The Polyhedron-Hitting Problem

We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC '80 and JACM '86)---determining whether a target polyhedron V may be reached from a starting point x under repeated applications of a linear transformation A in an ambient vector space Q^m. In the context of program verification, very similar reachability questions were also considered and left open by Lee and Yannakakis in (STOC '92). We present what amounts to a complete characterisation of the decidability landscape for the Polyhedron-Hitting Problem, expressed as a function of the dimension m of the ambient space, together with the dimension of the polyhedral target V: more precisely, for each pair of dimensions, we either establish decidability, or show hardness for longstanding number-theoretic open problems

On the Polytope Escape Problem for Continuous Linear Dynamical Systems

The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and a convex polyhedron $\mathcal{P} \subseteq \mathbb{R}^{d}$, whether, for some initial point $\boldsymbol{x}_{0}$ in $\mathcal{P}$, the trajectory of the unique solution to the differential equation $\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t))$, $\boldsymbol{x}(0)=\boldsymbol{x}_{0}$, is entirely contained in $\mathcal{P}$. We show that this problem is decidable, by reducing it in polynomial time to the decision version of linear programming with real algebraic coefficients, thus placing it in $\exists \mathbb{R}$, which lies between NP and PSPACE. Our algorithm makes use of spectral techniques and relies among others on tools from Diophantine approximation.Comment: Accepted to HSCC 201

On Termination of Integer Linear Loops

A fundamental problem in program verification concerns the termination of simple linear loops of the form x := u ; while Bx >= b do {x := Ax + a} where x is a vector of variables, u, a, and c are integer vectors, and A and B are integer matrices. Assuming the matrix A is diagonalisable, we give a decision procedure for the problem of whether, for all initial integer vectors u, such a loop terminates. The correctness of our algorithm relies on sophisticated tools from algebraic and analytic number theory, Diophantine geometry, and real algebraic geometry. To the best of our knowledge, this is the first substantial advance on a 10-year-old open problem of Tiwari (2004) and Braverman (2006).Comment: Accepted to SODA1

On the Counting Complexity of the Skolem Problem

The Skolem Problem asks, given an integer linear recurrence sequence (LRS), to determine whether the sequence contains a zero term or not. Its decidability is a longstanding open problem in theoretical computer science and automata theory. Currently, decidability is only known for LRS of order at most 4. On the other hand, the sole known complexity result is NP-hardness, due to Blondel and Portier. A fundamental result in this area is the celebrated Skolem-Mahler-Lech theorem, which asserts that the zero set of any LRS is the union of a finite set and finitely many arithmetic progressions. This paper focuses on a computational perspective of the Skolem-Mahler-Lech theorem: we show that the problem of counting the zeros of a given LRS is #P-hard, and in fact #P-complete for the instances generated in our reduction

Polynomial Invariants for Affine Programs

We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate

Algebraic model checking for discrete linear dynamical systems

Model checking infinite-state systems is one of the central challenges in automated verification. In this survey we focus on an important and fundamental subclass of infinite-state systems, namely discrete linear dynamical systems. While such systems are ubiquitous in mathematics, physics, engineering, etc., in the present context our motivation stems from their relevance to the formal analysis and verification of program loops, weighted automata, hybrid systems, and control systems, amongst many others. Our main object of study is the problem of model checking temporal properties on the infinite orbit of a linear dynamical system, and our principal contribution is to show that for a rich class of properties this problem can be reduced to certain classical decision problems on linear recurrence sequences, notably the Skolem Problem. This leads us to discuss recent advances on the latter and to highlight the prospects for further progress on charting the algorithmic landscape of linear recurrence sequences and linear dynamical systems