Keywords

1 Introduction

Dynamical systems are a fundamental modelling paradigm in many branches of science, and have been the subject of extensive research for many decades. A (rational) discrete linear dynamical system (LDS) in ambient space \(\mathbb {R}^d\) is given by a square \(d \times d\) matrix M with rational entries, together with a starting point \(x \in \mathbb {Q}^d\). The orbit of (M, x) is the infinite trajectory \(\mathcal O(M,x) := \langle x, Mx, M^2x,\dots \rangle \). An example of a two-dimensional LDS is given in Fig. 1. A central concern in the computational theory of dynamical systems is the task of devising algorithms enabling one to decide various kinds of assertions on dynamical-system orbits.

Fig. 1.
figure 1

A two-dimensional discrete linear dynamical system.

Full size image

One of the most natural and fundamental computational questions concerning linear dynamical systems is the Point-to-Point Reachability Problem, also known as the Kannan-Lipton Orbit Problem: given a d-dimensional LDS (M, x) together with a point target \(y \in \mathbb {Q}^d\), does the orbit of the LDS ever hit the target? The decidability of this question was settled affirmatively in the 1980s in the seminal work of Kannan and Lipton [28, 29]. In fact, Kannan and Lipton showed that this problem is solvable in polynomial time, answering an earlier open problem of Harrison from the 1960s on reachability for linear sequential machines [26].

Interestingly, one of Kannan and Lipton’s motivations was to propose a line of attack to the well-known Skolem Problem, which had itself been famously open since the 1930s. The Skolem Problem remains unsolved to this day, although substantial advances have recently been made—more on this shortly. Phrased in the language of linear dynamical systems, the Skolem Problem asks whether it is decidable, given (M, x) as above, together with a \((d-1)\)-dimensional subspace H of \(\mathbb {R}^d\), to determine if the orbit of (M, x) ever hits H. Kannan and Lipton suggested that, in ambient space \(\mathbb {R}^d\) of arbitrary dimension, the problem of hitting a low-dimensional subspace might be decidable. Indeed, this was eventually substantiated by Chonev et al. for linear subspaces of dimension at most 3 [17, 19].

Subsequent research focussed on the decidability of hitting targets of increasing complexity, such as half-spaces [25, 33, 36,37,38], polytopes [3, 18, 42], and semialgebraic sets [4, 5]. It is also worth noting that discrete linear dynamical systems can equivalently be viewed as linear (or affine) simple, branching-free while loops, where reachability corresponds to loop termination. There is a voluminous literature on the topic, albeit largely focussing on heuristics and semi-algorithms (via spectral methods or the synthesis of ranking functions), rather than exact decidability results. Relevant papers include [6,7,8,9, 13, 14, 16, 21, 27, 40, 41, 44]. Several of these approaches have moreover been implemented in software verification tools, such as Microsoft’s Terminator [22, 23].

In recent years, motivated in part by verification problems for stochastic systems and linear loops, researchers have begun investigating more sophisticated specification formalisms than mere reachability: for example, the paper [1] studies approximate LTL model checking of Markov chains (which themselves can be viewed as particular kinds of linear dynamical systems), whereas [32] focuses on LTL model checking of low-dimensional linear dynamical systems with semialgebraic predicates.Footnote 1 In [2], the authors solve the semialgebraic model-checking problem for diagonalisable linear dynamical systems in arbitrary dimension against prefix-independent MSOFootnote 2 properties, whereas [31] investigates semialgebraic MSO model checking of linear dynamical systems in which the dimensions of predicates are constrained. For a comprehensive survey of the state of the art on model checking for linear dynamical systems, we refer the reader to [30].

