Comparing Two Multivariable Complexity Functions Using One-variable Complexity Classes

The comparison of algorithms complexities is very important both in theory and in practice. When we compare algorithms complexities we need to compare complexity functions. Usually we use one-variable complexity functions. Sometimes, we need multivariable complexity func-tions. In a previous paper we defined several one-variable complexity classes for multivariable complexity functions. Each complexity class of this type is a set of multivariable complexity functions, represented by a one-variable complexity function. In this paper we continue the work from that paper: we define new one-variable complexity classes and we prove several properties. The most important results are several criteria for two multivariable complexity functions to be comparable.Algorithm, One-Variable Complexity Class, One-Variable Complexity Function, Multivariable Complexity Function, Functions Comparison

Logspace self-reducibility

A definition of self-reducibility is proposed to deal with logarithmic space complexity classes. A general property derived from the definition is used to prove known results comparing uniform and nonuniform complexity classes below polynomial time, and to obtain novel ones regarding nondeterministic nonuniform classes and reducibility to context-free languages.Peer ReviewedPostprint (published version

Strong isomorphism reductions in complexity theory

We give the first systematic study of strong isomorphism reductions, a notion of reduction more appropriate than polynomial time reduction when, for example, comparing the computational complexity of the isomorphim problem for different classes of structures. We show that the partial ordering of its degrees is quite rich. We analyze its relationship to a further type of reduction between classes of structures based on purely comparing for every n the number of nonisomorphic structures of cardinality at most n in both classes. Furthermore, in a more general setting we address the question of the existence of a maximal element in the partial ordering of the degrees

An Object-Based Approach to Modelling and Analysis of Failure Properties

In protection systems, when traditional technology is replaced by software, the functionality and complexity of the system is likely to increase. The quantitative evidence normally provided for safety certification of traditional systems cannot be relied upon in software-based systems. Instead there is a need to provide qualitative evidence. As a basis for the required qualitative evidence, we propose an object-based approach that allows modelling of both the application and software domains. From the object class model of a system and a formal specification of the failure properties of its components, we generate a graph of failure propagation over object classes, which is then used to generate a graph in terms of object instances in order to conduct fault tree analysis. The model is validated by comparing the resulting minimal cut sets with those obtained from the fault tree analysis of the original system. The approach is illustrated on a case study based on a protection system from..

Model selection applied to reconstruction of the Primordial Power Spectrum

The preferred shape for the primordial spectrum of curvature perturbations is determined by performing a Bayesian model selection analysis of cosmological observations. We first reconstruct the spectrum modelled as piecewise linear in \log k between nodes in k-space whose amplitudes and positions are allowed to vary. The number of nodes together with their positions are chosen by the Bayesian evidence, so that we can both determine the complexity supported by the data and locate any features present in the spectrum. In addition to the node-based reconstruction, we consider a set of parameterised models for the primordial spectrum: the standard power-law parameterisation, the spectrum produced from the Lasenby & Doran (LD) model and a simple variant parameterisation. By comparing the Bayesian evidence for different classes of spectra, we find the power-law parameterisation is significantly disfavoured by current cosmological observations, which show a preference for the LD model.Comment: Minor changes to match version accepted by JCA