10 research outputs found
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
This article studies the expressive power of finite automata recognizing sets
of real numbers encoded in positional notation. We consider Muller automata as
well as the restricted class of weak deterministic automata, used as symbolic
set representations in actual applications. In previous work, it has been
established that the sets of numbers that are recognizable by weak
deterministic automata in two bases that do not share the same set of prime
factors are exactly those that are definable in the first order additive theory
of real and integer numbers. This result extends Cobham's theorem, which
characterizes the sets of integer numbers that are recognizable by finite
automata in multiple bases.
In this article, we first generalize this result to multiplicatively
independent bases, which brings it closer to the original statement of Cobham's
theorem. Then, we study the sets of reals recognizable by Muller automata in
two bases. We show with a counterexample that, in this setting, Cobham's
theorem does not generalize to multiplicatively independent bases. Finally, we
prove that the sets of reals that are recognizable by Muller automata in two
bases that do not share the same set of prime factors are exactly those
definable in the first order additive theory of real and integer numbers. These
sets are thus also recognizable by weak deterministic automata. This result
leads to a precise characterization of the sets of real numbers that are
recognizable in multiple bases, and provides a theoretical justification to the
use of weak automata as symbolic representations of sets.Comment: 17 page
Automatic sets of rational numbers
The notion of a k-automatic set of integers is well-studied. We develop a new
notion - the k-automatic set of rational numbers - and prove basic properties
of these sets, including closure properties and decidability.Comment: Previous version appeared in Proc. LATA 2012 conferenc
Decidability of definability issues in the theory of real addition
Given a subset of we can associate with every
point a vector space of maximal dimension with the
property that for some ball centered at , the subset coincides inside
the ball with a union of lines parallel with . A point is singular if
has dimension . In an earlier paper we proved that a -definable relation is actually definable in if and only if the number of singular points is finite and every rational
section of is -definable, where a rational section is
a set obtained from by fixing some component to a rational value. Here we
show that we can dispense with the hypothesis of being -definable by assuming that the components of the singular points
are rational numbers. This provides a topological characterization of
first-order definability in the structure . It also
allows us to deliver a self-definable criterion (in Muchnik's terminology) of
- and -definability for a
wide class of relations, which turns into an effective criterion provided that
the corresponding theory is decidable. In particular these results apply to the
class of recognizable relations on reals, and allow us to prove that it is
decidable whether a recognizable relation (of any arity) is
recognizable for every base .Comment: added sections 5 and 6, typos corrected. arXiv admin note: text
overlap with arXiv:2002.0428
Theories of real addition with and without a predicate for integers
We show that it is decidable whether or not a relation on the reals definable
in the structure can be defined
in the structure . This result is achieved
by obtaining a topological characterization of -definable relations in the family of -definable relations and then by following Muchnik's
approach of showing that the characterization of the relation can be
expressed in the logic of .
The above characterization allows us to prove that there is no intermediate
structure between and . We also show that a -definable relation is -definable if and only if its intersection with every -definable line is -definable. This gives a noneffective but simple characterization of
-definable relations
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
This article studies the expressive power of finite automata recognizing sets
of real numbers encoded in positional notation. We consider Muller automata as
well as the restricted class of weak deterministic automata, used as symbolic
set representations in actual applications. In previous work, it has been
established that the sets of numbers that are recognizable by weak
deterministic automata in two bases that do not share the same set of prime
factors are exactly those that are definable in the first order additive theory
of real and integer numbers. This result extends Cobham's theorem, which
characterizes the sets of integer numbers that are recognizable by finite
automata in multiple bases.
In this article, we first generalize this result to multiplicatively
independent bases, which brings it closer to the original statement of Cobham's
theorem. Then, we study the sets of reals recognizable by Muller automata in
two bases. We show with a counterexample that, in this setting, Cobham's
theorem does not generalize to multiplicatively independent bases. Finally, we
prove that the sets of reals that are recognizable by Muller automata in two
bases that do not share the same set of prime factors are exactly those
definable in the first order additive theory of real and integer numbers. These
sets are thus also recognizable by weak deterministic automata. This result
leads to a precise characterization of the sets of real numbers that are
recognizable in multiple bases, and provides a theoretical justification to the
use of weak automata as symbolic representations of sets