Orbit counting in conjugacy classes for free groups acting on trees

In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an irrationality condition on the edge lengths. We obtain an asymptotic for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most $T$. Under a mild extra assumption we also obtain a polynomial error term.Comment: 13 pages, additional section discusses error terms, revised expositio

Applications of transcendental number theory to decision problems for hypergeometric sequences

A rational-valued sequence is hypergeometric if it satisfies a first-order linear recurrence relation with polynomial coefficients. In this note we discuss two decision problems, the membership and threshold problems, for hypergeometric sequences. The former problem asks whether a chosen target is in the orbit of a given sequence, whilst the latter asks whether every term in a sequence is bounded from below by a given value. We establish decidability results for restricted variants of these two decision problems with an approach via transcendental number theory. Our contributions include the following: the membership and threshold problems are both decidable for the class of rational-valued hypergeometric sequences with Gaussian integer parameters

The Membership Problem for hypergeometric sequences with quadratic parameters

Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, a hypergeometric sequence ⟨⟩∞=0 is one that satisfies a recurrence of the form ()=()−1 where , ∈ Z[]. In this paper, we consider the Membership Problem for hypergeometric sequences: given a hypergeometric sequence ⟨⟩∞=0 and a target value ∈ Q, determine whether = for some index . We establish decidability of the Membership Problem under the assumption that either (i) and have distinct splitting fields or (ii) and are monic polynomials that both split over a quadratic extension of Q. Our results are based on an analysis of the prime divisors of polynomial sequences ⟨()⟩∞=1 and ⟨()⟩∞=1 appearing in the recurrence relation

On the Skolem Problem and Prime Powers

The Skolem Problem asks, given a linear recurrence sequence $(u_n)$, whether there exists $n\in\mathbb{N}$ such that $u_n=0$. In this paper we consider the following specialisation of the problem: given in addition $c\in\mathbb{N}$, determine whether there exists $n\in\mathbb{N}$ of the form $n=lp^k$, with $k,l\leq c$ and $p$ any prime number, such that $u_n=0$.Comment: 13 pages, ISSAC 202

(Un)Solvable Loop Analysis

Automatically generating invariants, key to computer-aided analysis of probabilistic and deterministic programs and compiler optimisation, is a challenging open problem. Whilst the problem is in general undecidable, the goal is settled for restricted classes of loops. For the class of solvable loops, introduced by Kapur and Rodr\'iguez-Carbonell in 2004, one can automatically compute invariants from closed-form solutions of recurrence equations that model the loop behaviour. In this paper we establish a technique for invariant synthesis for loops that are not solvable, termed unsolvable loops. Our approach automatically partitions the program variables and identifies the so-called defective variables that characterise unsolvability. Herein we consider the following two applications. First, we present a novel technique that automatically synthesises polynomials from defective monomials, that admit closed-form solutions and thus lead to polynomial loop invariants. Second, given an unsolvable loop, we synthesise solvable loops with the following property: the invariant polynomials of the solvable loops are all invariants of the given unsolvable loop. Our implementation and experiments demonstrate both the feasibility and applicability of our approach to both deterministic and probabilistic programs.Comment: Extended version of the conference paper `Solving Invariant Generation for Unsolvable Loops' published at SAS 2022 (see also the preprint arXiv:2206.06943). We extended both the text and results. 36 page

On the Skolem Problem for Reversible Sequences

Given an integer linear recurrence sequence ?X_n?, the Skolem Problem asks to determine whether there is a natural number n such that X_n = 0. Recent work by Lipton, Luca, Nieuwveld, Ouaknine, Purser, and Worrell proved that the Skolem Problem is decidable for a class of reversible sequences of order at most seven. Here we give an alternative proof of their result. Our novel approach employs a powerful result for Galois conjugates that lie on two concentric circles due to Dubickas and Smyth

Asymptotics in conjugacy classes for free groups

In this thesis we consider asymptotic counts of words in free groups. In particular, we establish results when we restrict the group elements to a non-trivial conjugacy class. We study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Our results require an irrationality condition on the edge lengths. We obtain an asymptotic formula for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most T. Under a mild extra assumption we also obtain a polynomial error term. Related to the above orbit counting result for metric trees, we also consider the spatial distribution of the lattice points of a given conjugacy class in the universal covering tree. We show that the lattice points of a fixed conjugacy class are asymptotically spatially distributed according to a Patterson-Sullivan measure supported on the boundary of the universal cover. For a class of functions on a free group with suitable symbolic properties we establish an asymptotic average and, subject to an appropriate normalisation, a central limit theorem when the elements of the free group are restricted to a non-trivial conjugacy class