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Serge Grigorieff

Publications and source records attributed to Serge Grigorieff.

At least 19 recordsLinked to original sources
arXiv2024

Congruence Preservation, Lattices and Recognizability

We study in general algebras Gratzer's notion of congruence preserving function, characterizing functions in terms of stability under inverse image of particular Boolean algebras of subsets generated from any subset of the algebra. Weakening Gratzer's notion to only finite index congruences, a similar result holds with lattices of sets. Genereralizing the notion to that of stable preorder preserving function, we extend these characterizations to Boolean algebras and lattices generated from any recognizable subset of the algebra. Our starting point is a result with related flavor on the additive algebra of natural integers which was obtained some years ago. All these results can be visualized in the diagram of Table 1. We finally consider some simple particular conditions on the algebra allowing to get a richer diagram.

math.LO↗
arXiv2021

Randomness and uniform distribution modulo one

We elaborate the notions of Martin-Löf and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform distribution of sequences. This extends the result proved by Avigad for sequences of linear functions with integer coefficients to the wider classical class of Koksma sequences of functions. And, by requiring equidistribution with respect to every computably enumerable open set (respectively, computably enumerable open set with computable measure) in the unit interval, we give a sufficient condition for Martin-Löf (respectively Schnorr) randomness.

math.LO↗
arXiv2021

The algebra of binary trees is affine complete

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. We show that on the algebra of binary trees whose leaves are labeled by letters of an alphabet containing at least three letters, a function is congruence preserving if and only if it is polynomial.

cs.FL↗
arXiv2020

Congruence Preservation, Lattices and Recognizability

Looking at some monoids and (semi)rings (natural numbers, integers and p-adic integers), and more generally, residually finite algebras (in a strong sense), we prove the equivalence of two ways for a function on such an algebra to behave like the operations of the algebra. The first way is to preserve congruences or stable preorders. The second way is to demand that preimages of recognizable sets belong to the lattice or the Boolean algebra generated by the preimages of recognizable sets by derived unary operation of the algebra (such as translations, quotients,. . . ).

cs.LO↗
arXiv2016

Congruence Preserving Functions on Free Monoids

A function on an algebra is congruence preserving if, for any congruence, it maps congruent elements to congruent elements. We show that, on a free monoid generated by at least 3 letters, a function from the free monoid into itself is congruence preserving %nonmonogenic if and only if it is of the form $x \mapsto w_0 x w_1 \cdots w_{n-1} x w_n$ for some finite sequence of words $w_0,\ldots,w_n$. We generalize this result to functions of arbitrary arity. This shows that a free monoid with at least three generators is a (noncommutative) affine complete algebra. Up to our knowledge, it is the first (nontrivial) case of a noncommutative affine complete algebra.

math.RA↗
arXiv2015

Characterizing congruence preserving functions $Z/nZ\to Z/mZ$ via rational polynomials

We introduce a basis of rational polynomial-like functions $P_0,\ldots,P_{n-1}$ for the free module of functions $Z/nZ\to Z/mZ$. We then characterize the subfamily of congruence preserving functions as the set of linear combinations of the functions $lcm(k)\,P_k$ where $lcm(k)$ is the least common multiple of $2,\ldots,k$ (viewed in $Z/mZ$). As a consequence, when $n\geq m$, the number of such functions is independent of $n$.

math.NT↗
arXiv2015

Arithmetical Congruence Preservation: from Finite to Infinite

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational polynomials (taking only integral values) and the function giving the least common multiple of $1,2,\ldots,k$. The tool used to obtain these characterizations is "lifting": if $π\colon X\to Y$ is a surjective morphism, and $f$ a function on $Y$ a lifting of $f$ is a function $F$ on $X$ such that $π\circ F=f\circπ$. In this paper we relate the finite and infinite notions by proving that the finite case can be lifted to the infinite one. For $p$-adic and profinite integers we get similar characterizations via lifting. We also prove that lattices of recognizable subsets of $Z$ are stable under inverse image by congruence preserving functions.

