Presburger arithmetic, rational generating functions, and quasi-polynomials

Presburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p=(p_1,...,p_n) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.Comment: revised, including significant additions explaining computational complexity results. To appear in Journal of Symbolic Logic. Extended abstract in ICALP 2013. 17 page

Complexity of short Presburger arithmetic

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

Automatic Equivalence Structures of Polynomial Growth

In this paper we study the class EqP of automatic equivalence structures of the form ?=(D, E) where the domain D is a regular language of polynomial growth and E is an equivalence relation on D. Our goal is to investigate the following two foundational problems (in the theory of automatic structures) aimed for the class EqP. The first is to find algebraic characterizations of structures from EqP, and the second is to investigate the isomorphism problem for the class EqP. We provide full solutions to these two problems. First, we produce a characterization of structures from EqP through multivariate polynomials. Second, we present two contrasting results. On the one hand, we prove that the isomorphism problem for structures from the class EqP is undecidable. On the other hand, we prove that the isomorphism problem is decidable for structures from EqP with domains of quadratic growth

Symbolic and analytic techniques for resource analysis of Java bytecode

Recent work in resource analysis has translated the idea of amortised resource analysis to imperative languages using a program logic that allows mixing of assertions about heap shapes, in the tradition of separation logic, and assertions about consumable resources. Separately, polyhedral methods have been used to calculate bounds on numbers of iterations in loop-based programs. We are attempting to combine these ideas to deal with Java programs involving both data structures and loops, focusing on the bytecode level rather than on source code

Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination

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 impossible with 2 or more parameters. Indeed (assuming \textbf{P} $\neq$ \textbf{NP}) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1, t_2}|$ is not even polynomial-time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_\mathbf{t} \subseteq \mathbb{Z}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_\mathbf{t}|$ is always polynomial-time computable in the size of $\mathbf{t}$, and in fact can be represented using the gcd and similar functions.Comment: 14 pages, 1 figur

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters t1,…,tk. A formula in this language defines a parametric set St⊆Zd as t varies in Zk, and we examine the counting function |St| as a function of t. For a single parameter, it is known that |St| 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 impossible with 2 or more parameters. Indeed (assuming P≠NP) we construct a parametric set St1,t2 such that |St1,t2| is not even polynomial‐time computable on input (t1,t2). In contrast, for parametric sets St⊆Zd with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that |St| is always polynomial‐time computable in the size of t, and in fact can be represented using the gcd and similar functions.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/151911/1/malq201800068_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151911/2/malq201800068.pd

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior, Discrete Analysis 2017:4, 34 pp. Let $T$ be a triangle with vertices $(0,0)$, $(0,1/3)$, and $(1,0)$, and let $t$ be a positive integer. Then it is not hard to check that there are three quadratics $q_1,q_2$ and $q_3$ such that the number of integer points in $tT$ is $q_i(t)$ if $t\equiv i$ mod 3. In a situation like this, we say that the number of integer points is _quasipolynomial_ with period 3. In 1962, Ehrhart proved that if $P$ is a polytope in $\mathbb Z^d$ defined by a finite set of linear inequalities of the form $a_i.x\leq b_i$, where each of the $a_i$ belong to $\mathbb Z^d$ and $b$ belongs to $\mathbb Z$, then the number of lattice points in $tP$ is quasipolynomial with period $m$, where $m$ is the smallest integer such that the vertices of $mP$ are all lattice points. Since then, the same conclusion has been established for other families of sets $S_t\subset \mathbb Z^d$ by Chen, Li and Sam, by Calegari and Walker, and by Roune and Woods. After these results, it was tempting to wonder whether _all_ families of sets, provided that they are sufficiently nice in some appropriate sense, exhibit this quasipolynomial behaviour. The constraints would have to be reasonably strong -- for example, the number of lattice points inside the unit sphere of radius $t$ is certainly not quasipolynomial (and indeed, estimating it is a famous problem) -- but one could still hope for a general theorem that would encompass the known results and give a number of further ones. It turns out that the right behaviour to look for in general is that the size of $S_t$ should be _eventually_ quasipolynomial -- that is, it should agree with a quasipolynomial for sufficiently large $t$. Woods conjectured that eventual quasipolynomial behaviour should occur whenever the family is definable in _parametric Presburger arithmetic_. Roughly what this means (for a more precise definition, see the paper) is that the family $S_t$ of subsets of $\mathbb Z^d$ can be defined using addition, inequalities, integer constants, Boolean operations, multiplication by $t$, and quantification over $\mathbb Z$. The polytopes discussed earlier are examples. For a somewhat different kind of example, let $S_t$ be the set of positive integers $n$ such that there do not exist non-negative integers $a,b,c$ with $n=at+b(t+1)+c(t+2)$. This example involves quantification over $\mathbb Z$, but again the number of points in $S_t$ turns out to be quasipolynomial: in fact, it is $\lfloor t^2/4 \rfloor$ (the paper also discusses a quasipolynomial formula for the maximum element of $S_t$). Note that it is crucial in this definition that multiplication, except by the parameter $t$, should not be allowed, since otherwise we would have the full power of Peano arithmetic, which is undecidable. The main result of this paper is a proof of this very appealing conjecture. The proof uses a series of reductions that make the family simpler and simpler until the result can be shown using previously developed methods. One of the reductions uses the well-known technique of quantifier elimination. However, this cannot be applied straightforwardly, owing to the multiplication-by-$t$ operation, which is not part of standard Presburger arithmetic (hence the word "parametric"). The paper also discusses the power of parametric Presburger arithmetic, which, considering the necessary restrictions, is greater than one might expect. Thus, it proves eventual quasipolynomial behaviour for an extremely wide class of families and is probably the most general result one could hope for along these lines. [Image created by Georgios Barmparis, Georgios Kopidakis and Ioannis Remediakis](http://www.mdpi.com/1996-1944/9/4/301/htm)</sup