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Non-monotone inductive definitions were studied in the late 1960's and early 1970's with the aim of understanding connections between the complexity of the formulas defining the induction steps and the size of the ordinals measuring the…
We prove exact formulas for weighted $2k$th moments of the Riemann zeta function for all integer $k\geq 1$ in terms of the analytic continuation of an auto-correlation function. This latter enjoys several functional equations. One of them,…
Fuzzy optimization deals with the problem of determining 'optimal'solutions of an optimization problem when some of the elements that appear in the problem are not precise. In real situations it is usual to have information, in systems…
Quantum algorithms are known for presenting more efficient solutions to certain computational tasks than any corresponding classical algorithm. It has been thought that the origin of the power of quantum computation has its roots in…
The Additive Transform of an arithmetic function represents a novel approach to examining the interplay between multiplicative arithmetic function and additive functions. This transform concept introduces a method to systematically generate…
We investigate the computational power of affine automata (AfAs) introduced in [4]. In particular, we present a simpler proof for how to change the cutpoint for any affine language and a method how to reduce error in bounded error case.…
A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field. When the degrees of these polynomials are bounded by $n$, the algorithm uses $O(n^{1.43})$ field operations, breaking…
This paper explores practical aspects of using a high-level functional language for GPU-based arithmetic on ``midsize'' integers. By this we mean integers of up to about a quarter million bits, which is sufficient for most practical…
In classical computation, a "write-only memory" (WOM) is little more than an oxymoron, and the addition of WOM to a (deterministic or probabilistic) classical computer brings no advantage. We prove that quantum computers that are augmented…
We introduce a model of register automata over infinite trees with extrema constraints. Such an automaton can store elements of a linearly ordered domain in its registers, and can compare those values to the suprema and infima of register…
Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer…
Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…
We show that Automatic Differentiation (AD) operators can be provided in a dynamic language without sacrificing numeric performance. To achieve this, general forward and reverse AD functions are added to a simple high-level dynamic…
In this paper, we investigate the problem of synthesizing computable functions of infinite words over an infinite alphabet (data omega-words). The notion of computability is defined through Turing machines with infinite inputs which can…
We investigate questions related to the presence of primitive words and Lyndon words in automatic and linearly recurrent sequences. We show that the Lyndon factorization of a k-automatic sequence is itself k-automatic. We also show that the…
We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…
It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical…
Primitive recursion is a mature, well-understood topic in the theory and practice of programming. Yet its dual, primitive corecursion, is underappreciated and still seen as exotic. We aim to put them both on equal footing by giving a…
We consider quantum learning machines -- quantum computers that modify themselves in order to improve their performance in some way -- that are trained to perform certain classical task, i.e. to execute a function which takes classical bits…
In this report, a novel approach to intelligence and learning is introduced, this approach is based on what we call 'perception logic'. Based on this logic, a computing mechanism and automata are introduced. Multi-resolution analysis of…