Ambiguity and Communication

The ambiguity of a nondeterministic finite automaton (NFA) N for input size n is the maximal number of accepting computations of N for an input of size n. For all k, r 2 N we construct languages Lr,k which can be recognized by NFA's with size k poly(r) and ambiguity O(nk), but Lr,k has only NFA's with exponential size, if ambiguity o(nk) is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, 1989, Leung, 1998)

Fast equality test for straight-line compressed strings

The paper describes a simple and fast randomized test for equality of grammar-compressed strings. The thorough running time analysis is done by applying a logarithmic cost measure. Keywords: randomized algorithms, straight line programs, grammar-based compressio

On the power of randomized multicounter machines

AbstractOne-way two-counter machines represent a universal model of computation. Here we consider the polynomial-time classes of multicounter machines with a constant number of reversals and separate the computational power of nondeterminism, randomization and determinism. For instance, we show that polynomial-time one-way multicounter machines, with error probability tending to zero with growing input length, can recognize languages that cannot be accepted by polynomial-time nondeterministic two-way multicounter machines with a bounded number of reversals. A similar result holds for the comparison of determinism and one-sided-error randomization, and of determinism and Las Vegas randomization

Yet harder knapsack problems

AbstractAlready 30 years ago, Chvátal has shown that some instances of the zero-one knapsack problem cannot be solved in polynomial time using a particular type of branch-and-bound algorithms based on relaxations of linear programs together with some rudimentary cutting-plane arguments as bounding rules. We extend this result by proving an exponential lower bound in a more general class of branch-and-bound and dynamic programming algorithms which are allowed to use memoization and arbitrarily powerful bound rules to detect and remove subproblems leading to no optimal solution

Minimizing nfa's and regular expressions

AbstractWe show inapproximability results concerning minimization of nondeterministic finite automata (nfa's) as well as of regular expressions relative to given nfa's, regular expressions or deterministic finite automata (dfa's).We show that it is impossible to efficiently minimize a given nfa or regular expression with n states, transitions, respectively symbols within the factor o(n), unless P=PSPACE. For the unary case, we show that for any δ>0 it is impossible to efficiently construct an approximately minimal nfa or regular expression within the factor n1−δ, unless P=NP.Our inapproximability results for a given dfa with n states are based on cryptographic assumptions and we show that any efficient algorithm will have an approximation factor of at least npoly(logn). Our setup also allows us to analyze the minimum consistent dfa problem

On the limits of the communication complexity technique for proving lower bounds on the size of minimal NFA’s

AbstractIn contrast to the minimization of deterministic finite automata (DFA’s), the task of constructing a minimal nondeterministic finite automaton (NFA) for a given NFA is PSPACE-complete. Moreover, there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal NFA’s.Since one is unable to efficiently estimate the size of a minimal NFA in an efficient way, one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal NFA for a given regular language. Here we consider the robust and most successful lower bound proof technique that is based on communication complexity. In this paper it is proved that even a strong generalization of this method fails for some concrete regular languages.“To fail” is considered here in a very strong sense. There is an exponential gap between the size of a minimal NFA and the achievable lower bound for a specific sequence of regular languages.The generalization of the concept of communication protocols is also strong here. It is shown that cutting the input word into 2O(n1/4) pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata. It seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head

08381 Abstracts Collection -- Computational Complexity of Discrete Problems

From the 14th of September to the 19th of September, the Dagstuhl Seminar 08381 ``Computational Complexity of Discrete Problems\u27\u27 was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work as well as open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this report. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Lower Bounds on Communication Complexity

We prove the following four results on communication complexity: 1) For every k ≥ 2, the language Lk of encodings of directed graphs of out degree one that contain a path of length k+1 from the first vertex to the last vertex and can be recognized by exchanging O(k log n) bits using a simple k-round protocol requires exchanging Ω(n1/2/k4log3n) bits if any (k-1)- round protocol is used. 2) For every k ≥ 1 and for infinitely many n ≥ 1, there exists a collection of sets Lnk ⊆ {0,1}2n that can be recognized by exchanging O(k log n) bits using a k-round protocol, and any (k-1)-round protocol recognizing Lnk requires exchanging Ω(n/k) bits. 3) Given a set L ⊆ {0,1}2n, there is a set L ⊆{0,1}8n such that any (k-round) protocol recognizing L can be transformed to a (k-round) fixed partition protocol recognizing L with the same communication complexity, and vice versa. 4) For every integer function f, 1 ≤ f(n) ≤ n, there are languages recognized by a one round deterministic protocol exchanging f(n) bits, but not by any nondeterministic protocol exchanging f(n)-1 bits. The first two results show in an incomparable way an exponential gap between (k-1)-round and k-round protocols, settling a conjecture by Papadimitriou and Sipser. The third result shows that as long as we are interested in existence proofs, a fixed partition of the input is not a restriction. The fourth result extends a result by Papadimitriou and Sipser who showed that for every integer function f, 1 ≤ f(n) ≤ n, there is a language accepted by a deterministic protocol exchanging f(n) bits but not by any deterministic protocol exchanging f(n) - 1 bits

