Stashing And Parallelization Pentagons

Parallelization is an algebraic operation that lifts problems to sequences in a natural way. Given a sequence as an instance of the parallelized problem, another sequence is a solution of this problem if every component is instance-wise a solution of the original problem. In the Weihrauch lattice parallelization is a closure operator. Here we introduce a dual operation that we call stashing and that also lifts problems to sequences, but such that only some component has to be an instance-wise solution. In this case the solution is stashed away in the sequence. This operation, if properly defined, induces an interior operator in the Weihrauch lattice. We also study the action of the monoid induced by stashing and parallelization on the Weihrauch lattice, and we prove that it leads to at most five distinct degrees, which (in the maximal case) are always organized in pentagons. We also introduce another closely related interior operator in the Weihrauch lattice that replaces solutions of problems by upper Turing cones that are strong enough to compute solutions. It turns out that on parallelizable degrees this interior operator corresponds to stashing. This implies that, somewhat surprisingly, all problems which are simultaneously parallelizable and stashable have computability-theoretic characterizations. Finally, we apply all these results in order to study the recently introduced discontinuity problem, which appears as the bottom of a number of natural stashing-parallelization pentagons. The discontinuity problem is not only the stashing of several variants of the lesser limited principle of omniscience, but it also parallelizes to the non-computability problem. This supports the slogan that "non-computability is the parallelization of discontinuity".

Order-Independent Algorithm for the Asymptotic Stability of Complex Polynomials

The Extended Routh Array (ERA) settles the asymptotic stability of complex polynomials. The ERA is a natural extension of the Routh Array which applies only to real polynomials. Although the ERA is a nice theoretical algorithm for stability testing, it has its limitations. Unfortunately, as the order of the polynomial increases, the size of calculations increases dramatically as will be shown below. In the current work, we offer an alternative algorithm which is basically equivalent to the ERA, but has the extra advantage of being simpler, more efficient, and easy to apply even to large order polynomials. In all the steps required in the construction of the new algorithm, only one single and simple algebraic operation is needed, which makes it a polynomial order-independent algorithm.

A Set of Integral Grid-Coding Algebraic Operations Based on GeoSOT-3D

As the amount of collected spatial information (2D/3D) increases, the real-time processing of these massive data is among the urgent issues that need to be dealt with. Discretizing the physical earth into a digital gridded earth and assigning an integral computable code to each grid has become an effective way to accelerate real-time processing. Researchers have proposed optimization algorithms for spatial calculations in specific scenarios. However, a complete set of algorithms for real-time processing using grid coding is still lacking. To address this issue, a carefully designed, integral grid-coding algebraic operation framework for GeoSOT-3D (a multilayer latitude and longitude grid model) is proposed. By converting traditional floating-point calculations based on latitude and longitude into binary operations, the complexity of the algorithm is greatly reduced. We then present the detailed algorithms that were designed, including basic operations, vector operations, code conversion operations, spatial operations, metric operations, topological relation operations, and set operations. To verify the feasibility and efficiency of the above algorithms, we developed an experimental platform using C++ language (including major algorithms, and more algorithms may be expanded in the future). Then, we generated random data and conducted experiments. The experimental results show that the computing framework is feasible and can significantly improve the efficiency of spatial processing. The algebraic operation framework is expected to support large geospatial data retrieval and analysis, and experience a revival, on top of parallel and distributed computing, in an era of large geospatial data.

Constraint satisfaction problems in the logic LFP +rank

Constraint satisfaction problems (CSPs) are one of the central topics in theoretical computer science, in particular, in the area of artificial intelligence. Their computational complexity is due to relatively recent results from areas of mathematics, including finite-model-theory, algebra and graph homomorphisms. The main conjecture by Feder and Vardi states that any CSP over a finite relational template is either in P or is NP-complete. Further, it amounts to showing that every non NP-complete CSP can be expressed as an extension of first-order logic. A finite template is Mal'tsev, a compatible algebraic operation, which is closely related to an affine space over a finite field. The so-called Bulatov-Dalmau algorithm, a natural generalization of the Gaussian elimination on vector spaces, shows such CSPs are tractable. In this work, we prove that CSPs described over a finite template Mal'tsev are expressible in logic LFP+rnk, providing a logical proof that such CSPs are tractable.

