Green's Relations in Finite Transformation Semigroups

We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.Comment: Full version of a paper submitted to CSR 2017 on 2016-12-1

Effective Theories for Circuits and Automata

Abstracting an effective theory from a complicated process is central to the study of complexity. Even when the underlying mechanisms are understood, or at least measurable, the presence of dissipation and irreversibility in biological, computational and social systems makes the problem harder. Here we demonstrate the construction of effective theories in the presence of both irreversibility and noise, in a dynamical model with underlying feedback. We use the Krohn-Rhodes theorem to show how the composition of underlying mechanisms can lead to innovations in the emergent effective theory. We show how dissipation and irreversibility fundamentally limit the lifetimes of these emergent structures, even though, on short timescales, the group properties may be enriched compared to their noiseless counterparts.Comment: 11 pages, 9 figure

Finding All Solutions of Equations in Free Groups and Monoids with Involution

The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups as well as the set of all solutions of equations in free monoids with involution in the presence of rational constraints. This became possible due to the recently invented emph{recompression} technique of the second author. He successfully applied the recompression technique for pure word equations without involution or rational constraints. In particular, his method could not be used as a black box for free groups (even without rational constraints). Actually, the presence of an involution (inverse elements) and rational constraints complicates the situation and some additional analysis is necessary. Still, the recompression technique is general enough to accommodate both extensions. In the end, it simplifies proofs that solving word equations is in PSPACE (Plandowski 1999) and the corresponding result for equations in free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As a byproduct we obtain a direct proof that it is decidable in PSPACE whether or not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk at CSR 2014 in Moscow, June 7 - 11, 201

Reducing Computational Costs in the Basic Perturbation Lemma

Homological Perturbation Theory [11, 13] is a well-known general method for computing homology, but its main algorithm, the Basic Perturbation Lemma, presents, in general, high computational costs. In this paper, we propose a general strategy in order to reduce the complexity in some important formulas (those following a specific pattern) obtained by this algorithm. Then, we show two examples of application of this methodology.

The Computational Complexity of Symbolic Dynamics at the Onset of Chaos

In a variety of studies of dynamical systems, the edge of order and chaos has been singled out as a region of complexity. It was suggested by Wolfram, on the basis of qualitative behaviour of cellular automata, that the computational basis for modelling this region is the Universal Turing Machine. In this paper, following a suggestion of Crutchfield, we try to show that the Turing machine model may often be too powerful as a computational model to describe the boundary of order and chaos. In particular we study the region of the first accumulation of period doubling in unimodal and bimodal maps of the interval, from the point of view of language theory. We show that in relation to the ``extended'' Chomsky hierarchy, the relevant computational model in the unimodal case is the nested stack automaton or the related indexed languages, while the bimodal case is modeled by the linear bounded automaton or the related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of manuscrip

Neutrophil gelatinase associated lipocalin (NGAL) is elevated in type 2 diabetics with carotid artery stenosis and reduced under metformin treatment

Abstract Background Neutrophil gelatinase-associated lipocalin (NGAL), an acute phase protein released by neutrophils, has been described as biomarker of inflammatory states. Type 2 diabetes mellitus (T2DM) is characterized by increased inflammation and an elevated risk for embolization of carotid artery stenosis (CAS). We aimed to explore the role of NGAL systemically and in plaques of diabetics undergoing carotid endarterectomy. Moreover, the potential anti-inflammatory effect of metformin on NGAL was addressed in diabetics. Methods Serum NGAL and matrix metalloproteinase (MMP)-9/NGAL levels were measured in 136 patients (67 with T2DM vs. 69 non-diabetics) by specific ELISA. Endarterectomy samples were graded histologically according to the American Heart Association´s classification. NGAL mRNA expression was detected using RealTime-PCR in carotid endarterectomy specimens. Results Serum NGAL [median 107.4 ng/ml (quartiles: 75.2–145.0) vs. 64.4 (50.4 –81.3), p < 0.0001] and MMP-9/NGAL [41.5 ng/ml (20.8–63.9) vs. 27.6 (16.0–42.4), p = 0.017] were significantly elevated in diabetics compared to non-diabetics, as were leukocytes, neutrophils, C-reactive protein and fibrinogen (all p < 0.05). In patients with symptomatic and asymptomatic CAS diabetics had higher NGAL levels compared to non-diabetics [128.8 ng/ml (100.8–195.6) vs. 64.8 (48.9–82.2] and [101.6 ng/ml (70.1–125.3) vs. 63.8 (51.0–81.3), respectively, both p < 0.0001]. Presence of T2DM and type VI plaques (with surface defect, hemorrhage or thrombus) had a profound impact on NGAL levels (both p < 0.01) in multiple linear regression analysis. NGAL mRNA was detectable in 95% of analyzed carotid artery lesions of diabetics compared to 5% of non-diabetics (p < 0.0001). Accordingly, cerebral embolization was more frequent in diabetics (52.2% vs. 29%, p = 0.006). Metformin treatment was associated with decreased NGAL [60.7 ng/ml (51.9–69.2) vs. 121.7 (103.7–169.9), p < 0.0001] and MMP-9/NGAL [20.8 ng/ml (12.1–26.5) vs. 53.7 (27.4–73.4), p = 0.007] in diabetics and reduced leukocyte infiltration in carotid lesions of diabetics. Conclusions Higher NGAL levels in serum and plaques are associated with T2DM in patients with CAS. Metformin significantly reduced the inflammatory burden including NGAL in diabetics. Early treatment of these patients may be recommended, as elevated NGAL levels were linked with vulnerable plaques prone for embolization

Quantum Picturalism

The quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. In this review we present steps towards a diagrammatic `high-level' alternative for the Hilbert space formalism, one which appeals to our intuition. It allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the no-cloning theorem, and phenomena such as quantum teleportation. As a logic, it supports `automation'. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required step-stone towards a deeper conceptual understanding of quantum theory, as well as its unification with other physical theories. Specific applications discussed here are purely diagrammatic proofs of several quantum computational schemes, as well as an analysis of the structural origin of quantum non-locality. The underlying mathematical foundation of this high-level diagrammatic formalism relies on so-called monoidal categories, a product of a fairly recent development in mathematics. These monoidal categories do not only provide a natural foundation for physical theories, but also for proof theory, logic, programming languages, biology, cooking, ... The challenge is to discover the necessary additional pieces of structure that allow us to predict genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures, some colo

Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories

Algebraic description of spacetime foam

A mathematical formalism for treating spacetime topology as a quantum observable is provided. We describe spacetime foam entirely in algebraic terms. To implement the correspondence principle we express the classical spacetime manifold of general relativity and the commutative coordinates of its events by means of appropriate limit constructions.Comment: 34 pages, LaTeX2e, the section concerning classical spacetimes in the limit essentially correcte