Minimization via duality

We show how to use duality theory to construct minimized versions of a wide class of automata. We work out three cases in detail: (a variant of) ordinary automata, weighted automata and probabilistic automata. The basic idea is that instead of constructing a maximal quotient we go to the dual and look for a minimal subalgebra and then return to the original category. Duality ensures that the minimal subobject becomes the maximally quotiented object

On the AGN radio luminosity distribution and the black hole fundamental plane

We have studied the dependence of the AGN nuclear radio (1.4 GHz) luminosity on both the AGN 2-10 keV X-ray and the host-galaxy K-band luminosity. A complete sample of 1268 X-ray selected AGN (both type 1 and type 2) has been used, which is the largest catalogue of AGN belonging to statistically well defined samples where radio, X and K band information exists. At variance with previous studies, radio upper limits have been statistically taken into account using a Bayesian Maximum Likelihood fitting method. It resulted that a good fit is obtained assuming a plane in the 3D L_R-L_X-L_K space, namely logL_R= xi_X logL_X + xi_K logL_K + xi_0, having a ~1 dex wide (1 sigma) spread in radio luminosity. As already shown, no evidence of bimodality in the radio luminosity distribution was found and therefore any definition of radio loudness in AGN is arbitrary. Using scaling relations between the BH mass and the host galaxy K-band luminosity, we have also derived a new estimate of the BH fundamental plane (in the L_5GHz -L_X-M_BH space). Our analysis shows that previous measures of the BH fundamental plane are biased by ~0.8 dex in favor of the most luminous radio sources. Therefore, many AGN studies, where the BH fundamental plane is used to investigate how AGN regulate their radiative and mechanical luminosity as a function of the accretion rate, or many AGN/galaxy co-evolution models, where radio-feedback is computed using the AGN fundamental plane, should revise their conclusions.Comment: Submitted to MNRAS. Revised version after minor referee comments. 12 pages, 12 figure

The Calculus of Signal Flow Diagrams I: Linear relations on streams

We introduce a graphical syntax for signal flow diagrams based on the language of symmetric monoidal categories. Using universal categorical constructions, we provide a stream semantics and a sound and complete axiomatisation. A certain class of diagrams captures the orthodox notion of signal flow graph used in control theory; we show that any diagram of our syntax can be realised, via rewriting in the equational theory, as a signal flow graph

Borrowed contexts for attributed graphs

Borrowed context graph transformation is a simple and powerful technique developed by Ehrig and König that allow us to derive labeled transitions and bisimulation congruences for graph transformation systems or, in general, for pocess calculi that can be defined in terms of graph transformation systems. Moreover, the same authors have also shown how to use this technique for the verification of bisimilarity. In principle, the main results about borrowed context transformation do not apply only to plain graphs, but they are generic in the sense that they apply to all categories tha satisfy certain properties related to the notion of adhesivity. In particular, this is the case of attributed graphs. However, as we show in the paper, the techniques used for checking bisimilarity are not equally generic and, in particular they fail, if we want to apply them to attributed graphs. To solve this problem, in this paper, we define a special notion of symbolic graph bisimulation and show how it can be used to check bisimilarity of attributed graphs.Postprint (published version

The theory of traces for systems with nondeterminism and probability

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories

From Farkas’ lemma to linear programming: An exercise in diagrammatic algebra

Farkas’ lemma is a celebrated result on the solutions of systems of linear inequalities, which finds application pervasively in mathematics and computer science. In this work we show how to formulate and prove Farkas’ lemma in diagrammatic polyhedral algebra, a sound and complete graphical calculus for polyhedra. Furthermore, we show how linear programs can be modeled within the calculus and how some famous duality results can be proved

Bialgebraic foundations for the operational semantics of string diagrams

Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) is compositional. In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their semantics as a distributive law for that monad. As a proof of concept, we provide bialgebraic semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory)

Rewriting modulo symmetric monoidal structure

String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory. An important role in many such approaches is played by equational theories of diagrams, typically oriented and applied as rewrite rules. This paper lays a comprehensive foundation for this form of rewriting. We interpret diagrams combinatorially as typed hypergraphs and establish the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hypergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure. We illustrate our approach with a proof of termination for the theory of non-commutative bimonoids

gPrune: A Constraint Pushing Framework for Graph Pattern Mining

Abstract. In graph mining applications, there has been an increasingly strong urge for imposing user-specified constraints on the mining results. However, unlike most traditional itemset constraints, structural constraints, such as density and diameter of a graph, are very hard to be pushed deep into the mining process. In this paper, we give the first comprehensive study on the pruning properties of both traditional and structural constraints aiming to reduce not only the pattern search space but the data search space as well. A new general framework, called gPrune, is proposed to incorporate all the constraints in such a way that they recursively reinforce each other through the entire mining process. A new concept, Pattern-inseparable Data-antimonotonicity, is proposed to handle the structural constraints unique in the context of graph, which, combined with known pruning properties, provides a comprehensive and unified classification framework for structural constraints. The exploration of these antimonotonicities in the context of graph pattern mining is a significant extension to the known classification of constraints, and deepens our understanding of the pruning properties of structural graph constraints.

How to kill epsilons with a dagger: a coalgebraic take on systems with algebraic label structure

We propose an abstract framework for modeling state-based systems with internal behavior as e.g. given by silent or ϵ-transitions. Our approach employs monads with a parametrized fixpoint operator † to give a semantics to those systems and implement a sound procedure of abstraction of the internal transitions, whose labels are seen as the unit of a free monoid. More broadly, our approach extends the standard coalgebraic framework for state-based systems by taking into account the algebraic structure of the labels of their transitions. This allows to consider a wide range of other examples, including Mazurkiewicz traces for concurrent systems.Funded by the ERDF through the Programme COMPETE and by the Portuguese Foundation for Science and Technology, project ref. FCOMP-01-0124-FEDER-020537 and SFRH/BPD/71956/2010. Acknowledge support by project ANR 12IS0 2001 PACE