Graph Spectral Properties of Deterministic Finite Automata

We prove that a minimal automaton has a minimal adjacency matrix rank and a minimal adjacency matrix nullity using equitable partition (from graph spectra theory) and Nerode partition (from automata theory). This result naturally introduces the notion of matrix rank into a regular language L, the minimal adjacency matrix rank of a deterministic automaton that recognises L. We then define and focus on rank-one languages: the class of languages for which the rank of minimal automaton is one. We also define the expanded canonical automaton of a rank-one language.Comment: This paper has been accepted at the following conference: 18th International Conference on Developments in Language Theory (DLT 2014), August 26 - 29, 2014, Ekaterinburg, Russi

Annotated nonmonotonic rule systems

AbstractAnnotated logics were proposed by Subrahmanian as a unified paradigm for representing a wide variety of reasoning tasks including reasoning with uncertainty within a single theoretical framework. Subsequently, Marek, Nerode and Remmel have shown how to provide nonmonotonic extensions of arbitrary languages through their notion of a nonmonotonic rule systems. The primary aim of this paper is to define annotated nonmonotonic rule systems which merge these two frameworks into a general purpose nonmonotonic reasoning framework over arbitrary multiple-valued logics. We then show how Reiter's normal default theories may be generalized to the framework of annotated nonmonotonic rule systems

Extracting winning strategies in update games

This paper investigates algorithms for extracting winning strategies in two-player games played on nite graphs. We focus on a special class of games called update games. We present a procedure for extracting winning strategies in update games by constructing strategies explicitly. This is based on an algorithm that solves update games in quadratic time. We also show that solving update games with a bounded number of nondeterministic nodes takes linear time

Unstable Attractors: Existence and Robustness in Networks of Oscillators With Delayed Pulse Coupling

We consider unstable attractors; Milnor attractors $A$ such that, for some neighbourhood $U$ of $A$, almost all initial conditions leave $U$. Previous research strongly suggests that unstable attractors exist and even occur robustly (i.e. for open sets of parameter values) in a system modelling biological phenomena, namely in globally coupled oscillators with delayed pulse interactions. In the first part of this paper we give a rigorous definition of unstable attractors for general dynamical systems. We classify unstable attractors into two types, depending on whether or not there is a neighbourhood of the attractor that intersects the basin in a set of positive measure. We give examples of both types of unstable attractor; these examples have non-invertible dynamics that collapse certain open sets onto stable manifolds of saddle orbits. In the second part we give the first rigorous demonstration of existence and robust occurrence of unstable attractors in a network of oscillators with delayed pulse coupling. Although such systems are technically hybrid systems of delay differential equations with discontinuous `firing' events, we show that their dynamics reduces to a finite dimensional hybrid system system after a finite time and hence we can discuss Milnor attractors for this reduced finite dimensional system. We prove that for an open set of phase resetting functions there are saddle periodic orbits that are unstable attractors.Comment: 29 pages, 8 figures,submitted to Nonlinearit

Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods

Quotient Complexity of Regular Languages

The past research on the state complexity of operations on regular languages is examined, and a new approach based on an old method (derivatives of regular expressions) is presented. Since state complexity is a property of a language, it is appropriate to define it in formal-language terms as the number of distinct quotients of the language, and to call it "quotient complexity". The problem of finding the quotient complexity of a language f(K,L) is considered, where K and L are regular languages and f is a regular operation, for example, union or concatenation. Since quotients can be represented by derivatives, one can find a formula for the typical quotient of f(K,L) in terms of the quotients of K and L. To obtain an upper bound on the number of quotients of f(K,L) all one has to do is count how many such quotients are possible, and this makes automaton constructions unnecessary. The advantages of this point of view are illustrated by many examples. Moreover, new general observations are presented to help in the estimation of the upper bounds on quotient complexity of regular operations

Reverse mathematics of matroids

Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for \Sigma^0_2 formulas

Parameterized Synthesis with Safety Properties

Parameterized synthesis offers a solution to the problem of constructing correct and verified controllers for parameterized systems. Such systems occur naturally in practice (e.g., in the form of distributed protocols where the amount of processes is often unknown at design time and the protocol must work regardless of the number of processes). In this paper, we present a novel learning based approach to the synthesis of reactive controllers for parameterized systems from safety specifications. We use the framework of regular model checking to model the synthesis problem as an infinite-duration two-player game and show how one can utilize Angluin's well-known L* algorithm to learn correct-by-design controllers. This approach results in a synthesis procedure that is conceptually simpler than existing synthesis methods with a completeness guarantee, whenever a winning strategy can be expressed by a regular set. We have implemented our algorithm in a tool called L*-PSynth and have demonstrated its performance on a range of benchmarks, including robotic motion planning and distributed protocols. Despite the simplicity of L*-PSynth it competes well against (and in many cases even outperforms) the state-of-the-art tools for synthesizing parameterized systems.Comment: 18 page