Continuous-time Computation Research Articles

Measuring the Rate of Information Exchange in Point-Process Data With Application to Cardiovascular Variability.

The amount of information exchanged per unit of time between two dynamic processes is an important concept for the analysis of complex systems. Theoretical formulations and data-efficient estimators have been recently introduced for this quantity, known as the mutual information rate (MIR), allowing its continuous-time computation for event-based data sets measured as realizations of coupled point processes. This work presents the implementation of MIR for point process applications in Network Physiology and cardiovascular variability, which typically feature short and noisy experimental time series. We assess the bias of MIR estimated for uncoupled point processes in the frame of surrogate data, and we compensate it by introducing a corrected MIR (cMIR) measure designed to return zero values when the two processes do not exchange information. The method is first tested extensively in synthetic point processes including a physiologically-based model of the heartbeat dynamics and the blood pressure propagation times, where we show the ability of cMIR to compensate the negative bias of MIR and return statistically significant values even for weakly coupled processes. The method is then assessed in real point-process data measured from healthy subjects during different physiological conditions, showing that cMIR between heartbeat and pressure propagation times increases significantly during postural stress, though not during mental stress. These results document that cMIR reflects physiological mechanisms of cardiovascular variability related to the joint neural autonomic modulation of heart rate and arterial compliance.

The [formula omitted] conjecture in the context of real and complex analysis

In this paper, we aim at an analog characterization of the classical P ≠ NP conjecture of Structural Complexity. We consider functions over continuous real and complex valued variables. Subclasses of functions can be defined using Laplace transforms adapted to continuous-time computation, introducing analog classes DAnalog and NAnalog. We then show that if DAnalog ≠ NAnalog then P ≠ NP .

Type-2 Computability and Moore's Recursive Functions

Regarding effectivity of functions on the reals, there have been several proposed models of analog, continuous-time computation, as opposed to the digital, discrete nature of the type-2 computability. We study one of them, Moore's real (primitive) recursive functions, whose definition mimics the classical characterization of recursive functions on N by the closure properties. We show that the class of type-2 computable real functions falls between Moore's classes of primitive recursive and recursive functions.

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.

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.

Recursion theory on the reals and continuous-time computation

We define a class of recursive functions on the reals analogous to the classical recursive functions on the natural numbers, corresponding to a conceptual analog computer that operates in continuous time. This class turns out to be surprisingly large, and includes many functions which are uncomputable in the traditional sense. We stratify this class of functions into a hierarchy, according to the number of uses of the zero-finding operator μ. At the lowest level are continuous functions that are differentially algebraic, and computable by Shannon's general purpose analog computer. At higher levels are increasingly discontinuous and complex functions. We relate this μ-hierarchy to the arithmetical and analytical hierarchies of classical recursion theory.