Presburger Arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger Arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture by A. Visser. In order to prove the result we show that all linear orderings that are interpretable in $(\mathbb{N};+)$ are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations.
This paper is devoted to understand groups definable in Presburger arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded group definable in a model (Z,+,<) of Presburger Arithmetic is definably isomorphic to (Z, +)^{n} mod out by a lattice.
Presburger Arithmetic $\mathop{\mathbf{PrA}}\nolimits$ is the true theory of natural numbers with addition. We consider linear orderings interpretable in Presburger Arithmetic and establish various necessary and sufficient conditions for interpretability depending on dimension $n$ of interpretation. We note this problem is relevant to the interpretations of Presburger Arithmetic in itself, as well as the characterization of automatic orderings. For $n=2$ we obtain the complete criterion of interpretability.
Let $G$ be a model of Presburger arithmetic. Let $\mathcal{L}$ be an expansion of the language of Presburger $\mathcal{L}_{Pres}$. In this paper we prove that the $\mathcal{L}$-theory of $G$ is $\mathcal{L}_{Pres}$-minimal iff it has the exchange property and any bounded definable set has a maximum.
We generalize Cooper's method of quantifier elimination for classical Presburger arithmetic to give a new proof that all parametric Presburger families (as defined by Kevin Woods) are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f(t) for every polynomial f over the integers. In fact, this quantifier bounding method works more generally in expansions of Presburger arithmetic with multiplication by scalars from any ring of functions into Z.
University of Ghent（根特大学）；Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；National Research University Higher School of Economics（国立高等经济大学）
We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.
We show that if a first-order structure $\mathcal{M}$, with universe $\mathbb{Z}$, is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,<,0)$, then $\mathcal{M}$ must be interdefinable with $(\mathbb{Z},+,0)$ or $(\mathbb{Z},+,<,0)$.
Presburger arithmetic PrA is the true theory of natural numbers with addition. We study interpretations of PrA in itself. We prove that all one-dimensional self-interpretations are definably isomorphic to the identity self-interpretation. In order to prove the results we show that all linear orders that are interpretable in (N,+) are scattered orders with the finite Hausdorff rank and that the ranks are bounded in terms of the dimension of the respective interpretations. From our result about self-interpretations of PrA it follows that PrA isn't one-dimensionally interpretable in any of its fi…
We present a description of rigid models of Presburger arithmetic (i.e., Z-groups). In particular, we show that Presburger arithmetic has rigid models of all infinite cardinalities up to the continuum, but no larger.
In 1996, Michaux and Villemaire considered integer relations $R$ which are not definable in Presburger Arithmetic. That is, not definable in first-order logic over integers with the addition function and the order relation (FO[N,+,<]-definable relations). They proved that, for each such $R$, there exists a FO[N,+,<,$R$]-formula $ν_{R}(x)$ which defines a set of integers which is not ultimately periodic, i.e. not FO[N,+,<]-definable. It is proven in this paper that the formula $ν(x)$ can be chosen such that it does not depend on the interpretation of $R$. It is furthermore proven that $ν(x)$ ca…
We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic ($\sin$-PA), and systematically study decision problems for sets of sentences in $\sin$-PA. In particular, we detail a decision algorithm for existential $\sin$-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of $\sin$-PA sent…
We study complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integers involved in the inequalities. We prove that assuming Kannan's partition can be found in polynomial time, the satisfiability of Short-PA sentences can be decided in polynomial time. Furthermore, under the same assumption, we show that the numbers of satisfying assignments of short Presburger sentences can also be computed in polynomial time.
Parametric Presburger arithmetic concerns families of sets S_t in Z^d, for t in N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication by t, and quantifiers on variables ranging over Z. That is, such families are defined using quantifiers and Boolean combinations of formulas of the form a(t) x <= b(t), where a(t) is in Z[t]^d, b(t) in Z[t]. A function g: N -> Z is a quasi-polynomial if there exists a period m and polynomials f_0, ..., f_{m-1} in Q[t] such that g(t)=f_i(t) for t congruent to i (mod m.) Recent results of Chen, Li, Sam; Calegari, Walk…
Let $f_1(n), \ldots, f_k(n)$ be polynomial functions of $n$. For fixed $n\in\mathbb{N}$, let $S_n\subseteq \mathbb{N}$ be the numerical semigroup generated by $f_1(n),\ldots,f_k(n)$. As $n$ varies, we show that many invariants of $S_n$ are eventually quasi-polynomial in $n$, such as the Frobenius number, the type, the genus, and the size of the $Δ$-set. The tool we use is expressibility in the logical system of parametric Presburger arithmetic. Generalizing to higher dimensional families of semigroups, we also examine affine semigroups $S_n\subseteq \mathbb{N}^m$ generated be vectors whose coo…
We study the computational complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of Short-PA sentences with $m+2$ alternating quantifiers is $Σ_{P}^m$-complete or $Π_{P}^m$-complete, when the first quantifier is $\exists$ or $\forall$, respectively. Counting versions and restricted systems are also analyzed. Further application are given to hardne…
We study VC-dimension of short formulas in Presburger Arithmetic, defined to have a bounded number of variables, quantifiers and atoms. We give both lower and upper bounds, which are tight up to a polynomial factor in the bit length of the formula.
We consider an expansion of Presburger arithmetic which allows multiplication by $k$ parameters $t_1,\ldots,t_k$. A formula in this language defines a parametric set $S_\mathbf{t} \subseteq \mathbb{Z}^{d}$ as $\mathbf{t}$ varies in $\mathbb{Z}^k$, and we examine the counting function $|S_\mathbf{t}|$ as a function of $\mathbf{t}$. For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi-polynomial (there is a period $m$ such that, for sufficiently large $t$, the function is polynomial on each of the residue classes mod $m$). We show that such a nice expression is …
Büchi arithmetics BA_n, n >= 2, are extensions of Presburger arithmetic with an unary functional symbol V_n(x) denoting the largest power of n that divides x. Definability of a set in BA_n is equivalent to its recognizability by a finite automaton receiving numbers in their n-ary expansion. We show that Büchi arithmetics BA_n and BA_m are bi-interpretable for any n,m. Furthermore, we establish that any interpretation of some structure in BA_n is isomorphic to some one-dimensional interpretation; however, this isomorphism must not be BA_n-definable.
