313 research outputs found
Quantifier elimination for the reals with a predicate for the powers of two
In 1985, van den Dries showed that the theory of the reals with a predicate
for the integer powers of two admits quantifier elimination in an expanded
language, and is hence decidable. He gave a model-theoretic argument, which
provides no apparent bounds on the complexity of a decision procedure. We
provide a syntactic argument that yields a procedure that is primitive
recursive, although not elementary. In particular, we show that it is possible
to eliminate a single block of existential quantifiers in time ,
where is the length of the input formula and denotes -fold
iterated exponentiation
Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers
We prove decidability of univariate real algebra extended with predicates for
rational and integer powers, i.e., and . Our decision procedure combines computation over real algebraic
cells with the rational root theorem and witness construction via algebraic
number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated
Deduction, 2015. Proceedings to be published by Springer-Verla
The dynamical hierarchy for Roelcke precompact Polish groups
We study several distinguished function algebras on a Polish group , under
the assumption that is Roelcke precompact. We do this by means of the
model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate
the dynamics of -categorical metric structures under the action of
their automorphism group. We show that, in this context, every strongly
uniformly continuous function (in particular, every Asplund function) is weakly
almost periodic. We also point out the correspondence between tame functions
and NIP formulas, deducing that the isometry group of the Urysohn sphere is
\Tame\cap\UC-trivial.Comment: 25 page
Combining decision procedures for the reals
We address the general problem of determining the validity of boolean
combinations of equalities and inequalities between real-valued expressions. In
particular, we consider methods of establishing such assertions using only
restricted forms of distributivity. At the same time, we explore ways in which
"local" decision or heuristic procedures for fragments of the theory of the
reals can be amalgamated into global ones. Let Tadd[Q] be the
first-order theory of the real numbers in the language of ordered groups, with
negation, a constant 1, and function symbols for multiplication by
rational constants. Let Tmult[Q] be the analogous theory for the
multiplicative structure, and let T[Q] be the union of the two. We
show that although T[Q] is undecidable, the universal fragment of
T[Q] is decidable. We also show that terms of T[Q]can
fruitfully be put in a normal form. We prove analogous results for theories in
which Q is replaced, more generally, by suitable subfields F
of the reals. Finally, we consider practical methods of establishing
quantifier-free validities that approximate our (impractical) decidability
results.Comment: Will appear in Logical Methods in Computer Scienc
The real field with an irrational power function and a dense multiplicative subgroup
This paper provides a first example of a model theoretically well behaved
structure consisting of a proper o-minimal expansion of the real field and a
dense multiplicative subgroup of finite rank. Under certain Schanuel
conditions, a quantifier elimination result will be shown for the real field
with an irrational power function and a dense multiplicative subgroup of finite
rank whose elements are algebraic over the field generated by the irrational
power. Moreover, every open set definable in this structure is already
definable in the reduct given by just the real field and the irrational power
function
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