A heuristic approach for multiple restricted multiplication

Published versio

Decidability and Complexity of Tree Share Formulas

Fractional share models are used to reason about how multiple actors share ownership of resources. We examine the decidability and complexity of reasoning over the "tree share" model of Dockins et al. using first-order logic, or fragments thereof. We pinpoint a connection between the basic operations on trees union, intersection, and complement and countable atomless Boolean algebras, allowing us to obtain decidability with the precise complexity of both first-order and existential theories over the tree share model with the aforementioned operations. We establish a connection between the multiplication operation on trees and the theory of word equations, allowing us to derive the decidability of its existential theory and the undecidability of its full first-order theory. We prove that the full first-order theory over the model with both the Boolean operations and the restricted multiplication operation (with constants on the right hand side) is decidable via an embedding to tree-automatic structures

Fast arithmetic computing with neural networks

The authors introduce a restricted model of a neuron which is more practical as a model of computation then the classical model of a neuron. The authors define a model of neural networks as a feedforward network of such neurons. Whereas any logic circuit of polynomial size (in n) that computes the product of two n-bit numbers requires unbounded delay, such computations can be done in a neural network with constant delay. The authors improve some known results by showing that the product of two n-bit numbers and sorting of n n-bit numbers can both be computed by a polynomial size neural network using only four unit delays, independent of n . Moreover, the weights of each threshold element in the neural networks require only O(log n)-bit (instead of n-bit) accuracy

Period polynomial relations between double zeta values

The even weight period polynomial relations in the double shuffle Lie algebra $\mathfrak{ds}$ were discovered by Ihara, and completely classified by the second author by relating them to restricted even period polynomials associated to cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. In an article published in the same year, Gangl, Kaneko and Zagier displayed certain linear combinations of odd-component double zeta values which are equal to scalar multiples of simple zeta values in even weight, and also related them to restricted even period polynomials. In this paper, we relate the two sets of relations, showing how they can be deduced from each other by duality.Comment: 13 page