45 research outputs found
Expansions of MSO by cardinality relations
We study expansions of the Weak Monadic Second Order theory of (N,<) by
cardinality relations, which are predicates R(X1,...,Xn) whose truth value
depends only on the cardinality of the sets X1, ...,Xn. We first provide a
(definable) criterion for definability of a cardinality relation in (N,<), and
use it to prove that for every cardinality relation R which is not definable in
(N,<), there exists a unary cardinality relation which is definable in (N,<,R)
and not in (N,<). These results resemble Muchnik and Michaux-Villemaire
theorems for Presburger Arithmetic. We prove then that + and x are definable in
(N,<,R) for every cardinality relation R which is not definable in (N,<). This
implies undecidability of the WMSO theory of (N,<,R). We also consider the
related satisfiability problem for the class of finite orderings, namely the
question whether an MSO sentence in the language {<,R} admits a finite model M
where < is interpreted as a linear ordering, and R as the restriction of some
(fixed) cardinality relation to the domain of M. We prove that this problem is
undecidable for every cardinality relation R which is not definable in (N,<).Comment: to appear in LMC
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
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
Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy
It is shown that for any fixed , the -fragment of
Presburger arithmetic, i.e., its restriction to quantifier alternations
beginning with an existential quantifier, is complete for
, the -th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
and . This result completes the
computational complexity landscape for Presburger arithmetic, a line of
research which dates back to the seminal work by Fischer & Rabin in 1974.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
-fragment of Presburger arithmetic: given a -formula
, it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure
A Survey of Arithmetical Definability
We survey definability and decidability issues related to first-order fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions