Deciding ω-regular properties on linear recurrence sequences

Abstract

We consider the problem of deciding ω-regular properties on infinite traces produced by linear loops. Here we think of a given loop as producing a single infinite trace that encodes information about the signs of program variables at each time step. Formally, our main result is a procedure that inputs a prefix-independent ω-regular property and a sequence of numbers satisfying a linear recurrence, and determines whether the sign description of the sequence (obtained by replacing each positive entry with “+”, each negative entry with “−”, and each zero entry with “0”) satisfies the given property. Our procedure requires that the recurrence be simple, i.e., that the update matrix of the underlying loop be diagonalisable. This assumption is instrumental in proving our key technical lemma: namely that the sign description of a simple linear recurrence sequence is almost periodic in the sense of Muchnik, Sem'enov, and Ushakov. To complement this lemma, we give an example of a linear recurrence sequence whose sign description fails to be almost periodic. Generalising from sign descriptions, we also consider the verification of properties involving semi-algebraic predicates on program variables.

What's Decidable about Discrete Linear Dynamical Systems?

“…To sidestep these obstacles, in [4] the authors restrict ϕ to formulas that define prefix-independent properties. Similarly to liveness specifications, a property is prefix-independent if the infinite words that satisfy it are closed under the operations of insertion and deletion of finitely many letters.…”

“…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. 3 In [4], the authors solve the semialgebraic modelchecking problem for diagonalisable linear dynamical systems in arbitrary dimension against prefix-independent MSO 4 properties, whereas [31] investigates semialgebraic MSO model checking of linear dynamical systems in which the dimensions of predicates are constrained. To illustrate this last approach, recall the dynamical system (M, x) from Figure 1, and consider the following three semialgebraic predicates:…”

“…Semialgebraic predicates are Boolean combinations of polynomial equalities and inequalities 4. Monadic Second-Order Logic (MSO) is a highly expressive specification formalism that subsumes the vast majority of temporal logics employed in the field of automated verification, such as Linear Temporal Logic (LTL).…”

Skolem Meets Schanuel

“…As explained in Section 3, there exists a prime p and a positive integer L such that u Ln = f (n) for all n ∈ Z, for the p-adic power series f (X) = ∞ j=0 a j X j whose coefficients are given by the formula (4). Recall that in this formula the λ i are the characteristic roots of u and the Q i are the coefficients appearing in the exponential polynomial formula (3).…”

“…The Skolem Problem, along with closely related questions such as the Positivity Problem, is intimately connected to various fundamental topics in program analysis and automated verification, such as the termination and model checking of simple while loops [3,18,27] or the algorithmic analysis of stochastic systems [2,1,5,13,28]. It also appears in a variety of other contexts, such as formal power series [29,33] and control theory [9,16].…”