# WORReLL'23

## Workshop On Reachability, Recurrences, and Loops

### Satellite Workshop at ICALP 2023

#### Heinz Nixdorf MuseumsForum (HNF), Seminar Room 5

#### Paderborn, Germany

#### Monday 10th July 2023

#### Workshop Registration Link

## Topic

Recursively defined sequences are foundational objects of study in the computational sciences; they arise naturally in areas such as: the analysis of algorithms, weighted automata, loop termination, and probabilistic models.

## Aim

The aim of WORReLL'23 is to bring together researchers from the community and showcase cutting-edge research. The one-day workshop will also celebrate the research contributions of Professor James Worrell (also known as Ben). For context, Ben is giving an invited talk at ICALP this year.

Coincidentally, the workshop is a few days prior to Ben’s birthday. We plan to celebrate with a birthday dinner on Sunday 09th July (the evening before the workshop), so please take this into consideration for planning your travel to Paderborn.

## Participation

To participate in-person, please register for WORReLL'23 through ICALP’s registration portal. To participate remotely over Zoom, please contact the organisers for further details.

## Speakers, Talks, and Slides

(Click on a talk title for the abstract and slides.)

#### Nikhil Balaji (IIT Delhi)

*On the Complexity of the Sum of Square Roots Problem*

Abstract: Given positive integers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_m$, the Sum of Square Roots problem (SSR) is the computational problem of checking if $$\sqrt{a_1} + \sqrt{a_2} + \dots + \sqrt{a_n} > \sqrt{b_1} + \sqrt{b_2} + \dots + \sqrt{b_m}.$$ It is an essential primitive in Computational Geometry where efficient algorithms for several geometric problems are known relative to SSR. However the best-known complexity upper bound for SSR is in the Counting Hierarchy (CH), and no non-trivial lower bounds are known. Over the last two decades, SSR-hardness has emerged as a useful tool to capture the “numerical hardness” of problems, especially in Game Theory and Verification. In this talk, I will introduce a variant of SSR and show efficient “non-uniform” algorithms for this problem.

#### Paul Bell (Keele)

*From Quantum Automata to Quaternions and Rational Pairing Functions*

Abstract: We investigate some recent results on injectivity for Quantum Finite Automata. We call a QFA injective if the acceptance probability of every input word is unique. We show the undecidability of this problem by using tools and results from linear algebra, quaternions, number theory, and in particular rational polynomial pairing functions. We note a huge disparity in the number of required states to show undecidability depending on whether the initial state vector is rational or real algebraic. We will draw parallels with interesting questions for other matrix theoretic problems, and introduce some new research directions and open problems.

#### ValÃ©rie BerthÃ© (CNRS, Paris)

*Recursive Sequences and Numeration*

Abstract: In this lecture we revisit recursive sequences from the view point of numeration systems and their associated dynamical systems. Linear recurrences allow indeed the definition of numerations for both integers and real numbers. We focus on the so-called finiteness property which means that the set of elements having a finite expansion forms a ring, and on its numerous applications in arithmetics, aperiodic order, fractal geometry etc.

#### Dmitry Chistikov (Warwick)

*Globe-Hopping*

Abstract: I will talk about the following geometric puzzle. Take a random point (“grasshopper”) from a set $L$ (“lawn”, on the plane or on the sphere) and move it by a fixed distance in a random direction, once. How do you choose the shape of $L$, assuming a fixed area, so as to maximise the probability that the reached point is also in the set $L$?

Surprisingly, in 2D the disc (circle) is known to be sub-optimal. In a spherical version of the problem, motivated by quantum information theory, the hemispherical lawn is or is not optimal, depending on the length of the jump: whether it is a rational or irrational multiple of $\pi$, and if rational, then on the numerator in lowest terms.

Joint work with Olga Goulko, Adrian Kent, and Mike Paterson.

#### Florian Luca (Witwatersrand)

*Twisted Rational Zeros of Linearly Recurrent Sequences*

Abstract: Let $( T_n )_{n} \in \mathbb{Z}$ be the Tribonacci numbers. Marques and Lengyel (2014) obtained a formula relating the $2$-adic valuation of $T_n$ with the $2$-adic valuation of a linear function of $n$ (which might be constant) according to the residue class of $n$ modulo $32$ and optimistically conjectured that a similar formula holds true for every prime $p$. In the first part of the talk, we will see that their conjecture was indeed far too optimistic; in particular, it fails for all but seven primes below $600$. Along the way, we are led to consider a certain twisted rational zero of a linearly recurrent sequence. We prove that non-degenerate linearly recurrent sequences have only finitely many such twisted rational zeros, which may be seen as an extension of the Skolem–Mahler–Lech theorem. While our method is, in general, not effective, in some cases we can compute all such twisted rational zeros. In particular, returning to the Tribonacci sequence we show that its only twisted rational zeros are the integral ones $-17,-4,-1,0$ together with the rational non-integral ones $1/3$ and $-5/3$.

