Birecurrent sets

arXiv (Cornell University), 2017

A set is called recurrent if its minimal automaton is strongly connected and birecurrent if it is recurrent as well as its reversal. We prove a series of results concerning birecurrent sets. It is already known that any birecurrent set is completely reducible (that is, such that the minimal representation of its characteristic series is completely reducible). The main result of this paper characterizes completely reducible sets as linear combinations of birecurrent sets

Cybernetics, 1979

On this basis it is easy to construct a unidirectional combinational element al such that the network Z 1 in which signals move from left to right and constructed of elements (~ provides at the main output of the rightmost element the number of the set Bi to which the input pattern belongs. Accordingly, the automata r 1 and r 2 can serve as a basis for constructing unidirectional elements a2 and a3 that recognize the languages N i and N 2. By an arbitrary matrix M i E QT one constructs an element 0 i that determines whether the hi-neighborhood of a given element of the input pattern coincides with the M i matrix or not. Such elements have been constructed in [4] for the case n i = 1. The circuit diagram of an element a that defines the system Z is shown in Fig. 2. Note that in a single-output network, ~2 and a3 are needed only in the rightmost column and lowermost row. The element 8 is constructed of 0 i elements and of counters [4] and decides if the input pattern has the form (T, QT). The side channels of element 0 and threshold signals are not shown in Fig. 2 in order to simplify the diagram. The symbols X and Z denote the main inputs and outputs of the elements. Note that in the definition of an automaton-definable class of matrices it is essential that all languages of the set B are finite-automaton languages. Consider the following example. Let ~ be a non-finite-automaton language in the alphabet A 2 = {0, 1}. Let ~ (~) denote the class of all matrices of ~t (A2) in which the first row is a word of ~ and the remaining rows contain all zeros. The proof that the class .~ (A~) is not recogp2zable is similar to the proof of Theorem 3 in [4]. The author thanks M. I. Kratko for suggesting the problem and for his guidance.

2019

In this paper, we introduce and study the notion of recurrent sets of operators and some of its variations on Banach spaces. As application, we study the recurrence of $C$-regularized group of operators. We show that there exists recurrent $C$-regularized group in each Banach space with finite or infinite dimensional. Moreover, we prove that if $(S(z))_{z\in\mathbb{C}}$ is a recurrent $C$-regularized group and $S(z_0)$ is an operator in this $C$-regularized group, then $S(z_0)$ is not necessarily recurrent.

1991

Abstract The authors study one-word-decreasing self-reducible sets, which are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query to a word lexicographically smaller than the input. It is first shown that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible.

Theoretical Computer …, 2009

There exist several works that study the class of reversible languages defined as the union closure of 0-reversible languages, their properties and suitable representations. In this work we define and study the class of locally reversible languages, defined as the union closure of k-reversible languages. We characterize the class and prove that it is a local (positive) variety of formal languages. We also extend the definition of quasi-reversible automata to deal with locally reversible languages and propose a polynomial algorithm to obtain, for any given locally k-reversible language, a quasi-k-reversible automaton.

Acta Informatica, 1990

Geffert has shown that each recursively enumerable language L over Z can be expressed in the form L={h(x)-ig(x)lx in A+}nZ * where A is an alphabet and g, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations of L. For instance, L = p(Lo)n Z* where Lo is a minimal linear language and p is the Dyck reduction a~ ~ e, A,4 ~ e.

Theory of Computing Systems, 2010

This paper studies the notions of autoreducibility and length-decreasing self-reducibility of functions and languages. Recently Glaßer et al. have shown that for many classes C, including PSPACE and NP, it holds that all nontrivial complete languages are polynomial-time many-one autoreducible. In contrast, this paper shows that for many classes C such that P ⊆ C (e.g., PSPACE and NP) some complete languages in C are not polynomial-time length-decreasing self-reducible unless C ⊆ P, and for classes C such that L ⊆ C ⊆ P (e.g., P and NL) some complete languages in C are not logarithmic-space length-decreasing self-reducible unless C ⊆ L. This paper also shows that contrast between autoreducibility and length-decreasing selfreducibility for the case of functions. In particular, the paper shows that many function complexity classes FC (including well-studied #P, SpanP, and GapP and not-so-well-studied but highly natural #PE and TotP) have the property that all complete functions in FC are polynomial-time Turing-autoreducible. For #P and TotP, the autoreductions can be made to be polynomial-time one-Turing (one query per input). These results show that, under reasonable assumptions, the notions of length-decreasing selfreducibility and autoreducibility differ both on complete languages and on complete functions. In a similar vein, this paper shows that under reasonable assumptions autoreducibility and random-self-reducibility differ with respect to functions. * This technical report subsumes URCS-TR 2005-874. 1 Some of these variants of "self-reducibility" go beyond what the seminal papers of Meyer and Paterson [MP79] and of Schnorr [Sch76] intended to capture. In their work self-reducibility refers to situations in which autoreducibility is established according to a partial order that gives "short" downward chains. Such a short-downward-chain property does not exist for some of the variants, in particular, for random-self-reducibility and word-decreasing self-reducibility. This lack seems to blur the distinction between "self-reducibility" and autoreducibility. All the self-reducibility notions we study are special cases of length-decreasing self-reducibility. Although the term "length-decreasing autoreducibility" adequately characterizes the property, we follow the convention and keep the word "self-reducibility" in the names.

Notre Dame Journal of Formal Logic, 2008

For an arbitrary finite algebra (A, f (·, ·), 0, 1) one defines a double sequence a(i, j) by a(i, 0) = a(0, j) = 1 and a(i, j) = f (a(i, j − 1), a(i − 1, j)). The problem if such recurrent double sequences are ultimately zero is undecidable, even if we restrict it to the class of commutative finite algebras. A.M.S.-Classification: 03D10.

In this article one builds a class of recursive sets, one establishes properties of these sets and one proposes applications. This article widens some results of.

2018

It is shown that languages definable by weak pebble automata are not closed under reversal. For the proof, we establish a kind of periodicity of an automaton’s computation over a specific set of words. The periodicity is partly due to the finiteness of the automaton description and partly due to the word’s structure. Using such a periodicity we can find a word such that during the automaton’s run on it there are two different, yet indistinguishable, configurations. This enables us to remove a part of that word without affecting acceptance. Choosing an appropriate language leads us to the desired result.