On Recurrent Reachability for Continuous Linear Dynamical Systems

The continuous evolution of a wide variety of systems, including continuous-time Markov chains and linear hybrid automata, can be described in terms of linear differential equations. In this paper we study the decision problem of whether the solution $\boldsymbol{x}(t)$ of a system of linear differential equations $d\boldsymbol{x}/dt=A\boldsymbol{x}$ reaches a target halfspace infinitely often. This recurrent reachability problem can equivalently be formulated as the following Infinite Zeros Problem: does a real-valued function $f:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ satisfying a given linear differential equation have infinitely many zeros? Our main decidability result is that if the differential equation has order at most $7$, then the Infinite Zeros Problem is decidable. On the other hand, we show that a decision procedure for the Infinite Zeros Problem at order $9$ (and above) would entail a major breakthrough in Diophantine Approximation, specifically an algorithm for computing the Lagrange constants of arbitrary real algebraic numbers to arbitrary precision.Comment: Full version of paper at LICS'1

On the Skolem Problem for Continuous Linear Dynamical Systems

The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems, such as linear hybrid automata and continuous-time Markov chains. Decidability of the problem is currently open---indeed decidability is open even for the sub-problem in which a zero is sought in a bounded interval. In this paper we show decidability of the bounded problem subject to Schanuel's Conjecture, a unifying conjecture in transcendental number theory. We furthermore analyse the unbounded problem in terms of the frequencies of the differential equation, that is, the imaginary parts of the characteristic roots. We show that the unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. We complete the picture by showing that decidability of the unbounded problem in the case of two (or more) rationally linearly independent frequencies would entail a major new effectiveness result in Diophantine approximation, namely computability of the Diophantine-approximation types of all real algebraic numbers.Comment: Full version of paper at ICALP'1

The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations

We study decidability and complexity questions related to a continuous analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of a linear recurrent sequence. In particular, we show that the continuous version of the nonnegativity problem is NP-hard in general and we show that the presence of a zero is decidable for several subcases, including instances of depth two or less, although the decidability in general is left open. The problems may also be stated as reachability problems related to real zeros of exponential polynomials or solutions to initial value problems of linear differential equations, which are interesting problems in their own right.Comment: 14 pages, no figur

Fluid Model Checking

In this paper we investigate a potential use of fluid approximation techniques in the context of stochastic model checking of CSL formulae. We focus on properties describing the behaviour of a single agent in a (large) population of agents, exploiting a limit result known also as fast simulation. In particular, we will approximate the behaviour of a single agent with a time-inhomogeneous CTMC which depends on the environment and on the other agents only through the solution of the fluid differential equation. We will prove the asymptotic correctness of our approach in terms of satisfiability of CSL formulae and of reachability probabilities. We will also present a procedure to model check time-inhomogeneous CTMC against CSL formulae

Positivity Problems for Low-Order Linear Recurrence Sequences

We consider two decision problems for linear recurrence sequences (LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem} (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, with complexity in the Counting Hierarchy for Positivity, and in polynomial time for Ultimate Positivity. Moreover, we show by way of hardness that extending the decidability of either problem to LRS of order 6 would entail major breakthroughs in analytic number theory, more precisely in the field of Diophantine approximation of transcendental numbers

On the information carried by programs about the objects they compute

In computability theory and computable analysis, finite programs can compute infinite objects. Presenting a computable object via any program for it, provides at least as much information as presenting the object itself, written on an infinite tape. What additional information do programs provide? We characterize this additional information to be any upper bound on the Kolmogorov complexity of the object. Hence we identify the exact relationship between Markov-computability and Type-2-computability. We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets

On Decidability of Time-Bounded Reachability in CTMDPs

We consider the time-bounded reachability problem for continuous-time Markov decision processes. We show that the problem is decidable subject to Schanuel's conjecture. Our decision procedure relies on the structure of optimal policies and the conditional decidability (under Schanuel's conjecture) of the theory of reals extended with exponential and trigonometric functions over bounded domains. We further show that any unconditional decidability result would imply unconditional decidability of the bounded continuous Skolem problem, or equivalently, the problem of checking if an exponential polynomial has a non-tangential zero in a bounded interval. We note that the latter problems are also decidable subject to Schanuel's conjecture but finding unconditional decision procedures remain longstanding open problems