This is joint work with Y. Bilu, J. Nieuwveld, J. Ouaknine and J. Worrell.

#### JoÃ«l Ouaknine (MPI-SWS)

*Universal Skolem Sets*

Abstract: The Skolem Problem asks to determine whether a given integer linear recurrence sequence (such as the Fibonacci numbers) has a zero term. This problem, whose decidability has been open for many decades, arises across a wide range of topics in computer science, including loop termination, probabilistic model checking, weighted automata theory, and dynamical systems, amongst many others. Recently, a new line of attack to the Skolem Problem was initiated via the notion of a Universal Skolem Set: a set S of positive integers such that it is decidable whether a given linear recurrence sequence has a zero in S. I will present an overview of this approach.

This is joint work with Florian Luca, James Maynard, Armand Noubissie, and James Worrell.

#### Veronika Pillwein (JKU)

*The Positivity Problem for Recurrent Sequences*

Abstract: Proving positivity of sequences that are defined by recurrences is very hard. Even when restricted to the seemingly simple class of sequences defined by linear recurrences with constant coefficients (C-finite sequences), decidability of the positivity problem is only known for orders up to five.

On the other hand, some computer algebra methods are able to tackle this problem. Obviously, they do not succeed with all types of input and even though correctness can be proven, termination is typically an issue. In this talk, we will give an overview on this topic, present some available methods and show cases where algorithmic proofs actually succeeded.

#### Igor Potapov (Liverpool)

*Reachability Problems on Matrices and Maps*

Abstract: A large number of naturally defined matrix problems are still unanswered, despite the long history of matrix theory. In this presentation, I will discuss a number of challenging computational problems for matrix semigroups and their connections to matrix equations, iterative maps and linear recurrence sequences. Some of these problems are still unsolved but at the same time identified as reference points in the algorithmic analysis of reachability problems. The cornerstones of these reachability problems are in the synergy of the questions in mathematics and computer science: computability, automata theory, matrix theory, combinatorics, abstract algebra, number theory, etc. The main objective of future research is in developing new concepts and methods combining symbolic and numerical techniques as well as unifying the results in automata, computability, number and matrix theories.

#### Mahsa Shirmohammadi (CNRS, Paris)

*The Membership Problem for Hypergeometric Sequences*

Abstract: This talk is on the Membership problem for rational-valued hypergeometric sequences. Such sequences $\langle u_n \rangle_{n=0}^{\infty}$ satisfies a first-order linear recurrence relation with polynomial coefficients; that is, a recurrence of the form $f(n)u_n = g(n)u_{n-1}$ where $f,g \in \mathbb{Z}[x]$. The Membership Problem asks, given a hypergeometric sequence $\langle u_n \rangle_{n=0}^{\infty}$ and a target value $t\in \mathbb{Q}$, determine whether $t$ occurs in the sequence. We give hints on recent decidability results of the Membership Problem under the assumption that either (i) $f$ and $g$ have distinct splitting fields, (ii) $f$ and $g$ split totally over $\mathbb{Q}$, and (iii) $f$ and $g$ are monic polynomials that both split over a quadratic extension of $\mathbb{Q}$.

This talk is based on two papers appearing in ISSAC'22 and ISSAC'23, and are in collaboration with George Kenison, Amaury Pouly, Klara Nosan, and James Worrell.

## Schedule

(All times are UTC+02:00)

*08:00–09:00 Registration*

09:00–09:45 ValÃ©rie BerthÃ©

09:45–10:30 Dmitry Chistikov

*Coffee Break*

11:00–11:45 Mahsa Shirmohammadi

11:45–12:30 Florian Luca

*Lunch*

14:05–14:25 Veronika Pillwein (Online)

14:25–14:45 Nikhil Balaji

14:45–15:30 Paul Bell

*Coffee Break*

16:00–16:45 Igor Potapov

16:45–17:30 JoÃ«l Ouaknine

*Workshop Ends. (ICALP food and drinks event begins.)*

## Organisers

- George Kenison (TU Wien)
- Mahsa Shirmohammadi (CNRS, Paris)

## Acknowledgements

We gratefully acknowledge the financial support of both the CNRS and the ANR via the International Emerging Actions grant (IEA'22) and the VeSyAM grant (ANR-22-CE48-0005), respectively.