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 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

Infinite-Duration Bidding Games

Two-player games on graphs are widely studied in formal methods as they model the interaction between a system and its environment. The game is played by moving a token throughout a graph to produce an infinite path. There are several common modes to determine how the players move the token through the graph; e.g., in turn-based games the players alternate turns in moving the token. We study the {\em bidding} mode of moving the token, which, to the best of our knowledge, has never been studied in infinite-duration games. The following bidding rule was previously defined and called Richman bidding. Both players have separate {\em budgets}, which sum up to $1$. In each turn, a bidding takes place: Both players submit bids simultaneously, where a bid is legal if it does not exceed the available budget, and the higher bidder pays his bid to the other player and moves the token. The central question studied in bidding games is a necessary and sufficient initial budget for winning the game: a {\em threshold} budget in a vertex is a value $t \in [0,1]$ such that if Player $1$'s budget exceeds $t$, he can win the game, and if Player $2$'s budget exceeds $1-t$, he can win the game. Threshold budgets were previously shown to exist in every vertex of a reachability game, which have an interesting connection with {\em random-turn} games -- a sub-class of simple stochastic games in which the player who moves is chosen randomly. We show the existence of threshold budgets for a qualitative class of infinite-duration games, namely parity games, and a quantitative class, namely mean-payoff games. The key component of the proof is a quantitative solution to strongly-connected mean-payoff bidding games in which we extend the connection with random-turn games to these games, and construct explicit optimal strategies for both players.Comment: A short version appeared in CONCUR 2017. The paper is accepted to JAC

On the Complexity of the Orbit Problem

We consider higher-dimensional versions of Kannan and Lipton's Orbit Problem---determining whether a target vector space V may be reached from a starting point x under repeated applications of a linear transformation A. Answering two questions posed by Kannan and Lipton in the 1980s, we show that when V has dimension one, this problem is solvable in polynomial time, and when V has dimension two or three, the problem is in NP^{RP}

LIPIcs

Two-player games on graphs are widely studied in formal methods as they model the interaction between a system and its environment. The game is played by moving a token throughout a graph to produce an infinite path. There are several common modes to determine how the players move the token through the graph; e.g., in turn-based games the players alternate turns in moving the token. We study the bidding mode of moving the token, which, to the best of our knowledge, has never been studied in infinite-duration games. Both players have separate budgets, which sum up to $1$. In each turn, a bidding takes place. Both players submit bids simultaneously, and a bid is legal if it does not exceed the available budget. The winner of the bidding pays his bid to the other player and moves the token. For reachability objectives, repeated bidding games have been studied and are called Richman games. There, a central question is the existence and computation of threshold budgets; namely, a value t\in [0,1] such that if\PO's budget exceeds $t$, he can win the game, and if\PT's budget exceeds 1-t, he can win the game. We focus on parity games and mean-payoff games. We show the existence of threshold budgets in these games, and reduce the problem of finding them to Richman games. We also determine the strategy-complexity of an optimal strategy. Our most interesting result shows that memoryless strategies suffice for mean-payoff bidding games

