315 research outputs found
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
Viability in hybrid systems
AbstractHybrid systems are interacting systems of digital automata and continuous plants subject to disturbances. The digital automata are used to force the state trajectory of the continuous plant to obey a performance specification. For the basic concepts and notation for hybrid systems, see Kohn and Nerode (1993), and other papers in the same volume. Here we introduce tools for analyzing enforcing viability of all possible plant state trajectories of a hybrid system by suitable choices of finite state control automata. Thus, the performance specification considered here is that the state of the plant remain in a prescribed viability set of states at all times (Aubin, 1991). The tools introduced are local viability graphs and viability graphs for hybrid systems. We construct control automata which guarantee viability as the fixpoints of certain operators on graphs. When control and state spaces are compact, the viability set is closed, and a non-empty closed subset of a viability graph is given with a sturdiness property, one can extract finite state automata guaranteeing viable trajectories. This paper is a sequel to Kohn and Nerode (1993), especially Appendix II
Infinite games played on finite graphs
AbstractThe concept of an infinite game played on a finite graph is perhaps novel in the context of an rather extensive recent literature in which infinite games are generally played on an infinite game tree. We claim two advantages for our model, which is admittedly more restrictive. First, our games have a more apparent resemblance to ordinary parlor games in spite of their infinite duration. Second, by distinguishing those nodes of the graph that determine the winning and losing of the game (winning-condition nodes), we are able to offer a complexity analysis that is useful in computer science applications
Unstable Attractors: Existence and Robustness in Networks of Oscillators With Delayed Pulse Coupling
We consider unstable attractors; Milnor attractors such that, for some
neighbourhood of , almost all initial conditions leave . 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
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
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
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