There is an intimate connection between linear dynamical systems and linear recurrence sequences. A (rational) linear recurrence sequence (LRS) \(\boldsymbol{u} = \langle u_n \rangle _{n=0}^\infty \) is an infinite sequence of rational numbers satisfying

$$\begin{aligned} u_{n+d} = c_1u_{n+d-1} + \cdots + c_{d-1}u_{n+1}+ c_du_{n} \end{aligned}$$
(1)

for all \(n \in \mathbb {N}\), where the coefficients \(c_1,\ldots ,c_d\) are rational numbers and \(c_d \ne 0\). We say that the above recurrence has order d. We moreover say that an LRS is simple if the characteristic polynomialFootnote 3 of its minimal-order recurrence has no repeated roots. The sequence of Fibonacci numbers \(\langle f_n \rangle _{n=0}^\infty = \langle 0,1,1,2,3,5,\ldots \rangle \), which obeys the recurrence \(f_{n+2}=f_{n+1}+f_n\), is perhaps the most emblematic LRS, and also happens to be simple. It is a straightforward exercise to show that the orbit \(\langle x, Mx, M^2x,\dots \rangle \) of the LDS from Fig. 1 consists precisely of successive pairs of consecutive Fibonacci numbers:

$$\begin{aligned} \langle x, Mx, M^2x,\dots \rangle = \bigg \langle \!\! \begin{pmatrix} 1 \\ 0 \end{pmatrix}\!, \begin{pmatrix} 1 \\ 1 \end{pmatrix}\!, \begin{pmatrix} 2 \\ 1 \end{pmatrix}\!, \ldots \!\! \bigg \rangle = \bigg \langle \!\! \begin{pmatrix} f_1 \\ f_0 \end{pmatrix}\!, \begin{pmatrix} f_2 \\ f_1 \end{pmatrix}\!, \begin{pmatrix} f_3 \\ f_2 \end{pmatrix}\!, \ldots \!\! \bigg \rangle \, . \end{aligned}$$
(2)

Let us now define the following two bivariate predicates:

$$\begin{aligned}&P(y,z) \mathrel {\overset{{{ \tiny \mathsf def}}}{=}}\left( y^2-yz-z^2 - 1 = 0\right) \end{aligned}$$
(3)
$$\begin{aligned}&Q(y,z) \mathrel {\overset{{{ \tiny \mathsf def}}}{=}}\left( y^2-yz-z^2 + 1 = 0\right) \, . \end{aligned}$$
(4)

Identifying P and Q with the respective subsets of \(\mathbb {R}^2\) that they represent, one can straightforwardly show that the orbit of (M, x) visits P precisely at even-valued indices, and Q at odd-valued indices (where the first element of the orbit is understood to have index 0). In other words, the LDS (M, x) satisfies the following LTL specification:

$$\begin{aligned} P \wedge \lnot Q \wedge \textbf{G} (P \Rightarrow \textbf{X}Q) \wedge \textbf{G} (Q \Rightarrow \textbf{X} P) \, . \end{aligned}$$
(5)

Of course, the general task of determining algorithmically whether a given LDS (in arbitrary dimension) meets a given specification would appear to be highly challenging. The principal goal of this paper is to delineate the extent to which this can be achieved automatically when the predicates are built from algebraic setsFootnote 4 and the specification formalism is either MSO, or its prefix-independent fragment. Before stating our key results, we need to take a brief detour through the Skolem landscape.

1.1 Skolem Oracles

The celebrated theorem of Skolem, Mahler, and Lech (see [24]) describes the structure of the set \(\{ n \in \mathbb {N} : u_n = 0\}\) of zero terms of an LRS as follows:

Theorem 1

Given a linear recurrence sequence \(\boldsymbol{u}=\langle u_n\rangle _{n=0}^\infty \), its set of zero terms is a semilinear set, i.e., it consists of a union of finitely many full arithmetic progressions,Footnote 5 together with a finite set.

As shown by Berstel and Mignotte [10], in the above one can effectively extract all of the arithmetic progressions; we refer herein to the corresponding procedure as the ‘Berstel-Mignotte algorithm’. Nevertheless, how to compute the leftover finite set of zeros remains open, and is easily seen to be equivalent to the Skolem Problem: given an LRS \(\boldsymbol{u}\), does \(\boldsymbol{u}\) contain a zero term?

