OUP accepted manuscript

The first order logic \(\mathcal{R}ing(0,+,*,<)\) for finite residue class rings with order is presented, and extensions of this logic with generalized quantifiers are given. It is shown that this logic and its extensions capture DLOGTIME-uniform circuit complexity classes ranging from AC 0 to TC 0. Separability results are obtained for the hierarchy of these logics when order is not present, and for \(\mathcal{R}ing(0,+,*,<)\) from the unordered version. These separations are obtained using tools from class field theory, adapting notions as the spectra of polynomials over finite fields to sets of sentences in this logic of finite rings, and studying asymptotic measures of these sets such as their relative densities. This framework of finite rings with order provides new algebraic tools and a novel perspective for descriptive complexity.

The universal binary neurons are a central point of this Chapter. The mathematical models of the universal binary neuron over the field of the complex numbers, the residue class ring and the finite field are considered. A notion of the P-realizable Boolean function, which may be considered as a generalization of a notion of the threshold Boolean function, is introduced. It is shown that the implementation with a single neuron the input/output mapping described by non-threshold Boolean functions is possible, if weights are complex, and activation function of the neuron is a function of the argument of the weighted sum (similar to multi-valued neuron). It is also possible to define an activation function on the residue class ring and the finite field in such a way that the implementation of non-threshold Boolean functions will be possible on the single neuron with weights from these sets. P-realization of multiple-valued function over finite algebras and residue class ring in particular is also considered. The general features of P-realizable Boolean functions are considered, also as the synthesis of universal binary neuron.

Linear dynamical systems over finite rings

On the ℓ-DLIPs of codes over finite commutative rings

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.

On the Circuit Complexity of Sigmoid Feedforward Neural Networks

We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We introduce compressibility measures as a useful tool to classify the intersection problem for certain classes of finite semigroups into circuit complexity classes and Turing machine complexity classes. Using this framework, we obtain a new and simple proof that for groups and commutative semigroups, the problem is contained in \(\mathsf {NP}\). We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove \(\mathsf {NP}\)-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply containment in the circuit complexity class \(\mathsf {qAC}^k\) (for some \(k \in \mathbb {N}\)) and decidability in quasi-polynomial time. We also establish connections to the monoid variant of the problem.

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

Let F q \mathbb {F}_q denote the finite field of characteristic p p and order q q . Let Z q \mathbb {Z}_q denote the unramified extension of the p p -adic rational integers Z p \mathbb {Z}_p with residue field F q \mathbb {F}_q . Given two positive integers m , n m,n , define a box B m \mathcal B_m to be a subset of Z q n \mathbb {Z}_q^n with q n m q^{nm} elements such that B m \mathcal B_m modulo p m p^m is equal to ( Z q / p m Z q ) n (\mathbb {Z}_q/p^m \mathbb {Z}_q)^n . For a collection of nonconstant polynomials f 1 , … , f s ∈ Z q [ x 1 , … , x n ] f_1,\dots ,f_s\in \mathbb {Z}_q[x_1,\ldots ,x_n] and positive integers m 1 , … , m s m_1,\dots ,m_s , define the set of common zeros inside the box B m \mathcal B_m to be V = { X ∈ B m : f i ( X ) ≡ 0 mod p m i for all 1 ≤ i ≤ s } . \begin{equation*} V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\text { for all } 1\leq i\leq s\}. \end{equation*} It is an interesting problem to give the sharp estimates for the p p -divisibility of | V | |V| . This problem has been partially solved for the three cases: (i) m = m 1 = ⋯ = m s = 1 m=m_1=\cdots =m_s=1 , which is just the Ax-Katz theorem, (ii) m = m 1 = ⋯ = m s &gt; 1 m=m_1=\cdots =m_s&gt;1 , which was solved by Katz [Proc. Amer. Math. Soc. 137 (2009), pp. 4065–4075; Amer. J. Math. 93 (1971), pp. 485–499], Marshal and Ramage [Proc. Amer. Math. Soc. 49 (1975), pp. 35–38], and (iii) m = 1 m=1 , and m 1 , … , m s ≥ 1 m_1,\dots ,m_s\geq 1 , which was recently solved by Cao, Wan [Finite Fields Appl. 91 (2023), p. 25] and Grynkiewicz [ A generalization of the Chevalley-Warning and Ax-Katz theorems with a view towards combinatorial number theory , Preprint, arXiv: 2208.12895 , 2022]. Based on the multi-fold addition and multiplication of the finite Witt rings over F q \mathbb {F}_q , we investigate the remaining unconsidered case of m &gt; 1 m&gt;1 and m ≠ m j m\neq m_j for some 1 ≤ j ≤ s 1\leq j\leq s , and finally provide a complete answer to this problem.

