Complexity of Classes of Structures

The main theme of this thesis is studying classes of structures with respect to various measurements of complexity. We will briefly discuss the notion of computable dimension, while the breadth of the paper will focus on calculating the Turing ordinal and the back-and-forth ordinal of various classes, along with an exploration of how these two ordinals are related in general. Computable structure theorists study which computable dimensions can be realized by structures from a given class. Using a structural characterization of the computably categorical equivalence structures due to Calvert, Cenzer, Harizanov and Morozov, we prove that the only possible computable dimension of an equivalence structure is 1 or ω. In 1994, Jockusch and Soare introduced the notion of the Turing ordinal of a class of structures. It was unknown whether every computable ordinal was the Turing ordinal of some class. Following the work of Ash, Jocksuch and Knight, we show that the answer is yes, but, as one might expect, the axiomatizations of these classes are complex. In 2009, Montalban defined the back-and-forth ordinal of a class using the back-and-forth relations. Montalban, following a result of Knight, showed that if the back-and-forth ordinal is n+1, then the Turing ordinal is at least n. We will prove a theorem stated by Knight that extends the previous result to all computable ordinals and show that if the back-and-forth ordinal is α (infinite) then the Turing ordinal is at least α. It is conjectured at present that if a class of structures is relatively nice then the Turing ordinal and the back-and-forth ordinal of the class differ by at most 1. We will present many examples of classes having axiomatizations of varying complexities that support this conjecture; however, we will show that this result does not hold for arbitrary Borel classes. In particular, we will prove that there is a Borel class with infinite Turing ordinal but finite back-and-forth ordinal and show that, for each positive integer d, there exists a Borel class of structures such that the Turing ordinal and the back-and-forth ordinal of the class are both finite and differ by at least d

Contraction and optimality properties of an adaptive Legendre-Galerkin method: the multi-dimensional case

We analyze the theoretical properties of an adaptive Legendre-Galerkin method in the multidimensional case. After the recent investigations for Fourier-Galerkin methods in a periodic box and for Legendre-Galerkin methods in the one dimensional setting, the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral/$hp$ discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in $H^1$, based on a quasi-orthonormalization procedure. This allows us to design an adaptive algorithm, to prove its convergence by a contraction argument, and to discuss its optimality properties (in the sense of non-linear approximation theory) in certain sparsity classes of Gevrey type

Stream/block ciphers, difference equations and algebraic attacks

In this paper we introduce a general class of stream and block ciphers that are defined by means of systems of (ordinary) explicit difference equations over a finite field. We call this class "difference ciphers". Many important ciphers such as systems of LFSRs, Trivium/Bivium and Keeloq are difference ciphers. To the purpose of studying their underlying explicit difference systems, we introduce key notions as state transition endomorphisms and show conditions for their invertibility. Reducible and periodic systems are also considered. We then propose general algebraic attacks to difference ciphers which are experimented by means of Bivium and Keeloq.Comment: 22 page

New developments in the theory of Groebner bases and applications to formal verification

We present foundational work on standard bases over rings and on Boolean Groebner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Groebner basis in the polynomial ring over Z/2n while the bit-level model leads to Boolean Groebner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Groebner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of Pure and Applied Algebr