Let us therefore introduce the notion of a Skolem oracle: given an LRS \(\boldsymbol{u}=\langle u_n\rangle _{n=0}^\infty \), such an oracle returns the finite set of indices of zeros of \(\boldsymbol{u}\) that do not already belong to some infinite arithmetic progression of zeros. Likewise, a Simple-Skolem oracle is a Skolem oracle restricted to simple LRS.

As mentioned earlier, the decidability of the Skolem Problem is a longstanding open question [24, 39], with a positive answer for LRS of order at most 4 known since the mid-1980s [43, 45]. Very recently, two major conditional advances on the Skolem Problem have been made, achieving decidability subject to certain classical number-theoretic conjectures: in [34], Lipton et al. established decidability for LRS of order 5 assuming the Skolem Conjecture (also known as the Exponential Local-Global Principle); and in [11], Bilu et al. showed decidability for simple LRS of arbitrary order, subject to both the Skolem Conjecture and the p-adic Schanuel Conjecture. It is interesting to note that in both cases, the procedures in question rely on the conjectures only for termination; correctness is unconditional. In fact, these procedures are certifying algorithms (in the sense of [35]) in that, upon termination, they produce an independent certificate (or witness) that their output is correct. Such a certificate can be checked algorithmically by a third party with no reliance on any unproven conjectures. The authors of [11] have implemented their algorithm within the skolem tool, available online.Footnote 6

In view of the above, Simple-Skolem oracles can be implemented with unconditional correctness, and guaranteed termination subject to the Skolem and p-adic Schanuel conjectures. Whether full Skolem oracles can be devised is the subject of active research; at the time of writing, to the best of our knowledge, no putative procedure is even conjectured in the general (non-simple) case.

To illustrate the applicability of Skolem oracles to model checking linear dynamical systems, let us return to our running example involving the LDS (M, x) from Fig. 1. Recall predicate P(y, z) from Eq. (3), and identify it with the polynomial it implicitly represents, namely \(P(y,z) = y^2-yz-z^2 - 1\). In view of Eq. (2), we can write the orbit of (M, x) as follows:

$$ \langle M^n x\rangle _{n=0}^\infty = \bigg \langle \!\! \begin{pmatrix} y_n \\ z_n \end{pmatrix} \!\! \bigg \rangle _{n=0}^\infty = \bigg \langle \!\! \begin{pmatrix} f_{n+1} \\ f_n \end{pmatrix} \!\! \bigg \rangle _{n=0}^\infty \, , $$

where the reader will recall that \(\langle f_n\rangle _{n=0}^\infty \) is the LRS of Fibonacci numbers. Evaluating the polynomial P(y, z) at each point of the orbit therefore yields the sequence \(\langle f_{n+1}^2 - f_{n+1}f_n - f_n^2 -1 \rangle _{n=0}^\infty \). Given that LRS (resp. simple LRS) are closed under addition and multiplication, the resulting sequence is immediately seen to be a (simple) LRS. Therefore the Berstel-Mignotte algorithm, together with a (Simple-)Skolem oracle, enable us to compute the set of zeros of this LRS as a semilinear set. In turn, this set is precisely the sequence of indices at which the predicate P holds, i.e., at which the orbit of the LDS (M, x) visits the set represented by P. Since one-dimensional semilinear sets are ultimately periodic, and since every step along the way was effective (assuming the existence of (Simple-)Skolem oracles), evaluating the predicate P on the orbit of (M, x) gives rise to an effectively ultimately periodic word. As already noted, this word is indeed in fact \(\langle TRUE , FALSE , TRUE , FALSE , TRUE , FALSE , TRUE ,\ldots \rangle \).

One can of course repeat the procedure with the predicate Q, so that both P and Q are effectively ultimately periodic. Since MSO over effectively ultimately periodic words is decidable, we have just outlined a general algorithmic process by which one can decide algebraic MSO specifications (such as (5)) on orbits of linear dynamical systems, assuming the existence of Skolem or Simple-Skolem oracles.

