A backbone of a boolean formula @math is a collection @math of its variables for which there is a unique partial assignment @math such that @math is satisfiable [MZK+99,WGS03]. This paper studies the nontransparency of backbones. We show that, under the widely believed assumption that integer factoring is hard, there exist sets of boolean formulas that have obvious, nontrivial backbones yet finding the values, @math , of those backbones is intractable. We also show that, under the same assumption, there exist sets of boolean formulas that obviously have large backbones yet producing such a backbone @math is intractable. Further, we show that if integer factoring is not merely worst-case hard but is frequently hard, as is widely believed, then the frequency of hardness in our two results is not too much less than that frequency.
Even in the theoretical computer science world, where Borodin and Demers's work is set, the work has been very rarely used. In particular, it has been used to get characterizations regarding unambiguous computation @cite_6 , and Rothe and his collaborators have used it in various contexts to study the complexity of certificates @cite_11 @cite_10 , see also and Valiant .
@cite_10 @cite_6 @cite_11
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We present a new method for inferring complexity properties for imperative programs with bounded loops. The properties handled are: polynomial (or linear) boundedness of computed values, as a function of the input; and similarly for the running time. It is well known that complexity properties are undecidable for a Turing-complete programming language. Much work in program analysis overcomes this obstacle by relaxing the correctness notion: one does not ask for an algorithm that correctly decides whether the property of interest holds or not, but only for "yes" answers to be sound. In contrast, we reshaped the problem by defining a "core" programming language that is Turing-incomplete, but strong enough to model real programs of interest. For this language, our method is the first to give a certain answer; in other words, our inference is both sound and complete. The essence of the method is that every command is assigned a "complexity certificate", which is a concise specification of dependencies of output values on input. These certificates are produced by inference rules that are compositional and efficiently computable. The approach is inspired by previous work by Niggl and Wunderlich and by Jones and Kristiansen, but use a novel, more expressive kind of certificates. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. Valiant introduced 20 years ago an algebraic complexity theory to study the complexity of polynomial families. The basic computation model used is the arithmetic circuit, which makes these classes very easy to define and open to combinatorial techniques. In this paper we gather known results and new techniques under a unifying theme, namely the restrictions imposed upon the gates of the circuit, building a hierarchy from formulas to circuits. As a consequence we get simpler proofs for known results such as the equality of the classes VNP and VNP"e or the completeness of the Determinant for VQP, and new results such as a characterization of the classes VQP and VP (which we can also apply to the Boolean class LOGCFL) or a full answer to a conjecture in Burgisser's book [Completeness and reduction in algebraic complexity theory, Algorithms and Computation in Mathematics, vol. 7, Springer, Berlin, 2000]. We also show that for circuits of polynomial depth and unbounded size these models all have the same expressive power and can be used to characterize a uniform version of VNP. The following paper is reproduced from a Russian journal of the character of our own Proceedings of the National Academy of Sciences, Comptes Rendus (Doklady) de I'Academie des Sciences de I'URSS, 1942, Volume XXXVII, No. 7-8. The author is one of the most distinguished of Russian mathematicians. He has made very important contributions in pure mathematics in the theory of functional analysis, and has made equally important contributions to applied mathematics in numerical analysis and the theory and practice of computation. Although his exposition in this paper is quite terse and couched in mathematical language which may be difficult for some readers of Management Science to follow, it is thought that this presentation will: (1) make available to American readers generally an important work in the field of linear programming, (2) provide an indication of the type of analytic work which has been done and is being done in connection with rational planning in Russia, (3) through the specific examples mentioned indicate the types of interpretation which the Russians have made of the abstract mathematics (for example, the potential and field interpretations adduced in this country recently by W. Prager were anticipated in this paper). It is to be noted, however, that the problem of determining an effective method of actually acquiring the solution to a specific problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.--A. Charnes, Northwestern Technological Institute and The Transportation Center.