Reachability problems for linear dynamical systems

The object of principal interest in this thesis is linear dynamical systems: deterministic systems which evolve under a linear operator. They are specified by an initial state set I, contained in &amp;Ropf;m, and a real m-by-m evolution matrix A. We distinguish two varieties of linear dynamical systems: discrete-time and continuous-time. In the discrete-time setting, the state x(n) of the system at time n for natural n is governed by the difference equation x(n)=Ax(n-1). Similarly, in the continuous case, the state x(t) at real, non-negative times t is determined by a system of first-order linear differential equations: x'(t) = Ax(t). In both cases, x(0) lies in I. Throughout this thesis, we will be interested in the Reachability Problem for linear dynamical systems, which may be formulated in a general way as follows: given a target set T contained in &amp;Ropf;m and a (discrete- or continuous-time) linear dynamical system specified by the evolution matrix A and the set of initial states I, determine whether for all x(0) in I, starting from x(0), the system will eventually be in a state which lies in T. In order to make the decision problem well-defined, one must first fix an admissible class of initial sets and, similarly, a class of target sets of interest. For the purposes of expressing the problem instance, it is also necessary to restrict the domain of the input data to a subset of the reals which may be represented effectively, such as the rational numbers or the algebraic numbers. As we vary the choice of domain, the types of initial and target sets under consideration and the discreteness of time, a rich landscape of decision problems emerges. The goal of the present thesis is to explore pointwise reachability problems, that is, reachability from a single initial state. Under the assumption that I consists of a single point in &amp;Ropf;m provided as part of the input data, we will study reachability to polyhedral targets, in the context of both discrete- and continuous-time linear dynamical systems. We prove both upper complexity bounds and hardness results, employing in the process a wide-ranging arsenal of techniques and mathematical tools. We rely on powerful number-theoretic results, such as Baker's Theorem on inhomogeneous linear forms of logarithms of algebraic numbers, Schanuel's Conjecture on the transcendence degree of certain field extensions of the rationals, and Kronecker's Theorem on simultaneous inhomogeneous Diophantine approximation. We draw interesting connections with the study of linear recurrence sequences and exponential polynomials, and relate pointwise reachability to open problems concerning the approximability by rationals of algebraic numbers and logarithms of algebraic numbers. Albeit a simple model, linear dynamical systems are of profound interest, both from a theoretical and a practical standpoint. Reachability problems for linear dynamical systems have recently elicited considerable attention, due to their frequent occurrence in practice and their deep and wide-ranging connections with other fascinating areas of study, such as problems on Markov chains (Akshay et al., 2015), quantum automata (Derksen et al., 2005), Lindenmayer systems (Salomaa and Soittola, 1978), linear loops (Braverman, 2006), linear recurrence sequences (Everest et al., 2003) and exponential polynomials (Bell et al., 2010).</p

Reachability problems for linear dynamical systems

The object of principal interest in this thesis is linear dynamical systems: deterministic systems which evolve under a linear operator. They are specified by an initial state set I, contained in &Ropf;m, and a real m-by-m evolution matrix A. We distinguish two varieties of linear dynamical systems: discrete-time and continuous-time. In the discrete-time setting, the state x(n) of the system at time n for natural n is governed by the difference equation x(n)=Ax(n-1). Similarly, in the continuous case, the state x(t) at real, non-negative times t is determined by a system of first-order linear differential equations: x'(t) = Ax(t). In both cases, x(0) lies in I. Throughout this thesis, we will be interested in the Reachability Problem for linear dynamical systems, which may be formulated in a general way as follows: given a target set T contained in &Ropf;m and a (discrete- or continuous-time) linear dynamical system specified by the evolution matrix A and the set of initial states I, determine whether for all x(0) in I, starting from x(0), the system will eventually be in a state which lies in T. In order to make the decision problem well-defined, one must first fix an admissible class of initial sets and, similarly, a class of target sets of interest. For the purposes of expressing the problem instance, it is also necessary to restrict the domain of the input data to a subset of the reals which may be represented effectively, such as the rational numbers or the algebraic numbers. As we vary the choice of domain, the types of initial and target sets under consideration and the discreteness of time, a rich landscape of decision problems emerges. The goal of the present thesis is to explore pointwise reachability problems, that is, reachability from a single initial state. Under the assumption that I consists of a single point in &Ropf;m provided as part of the input data, we will study reachability to polyhedral targets, in the context of both discrete- and continuous-time linear dynamical systems. We prove both upper complexity bounds and hardness results, employing in the process a wide-ranging arsenal of techniques and mathematical tools. We rely on powerful number-theoretic results, such as Baker's Theorem on inhomogeneous linear forms of logarithms of algebraic numbers, Schanuel's Conjecture on the transcendence degree of certain field extensions of the rationals, and Kronecker's Theorem on simultaneous inhomogeneous Diophantine approximation. We draw interesting connections with the study of linear recurrence sequences and exponential polynomials, and relate pointwise reachability to open problems concerning the approximability by rationals of algebraic numbers and logarithms of algebraic numbers. Albeit a simple model, linear dynamical systems are of profound interest, both from a theoretical and a practical standpoint. Reachability problems for linear dynamical systems have recently elicited considerable attention, due to their frequent occurrence in practice and their deep and wide-ranging connections with other fascinating areas of study, such as problems on Markov chains (Akshay et al., 2015), quantum automata (Derksen et al., 2005), Lindenmayer systems (Salomaa and Soittola, 1978), linear loops (Braverman, 2006), linear recurrence sequences (Everest et al., 2003) and exponential polynomials (Bell et al., 2010)