Remark 2

As hinted above, it is a general fact that the sequence of values obtained by evaluating a multivariate polynomial on the successive points of the orbit of an LDS is always an LRS; moreoever, whenever the LDS is diagonalisableFootnote 7, the corresponding LRS is always simple. We provide sketch justifications of these facts in Sect. 2.

1.2 Main Results

We require one final ingredient in order to state the main contributions of this paper. Fix the ambient space to be \(\mathbb {R}^d\), and define the collection \(\mathcal {C}\) of subsets of \(\mathbb {R}^d\) to be the smallest set containing all algebraic subsets of \(\mathbb {R}^d\), and which is closed under finite union, finite intersection, and complement. In algebraic geometry, \(\mathcal {C}\) is usually referred to as the collection of constructible subsets of \(\mathbb {R}^d\). We refer to MSO formulas over predicates from \(\mathcal {C}\) as algebraic MSO, and the corresponding model-checking problem as algebraic model checking.

Our main results are as follows (precise definitions and statements can be found in Sect. 2):

  1. 1.

    The algebraic model-checking problem for LDS is decidable in arbitrary dimension, subject to the existence of a Skolem oracle.

  2. 2.

    The algebraic model-checking problem for diagonalisable LDS is decidable in arbitrary dimension, subject to the existence of a Simple-Skolem oracle.

    • As an immediate corollary, decidability holds subject to the Skolem and p-adic Schanuel conjectures; moreover, correctness of the model-checking procedure is unconditional, and independent correctness certificates can always be produced upon termination.

  3. 3.

    The algebraic model-checking problem for LDS against prefix-independent specifications is (unconditionally) decidable.

Item 3 above follows from the fact that prefix-independent assertions depend only upon the ultimately periodic components of predicates (see, e.g., [2]), and the latter can be effectively extracted via the Berstel-Mignotte algorithm.

Three further remarks are in order: (i) in ambient space \(\mathbb {R}^3\), algebraic and even semialgebraic model checking for LDS become unconditionally decidable; this follows immediately from the results of [31], since every predicate in \(\mathbb {R}^3\) belongs to a 3-dimensional subspace (namely \(\mathbb {R}^3\)). However: (ii) in ambient space \(\mathbb {R}^4\), unconditional decidability of algebraic model checking in not known to hold even for diagonalisable LDS, as one can establish hardness for the Skolem Problem at order 5; see [18] for details. (iii) For simplicity, all our results are stated in terms of rational linear dynamical systems, living in ambient space \(\mathbb {R}^d\). Nevertheless, it is a straightforward corollary that we can extend our entire framework to complex-algebraicFootnote 8 linear dynamical systems, replacing the ambient space \(\mathbb {R}^d\) by \(\mathbb {C}^d\), \(\mathbb {Q}^d\) by \(\overline{\mathbb {Q}}^d\), and real constructible sets by complex constructible sets.Footnote 9 As we sketch in Sect. 2, our main results (as listed above) carry over easily to this more general complex setting. In this extension, it is noteworthy that our Skolem oracles however remain unchanged, i.e., are maintained to apply only to rational (rather than complex-algebraic) linear recurrence sequences.

Lastly, it is interesting to note that the algebraic model-checking problem for LDS subsumes not only the original Point-to-Point Reachability Problem, but also the Subspace Reachability Problem suggested by Kannan and Lipton [29] (along with its affine variants), as well as reachability for the glued vector spaces of [20].

In the next section, we present a slightly more formal treatment of our framework and results, along with justifications for some of our unsupported assertions. Section 3 concludes with a brief summary and directions for further research.

2 Algebraic Model Checking

Throughout this section, we assume familiarity with the elementary theory of linear recurrence sequences as well as the rudiments of Monadic Second-Order Logic (MSO); there are many excellent references for both topics, such as [24] and [12].

