A Myhill-Nerode theorem for automata with advice

  title={A Myhill-Nerode theorem for automata with advice},
  author={Alex Kruckman and Sasha Rubin and John Sheridan and Ben Zax},
  booktitle={International Symposium on Games, Automata, Logics and Formal Verification},
An automaton with advice is a finite state automaton which has access to an additional fixed infinite string called an advice tape. We refine the Myhill-Nerode theorem to characterize the languages of finite strings that are accepted by automata with advice. We do the same for tree automata with advice. 

Advice Automatic Structures and Uniformly Automatic Classes

It is proved that the class of all torsion-free Abelian groups of rank one is uniformly omega-automatic and that there is a uniform omega-tree-automatic presentation of the classof all Abelian Groups up to elementary equivalence and of theclass of all countable divisible Abelian Group groups.

On the Complexity of Infinite Advice Strings

The main results explore the connections between classes of advice automatic structures, MSO-transductions and two-way transducers and suggest a closer study of the resulting hierarchy over infinite words.

Comparing the power of advice strings: a notion of complexity for infinite words

The main results explore the connections between classes of advice automatic structures, MSO-transductions and two-way transducers and suggest a closer study of the resulting hierarchy over infinite words.

Monadic Second-Order Logic with Arbitrary Monadic Predicates

We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic,

Algorithmic Solutions via Model Theoretic Interpretations

This dissertation introduces automatic presentations with advice for several automata models, which are closely related to monadic second-order interpretations over set variables, and investigates the efficiency of this approach by analysing the runtime of the generic automata-based model checking algorithm in terms of the complexity of the given presentation.

Automatic Structures: Twenty Years Later

This tutorial presents an introduction into the history and basic definitions of automatic structures, and surveys the achievements in the study of different variants ofautomatic structures.

On the Width of Regular Classes of Finite Structures

It is shown that the problem of counting satisfying assignments for a first-order logic formula in a structure of constant width is fixed parameter tractable when parameterized by the width parameter and can be solved in quadratic time with respect to the length of the input representation of the structure.



Automata Presenting Structures: A Survey of the Finite String Case

  • S. Rubin
  • Computer Science
    Bulletin of Symbolic Logic
  • 2008
The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

Regular sets over extended tree structures

On the Myhill-Nerode Theorem for Trees

This result generalizes in a straightforward way to automata on nite trees to conventional nite automata theory and it goes through quite neatly Myhill and Nerode s work.

Decidable Theories of the Ordering of Natural Numbers with Unary Predicates

Two characterizations of the decidable theories of this form are given, in terms of effectiveness conditions on two types of “homogeneous sets”, and it is shown that the first-order theory of successor with extra predicates is not covered by this approach.

The Monadic Theory of Morphic Infinite Words and Generalizations

A large class of predicates P is exhibited such that the monadic theory MTh(N, <, P) is decidable, which unifies and extends the previously known examples.

Transforming structures by set interpretations

This paper investigates the expressive power of finite sets interpretations applied to infinite deterministic trees and finds that they can be used in the study of automatic and tree-automatic structures.

Describing Groups

  • A. Nies
  • Mathematics
    Bulletin of Symbolic Logic
  • 2007
This paper surveys examples of FA-presentable groups, but also discusses theorems restricting this class, and gives examples of quasi-finitely axiomatizable groups, and considers the algebraic content of the notion.

A Hierarchy of Automatic $\omega$-Words having a Decidable MSO Theory

  • V. Bárány
  • Mathematics, Linguistics
    RAIRO Theor. Informatics Appl.
  • 2008
It is proved that lexicographic presentations of w-words are canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classes under MSO-definable recolorings, e.g. closure under deterministic sequential mappings.

Decidability and undecidability of extensions of second (first) order theory of (generalized) successor

In this chapter, certain first and second order theories which are semantically defined as the sets of all sentences true in certain given structures are studied.

The additive group of the rationals does not have an automatic presentation

  • T. Tsankov
  • Mathematics
    The Journal of Symbolic Logic
  • 2011
Abstract We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are