8 research outputs found
Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination
We consider an expansion of Presburger arithmetic which allows multiplication
by parameters . A formula in this language defines a
parametric set as varies
in , and we examine the counting function as a
function of . For a single parameter, it is known that can
be expressed as an eventual quasi-polynomial (there is a period such that,
for sufficiently large , the function is polynomial on each of the residue
classes mod ). We show that such a nice expression is impossible with 2 or
more parameters. Indeed (assuming \textbf{P} \textbf{NP}) we construct a
parametric set such that is not even
polynomial-time computable on input . In contrast, for parametric
sets with arbitrarily many parameters,
defined in a similar language without the ordering relation, we show that
is always polynomial-time computable in the size of
, 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
Some new results on decidability for elementary algebra and geometry
We carry out a systematic study of decidability for theories of (a) real
vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces,
Banach spaces and metric spaces, all formalised using a 2-sorted first-order
language. The theories for list (a) turn out to be decidable while the theories
for list (b) are not even arithmetical: the theory of 2-dimensional Banach
spaces, for example, has the same many-one degree as the set of truths of
second-order arithmetic.
We find that the purely universal and purely existential fragments of the
theory of normed spaces are decidable, as is the AE fragment of the theory of
metric spaces. These results are sharp of their type: reductions of Hilbert's
10th problem show that the EA fragments for metric and normed spaces and the AE
fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs
of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3
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 be a triangle with vertices , , and , and let be a positive integer. Then it is not hard to check that there are three quadratics and such that the number of integer points in is if 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 is a polytope in defined by a finite set of linear inequalities of the form , where each of the belong to and belongs to , then the number of lattice points in is quasipolynomial with period , where is the smallest integer such that the vertices of are all lattice points. Since then, the same conclusion has been established for other families of sets 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 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 should be _eventually_ quasipolynomial -- that is, it should agree with a quasipolynomial for sufficiently large . 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 of subsets of can be defined using addition, inequalities, integer constants, Boolean operations, multiplication by , and quantification over . The polytopes discussed earlier are examples. For a somewhat different kind of example, let be the set of positive integers such that there do not exist non-negative integers with . This example involves quantification over , but again the number of points in turns out to be quasipolynomial: in fact, it is (the paper also discusses a quasipolynomial formula for the maximum element of ). Note that it is crucial in this definition that multiplication, except by the parameter , 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- 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