Let us work in fixed ambient space \(\mathbb {R}^d\), and consider a d-dimensional LDS (M, x) (i.e., \(M \in \mathbb {Q}^{d\times d}\) and \(x \in \mathbb {Q}^d\)). Recall that the orbit \(\mathcal {O} = \mathcal {O}(M,x)\) of our LDS is the infinite sequence \(\langle x, Mx, M^2x, \ldots \rangle \) in \(\mathbb {Q}^d\). Let us write \(\mathcal {O}[n]\) for the nth term of the orbit, and, for \(1 \le i \le d\), \(\mathcal {O}[n]_i\) for the ith entry of the point \(\mathcal {O}[n] \in \mathbb {Q}^d\).

Lemma 3

Let (M, x) be as above. For any fixed \(i \in \{1, \ldots , d\}\), the sequence \(\langle \mathcal {O}[n]_i \rangle _{n=0}^\infty \) is an LRS whose characteristic polynomial divides the minimal polynomial of M.

Proof

The fact that the minimal polynomial of M is associated with a recurrence satisfied by the sequence \(\langle \mathcal {O}[n]_i \rangle _{n=0}^\infty \) is a straightforward linear-algebraic calculation. It follows that sequence \(\langle \mathcal {O}[n]_i \rangle _{n=0}^\infty \) is indeed an LRS having characteristic roots among the eigenvalues of M, with the multiplicity of each characteristic root at most the algebraic multiplicity of the corresponding eigenvalue. The result then immediately follows.    \(\square \)

Corollary 4

Let (M, x) be as above. If M is diagonalisable, then for each fixed \(i \in \{1, \ldots , d\}\), the LRS \(\langle \mathcal {O}[n]_i \rangle _{n=0}^\infty \) is simple.

Proof

This follows immediately from the well-known fact that a square matrix M is diagonalisable if and only if its minimal polynomial is a product of distinct linear factors (over \(\mathbb {C}\)).    \(\square \)

A set \(A \subseteq \mathbb {R}^d\) is algebraic if A can be written as a finite positive Boolean combination of polynomial equalities, where all polynomials involved have integer coefficients. The collection \(\mathcal {C} \subseteq 2^{\mathbb {R}^d}\) of constructible subsets of \(\mathbb {R}^d\) is the smallest set that includes all algebraic sets and is closed under finite Boolean combinations (including complementation). Any constructible set \(C \in \mathcal {C}\) can therefore be represented in conjunctive normal form, i.e., as an expression of the form \(C = \bigcap _{i=1}^a \bigcup _{j=1}^b B_{i,j}\), where each \(B_{i,j}\) is either an algebraic set or the complement of one.

Let \(\boldsymbol{C} = \{C_1, \ldots , C_\ell \}\) be a finite list of constructible sets (not necessarily disjoint), giving rise to an alphabet \(\varSigma \mathrel {\overset{{{ \tiny \mathsf def}}}{=}}2^{\boldsymbol{C}}\). The orbit \(\mathcal {O}\) of our LDS (M, x) then naturally gives rise to an infinite characteristic word \(w \in \varSigma ^\omega \), as follows: writing w[n] for the nth letter of w, we have \(C_i \in w[n]\) iff \(M^nx \in C_i\).

Proposition 5

Characteristic words over constructible predicates are ultimately periodic.

Proof

Let us write \(\mathcal {O} = \langle x_n \rangle _{n=0}^\infty \) to denote the orbit of our LDS (M, x), and let us further write each \(x_n\) as \((x_n^{(1)},\ldots ,x_n^{(d)})^T\). Fix a polynomial \(f \in \mathbb {Z}[X_1,\ldots ,X_d]\), and consider the sequence \(\langle f(x_n) \rangle _{n=0}^\infty \). Since sums and products of LRS are LRS, by Lemma 3 the sequence \(\langle f(x_n) \rangle _{n=0}^\infty \) is an LRS, and by the Skolem-Mahler-Lech theorem its set of zeros is therefore semilinear, hence ultimately periodic. Since pointwise Boolean combinations (including complementation) of ultimately periodic words are ultimately periodic, and taking account of the fact that any constructible set is a finite Boolean combination of algebraic sets, the result immediately follows.    \(\square \)

We say that a word is effectively ultimately periodic if one can compute an integer threshold beyond which the word in question becomes periodic. Stringing everything together:

Corollary 6

  1. 1.

