arXiv subjects

James Worrell

Publications and source records attributed to James Worrell.

At least 19 recordsLinked to original sources

arXiv2026

On Periodic and Aperiodic Optimal Strategies in Solvency Games

Solvency games are a gambling problem on infinite-state Markov decision processes in which the state $n \in \mathbb{N}$ represents an investor's fortune. In every round, the investor chooses an action from a finite action set, and every action yields a distribution over integer-valued gains in an interval $\{-\ell,\ldots,m\}$. The risk-averse investor wants to minimise the probability of eventual ruin (reaching a fortune $\le 0$). It was shown in [Berger et al.] that memoryless deterministic optimal strategies exist, but they are not eventually constant in general. Even in the special case of gains in $\{-2,\ldots,1\}$, the optimal strategy may need to make use of two different actions at arbitrarily high fortunes. We show that optimal strategies in solvency games need not be ultimately periodic in general (thus disproving a 2012 conjecture of Kučera). Already in the case of gains in $\{-3,\ldots,1\}$, it is possible for the optimal strategy to be unique but aperiodic. For gains in $\{-2,\ldots,1\}$, there always exists an ultimately periodic optimal strategy whose tail is constant or alternates between two actions. Finally, we show that the optimal strategy is computable if it is unique. Moreover, (some) optimal strategy can always be computed in the case of gains in $\{-\ell,\ldots,1\}$ for any $\ell \in \mathbb{N}$. Computability in the general case however remains open.

cs.GT↗

arXiv2026

Skolem Meets Bateman-Horn

The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and control theory. Decidability is notoriously open; the state of the art is a decision procedure for recurrences of order at most 4: an advance achieved some 40 years ago, based on Baker's theorem on linear forms in logarithms of algebraic numbers. A new approach to the Skolem Problem was recently initiated in [LOW21, LOW22] via the notion of a Universal Skolem Set -- a set $S$ of positive integers such that it is decidable whether a given non-degenerate linear recurrence sequence has a zero in $S$. Clearly, proving decidability of the Skolem Problem is equivalent to showing that $\mathbb{N}$ itself is a Universal Skolem Set. The main contribution of the present paper is to construct a Universal Skolem Set that has lower density at least $1/8$. We show moreover that this set has density $1$ subject to Martin's uniform formulation of the Bateman--Horn conjecture. The latter is a far-reaching quantitative hypothesis concerning the frequency of primes among the values of systems of polynomials.

cs.DM↗

arXiv2026

Positivity of Nearly Linearly Recurrent Sequences

Nearly linear recurrences generalise linear recurrences and can be represented as special cases of both linear time-invariant systems in control theory and linear-constraint loops in program analysis. We formulate the Positivity Problem for such recurrences: given a recurrence and initial values, decide whether every sequence satisfying the recurrence is termwise nonnegative. This problem generalises Positivity for linear recurrence sequences and is a special case of halfspace non-reachability for linear time-invariant systems. Our main result is a decision procedure for order-2 recurrences. The termination of the procedure relies on a transcendence theorem of independent interest: we prove that certain convergent series obtained by summing the absolute values of terms of algebraic linear recurrence sequences are transcendental.

math.DS↗

arXiv2026

Termination Analysis of Linear-Constraint Programs

This paper provides an overview of techniques in termination analysis for programs with numerical variables and transitions defined by linear constraints. This subarea of program analysis is challenging due to the existence of undecidable problems, and this Survey systematically explores approaches that mitigate this inherent difficulty. These include foundational decidability results, the use of ranking functions, and disjunctive well-founded transition invariants. The Survey also discusses non-termination witnesses, used to prove that a program will not halt. We examine the algorithmic and complexity aspects of these methods, showing how different approaches offer a trade-off between expressive power and computational complexity. The Survey does not discuss how termination analysis is performed on real-world programming languages, nor does it consider more expressive abstract models that include non-linear arithmetic, probabilistic choice, or term rewriting systems.

