output and the value of the unknown wedge position can be bounded depending on the choice of the time schedule. For the moment we shall state some nice properties of the above algorithm. Proposition 1.1. Let s be the result of Shoot(IP )(m l ), for an unknown wedge position y and time schedule T = λl.c2 (n 1)l. Then, 2 The reader should understand that the choice of this particular time schedule serves to simplify some of the following proofs. In fact, any exponential time schedule would suffice to produce good approximations, possibly with a polynomial slowdown. 11

28 1. If s = LEFT, then m < y; 2. If s = RIGHT, then m > y; 3. If s = TIMEOUT, then y m < 2 l. Proof: Cases 1 and 2 are obvious. For case 3, observe that we get a timeout whenever the experimental time t exp (m) exceeds the scheduled time T (l). This means that (using Definition 1.3) C m y n 1 > t exp(m) T (l) = C2 (n 1)l m y < 2 l. Proposition 1.2. For any unknown wedge position y and time schedule T = λl.c2 (n 1)l, 1. The time complexity of the algorithm BinarySearch(IP ) for input l is O(lT (l)); 2. The output of algorithm BinarySearch(IP ) for input l is a dyadic rational m such that y m < 2 l. Lemma 1.3. The setup (SmSM E, IP ) permits measurements in exponential time. Proof: This is a consequence of Proposition 1.2. We consider the oracle Turing machine that for input 1 l makes a call to BinarySearch(IP )(l) and the time schedule T = λl.c2 (n 1)l. For any real value r, we take SmSM E(r) as the intended experiment. By Proposition 1.2, the output of the computation is a dyadic rational m such that r m < 2 l. Since T (l) O(2 (n 1)l ), the number of steps of the AD machine is bounded by a function in O(l2 (n 1)l ) O(2 nl ). 1.6 Feasibility of the Physical Experiments Some readers may raise objections to the feasibility of the above experiments. One could argue that any computational power gained by coupling these physical experiments (for example, uncomputable power) is simply the result of the (un)computable character of the real number that denotes the wedge position, and that it would be impossible to place this wedge on any position in a finite amount of time. It could also be said that the measuring algorithm is itself infeasible, since the binary search procedure would require the ability to place the cannon at any dyadic position, regardless of the dyadic rational size. However, for query words with just a couple dozens of digits the precision required to do so will surpass the subatomic level, and thus it would be impossible to place the cannon with such accuracy. In this section we will attempt to provide an answer to these questions. First, we observe that this experiment is just a model of computation. We do not wish to imply that we should attempt to build AD machines to improve the computational power. Actually, what we can do is to obtain constraints to the classes of sets that we could compute with such machines. Regarding the fact that it is impossible to place the wedge on any real position in a finite amount of time, we will observe that for any halting computation only a finite amount of the digits of the wedge position are necessary to define the computation path, and so for input words bounded by some size n a similar machine with the wedge shifted by some small amount would accept and reject those words in the same way as the original machine did. Second, we observe that the computational power that we gain is also limited. We will find upper bounds for the classes of sets decided by AD machines in polynomial time, and we will see that we cannot get past the barrier of P/poly. So it makes sense to study this kind of machines since they can at least provide us some negative results concerning the computational power gained in polynomial time. 12