The \mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity

It is known that the alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict. However, the strictness of the alternation hierarchy does not necessarily carry over when considering restricted classes of structures. A prominent instance is the class of infinite words over which the alternation-free fragment is already as expressive as the full mu-calculus. Our current understanding of when and why the mu-calculus alternation hierarchy is not strict is limited. This paper makes progress in answering these questions by showing that the alternation hierarchy of the mu-calculus collapses to the alternation-free fragment over some classes of structures, including infinite nested words and finite graphs with feedback vertex sets of a bounded size. Common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the structure's graph decomposes it into graphs in which all paths are of finite length. Our collapse results are obtained in an automata-theoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the mu-calculus over restricted classes of structures.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Towards a Minimal Stabilizer ZX-calculus

The stabilizer ZX-calculus is a rigorous graphical language for reasoning about quantum mechanics. The language is sound and complete: one can transform a stabilizer ZX-diagram into another one using the graphical rewrite rules if and only if these two diagrams represent the same quantum evolution or quantum state. We previously showed that the stabilizer ZX-calculus can be simplified by reducing the number of rewrite rules, without losing the property of completeness [Backens, Perdrix & Wang, EPTCS 236:1--20, 2017]. Here, we show that most of the remaining rules of the language are indeed necessary. We do however leave as an open question the necessity of two rules. These include, surprisingly, the bialgebra rule, which is an axiomatisation of complementarity, the cornerstone of the ZX-calculus. Furthermore, we show that a weaker ambient category -- a braided autonomous category instead of the usual compact closed category -- is sufficient to recover the meta rule 'only connectivity matters', even without assuming any symmetries of the generators.Comment: 29 pages, minor updates for v

Connectivity calculus of fractal polyhedrons

The paper analyzes the connectivity information (more precisely, numbers of tunnels and their homological (co)cycle classification) of fractal polyhedra. Homology chain contractions and its combinatorial counterparts, called homological spanning forest (HSF), are presented here as an useful topological tool, which codifies such information and provides an hierarchical directed graph-based representation of the initial polyhedra. The Menger sponge and the Sierpiński pyramid are presented as examples of these computational algebraic topological techniques and results focussing on the number of tunnels for any level of recursion are given. Experiments, performed on synthetic and real image data, demonstrate the applicability of the obtained results. The techniques introduced here are tailored to self-similar discrete sets and exploit homology notions from a representational point of view. Nevertheless, the underlying concepts apply to general cell complexes and digital images and are suitable for progressing in the computation of advanced algebraic topological information of 3-dimensional objects

A streamlined proof of the convergence of the Taylor tower for embeddings in Rn\mathbb R^n

Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the theory produces a Taylor tower whose purpose is to approximate this space in a suitable sense. Central to the story are deep theorems about the convergence of this tower. We provide an exposition of the convergence results in the special case of embeddings into $\mathbb R^n$, which has been the case of primary interest in applications. We try to use as little machinery as possible and give several improvements and restatements of existing arguments used in the proofs of the main results.Comment: Minor changes, final versio

Proof Diagrams for Multiplicative Linear Logic

The original idea of proof nets can be formulated by means of interaction nets syntax. Additional machinery as switching, jumps and graph connectivity is needed in order to ensure correspondence between a proof structure and a correct proof in sequent calculus. In this paper we give an interpretation of proof nets in the syntax of string diagrams. Even though we lose standard proof equivalence, our construction allows to define a framework where soundness and well-typeness of a diagram can be verified in linear time.Comment: In Proceedings LINEARITY 2016, arXiv:1701.0452

The Impacts of Privacy Rules on Users' Perception on Internet of Things (IoT) Applications: Focusing on Smart Home Security Service

Department of Management EngineeringAs communication and information technologies advance, the Internet of Things (IoT) has changed the way people live. In particular, as smart home security services have been widely commercialized, it is necessary to examine consumer perception. However, there is little research that explains the general perception of IoT and smart home services. This article will utilize communication privacy management theory and privacy calculus theory to investigate how options to protect privacy affect how users perceive benefits and costs and how those perceptions affect individuals??? intentions to use of smart home service. Scenario-based experiments were conducted, and perceived benefits and costs were treated as formative second-order constructs. The results of PLS analysis in the study showed that smart home options to protect privacy decreased perceived benefits and increased perceived costs. In addition, the perceived benefits and perceived costs significantly affected the intention to use smart home security services. This research contributes to the field of IoT and smart home research and gives practitioners notable guidelines.ope

General Connectivity Distribution Functions for Growing Networks with Preferential Attachment of Fractional Power

We study the general connectivity distribution functions for growing networks with preferential attachment of fractional power, $\Pi_{i} \propto k^{\alpha}$, using the Simon's method. We first show that the heart of the previously known methods of the rate equations for the connectivity distribution functions is nothing but the Simon's method for word problem. Secondly, we show that the case of fractional $\alpha$ the $Z$-transformation of the rate equation provides a fractional differential equation of new type, which coincides with that for PA with linear power, when $\alpha = 1$. We show that to solve such a fractional differential equation we need define a transidental function $\Upsilon (a,s,c;z)$ that we call {\it upsilon function}. Most of all previously known results are obtained consistently in the frame work of a unified theory.Comment: 10 page