cs.PL↗

arXiv2026

On the Complexity of the Skolem Problem at Low Orders

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) $\langle u_n \rangle_{n=0}^\infty$ over the integers has a zero term, that is, whether there exists $n$ such that $u_n = 0$. Decidability of the problem is open in general, with the most notable positive result being a decision procedure for LRS of order at most 4. In this paper we consider a bounded version of the Skolem Problem, in which the input consists of an LRS $\langle u_n \rangle_{n=0}^\infty$ and a bound $N \in \mathbb N$ (with all integers written in binary), and the task is to determine whether there exists $n\in\{0,\ldots,N\}$ such that $u_n=0$. We give a randomised algorithm for this problem that, for all $d\in \mathbb N$, runs in polynomial time on the class of LRS of order at most $d$. As a corollary we show that the (unrestricted) Skolem Problem for LRS of order at most 4 lies in $\mathsf{coRP}$, improving the best previous upper bound of $\mathsf{NP}^{\mathsf{RP}}$. The running time of our algorithm is exponential in the order of the LRS -- a dependence that appears necessary in view of the $\mathsf{NP}$-hardness of the Bounded Skolem Problem. However, even for LRS of a fixed order, the problem involves detecting zeros within an exponentially large range. For this, our algorithm relies on results from $p$-adic analysis to isolate polynomially many candidate zeros and then test in randomised polynomial time whether each candidate is an actual zero by reduction to arithmetic-circuit identity testing.

cs.CC↗

arXiv2026

Conjectural Decidability of the Skolem Problem

The Skolem Problem asks to determine whether a given integer linear recurrence sequence (LRS) 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, formal languages, automata theory, and probabilistic model checking, amongst many others. In the present paper, we introduce a notion of "large" zeros of (non-degenerate) linear recurrence sequences, i.e., zeros occurring at an index larger than a double exponential of the magnitude of the data defining the given LRS. We establish two main results. First, we define an infinite set of prime numbers, termed "good", having density one amongst all prime numbers, with the following property: for any large zero of a given LRS, there is an interval around the large zero together with an upper bound on the number of good primes possibly present in that interval. The bound in question is much lower than one would expect if good primes were distributed similarly as ordinary prime numbers, as per the Cramér model in number theory. We therefore conclude, conditionally on a strengthening of the classical Cramér conjecture, that large zeros do not exist, which would entail decidability of the Skolem Problem. Second, we show unconditionally that large zeros are very sparse: the set of positive integers that can possibly arise as large zeros of some LRS has null density. This in turn immediately yields a Universal Skolem Set of density one, answering a question left open in the literature.

cs.DM↗

arXiv2026

On the $p$-adic Skolem Problem

The Skolem Problem asks to determine whether a given linear recurrence sequence (LRS) has a zero term. Showing decidability of this problem is equivalent to giving an effective proof of the Skolem-Mahler-Lech Theorem, which asserts that a non-degenerate LRS has finitely many zeros. The latter result was proven over 90 years ago via an ineffective method showing that such an LRS has only finitely many $p$-adic zeros. In this paper we consider the problem of determining whether a given LRS has a $p$-adic zero, as well as the corresponding function problem of computing exact representations of all $p$-adic zeros. We present algorithms for both problems and report on their implementation. The output of the algorithms is unconditionally correct, and termination is guaranteed subject to the $p$-adic Schanuel Conjecture (a standard number-theoretic hypothesis concerning the $p$-adic exponential function). While these algorithms do not solve the Skolem Problem, they can be exploited to find natural-number and rational zeros under additional hypotheses. To illustrate this, we apply our results to show decidability of the Simultaneous Skolem Problem (determine whether two coprime linear recurrences have a common natural-number zero), again subject to the $p$-adic Schanuel Conjecture.

cs.LO↗

arXiv2026

On the Decidability of Monadic Theories of Arithmetic Predicates

We investigate the decidability of the monadic second-order (MSO) theory of the structure $\langle \mathbb{N};<,P_1, \ldots,P_d \rangle$, for various unary predicates $P_1,\ldots,P_d \subseteq \mathbb{N}$. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers $k^{\mathbf{N}} = \{k^n : n \in \mathbb{N}\}$, $k$-th powers $\mathbf{N}^k = \{n^k : n \in \mathbb{N}\}$, and the set of terms of the Fibonacci sequence $\mathsf{Fib} = \{0,1,2,3,5,8,13,\ldots\}$ (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following: $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, \mathsf{Fib} \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, 3^{\mathbf{N}}, 6^{\mathbf{N}} \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, 3^{\mathbf{N}}, 5^{\mathbf{N}} \rangle$ is decidable assuming Schanuel's conjecture; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 4^{\mathbf{N}}, \mathbf{N}^2 \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, \mathbf{N}^2 \rangle$ is Turing-equivalent to the MSO theory of $\langle \mathbb{N};<,S \rangle$, where $S$ is the predicate corresponding to the binary expansion of $\sqrt{2}$. (As the binary expansion of $\sqrt{2}$ is widely believed to be normal, the corresponding MSO theory is in turn expected to be decidable.) These results are obtained by exploiting and combining techniques from dynamical systems, number theory, and automata theory.

