What’s decidable about linear loops?

Abstract

We consider the MSO model-checking problem for simple linear loops, or equivalently discrete-time linear dynamical systems, with semialgebraic predicates (i.e., Boolean combinations of polynomial inequalities on the variables). We place no restrictions on the number of program variables, or equivalently the ambient dimension. We establish decidability of the model-checking problem provided that each semialgebraic predicate either has intrinsic dimension at most 1, or is contained within some three-dimensional subspace. We also note that lifting either of these restrictions and retaining decidability would necessarily require major breakthroughs in number theory.

On eventual non-negativity and positivity for the weighted sum of powers of matrices

“…Finally, logics to reason about temporal properties of linear loops have been studied, although decidability is known only in restrictive settings, e.g. when each predicate defines a semi-algebraic set contained in some 3-dimensional subspace, or has intrinsic dimension 1 [23].…”

What's Decidable about Discrete Linear Dynamical Systems?

“…The reader will probably agree that whether or not the above assertion holds for our LDS (M, x) is not immediately obvious to determine (even, arguably, in principle). Nevertheless, this example falls within the scope of [31], as the semialgebraic predicates P 1 , P 2 , and P 3 are admissible, i.e., they are each either contained in some three-dimensional subspace (this is the case for P 1 and P 2 ), or have intrinsic dimension at most 1 (this is the case of P 3 , which is 'string-like', or a curve, as a subset of R 4 ). Naturally, we shall return to these notions in due course, and articulate the relevant results in full details.…”

“…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:…”

The Pseudo-Reachability Problem for Diagonalisable Linear Dynamical Systems

“…of points in Q d . Orbits of LDS arise in many areas of computer science and mathematics, including verification of linear loops [10], automata theory [3], and the theory of linear recurrence sequences [17].…”