On the power of ordering in linear arithmetic theories

We study the problems of deciding whether a relation definable by a first-order formula in linear rational or linear integer arithmetic with an order relation is definable in absence of the order relation. Over the integers, this problem was shown decidable by Choffrut and Frigeri [Discret. Math. Theor. C., 12(1), pp. 21 - 38, 2010], albeit with non-elementary time complexity. Our contribution is to establish a full geometric characterisation of those sets definable without order which in turn enables us to prove coNP-completeness of this problem over the rationals and to establish an elementary upper bound over the integers. We also provide a complementary ??^P lower bound for the integer case that holds even in a fixed dimension. This lower bound is obtained by showing that universality for ultimately periodic sets, i.e., semilinear sets in dimension one, is ??^P-hard, which resolves an open problem of Huynh [Elektron. Inf.verarb. Kybern., 18(6), pp. 291 - 338, 1982]

On the Complexity of Quantified Integer Programming

Quantified integer programming is the problem of deciding assertions of the form Q_k x_k ... forall x_2 exists x_1 : A * x >= c where vectors of variables x_k,..,x_1 form the vector x, all variables are interpreted over N (alternatively, over Z), and A and c are a matrix and vector over Z of appropriate sizes. We show in this paper that quantified integer programming with alternation depth k is complete for the kth level of the polynomial hierarchy

Re-pairing brackets

Consider the following one-player game. Take a well-formed sequence of opening and closing brackets (a Dyck word). As a move, the player can pair any opening bracket with any closing bracket to its right, erasing them. The goal is to re-pair (erase) the entire sequence, and the cost of a strategy is measured by its width: the maximum number of nonempty segments of symbols (separated by blank space) seen during the play. For various initial sequences, we prove upper and lower bounds on the minimum width sufficient for re-pairing. (In particular, the sequence associated with the complete binary tree of height n admits a strategy of width sub-exponential in log n.) Our two key contributions are (1) lower bounds on the width and (2) their application in automata theory: quasi-polynomial lower bounds on the translation from one-counter automata to Parikh-equivalent nondeterministic finite automata. The latter result answers a question by Atig et al. (2016)

On Restricted Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. Restricted NMF requires in addition that the column spaces of $M$ and $W$ coincide. Finding the minimal inner dimension $d$ is known to be NP-hard, both for NMF and restricted NMF. We show that restricted NMF is closely related to a question about the nature of minimal probabilistic automata, posed by Paz in his seminal 1971 textbook. We use this connection to answer Paz's question negatively, thus falsifying a positive answer claimed in 1974. Furthermore, we investigate whether a rational matrix $M$ always has a restricted NMF of minimal inner dimension whose factors $W$ and $H$ are also rational. We show that this holds for matrices $M$ of rank at most $3$ and we exhibit a rank-$4$ matrix for which $W$ and $H$ require irrational entries.Comment: Full version of an ICALP'16 pape

O-Minimal Invariants for Linear Loops

The termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Such deceptively simple questions also relate to a number of deep open problems, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this paper, we introduce the class of o-minimal invariants, which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture

Asymmetric distances for approximate differential privacy

Differential privacy is a widely studied notion of privacy for various models of computation, based on measuring differences between probability distributions. We consider (epsilon,delta)-differential privacy in the setting of labelled Markov chains. For a given epsilon, the parameter delta can be captured by a variant of the total variation distance, which we call lv_{alpha} (where alpha = e^{epsilon}). First we study lv_{alpha} directly, showing that it cannot be computed exactly. However, the associated approximation problem turns out to be in PSPACE and #P-hard. Next we introduce a new bisimilarity distance for bounding lv_{alpha} from above, which provides a tighter bound than previously known distances while remaining computable with the same complexity (polynomial time with an NP oracle). We also propose an alternative bound that can be computed in polynomial time. Finally, we illustrate the distances on case studies

Synchronizing automata over nested words

We extend the concept of a synchronizing word from deterministic finite-state automata (DFA) to nested word automata (NWA): A well-matched nested word is called synchronizing if it resets the control state of any configuration, i. e., takes the NWA from all control states to a single control state. We show that although the shortest synchronizing word for an NWA, if it exists, can be (at most) exponential in the size of the NWA, the existence of such a word can still be decided in polynomial time. As our main contribution, we show that deciding the existence of a short synchronizing word (of at most given length) becomes PSPACE-complete (as opposed to NP-complete for DFA). The upper bound makes a connection to pebble games and Strahler numbers, and the lower bound goes via small-cost synchronizing words for DFA, an intermediate problem that we also show PSPACE-complete. We also characterize the complexity of a number of related problems, using the observation that the intersection nonemptiness problem for NWA is EXP-complete