cs.LO↗

arXiv2025

Determination Problems for Orbit Closures and Matrix Groups

Computational problems concerning the orbit of a point under the action of a matrix group occur throughout computer science, including in program analysis, complexity theory, quantum computation, and automata theory. In many cases the focus extends beyond orbits proper to orbit closures under a suitable topology. Typically one starts from a group and a set of points and asks questions about the orbit closure of the set under the action of the group, e.g., whether two given orbit closures intersect. In this paper we consider a collection of what we call determination problems concerning matrix groups and orbit closures. These problems begin with a given variety and seek to understand whether and how it arises either as an algebraic matrix group or as an orbit closure. The how question asks whether the underlying group is $s$-generated, meaning it is topologically generated by $s$ matrices for a given number $s$. Among other applications, problems of this type have recently been studied in the context of synthesising loops subject to certain specified invariants on program variables. Our main result is a polynomial-space procedure that inputs a variety and a number $s$ and determines whether the given variety arises as an orbit closure of a point under an $s$-generated commutative algebraic matrix group. The main tools in our approach are structural properties of commutative algebraic matrix groups and module theory. We leave open the question of determining whether a variety is an orbit closure of a point under an $s$-generated algebraic matrix group (without the requirement of commutativity).

cs.CC↗

arXiv2025

Algebraic Closure of Matrix Sets Recognized by 1-VASS

It is known how to compute the Zariski closure of a finitely generated monoid of matrices and, more generally, of a set of matrices specified by a regular language. This result was recently used to give a procedure to compute all polynomial invariants of a given affine program. Decidability of the more general problem of computing all polynomial invariants of affine programs with recursive procedure calls remains open. Mathematically speaking, the core challenge is to compute the Zariski closure of a set of matrices defined by a context-free language. In this paper, we approach the problem from two sides: Towards decidability, we give a procedure to compute the Zariski closure of sets of matrices given by one-counter languages (that is, languages accepted by one-dimensional vector addition systems with states and zero tests), a proper subclass of context-free languages. On the other side, we show that the problem becomes undecidable for indexed languages, a natural extension of context-free languages corresponding to nested pushdown automata. One of our main technical tools is a novel adaptation of Simon's factorization forests to infinite monoids of matrices.

cs.FL↗

arXiv2025

Multiple Reachability in Linear Dynamical Systems

We consider reachability decision problems for linear dynamical systems: Given a linear map on $\mathbb{R}^d$ , together with source and target sets, determine whether there is a point in the source set whose orbit, obtained by repeatedly applying the linear map, enters the target set. When the source and target sets are semialgebraic, this problem can be reduced to a point-to-polytope reachability question. The latter is generally believed not to be substantially harder than the well-known Skolem and Positivity Problems. The situation is markedly different for multiple reachability, i.e. the question of whether the orbit visits the target set at least m times, for some given positive integer m. In this paper, we prove that when the source set is semialgebraic and the target set consists of a hyperplane, multiple reachability is undecidable; in fact we already obtain undecidability in ambient dimension d = 10 and with fixed m = 9. Moreover, as we observe that procedures for dimensions 3 up to 9 would imply strong results pertaining to effective solutions of Diophantine equations, we mainly focus on the affine plane ($\mathbb{R}^2$). We obtain two main positive results. We show that multiple reachability is decidable for halfplane targets, and that it is also decidable for general semialgebraic targets, provided the linear map is a rotation. The latter result involves a new method, based on intersections of algebraic subgroups with subvarieties, due to Bombieri and Zannier.

cs.LO↗

arXiv2025

On the growth of hypergeometric sequences

Hypergeometric sequences obey first-order linear recurrence relations with polynomial coefficients and are commonplace throughout the mathematical and computational sciences. For certain classes of hypergeometric sequences, we prove linear growth estimates on their Weil heights. We give an application of our effective results towards the Membership Problem from Computer Science. Recall that Membership asks to procedurally determine whether a specified target is an element of a given recurrence sequence.

math.NT↗

arXiv2025

The Monadic Theory of Toric Words

For which unary predicates $P_1, \ldots, P_m$ is the MSO theory of the structure $\langle \mathbb{N}; <, P_1, \ldots, P_m \rangle$ decidable? We survey the state of the art, leading us to investigate combinatorial properties of almost-periodic, morphic, and toric words. In doing so, we show that if each $P_i$ can be generated by a toric dynamical system of a certain kind, then the attendant MSO theory is decidable.

