Manuel Lameiras Campagnolo

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.

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.

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature.

Robust simulations of turing machines with analytic maps and flows

In this paper, we show that closed-form analytic maps and flows can simulate Turing machines in an error-robust manner. The maps and ODEs defining the flows are explicitly obtained and the simulation is performed in real time.

The Complexity of Real Recursive Functions

We explore recursion theory on the reals, the analog counterpart of recursive function theory. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions, such as composition and various forms of differential equations. We define classes of real recursive functions, in a manner similar to the classical approach in recursion theory, and we study their complexity. In particular, we prove both upper and lower bounds for several classes of real recursive functions, which lie inside the primitive recursive functions and, therefore, can be characterized in terms of standard computational complexity.

The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation

In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPAC-computability in a natural way, we compute exactly all real computable functions (in the sense of computable analysis). Moreover, since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions can be defined by such models.

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[@p]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines. We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56. Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.

Continuous-time computation with restricted integration capabilities

Recursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuous-time computation inspired by the models of Classical Physics. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions such as composition and various forms of differential equations like indefinite integrals, linear differential equations and more general Cauchy problems. We define classes of real recursive functions in a manner similar to the standard recursion theory and we study their complexity. We prove both upper and lower bounds for several classes of real recursive functions, which lie inside the elementary functions, and can be characterized in terms of space complexity. In particular, we show that hierarchies of real recursive classes closed under restricted integration operations are related to the exponential space hierarchy. The results in this paper, combined with earlier results, suggest that there is a close connection between analog complexity classes and subrecursive classes, at least in the region between FLINSPACE and the primitive recursive functions.

Upper and Lower Bounds on Continuous-Time Computation

We consider various extensions and modifications of Shannons General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive functions, the elementary functions, the levels of the Grzegorczyk hierarchy, and the arithmetical and analytical hierarchies.

Characterizing Computable Analysis with Differential Equations

The functions of Computable Analysis are defined by enhancing the capacities of normal Turing Machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as Real Recursive Functions. Bournez and Hainry 2006 [Bournez, O. and E. Hainry, Recursive analysis characterized as a class of real recursive functions, Fundamenta Informaticae 74 (2006), pp. 409-433] used a function algebra to characterize the twice continuously differentiable functions of Computable Analysis, restricted to certain compact domains. In a similar model, Shannons General Purpose Analog Computer, Bournez et. al. 2007 [Bournez, O., M. L. Campagnolo, D. S. Graca and E. Hainry, Polynomial differential equations compute all real computable functions on computable compact intervals, Journal of Complexity 23 (2007), pp. 317-335] also characterize the functions of Computable Analysis. We combine the results of [Bournez, O. and E. Hainry, Recursive analysis characterized as a class of real recursive functions, Fundamenta Informaticae 74 (2006), pp. 409-433] and Graca et. al. [Graca, D. S., N. Zhong and J. Buescu, Computability, noncomputability and undecidability of maximal intervals of IVPs, Transactions of the American Mathematical Society (2007), to appear], to show that a different function algebra also yields Computable Analysis. We believe that our function algebra is an improvement due to its simple definition and because the operations in our algebra are less obviously designed to mimic the operations in the usual definition of the recursive functions using the primitive recursion and minimization operators.