Related papers: A Circuit Complexity Formulation of Algorithmic In…
Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one…
In the paper we present results to develop an irreducible theory of complex systems in terms of self-organization processes of prime integer relations. Based on the integers and controlled by arithmetic only the self-organization processes…
It is known a method for converting a system of Boolean polynomial equations to a single Boolean polynomial equation with less variables. In this paper, we show a formula for systems of Boolean polynomial equations which is based on the…
This work reviews how database theory uses tractable circuit classes from knowledge compilation. We present relevant query evaluation tasks, and notions of tractable circuits. We then show how these tractable circuits can be used to address…
One of the most fundamental problems in science is to define {\it quantitatively} the complexity of organized matters, i.e., {\it organized complexity}. Although many measures have been proposed toward this aim in previous decades, there is…
In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\textrm{NC}^1$, $\textrm{SAC}^1$ and $\textrm{AC}^1$ as…
We initiate the study of a new model of query complexity of Boolean functions where, in addition to 0 and 1, the oracle can answer queries with ``unknown''. The query algorithm is expected to output the function value if it can be…
We formally define algorithmic capture of combinatorial tasks as the ability of a transformer to extrapolate to arbitrary task sizes with controllable error and logarithmic sample adaptation, providing a sharp scaling criterion for…
Training data is always finite, making it unclear how to generalise to unseen situations. But, animals do generalise, wielding Occam's razor to select a parsimonious explanation of their observations. How they do this is called their…
We introduce a new notion of complexity of functions and we show that it has the following properties: (i) it governs a PAC Bayes-like generalization bound, (ii) for neural networks it relates to natural notions of complexity of functions…
This paper proposes a quantum circuit for computing the mean value from a given set of quantum states. The circuit consults a Quantum Random Access Memory to get the values of the set, and by using superposition, interference and…
Circuit representations are becoming the lingua franca to express and reason about tractable generative and discriminative models. In this paper, we show how complex inference scenarios for these models that commonly arise in machine…
In this paper, we present a method for the Hamiltonian simulation in the context of eigenvalue estimation problems which improves earlier results dealing with Hamiltonian simulation through the truncated Taylor series. In particular, we…
Boolean and quantum circuits have commonalities and differences. To formalize the syntactical commonality we introduce syntactic circuits where the gates are black boxes. Syntactic circuits support various semantics. One semantics is…
We show new upper and lower bounds for the complexity of implementation of a sequence of Boolean matrices proposed by Kaski et al. (arXiv:1208.0554) with additive circuits.
With the advent of high-performance computing, Bayesian methods are increasingly popular tools for the quantification of uncertainty throughout science and industry. Since these methods impact the making of sometimes critical decisions in…
Learning the topology of higher-order networks from data is a fundamental challenge in many signal processing and machine learning applications. Simplicial complexes provide a principled framework for modeling multi-way interactions, yet…
The proliferation of agentic systems has thrust the reasoning capabilities of AI into the forefront of contemporary machine learning. While it is known that there \emph{exist} neural networks which can reason through any Boolean task…
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by F\"{u}rer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg…
The celebrated result of Kabanets and Impagliazzo (Computational Complexity, 2004) showed that PIT algorithms imply circuit lower bounds, and vice versa. Since then it has been a major challenge to understand the precise connections between…