Aperiodic String Transducers

Regular string-to-string functions enjoy a nice triple characterization through deterministic two-way transducers (2DFT), streaming string transducers (SST) and MSO definable functions. This result has recently been lifted to FO definable functions, with equivalent representations by means of aperiodic 2DFT and aperiodic 1-bounded SST, extending a well-known result on regular languages. In this paper, we give three direct transformations: i) from 1-bounded SST to 2DFT, ii) from 2DFT to copyless SST, and iii) from k-bounded to 1-bounded SST. We give the complexity of each construction and also prove that they preserve the aperiodicity of transducers. As corollaries, we obtain that FO definable string-to-string functions are equivalent to SST whose transition monoid is finite and aperiodic, and to aperiodic copyless SST

Verifying proofs in constant depth

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC proof systems for a variety of languages ranging from regular to NP-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC proof systems. We also present a general construction of proof systems for regular languages with strongly connected NFA's

The Complexity of Aggregates over Extractions by Regular Expressions

Regular expressions with capture variables, also known as "regex-formulas", extract relations of spans (intervals identified by their start and end indices) from text. In turn, the class of regular document spanners is the closure of the regex formulas under the Relational Algebra. We investigate the computational complexity of querying text by aggregate functions, such as sum, average, and quantile, on top of regular document spanners. To this end, we formally define aggregate functions over regular document spanners and analyze the computational complexity of exact and approximate computation. More precisely, we show that in a restricted case, all studied aggregate functions can be computed in polynomial time. In general, however, even though exact computation is intractable, some aggregates can still be approximated with fully polynomial-time randomized approximation schemes (FPRAS)

Bounded time computation on metric spaces and Banach spaces

We extend the framework by Kawamura and Cook for investigating computational complexity for operators occurring in analysis. This model is based on second-order complexity theory for functions on the Baire space, which is lifted to metric spaces by means of representations. Time is measured in terms of the length of the input encodings and the required output precision. We propose the notions of a complete representation and of a regular representation. We show that complete representations ensure that any computable function has a time bound. Regular representations generalize Kawamura and Cook's more restrictive notion of a second-order representation, while still guaranteeing fast computability of the length of the encodings. Applying these notions, we investigate the relationship between purely metric properties of a metric space and the existence of a representation such that the metric is computable within bounded time. We show that a bound on the running time of the metric can be straightforwardly translated into size bounds of compact subsets of the metric space. Conversely, for compact spaces and for Banach spaces we construct a family of admissible, complete, regular representations that allow for fast computation of the metric and provide short encodings. Here it is necessary to trade the time bound off against the length of encodings

Husimi-Wigner representation of chaotic eigenstates

Just as a coherent state may be considered as a quantum point, its restriction to a factor space of the full Hilbert space can be interpreted as a quantum plane. The overlap of such a factor coherent state with a full pure state is akin to a quantum section. It defines a reduced pure state in the cofactor Hilbert space. The collection of all the Wigner functions corresponding to a full set of parallel quantum sections defines the Husimi-Wigner reresentation. It occupies an intermediate ground between drastic suppression of nonclassical features, characteristic of Husimi functions, and the daunting complexity of higher dimensional Wigner functions. After analysing these features for simpler states, we exploit this new representation as a probe of numerically computed eigenstates of chaotic Hamiltonians. The individual two-dimensional Wigner functions resemble those of semiclassically quantized states, but the regular ring pattern is broken by dislocations.Comment: 21 pages, 7 figures (6 color figures), submitted to Proc. R. Soc.

On Optimality of Myopic Policy for Restless Multi-armed Bandit Problem with Non i.i.d. Arms and Imperfect Detection

We consider the channel access problem in a multi-channel opportunistic communication system with imperfect channel sensing, where the state of each channel evolves as a non independent and identically distributed Markov process. This problem can be cast into a restless multi-armed bandit (RMAB) problem that is intractable for its exponential computation complexity. A natural alternative is to consider the easily implementable myopic policy that maximizes the immediate reward but ignores the impact of the current strategy on the future reward. In particular, we develop three axioms characterizing a family of generic and practically important functions termed as $g$-regular functions which includes a wide spectrum of utility functions in engineering. By pursuing a mathematical analysis based on the axioms, we establish a set of closed-form structural conditions for the optimality of myopic policy.Comment: Second version, 16 page

Dynamic Complexity of Formal Languages

The paper investigates the power of the dynamic complexity classes DynFO, DynQF and DynPROP over string languages. The latter two classes contain problems that can be maintained using quantifier-free first-order updates, with and without auxiliary functions, respectively. It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation. This enables lower bounds for DynPROP and separates DynPROP from DynQF and DynFO. Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF. Furthermore, the dynamic complexity of regular tree languages is investigated and some results concerning arbitrary structures are obtained: there exist first-order definable properties which are not maintainable in DynPROP. On the other hand any existential first-order property can be maintained in DynQF when allowing precomputation.Comment: Contains the material presenten at STACS 2009, extendes with proofs and examples which were omitted due lack of spac

A polynomial lower bound for testing monotonicity

We show that every algorithm for testing n-variate Boolean functions for monotonicity has query complexity Ω(n1/4). All previous lower bounds for this problem were designed for nonadaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only Ω(logn). Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contrast, we show that there is an exponential gap between the query complexity of adaptive and non-adaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015) recently showed that non-adaptive algorithms require almost Ω(n1/2) queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity O(logn) when the input is a regular LTF