Reachability problems for linear dynamical systems

The object of principal interest in this thesis is linear dynamical systems: deterministic systems which evolve under a linear operator. They are specified by an initial state set I, contained in &Ropf;m, and a real m-by-m evolution matrix A. We distinguish two varieties of linear dynamical systems: discrete-time and continuous-time. In the discrete-time setting, the state x(n) of the system at time n for natural n is governed by the difference equation x(n)=Ax(n-1). Similarly, in the continuous case, the state x(t) at real, non-negative times t is determined by a system of first-order linear differential equations: x'(t) = Ax(t). In both cases, x(0) lies in I. Throughout this thesis, we will be interested in the Reachability Problem for linear dynamical systems, which may be formulated in a general way as follows: given a target set T contained in &Ropf;m and a (discrete- or continuous-time) linear dynamical system specified by the evolution matrix A and the set of initial states I, determine whether for all x(0) in I, starting from x(0), the system will eventually be in a state which lies in T. In order to make the decision problem well-defined, one must first fix an admissible class of initial sets and, similarly, a class of target sets of interest. For the purposes of expressing the problem instance, it is also necessary to restrict the domain of the input data to a subset of the reals which may be represented effectively, such as the rational numbers or the algebraic numbers. As we vary the choice of domain, the types of initial and target sets under consideration and the discreteness of time, a rich landscape of decision problems emerges. The goal of the present thesis is to explore pointwise reachability problems, that is, reachability from a single initial state. Under the assumption that I consists of a single point in &Ropf;m provided as part of the input data, we will study reachability to polyhedral targets, in the context of both discrete- and continuous-time linear dynamical systems. We prove both upper complexity bounds and hardness results, employing in the process a wide-ranging arsenal of techniques and mathematical tools. We rely on powerful number-theoretic results, such as Baker's Theorem on inhomogeneous linear forms of logarithms of algebraic numbers, Schanuel's Conjecture on the transcendence degree of certain field extensions of the rationals, and Kronecker's Theorem on simultaneous inhomogeneous Diophantine approximation. We draw interesting connections with the study of linear recurrence sequences and exponential polynomials, and relate pointwise reachability to open problems concerning the approximability by rationals of algebraic numbers and logarithms of algebraic numbers. Albeit a simple model, linear dynamical systems are of profound interest, both from a theoretical and a practical standpoint. Reachability problems for linear dynamical systems have recently elicited considerable attention, due to their frequent occurrence in practice and their deep and wide-ranging connections with other fascinating areas of study, such as problems on Markov chains (Akshay et al., 2015), quantum automata (Derksen et al., 2005), Lindenmayer systems (Salomaa and Soittola, 1978), linear loops (Braverman, 2006), linear recurrence sequences (Everest et al., 2003) and exponential polynomials (Bell et al., 2010).</p

The Orbit Problem in Higher Dimensions

We consider higher-dimensional versions of Kannan and Lipton’s Orbit Problem—determining whether a target vector space V may be reached from a starting point x under repeated applications of a linear transformation A. Answering two questions posed by Kannan and Lipton in the 1980s, we show that when V has dimension one, this problem is solvable in polynomial time, and when V has dimension two or three, the problem is in NP RP. Categories and Subject Descriptor