    Assume the existence of a Skolem oracle. Then characteristic words over constructible predicates are effectively ultimately periodic.

  2. 2.

    Suppose (M, x) is a diagonalisable LDS, and assume the existence of a Simple-Skolem oracle. Then any characteristic word associated with the orbit of (M, x) over constructible predicates is effectively ultimately periodic.

Proof

The first item follows directly from Proposition 5, together with the Berstel-Mignotte algorithm. So does the second item, invoking in addition Corollary 4, and taking account of the fact that simple LRS are closed under sums and products, along with the fact that constant sequences are themselves simple LRS.    \(\square \)

Let (M, x) and \(\boldsymbol{C}\) be as above, and let \(\varphi \) be an MSO formula with atomic predicates drawn from \(\boldsymbol{C}\)—we refer to such formulas as algebraic MSO specifications. The question of whether the characteristic word w associated with the orbit of (M, x) satisfies specification \(\varphi \)—which is usually written as \((M,x) \vDash \varphi \)—is the algebraic model-checking problem for discrete linear dynamical systems.

In addition, we say that \(\varphi \) is prefix-independent if the infinite words that satisfy it are closed under the operations of insertion and deletion of finitely many letters. Prefix-independent properties can be used to describe asymptotic behaviours (e.g., “does the orbit enter \(C_1\) infinitely often?”), but not reachability.

We are now ready to formally state our main results:

Theorem 7

  1. 1.

    The algebraic model-checking problem for LDS is decidable, subject to the existence of a Skolem oracle.

  2. 2.

    The algebraic model-checking problem for diagonalisable LDS is decidable, subject to the existence of a Simple-Skolem oracle.

  3. 3.

    The algebraic model-checking problem for LDS against prefix-independent specifications is (unconditionally) decidable.

Proof

Item 1 is an immediate consequence of Büchi’s seminal work [15] establishing the decidability of MSO, together with Corollary 6(1) and the observation that effectively ultimately periodic predicates can be algorithmically translated to ordinary MSO by encoding the predicates as formulas.

The same holds for Item 2, invoking Corollary 6(2) in lieu of Corollary 6(1).

Finally, Item 3 follows from the fact that prefix-independent assertions depend only upon the ultimately periodic components of predicates (see, e.g., [2]), and the latter can be effectively extracted via the Berstel-Mignotte algorithm.    \(\square \)

As noted earlier, since Simple-Skolem oracles can be implemented into provably correct certifying procedures which terminate subject to classical number-theoretic conjectures [11], we have:

Corollary 8

The algebraic model-checking problem for diagonalisable LDS is decidable, assuming the Skolem Conjecture and the p-adic Schanuel Conjecture. Moreover, correctness of the attendant procedure is unconditional, and independent correctness certificates can be produced upon termination.

Corollary 8 is arguably our most consequential and interesting contribution. Given that promising experimental results are reported in [11] regarding the implementation of a Simple-Skolem algorithm, it appears rather plausible that one could likewise build an efficient and practical model checker for diagonalisable LDS against algebraic specifications.

Finally, let us record, as already noted in the Introduction, that both Theorem 7 and Corollary 8 can be extended mutatis mutandis to complex-algebraic linear dynamical systems, whilst only invoking Skolem oracles for rational linear recurrence sequences. We sketch a short justification of this claim below.

