The Spinning Cube: An Algebraic Excursion

A Short Excursion into Semi-Algebraic Hierarchies

This brief survey gives a (roughly) self-contained overview of some complexity theoretic results about semi-algebraic proof systems and related hierarchies and the strong connections between them. The article is not intended to be a detailed survey on “Lift and Project” type optimization hierarchies (cf. Chlamtac and Tulsiani [12]) or related proof systems (cf. Grigoriev et al. [16]).

Algebraic Properties of the Cube Attack

Cube attacks can be used to analyse and break cryptographic primitives that have an easy algebraic description. One example for such a primitive is the stream cipher Trivium. In this article we give a new framework for cubes that are useful in the cryptanalytic context. In addition, we show how algebraic modelling of a cipher can greatly be improved when taking both cubes and linear equivalence...

AN ALGEBRAIC STRUCTURE FOR INTUITIONISTIC FUZZY LOGIC

In this paper we extend the notion of  degrees of membership and non-membership of intuitionistic fuzzy sets to lattices and  introduce a residuated lattice with appropriate operations to serve as semantics of intuitionistic fuzzy logic. It would be a step forward to find an algebraic counterpart for intuitionistic fuzzy logic. We give the main properties of the operations defined and prove som...

An Excursion to the Kolmogorov Random Strings

We study the set of resource bounded Kolmogorov random strings: R t = fx j K t (x) jxjg for t a time constructible function such that t(n) 2 n 2 and t(n) 2 2 n O(1). We show that the class of sets that Tur-ing reduce to R t has measure 0 in EXP with respect to the resource-bounded measure introduced by 17]. From this we conclude that R t is not Turing-complete for EXP. This contrasts the resour...

An excursion into geometric analysis