This paper examines the circuit complexity of feedforward neural networks having sigmoid activation function. The starting point is the complexity class NN k defined in [18]. First two additional complexity classes N N Δ k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315784076/1ea703af-0c66-41e4-8ff3-9e9c700ac445/content/eq1772.tif/> and N N Δ , e k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315784076/1ea703af-0c66-41e4-8ff3-9e9c700ac445/content/eq1773.tif/> having less restrictive conditions (than NN k ) concerning fan-in and accuracy are defined. We then prove several relations among these three classes and well established circuit complexity classes. All the proofs are constructive as being built around binary adders. By relaxing the fan-in condition from logarithmic [18] to (almost) polynomial, the results show that substantially larger classes of sigmoid activation feedforward neural networks can be implemented in polynomial size Boolean circuits. This is done on the expense of a logarithmic factor increase in the number of layers, but having constant fan-in (of 2). The main conclusion is that there are interesting fan-in dependent depth-size tradeoffs when trying to digitally implement sigmoid activation neural networks.

We give a decomposition of total graphs on some finite commutative rings R = Zm, where the set of zero-divisors of R is not an ideal. In particular, we study the total graph T(􀀀(Z2npm))where p is a prime and m and n are positive integers and investigate some graph theoretical properties with some of its fundamental subgraphs.

For R a Galois ring and m 1, . . . , m l positive integers, a generalized quasi-cyclic (GQC) code over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ is an R[x]-submodule of $${R[x]/(x^{m_1}-1)\times\cdots \times R[x]/(x^{m_l}-1)}$$. Suppose m 1, . . . , m l are all coprime to the characteristic of R and let {g 1, . . . , g t } be the set of all monic basic irreducible polynomials in the factorizations of $${x^{m_i}-1}$$ (1 ≤ i ≤ l). Then the GQC codes over R of block lengths (m 1, m 2, . . . , m l ) and length $${\sum_{i=1}^lm_i}$$ are identified with $${{\mathcal G}_1\times\cdots\times {\mathcal G}_t}$$, where $${{\mathcal G}_j}$$ is an R[x]/(g j )-submodule of $${(R[x]/(g_j))^{n_j}}$$, where n j is the number of i for which g j appears in the factorization of $${x^{m_i}-1}$$ into monic basic irreducible polynomials. This identification then leads to an enumeration of such GQC codes. An analogous result is also obtained for the 1-generator GQC codes. A notion of a parity-check polynomial is given when R is a finite field, and the number of GQC codes with a given parity-check polynomial is determined. Finally, an algorithm is given to compute the number of GQC codes of given block lengths.

Olga Taussky’s interest in class field theory began when she was a student with Ph. Furtwangler at the University of Vienna (she received her doctorate under his supervision in 1930). At this time, E. Artin, using his general reciprocity law, reformulated the “Principal Ideal Theorem” as a purely grouptheoretic question regarding the triviality of the “Verlagerung” homomorphism from a finite group to its commutator subgroup. Since Furtwangler had been developing group theoretic methods in his own number theoretic investigations, Artin communicated this new idea to him. This was considered to be one of the central problems in the subject at the time, and so there was a good deal of excitement when Furtwangler eventually succeeded in proving the result using this new approach. Olga Taussky had been looking for a thesis topic and Furtwangler suggested several questions related to the finer structure of the Principal Ideal Theorem. The optimistic atmosphere, and the prospect of contributing at the frontiers of class field theory, sparked in Olga a deep interest in this problem which lasted her entire career. She returned to the questions of “capitulation”, a term coined by one of her co-authors Arnold Scholz, several times in her life, always with the sense that these were questions of deep arithmetic significance. I believe that it was the experience of her early contact with class field theory which dominated her research interests in number theory and which provided her a fertile source of inspiration for most of her career. I will describe some of her work in this area below. I would like to take this opportunity to thank the referee for many valuable suggestions. For a number field F , (i.e., an extension of finite degree over the rational field Q) its Hilbert class field H = H(F ) is the maximal Galois extension of F which is everywhere unramified and whose Galois group Gal(H/F ) is abelian. It is a consequence of Artin’s reciprocity law that Gal(H/F ) is isomorphic to the ideal class group C(F ) of F. The Principal Ideal Theorem is the statement that every ideal of F becomes a principal ideal when considered

A word-to-word function is continuous for a class of languages~$\mathcal{V}$ if its inverse maps $\mathcal{V}$_languages to~$\mathcal{V}$. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. To this end, we develop a robust theory rooted in the standard profinite analysis of regular languages. Since previous algebraic studies of transducers have focused on the sole structure of the underlying input automaton, we also compare the two algebraic approaches. We focus on two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?

Let m, e be positive integers, p a prime number, $$\mathbb {F}_{p^m}$$ be a finite field of $$p^m$$ elements and $$R=\mathbb {F}_{p^m}[u]/\langle u^e\rangle $$ which is a finite chain ring. For any $$\omega \in R^\times $$ and positive integers k, n satisfying $$\mathrm{gcd}(p,n)=1$$ , we prove that any $$(1+\omega u)$$ -constacyclic code of length $$p^kn$$ over R is monomially equivalent to a matrix-product code of a nested sequence of $$p^k$$ cyclic codes with length n over R and a $$p^k\times p^k$$ matrix $$A_{p^k}$$ over $$\mathbb {F}_p$$ . Using the matrix-product structures, we give an iterative construction of every $$(1+\omega u)$$ -constacyclic code by $$(1+\omega u)$$ -constacyclic codes of shorter lengths over R.