Let (M, x) be a complex-algebraic LDS in ambient space \(\mathbb {C}^d\), i.e., \(M \in \overline{\mathbb {Q}}^{d \times d}\) and \(x \in \overline{\mathbb {Q}}^d\), and let \(f \in \overline{\mathbb {Q}}[X_1,\ldots ,X_d]\) be an arbitrary polynomial with complex-algebraic coefficients. Let \(Z \mathrel {\overset{{{ \tiny \mathsf def}}}{=}}\{ n \in \mathbb {N}: f(M^nx) = 0 \}\) be the set of indices at which the orbit of (M, x) lies in the algebraic set \(f^{-1}(0)\). The crux of our claim boils down to the following lemma:

Lemma 9

Let (M, x), f, and Z be as above. Then one can effectively construct a rational LRS \(\boldsymbol{u}=\langle u_n\rangle _{n=0}^\infty \) such that Z is precisely the zero set of \(\boldsymbol{u}\). Moreover, if M is diagonalisable, then \(\boldsymbol{u}\) is a simple rational LRS.

Proof

Let \(\mathbb {K}\subset \overline{\mathbb {Q}}\) be the smallest number field containing all the entries of M and x, as well as all the coefficients of f. As in the proof of Proposition 5, one easily shows that the sequence \(\boldsymbol{v} = \langle f(M^n(x)) \rangle _{n=0}^\infty \) is an LRS lying entirely in \(\mathbb {K}\).

Recall the notion of norm \(\mathcal {N_\mathbb {K}}: \mathbb {K}\rightarrow \mathbb {Q}\) from algebraic number theory, defined as \(\mathcal {N}_\mathbb {K}(x) = \prod \limits _{\sigma : \mathbb {K}\rightarrow \mathbb {C}} \sigma (x) \),where the product is indexed by the collection of field embeddings of \(\mathbb {K}\) into \(\mathbb {C}\). Since \(\boldsymbol{v}\) is an LRS, then so is \(\sigma (\boldsymbol{v}) = \langle \sigma (v_n) \rangle _{n=0}^\infty \) for any embedding \(\sigma : \mathbb {K}\rightarrow \mathbb {C}\), and moreover such applications will also preserve simplicity of LRS (this can be seen by inspecting the effect of \(\sigma \) on the exponential-polynomial closed-form representation of \(\boldsymbol{v}\)). Write \(\boldsymbol{u} \mathrel {\overset{{{ \tiny \mathsf def}}}{=}}\mathcal {N}_\mathbb {K}(\boldsymbol{v}) = \langle \mathcal {N}_\mathbb {K}( v_n) \rangle _{n=0}^\infty \). Since products of (simple) LRS are again (simple) LRS, we have that \(\boldsymbol{u}\) is a (simple) LRS lying entirely in \(\mathbb {Q}\). Moreover, since field embeddings fix 0, \(\boldsymbol{u}\) and \(\boldsymbol{v}\) have precisely the same zero set, as required.    \(\square \)

3 Conclusion

This paper has demonstrated that solving the Skolem Problem is key to model checking a rich class of algebraic properties on linear dynamical systems. We have formulated our results in terms of the existence of oracles for the Skolem Problem and a special subcase, the Simple-Skolem Problem. Implementing such oracles is the subject of ongoing research. As remarked earlier, we have recently devised an algorithm for the Simple-Skolem Problem whose output comes with an easily checkable correctness certificate (namely a set of zeros of the given sequence and a certificate that the remaining terms of the sequence are non-zero) and which terminates subject to certain classical number-theoretic conjectures. We are currently investigating whether a similar approach can be devised in the general (non-simple) case.

In this note we have concentrated on logical specifications built over constructible predicates, that is, those that are defined by Boolean combinations of polynomial equalities. In many applications, one is also interested in the more general class of semialgebraic predicates, that is, those defined by Boolean combinations of polynomial inequalities. The task of model checking MSO formulas over such predicates appears vastly more complex. Already the question of whether the orbit of an LDS remains within a prescribed halfspace—or equivalently whether all terms of an LRS are non-negative, known as the Positivity Problem—is highly challenging: decidability is known only for sequences of order at most 5, whereas for sequences of order 6 a solution to the Positivity Problem would entail major breakthroughs in number theory [37, 39].