Algorithmic Problems for Linear Recurrence Sequences
Joris Nieuwveld
Max Planck Institute for Software Systems
05 Sep 2025, 3:00 pm - 4:00 pm
Saarbrücken building E1 5, room 029
SWS Student Defense Talks - Thesis Defense
Linear recurrence sequences (LRS) are among the most fundamental and
easily definable classes of number sequences, encompassing many
classical sequences such as polynomials, powers of two, and the
Fibonacci numbers. They also describe the dynamics of iterated linear
maps and arise naturally in numerous contexts within computer science,
mathematics, and other quantitive sciences. However, despite their
simplicity, many easy-to-state decision problems for LRS have stubbornly
remained open for decades despite considerable and sustained attention.
Chief among these are the Skolem problem and the Positivity problem, ...
Linear recurrence sequences (LRS) are among the most fundamental and
easily definable classes of number sequences, encompassing many
classical sequences such as polynomials, powers of two, and the
Fibonacci numbers. They also describe the dynamics of iterated linear
maps and arise naturally in numerous contexts within computer science,
mathematics, and other quantitive sciences. However, despite their
simplicity, many easy-to-state decision problems for LRS have stubbornly
remained open for decades despite considerable and sustained attention.
Chief among these are the Skolem problem and the Positivity problem,
which ask to determine, for a given LRS, whether it contains a zero term
and whether it contains only positive terms, respectively. For both
problems, decidability is currently open, i.e., whether they are
algorithmically solvable.
In this thesis, we present the following results. For the Skolem problem, we introduce an algorithm for simple LRS whose correctness is unconditional but whose termination relies on two classical, widely-believed number-theoretic conjectures. This algorithm is implementable in practice, and we report on experimental results. For the Positivity problem, we introduce the notion of reversible LRS, which enables us to carve out a large decidable class of sequences. We also examine various expansions of classical logics by predicates obtained from LRS. In particular, we study expansions of monadic second-order logic of the natural numbers with order and present major advances over the seminal results of Büchi, Elgot, and Rabin from the early 1960s. Finally, we investigate fragments of Presburger arithmetic, where, among others, we establish the decidability of the existential fragment of Presburger arithmetic expanded with powers of 2 and 3.
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In this thesis, we present the following results. For the Skolem problem, we introduce an algorithm for simple LRS whose correctness is unconditional but whose termination relies on two classical, widely-believed number-theoretic conjectures. This algorithm is implementable in practice, and we report on experimental results. For the Positivity problem, we introduce the notion of reversible LRS, which enables us to carve out a large decidable class of sequences. We also examine various expansions of classical logics by predicates obtained from LRS. In particular, we study expansions of monadic second-order logic of the natural numbers with order and present major advances over the seminal results of Büchi, Elgot, and Rabin from the early 1960s. Finally, we investigate fragments of Presburger arithmetic, where, among others, we establish the decidability of the existential fragment of Presburger arithmetic expanded with powers of 2 and 3.