Lapses in Responsiveness: Characteristics and Detection from the EEG

Performance lapses in occupations where public safety is paramount can have disastrous consequences, resulting in accidents with multiple fatalities. Drowsy individuals performing an active task, like driving, often cycle rapidly between periods of wake and sleep, as exhibited by cyclical variation in both EEG power spectra and task performance measures. The aim of this project was to identify reliable physiological cues indicative of lapses, related to behavioural microsleep episodes, from the EEG, which could in turn be used to develop a real-time lapse detection (or better still, prediction) system. Additionally, the project also sought to achieve an increased understanding of the characteristics of lapses in responsiveness in normal subjects. A study was conducted to determine EEG and/or EOG cues (if any) that expert raters use to detect lapses that occur during a psychomotor vigilance task (PVT), with the subsequent goal of using these cues to design an automated system. A previously-collected dataset comprising physiological and performance data of 10 air traffic controllers (ATCs) was used. Analysis showed that the experts were unable to detect the vast majority of lapses based on EEG and EOG cues. This suggested that, unlike automated sleep staging, an automated lapse detection system needed to identify features not generally visible in the EEG. Limitations in the ATC dataset led to a study where more comprehensive physiological and performance data were collected from normal subjects. Fifteen non-sleep-deprived male volunteers aged 18-36 years were recruited. All performed a 1-D continuous pursuit visuomotor tracking task for 1 hour during each of two sessions that occurred between 1 and 7 weeks apart. A video camera was used to record head and facial expressions of the subject. EEG was recorded from electrodes at 16 scalp locations according to the 10-20 system at 256 Hz. Vertical and horizontal EOG was also recorded. All experimental sessions were held between 12:30 and 17:00 hours. Subjects were asked to refrain from consuming stimulants or depressants, for 4 h prior to each session. Rate and duration were estimated for lapses identified by a tracking flat spot and/or video sleep. Fourteen of the 15 subjects had one or more lapses, with an overall rate of 39.3 ± 12.9 lapses per hour (mean ± SE) and a lapse duration of 3.4 ± 0.5 s. The study also showed that lapsing and tracking error increased during the first 30 or so min of a 1-h session, then decreased during the remaining time, despite the absence of external temporal cues. EEG spectral power was found to be higher during lapses in the delta, theta, and alpha bands, and lower in the beta, gamma, and higher bands, but correlations between changes in EEG power and lapses were low. Thus, complete lapses in responsiveness are a frequent phenomenon in normal subjects - even when not sleep-deprived - undertaking an extended, monotonous, continuous visuomotor task. This is the first study to investigate and report on the characteristics of complete lapses of responsiveness during a continuous tracking task in non-sleep-deprived subjects. The extent to which non-sleep-deprived subjects experience complete lapses in responsiveness during normal working hours was unexpected. Such findings will be of major concern to individuals and companies in various transport sectors. Models based on EEG power spectral features, such as power in the traditional bands and ratios between bands, were developed to detect the change of brain state during behavioural microsleeps. Several other techniques including spectral coherence and asymmetry, fractal dimension, approximate entropy, and Lempel-Ziv (LZ) complexity were also used to form detection models. Following the removal of eye blink artifacts from the EEG, the signal was transformed into z-scores relative to the baseline of the signal. An epoch length of 2 s and an overlap of 1 s (50%) between successive epochs were used for all signal processing algorithms. Principal component analysis was used to reduce redundancy in the features extracted from the 16 EEG derivations. Linear discriminant analysis was used to form individual classification models capable of detecting lapses using data from each subject. The overall detection model was formed by combining the outputs of the individual models using stacked generalization with constrained least-squares fitting used to determine the optimal meta-learner weights of the stacked system. The performance of the lapse detector was measured both in terms of its ability to detect lapse state (in 1-s epochs) and lapse events. Best performance in lapse state detection was achieved using the detector based on spectral power (SP) features (mean correlation of φ = 0.39 ± 0.06). Lapse event detection performance using SP features was moderate at best (sensitivity = 73.5%, selectivity = 25.5%). LZ complexity feature-based detector showed the highest performance (φ = 0.28 ± 0.06) out of the 3 non-linear feature-based detectors. The SP+LZ feature-based model had no improvement in performance over the detector based on SP alone, suggesting that LZ features contributed no additional information. Alpha power contributed the most to the overall SP-based detection model. Analysis showed that the lapse detection model was detecting phasic, rather than tonic, changes in the level of drowsiness. The performance of these EEG-based lapse detection systems is modest. Further research is needed to develop more sensitive methods to extract cues from the EEG leading to devices capable of detecting and/or predicting lapses

Min-Rank Conjecture for Log-Depth Circuits

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by setting all *-entries to constants 0 or 1. A system of semi-linear equations over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate of which can only depend on variables corresponding to *-entries in the i-th row of A. We conjecture that no such system can have more than 2^{n-c\cdot mr(A)} solutions, where c>0 is an absolute constant and mr(A) is the smallest rank over GF(2) of a completion of A. The conjecture is related to an old problem of proving super-linear lower bounds on the size of log-depth boolean circuits computing linear operators x --> Mx. The conjecture is also a generalization of a classical question about how much larger can non-linear codes be than linear ones. We prove some special cases of the conjecture and establish some structural properties of solution sets.Comment: 22 pages, to appear in: J. Comput.Syst.Sci