Width of Non-deterministic Automata

International audienceWe introduce a measure called width, quantifying the amount of nondeterminism in automata. Width generalises the notion of good-for-games (GFG) automata, that correspond to NFAs of width 1, and where an accepting run can be built on-the-fly on any accepted input. We describe an incremental determinisation construction on NFAs, which can be more efficient than the full powerset determinisation, depending on the width of the input NFA. This construction can be generalised to infinite words, and is particularly well-suited to coBüchi automata in this context. For coBüchi automata, this procedure can be used to compute either a deterministic automaton or a GFG one, and it is algorithmically more efficient in this last case. We show this fact by proving that checking whether a coBüchi automaton is determinisable by pruning is NP-complete. On finite or infinite words, we show that computing the width of an automaton is PSPACE-hard. 1 Introduction Determinisation of non-deterministic automata (NFAs) is one of the cornerstone problems of automata theory, with countless applications in verification. There is a very active field of research for optimizing or approximating determinisation, or circumventing it in contexts like inclusion of NFA or Church Synthesis. Indeed, determinisation is a costly operation, as the state space blow-up is in O(2 n) on finite words, O(3 n) for coBüchi automata [16], and 2 O(n log(n)) for Büchi automata [17]. If A and B are NFAs, the classical way of checking the inclusion L(A) ⊆ L(B) is to determinise B, complement it, and test emptiness of L(A) ∩ L(B). To circumvent a full determinisation, the recent algorithm from [3] proved to be very efficient, as it is likely to explore only a part of the powerset construction. Other approaches use simulation games to approximate inclusion at a cheaper cost, see for instance [8]. Another approach consists in replacing determinism by a weaker constraint that suffices in some particular context. In this spirit, Good-for-Games automata (GFG for short) were introduced in [9], as a way to solve the Church synthesis problem. This problem asks, given a specification L, typically given by an LTL formula, over an alphabet of inputs and outputs, whether there is a reactive system (transducer) whose behaviour is included in L. The classical solution computes a deterministic automaton for L, and solves a game defined on this automaton. It turns out that replacing determinism by the weaker constraint of being GFG is sufficient in this context. Intuitively, GFG automata are non-deterministic * This work was supported by the grant PALSE Impulsion

On the Composability of Statistically Secure Random Oblivious Transfer

We show that random oblivious transfer protocols that are statistically secure according to a definition based on a list of information-theoretical properties are also statistically universally composable. That is, they are simulatable secure with an unlimited adversary, an unlimited simulator, and an unlimited environment machine. Our result implies that several previous oblivious transfer protocols in the literature that were proven secure under weaker, non-composable definitions of security can actually be used in arbitrary statistically secure applications without lowering the security

Optimal Dynamic Distributed MIS

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a $3$-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.Comment: 19 pages including appendix and reference

Computing the Width of Non-deterministic Automata

International audienceWe introduce a measure called width, quantifying the amount of nondetermin-ism in automata. Width generalises the notion of good-for-games (GFG) automata, that correspond to NFAs of width 1, and where an accepting run can be built on-the-fly on any accepted input. We describe an incremental determinisation construction on NFAs, which can be more efficient than the full powerset determinisation, depending on the width of the input NFA. This construction can be generalised to infinite words, and is particularly well-suited to coBüchi automata. For coBüchi automata, this procedure can be used to compute either a deterministic automaton or a GFG one, and it is algorithmically more efficient in the last case. We show this fact by proving that checking whether a coBüchi automaton is determinisable by pruning is NP-complete. On finite or infinite words, we show that computing the width of an automaton is EXPTIME-complete. This implies EXPTIME-completeness for multipebble simulation games on NFAs

Hardness of Sparse Sets and Minimal Circuit Size Problem

We develop a polynomial method on finite fields to amplify the hardness of spare sets in nondeterministic time complexity classes on a randomized streaming model. One of our results shows that if there exists a $2^{n^{o(1)}}$-sparse set in $NTIME(2^{n^{o(1)}})$ that does not have any randomized streaming algorithm with $n^{o(1)}$ updating time, and $n^{o(1)}$ space, then $NEXP\not=BPP$, where a $f(n)$-sparse set is a language that has at most $f(n)$ strings of length $n$. We also show that if MCSP is $ZPP$-hard under polynomial time truth-table reductions, then $EXP\not=ZPP$

Hardness of Sparse Sets and Minimal Circuit Size Problem

We study the magnification of hardness of sparse sets in nondeterministic time complexity classes on a randomized streaming model. One of our results shows that if there exists a 2no(1) -sparse set in NDTIME(2no(1)) that does not have any randomized streaming algorithm with no(1) updating time, and no(1) space, then NEXP≠BPP , where a f(n)-sparse set is a language that has at most f(n) strings of length n. We also show that if MCSP is ZPP -hard under polynomial time truth-table reductions, then EXP≠ZPP

A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is $\Theta(\sqrt 2^k)$ and used this to give an $O^*((2+\sqrt{2})^{\mathsf{pw}})$ time algorithm for counting Hamiltonian cycles modulo $2$ on graphs of pathwidth $\mathsf{pw}$. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within $\mathbf{M}_k$, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of $\mathbf{M}_k$ is given; no stronger structural insights such as the existence of large permutation submatrices in $\mathbf{M}_k$ are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes $p$) parameterized by pathwidth. To apply this technique, we prove that the rank of $\mathbf{M}_k$ over the rationals is $4^k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_p$ is $\Omega(1.97^k)$ for any prime $p\neq 2$ and even $\Omega(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-\epsilon)^{\mathsf{pw}})$ for any $\epsilon>0$ unless SETH fails. This bound is tight due to a $O^*(6^{\mathsf{pw}})$ time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes $p\neq 2$ in time $O^*(3.97^\mathsf{pw})$, indicating that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in SODA 201