Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.

Distributed computing and the multicore revolution

Abstract: Changes in technology can have far-reaching effects on theory. For example, while Turing's work on computability predated the first electronic computers, complexity theory flowered only after computers became a reality. After all, an algorithm's complexity may not matter much in a mathematics journal, but matters quite a bit in a FORTRAN program. We argue that something similar is going on with parallel and concurrent computation: after decades of being respected but not taken seriously, research on multiprocessor algorithms and data structures is going mainstream.

Polynomial time query processing in temporal deductive databases

Abstract: We study conditions guaranteeing polynomial time computability of queries in temporal deductive databases. We show that if for a given set of temporal rules, the period of its least models is bounded from the above by a polynomial in the database size, then also the time to process yes-no queries (as well as to compute finite representations of all query answers) can be polynomially bounded. We present a bottom-up query processing algorithm BT that is guaranteed to terminate in polynomial time if the periods are polynomially bounded. Polynomial periodicity is our most general criterion, however it can not be directly applied. Therefore, we exhibit two weaker criteria, defining inflationary and I-periodic sets of temporal rules. We show that it can be decided whether a set of temporal rules is inflationary. I-periodicity is undecidable (as we show), but it can be closely approximated by a syntactic notion of multi-separability.

Height-deterministic pushdown automata

Abstract: We define the notion of height-deterministic pushdown automata, a model where for any given input string the stack heights during any (nondeterministic) computation on the input are a priori fixed. Different subclasses of height-deterministic pushdown automata, strictly containing the class of regular languages and still closed under boolean language operations, are considered. Several such language classes have been described in the literature. Here, we suggest a natural and intuitive model that subsumes all the formalisms proposed so far by employing height-deterministic pushdown automata. Decidability and complexity questions are also considered.

Open problems about regular languages

Abstract: Publisher Summary The theory of regular languages and finite automata was developed in the early 1950s and is one of the oldest branches of theoretical computer science. Regular languages constitute the best known family of formal languages, and finite automata constitute the best known family of abstract machine models. The concepts of regular languages and finite automata appear frequently in theoretical computer science and have several important applications. There is a vast literature on these subjects. Despite the fact that many researchers have worked in this field, there remain several difficult open problems. The chapter discusses six of these problems. These problems are of fundamental importance and considerable difficulty. Most of them are intimately involved with the fundamental property of finite automata, namely finiteness. In a monograph published in 1971, McNaughton and Papert included a collection of open problems concerning regular languages. Their list is headed by the star height problem and until now, no progress has been made on such an intriguing question. The bounds on star height apply only to languages whose syntactic monoids are groups. In that case, the corresponding semiautomata are permutation semiautomata.

Estimation in inverse problems and second-generation wavelets

Abstract: We consider the problem of recovering a function f when we receive a blurred (by a linear operator) and noisy version: Y�A = Kf +�AW. . We will have as guides 2 famous examples of such inverse problems: the deconvolution and theWicksell problem. The direct problem (K is the identity) isolates the denoising operation. It cannot be solved unless accepting to estimate a smoothed version of f : for instance, if f has an expansion on a basis, this smoothing might correspond to stopping the expansion at some stage m. Then a crucial problem lies in finding an equilibrium for m, considering the fact that for m large, the difference between f and its smoothed version is small, whereas the random effect introduces an error which is increasing withm. In the true inverse problem, in addition to denoising, we have to �einverse the operator�fK, an operation which not only creates the usual difficulties, but also introduces the necessity to control the additional instability due to the inversion of the random noise. Our purpose here is to emphasize the fact that in such a problem there generally exists a basis which is fully adapted to the problem, where for instance the inversion remains very stable: this is the singular value decomposition basis. On the other hand, the SVD basis might be difficult to determine and to numerically manipulate. It also might not be appropriate for the accurate description of the solution with a small number of parameters. Moreover, in many practical situations the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases, other bases (in particular, localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. Our approach here will be to produce estimation procedures keeping the advantages of a localisation properly without loosing the stability and computability of SVD decompositions. We will especially consider two cases. In the first one (which is the case of the deconvolution example) we show that a fairly simple algorithm (WAVE-VD), using an appropriate thresholding technique performed on a standard wavelet system, enables us to estimate the object with rates which are almost optimal up to logarithmic factors for any Lp loss function and on the whole range of Besov spaces. In the second case (which is the case of the Wicksell example where the SVD basis lies in the range of Jacobi polynomials) we prove that a similar algorithm (NEED-VD) can be performed provided one replaces the standard wavelet system by a second generation wavelet-type basis: the needlets. We use here the construction (essentially following the work of Petrushev and co-authors) of a localised frame linked with a prescribed basis (here Jacobi polynomials) using a Littlewood.Paley decomposition combined with a cubature formula. Section 5 describes the direct case (K = I ). It has its own interest and will act as a guide for understanding the �etrue�f inverse models for a reader who is not familiar with nonparametric statistical estimation. It can be read first. Section 1 introduces the general inverse problem and describes the examples of deconvolution and Wicksell�fs problem. A review of standard methods is given with a special focus on SVD methods. Section 2 describes the WAVE-VD procedure. Section 3 and 4 give a description of the needlets constructions and the performances of the NEED-VD procedure.