Constraint satisfaction problems in the logic LFP +rank

Constraint satisfaction problems (CSPs) are one of the central topics in theoretical computer science, in particular, in the area of artificial intelligence. Their computational complexity is due to relatively recent results from areas of mathematics, including finite-model-theory, algebra and graph homomorphisms. The main conjecture by Feder and Vardi states that any CSP over a finite relational template is either in P or is NP-complete. Further, it amounts to showing that every non NP-complete CSP can be expressed as an extension of first-order logic. A finite template is Mal'tsev, a compatible algebraic operation, which is closely related to an affine space over a finite field. The so-called Bulatov-Dalmau algorithm, a natural generalization of the Gaussian elimination on vector spaces, shows such CSPs are tractable. In this work, we prove that CSPs described over a finite template Mal'tsev are expressible in logic LFP+rnk, providing a logical proof that such CSPs are tractable.

Identification of nonlinear dynamical systems with time delay

This paper proposes a technique to identify nonlinear dynamical systems with time delay. The sparse optimization algorithm is extended to nonlinear systems with time delay. The proposed algorithm combines cross-validation techniques from machine learning for automatic model selection and an algebraic operation for preprocessing signals to filter the noise and for removing the dependence on initial conditions. We further integrate the bootstrapping resampling technique with the sparse regression to obtain the statistical properties of estimation. We use Taylor expansion to parameterize time delay. The proposed algorithm in this paper is computationally efficient and robust to noise. A nonlinear Duffing oscillator is simulated to demonstrate the efficiency and accuracy of the proposed technique. An experimental example of a nonlinear rotary flexible joint is presented to further validate the proposed method.

One Step in-Memory Solution of Inverse Algebraic Problems

Machine learning requires to process large amount of irregular data and extract meaningful information. Von-Neumann architecture is being challenged by such computation, in fact a physical separation between memory and processing unit limits the maximum speed in analyzing lots of data and the majority of time and energy are spent to make information travel from memory to the processor and back. In-memory computing executes operations directly within the memory without any information travelling. In particular, thanks to emerging memory technologies such as memristors, it is possible to program arbitrary real numbers directly in a single memory device in an analog fashion and at the array level, execute algebraic operation in-memory and in one step. In this chapter the latest results in accelerating inverse operation, such as the solution of linear systems, in-memory and in a single computational cycle will be presented.

Topological Analysis of Fibrations in Multidimensional (C, R) Space

A holomorphically fibred space generates locally trivial bundles with positive dimensional fibers. This paper proposes two varieties of fibrations (compact and non-compact) in the non-uniformly scalable quasinormed topological (C, R) space admitting cylindrically symmetric continuous functions. The projective base space is dense, containing a complex plane, and the corresponding surjective fiber projection on the base space can be fixed at any point on real subspace. The contact category fibers support multiple oriented singularities of piecewise continuous functions within the topological space. A composite algebraic operation comprised of continuous linear translation and arithmetic addition generates an associative magma in the non-compact fiber space. The finite translation is continuous on complex planar subspace under non-compact projection. Interestingly, the associative magma resists transforming into a monoid due to the non-commutativity of composite algebraic operation. However, an additive group algebraic structure can be admitted in the fiber space if the fibration is a non-compact variety. Moreover, the projection on base space supports additive group structure, if and only if the planar base space passes through the real origin of the topological (C, R) space. The topological analysis shows that outward deformation retraction is not admissible within the dense topological fiber space. The comparative analysis of the proposed fiber space with respect to Minkowski space and Seifert fiber space illustrates that the group algebraic structures in each fiber spaces are of different varieties. The proposed topological fiber bundles are rigid, preserving sigma-sections as compared to the fiber bundles on manifolds.

A Matrix Approach for Analyzing Signal Flow Graph

Mason’s gain formula can grow factorially because of growth in the enumeration of paths in a directed graph. Each of the (n − 2)! permutation of the intermediate vertices includes a path between input and output nodes. This paper presents a novel method for analyzing the loop gain of a signal flow graph based on the transform matrix approach. This approach only requires matrix determinant operations to determine the transfer function with complexity O(n3) in the worst case, therefore rendering it more efficient than Mason’s gain formula. We derive the transfer function of the signal flow graph to the ratio of different cofactor matrices of the augmented matrix. By using the cofactor expansion, we then obtain a correspondence between the topological operation of deleting a vertex from a signal flow graph and the algebraic operation of eliminating a variable from the set of equations. A set of loops sharing the same backward edges, referred to as a loop group, is used to simplify the loop enumeration. Two examples of feedback networks demonstrate the intuitive approach to obtain the transfer function for both numerical and computer-aided symbolic analysis, which yields the same results as Mason’s gain formula. The transfer matrix offers an excellent physical insight, because it enables visualization of the signal flow.