José Félix Costa

Analog computers and recursive functions over the reals

In this paper we show that Shannons general purpose analog computer (GPAC) is equivalent to a particular class of recursive functions over the reals with the flavour of Kleenes classical recursive function theory.We first consider the GPAC and several of its extensions to show that all these models have drawbacks and we introduce an alternative continuous-time model of computation that solves these problems. We also show that this new model preserves all the significant relations involving the previous models (namely, the equivalence with the differentially algebraic functions).We then continue with the topic of recursive functions over the reals, and we show full connections between functions generated by the model introduced so far and a particular class of recursive functions over the reals.

An analog characterization of the Grzegorczyk hierarchy

We study a restricted version of Shannons general purpose analog computer in which we only allow the machine to solve linear differential equations. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions, the smallest known recursive class closed under time and space complexity. Furthermore, we show that if the machine has access to a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n - 3 non-linear differential equations of a certain kind. Therefore, we claim that, at least in this region of the complexity hierarchy, there is a close connection between analog complexity classes, the dynamical systems that compute them, and classical sets of subrecursive functions.

Real recursive functions and their hierarchy

In the last years, recursive functions over the reals (Theoret. Comput. Sci. 162 (1996) 23) have been considered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes (Unconventional Models of Computation, UMC 2002, Lecture Notes in Computer Science, Vol. 2509, Springer, Berlin, pp. 1-14). However, one of the operators introduced in the seminal paper by Moore (1996), the minimalization operator, has not been considered: (a) although differential recursion (the analog counterpart of classical recurrence) is, in some extent, directly implementable in the General Purpose Analog Computer of Claude Shannon, analog minimalization is far from physical realizability, and (b) analog minimalization was borrowed from classical recursion theory and does not fit well the analytic realm of analog computation. In this paper, we show that a most natural operator captured from analysis--the operator of taking a limit--can be used properly to enhance the theory of recursion over the reals, providing good solutions to puzzling problems raised by the original model.

Iteration, Inequalities, and Differentiability in Analog Computers

Shannons general purpose analog computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x)?G there is a function F(x, t)?G such that F(x, t)=ft(x) for nonnegative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions xk?(x) that sense inequalities in a differentiable way, the resulting class, which we call G+?k, is closed under iteration. Furthermore, G+?k includes all primitive recursive functions and has the additional closure property that if T(x) is in G+?k, then any function ofx computable by a Turing machine in T(x) time is also.

Computational complexity with experiments as oracles

We discuss combining physical experiments with machine computations and introduce a form of analogue–digital (AD) Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of AD machine are studied, in which physical parameters can be set exactly and approximately. Using non-uniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines.

Limits to measurement in experiments governed by algorithms

We pose the following question: If a physical experiment were to be completely controlled by an algorithm, what effect would the algorithm have on the physical measurements made possible by the experiment? In a programme to study the nature of computation possible by physical systems, and by algorithms coupled with physical systems, we have begun to analyse: (i) the algorithmic nature of experimental procedures; and (ii) the idea of using a physical experiment as an oracle to Turing Machines. To answer the question, we will extend our theory of experimental oracles so that we can use Turing machines to model the experimental procedures that govern the conduct of physical experiments. First, we specify an experiment that measures mass via collisions in Newtonian dynamics and examine its properties in preparation for its use as an oracle. We begin the classification of the computational power of polynomial time Turing machines with this experimental oracle using non-uniform complexity classes. Second, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on what can be measured using equipment. Indeed, the theorems suggest a new form of uncertainty principle for our knowledge of physical quantities measured in simple physical experiments. We argue that the results established here are representative of a huge class of experiments.

Computational complexity with experiments as oracles. II. Upper bounds

Earlier, to explore the idea of combining physical experiments with algorithms, we introduced a new form of analogue–digital (AD) Turing machine. We examined in detail 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 calls was counted and three forms of AD queries were studied, in which physical parameters can be set exactly and approximately. Here, in this sequel, we complete the classification of the computational power of these AD Turing machines and determine precisely what they can compute, using non-uniform complexity classes and probabilities.

Mirror, mirror in my hand: a duality between specifications and models of process behaviour

Since Pnueli’s seminal paper in 1977, Temporal Logic has been used as a formalism for specifying and verifying the correctness of reactive systems. In this paper, we show that, besides its expressive power, Temporal Logic enjoys a very strong structural property: it is categorical on processes. That is, we show how temporal specifications (as theories) can be embedded in categories of process behaviour, and out of this adjunction we build an institution that is categorical in the sense of Meseguer. This characterisation means that temporal logic is, in a sense, ‘sound and complete’ with respect to process specification and interconnection techniques.

Analysis of a traffic model for GSM/GPRS

A traffic model for GSM/GPRS, the hybrid radio resource allocation (HRRA) algorithm is evaluated. A dedicated number of GPRS channels plus idle periods between GSM voice calls are used for GPRS data packet transfers. A simulator was developed in order to evaluate the HRRA algorithm, which provides a reasonable forecast on the voice blocking probability and on packet delay for a single cell system. Since the major issue is the correct resource allocation, results are shown for the influence of some choices and assumptions on the overall performance. As expected, blocking probability can reach very high values if the number of dedicated channels increases too much. For the specific case of 4 carries and traffic of 20 Erl, 4 channels dedicated to GPRS still enable an affordable blocking probability, leading to a mean packet delay of 15 s. The results can be used to illustrate the fundamental options that need to be taken by an operator, when implementing GPRS.

The impact of models of a physical oracle on computational power

Using physical experiments as oracles for algorithms, we can characterise the computational power of classes of physical systems. Here we show that two different physical models of the apparatus for a single experiment can have different computational power. The experiment is the scatter machine experiment (SME), which was first presented in Beggs and Tucker (2007b). Our first physical model contained a wedge with a sharp vertex that made the experiment non-deterministic with constant runtime. We showed that Turing machines with polynomial time and an oracle based on a sharp wedge computed the non-uniform complexity class P/poly. Here we reconsider the experiment with a refined physical model where the sharp vertex of the wedge is replaced by any suitable smooth curve with vertex at the same point. These smooth models of the experimental apparatus are deterministic. We show that no matter what shape is chosen for the apparatus: the time of detection of the scattered particles increases at least exponentially with the size of the query; and Turing machines with polynomial time and an oracle based on a smooth wedge compute the non-uniform complexity class P/log* ? P/poly. We discuss evidence that many experiments that measure quantities have exponential runtimes and a computational power of P/log*.