Related papers: Presburger arithmetic, rational generating functio…
We construct an addition and a multiplication on the set of planar binary trees, closely related to addition and multiplication on the integers. This gives rise to a new kind of (noncommutative) arithmetic theory. The price to pay for this…
We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice…
We collect and organise known results and add some new ones of the following nature: if A is a bounded operator in a Hilbert or Banach space, does there exist a nonconstant polynomial p(z) such that p(A) is "simpler", "nicer" than A. The…
Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where non-termination happens, more structure. In this paper we consider…
This paper introduces a robust class of functions from finite words to integers that we call Z-polyregular functions. We show that it admits natural characterizations in terms of logics, Z-rational expressions, Z-rational series and…
A new characterization of provably recursive functions of first-order arithmetic is described. Its main feature is using only terms consisting of 0, the successor S and variables in the quantifier rules, namely, universal elimination and…
We study generating functions in the context of Rota-Baxter algebras. We show that exponential generating functions can be naturally viewed in a very special case of complete free commutative Rota-Baxter algebras. This allows us to use free…
Let P be an elementary closed semi-algebraic set in R^d, i.e., there exist real polynomials p_1,...,p_s such that P= \{x \in R^d : p_1(x) \ge 0, >..., p_s(x) \ge 0 \}; in this case p_1,...,p_s are said to represent P. Denote by $n$ the…
Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally…
We tackle the issue of representing infinite sets of real- valued vectors. This paper introduces an operator for combining integer and real sets. Using this operator, we decompose three well-known logics extending Presburger with reals. Our…
It is shown that several of Brafman's generating functions for the Gegenbauer polynomials are algebraic functions of their arguments, if the Gegenbauer parameter differs from an integer by one-fourth or one-sixth. Two examples are given,…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
Greene's rational function $\Psi_P({\bf x})$ is a sum of certain rational functions in ${\bf x}=(x_1, \ldots, x_n)$ over the linear extensions of the poset $P$ (which has $n$ elements), which he introduced in his study of the…
We define a notion of an arithmetic set in an arbitrary countable group and study properties of these sets in the cases of Abelian groups and non-abelian free groups.
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
We take two approaches to classifying the complexity of Presburger models: Scott analysis and degree spectra. In particular, we investigate the possible Scott sentence complexities and possible degree spectra of models of Presburger…
An approach to build Probabilistic Arithmetic in which initial values of all correlated random variables are known, but with varying degrees of accuracy. As a result of the proposed Probabilistic Arithmetic operations, variable values,…
In the present paper, we consider Presburger arithmetic PrA and the theory of real closed fields RCF. Due to quantifier elimination in these theories, there are two kinds of natural ways to axiomatize them. Namely, on one hand, PrA can be…
Craig interpolation has emerged as an effective means of generating candidate program invariants. We present interpolation procedures for the theories of Presburger arithmetic combined with (i) uninterpreted predicates (QPA+UP), (ii)…
The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…