cs.LO↗

arXiv2025

On the Decidability of Presburger Arithmetic Expanded with Powers

We prove that for any integers $α, β> 1$, the existential fragment of the first-order theory of the structure $\langle \mathbb{Z}; 0,1,<, +, α^{\mathbb{N}}, β^{\mathbb{N}}\rangle$ is decidable (where $α^{\mathbb{N}}$ is the set of positive integer powers of $α$, and likewise for $β^{\mathbb{N}}$). On the other hand, we show by way of hardness that decidability of the existential fragment of the theory of $\langle \mathbb{N}; 0,1, <, +, x\mapsto α^x, x \mapsto β^x\rangle$ for any multiplicatively independent $α,β> 1$ would lead to mathematical breakthroughs regarding base-$α$ and base-$β$ expansions of certain transcendental numbers.

cs.LO↗

arXiv2025

Computing transcendence and linear relations of 1-periods

A 1-period is a complex number given by the integral of a univariate algebraic function, where all data involved -- the integrand and the domain of integration -- are defined over algebraic numbers. We give an algorithm that, given a finite collection of 1-periods, computes the space of all linear relations among them with algebraic coefficients. In particular, the algorithm decides whether a given 1-period is transcendental, and whether two 1-periods are equal. This resolves, in the case of 1-periods, a problem posed by Kontsevich and Zagier, asking for an algorithm to decide equality of periods. The algorithm builds on the work of Huber and Wüstholz, who showed that all linear relations among 1-periods arise from 1-motives; we make this perspective effective by reducing the problem to divisor arithmetic on curves and providing the theoretical foundations for a practical and fully explicit algorithm. To illustrate the broader applicability of our methods, we also give an algorithmic classification of autonomous first-order (non-linear) differential equations.

math.AG↗

arXiv2025

Transcendence for Pisot Morphic Words over an Algebraic Base

It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $β$ such that $|β|>1$, the number $[\![\boldsymbol{u} ]\!]_β:=\sum_{n=0}^\infty \frac{u_n}{β^n}$ either lies in $\mathbb Q(β)$ or is transcendental. In this paper we show a similar rational-transcendental dichotomy for sequences defined by irreducible Pisot morphisms. Subject to the Pisot conjecture (an irreducible Pisot morphism has pure discrete spectrum), we generalise the latter result to arbitrary finite alphabets. In certain cases we are able to show transcendence of $[\![\boldsymbol{u}]\!]_β$ outright. In particular, for $k\geq 2$, if $\boldsymbol u$ is the $k$-bonacci word then $[\![\boldsymbol{u}]\!]_β$ is transcendental.

math.NT↗

arXiv2025

On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

We investigate the complexity of computing the Zariski closure of a finitely generated group of matrices. The Zariski closure was previously shown to be computable by Derksen, Jeandel, and Koiran, but the termination argument for their algorithm appears not to yield any complexity bound. In this paper we follow a different approach and obtain a bound on the degree of the polynomials that define the closure. Our bound shows that the closure can be computed in elementary time. We also obtain upper bounds on the length of chains of linear algebraic groups, where all the groups are generated over a fixed number field.

cs.CC↗

arXiv2025

Differential Tree Automata

A rationally dynamically algebraic (RDA) power series is one that arises as (a component of) the solution of a system of differential equations of the form $\boldsymbol{y}' = F(\boldsymbol{y})$, where $F$ is a vector of rational functions that is defined at $\boldsymbol{y}(0)$. RDA power series subsume algebraic power series and are a proper subclass of differentially algebraic power series (those that satisfy a univariate polynomial-differential equation). We give a combinatorial characterisation of RDA power series in terms of exponential generating functions of regular languages of labelled trees. Motivated by this connection, we define the notion of a differential tree automaton. Differential tree automata generalise weighted tree automata by allowing the transition weights to be rational functions of the tree size. Our main result is that the ordinary generating functions of the formal tree series recognised by differential tree automata are exactly the differentially algebraic power series. The proof of this result establishes a general form of recurrence satisfied by the sequence of coefficients of a differentially algebraic power series, generalising Reutenauer's matrix representation of polynomially recursive sequences. As a corollary we obtain a procedure for determining equality of differential tree automata.

cs.FL↗