Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

This paper reviews algorithmic information theory, which is an attempt to apply information-theoretic and probabilistic ideas to recursive function theory. Typical concerns in this approach are, for example, the number of bits of information required to specify an algorithm, or the probability that a program whose bits are chosen by coin flipping produces a given output. During the past few years the definitions of algorithmic information theory have been reformulated. The basic features of the new formalism are presented here and certain results of R. M. Solovay are reported.

Algorithmic Information Theory

Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification and is intractable to solve in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components by minimizing a linear combination of the l1 norm and the nuclear norm of the components. We develop a notion of rank-sparsity incoherence, expressed as an uncertainty principle between the sparsity pat- tern of a matrix and its row and column spaces, and we use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems.

http://www.optimization-online.org/DB_FILE/2009/06/2316.pdf

Rank-Sparsity Incoherence for Matrix Decomposition

Journal of the ACM

This work deals with several aspects concerning the formal verification of SN P systems and the computing power of some variants. A methodology based on the information given by the transition diagram associated with an SN P system is presented. The analysis of the diagram cycles codifies invariants formulae which enable us to establish the soundness and completeness of the system with respect to the problem it tries to resolve. We also study the universality of asynchronous and sequential SN P systems and the capability these models have to generate certain classes of languages. Further, by making a slight modification to the standard SN P systems, we introduce a new variant of SN P systems with a special I/O mode, called SN P modules, and study their computing power. It is demonstrated that, as string language acceptors and transducers, SN P modules can simulate several types of computing devices such as finite automata, a-finite transducers, and systolic trellis automata.

On spiking neural P systems

In the second part of this paper we discuss mainly various programming techniques for systolic automata. These techniques are used to derive theoretical results e.g. the closure of (homogeneous) systolic automaton under Boolean operations, or that every linear context free language can be accepted by a homogeneous systolic automaton and that their languages have time complexity 0(n2)

Systolic trellis automatat

Descriptional Complexity of Context-Free Languages.

Mathematical Foundations of Computer Science 1998

A characterization of context-free languages

In the paper, we propose a new edge detection schemes, based on a single-pixel imaging in the frequency domain. In SCHEME-I, special sinusoidal patterns for the x-direction edge and also y-direction edge of the unknown object are first designed. The frequency spectrum for the edge is then obtained using the a four-step phase-shifting technique with the designed sinusoidal patterns in a single-pixel imaging system. In SCHEME-II, the frequency spectrum of the unknown object is first obtained, then the frequency spectrum for the edge is obtained by calculations. The resulting edges are finally obtained by the inverse Fourier transform on their frequency spectrum. We have also verified the proposed schemes by experiments and numerical simulations. The results show that the proposed schemes can produce higher quality edges of character and also image objects. Comparing with SCHEME-II, the application of SCHEME-I to high frequency components has greatly improved signal-to-noise ratio of the received data in the bucket detector, resulting in better experimental results. Comparing with the edge detection scheme by speckle-shifting in ghost imaging systems, the proposed SCHEME-I shows an improvement in the signal-to-noise ratio. Since a single-pixel imaging system is used, the proposed schemes are capable of reconstructing edges from indirect measurements. The number of measurements required can be effectively reduced due to the sparsity of natural images in the Fourier domain and the conjugate symmetry of real-valued signals’ Fourier spectrum.

Jozef Gruska

Papers

Systolic trellis automatat

Descriptional Complexity of Context-Free Languages.

Mathematical Foundations of Computer Science 1998

A characterization of context-free languages

Edge detection based on single-pixel imaging.