math.NT↗
arXiv2013

Wadge Hardness in Scott Spaces and Its Effectivization

We prove some results on the Wadge order on the space of sets of natural numbers endowed with Scott topology, and more generally, on omega-continuous domains. Using alternating decreasing chains we characterize the property of Wadge hardness for the classes of the Hausdorff difference hierarchy (iterated differences of open sets). A similar characterization holds for Wadge one-to-one and finite-to-one completeness. We consider the same questions for the effectivization of the Wadge relation. We also show that for the space of sets of natural numbers endowed with the Scott topology, in each class of the Hausdorff difference hierarchy there are two strictly increasing chains of Wadge degrees of sets properly in that class. The length of these chains is the rank of the considered class, and each element in one chain is incomparable with all the elements in the other chain.

cs.LO↗
arXiv2013

Borel and Hausdorff Hierarchies in Topological Spaces of Choquet Games and Their Effectivization

What parts of classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Victor Selivanov in a series of papers centered on algebraic domains. And recently it has been considered by Matthew de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has a stationary strategy in the Choquet game. A natural proper subclass of approximation spaces coincides with the class of quasi-Polish spaces. We study the Borel and Hausdorff difference hierarchies in approximation spaces, revisiting the work done for the other topological spaces. We also consider the problem of effectivization of these results.

cs.LO↗
arXiv2013

Newton representation of functions over natural integers having integral difference ratios

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance all functions $x\mapsto\lfloor e^{1/a}\;a^x\;x!\rfloor$, with $a\in\Z\setminus\{0,1\}$, and a function equal to $\lfloor e\;x!\rfloor$ except on 0. Finally, to study the complement class, we look at functions $\N\to\RR$ which are not uniformly close to any function having integral difference ratios.

cs.DM↗
arXiv2010

Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume. Part I is dedicated to information theory and the mathematical formalization of randomness based on Kolmogorov complexity. This last application goes back to the 60's and 70's with the work of Martin-Löf, Schnorr, Chaitin, Levin, and has gained new impetus in the last years.

cs.LO↗
arXiv2010

ASMs and Operational Algorithmic Completeness of Lambda Calculus

We show that lambda calculus is a computation model which can step by step simulate any sequential deterministic algorithm for any computable function over integers or words or any datatype. More formally, given an algorithm above a family of computable functions (taken as primitive tools, i.e., kind of oracle functions for the algorithm), for every constant K big enough, each computation step of the algorithm can be simulated by exactly K successive reductions in a natural extension of lambda calculus with constants for functions in the above considered family. The proof is based on a fixed point technique in lambda calculus and on Gurevich sequential Thesis which allows to identify sequential deterministic algorithms with Abstract State Machines. This extends to algorithms for partial computable functions in such a way that finite computations ending with exceptions are associated to finite reductions leading to terms with a particular very simple feature.

cs.LO↗
arXiv2010

Evolving MultiAlgebras unify all usual sequential computation models

It is well-known that Abstract State Machines (ASMs) can simulate "step-by-step" any type of machines (Turing machines, RAMs, etc.). We aim to overcome two facts: 1) simulation is not identification, 2) the ASMs simulating machines of some type do not constitute a natural class among all ASMs. We modify Gurevich's notion of ASM to that of EMA ("Evolving MultiAlgebra") by replacing the program (which is a syntactic object) by a semantic object: a functional which has to be very simply definable over the static part of the ASM. We prove that very natural classes of EMAs correspond via "literal identifications" to slight extensions of the usual machine models and also to grammar models. Though we modify these models, we keep their computation approach: only some contingencies are modified. Thus, EMAs appear as the mathematical model unifying all kinds of sequential computation paradigms.

cs.FL↗
arXiv2008

Church, Cardinal and Ordinal Representations of Integers and Kolmogorov complexity

We consider classical representations of integers: Church's function iterators, cardinal equivalence classes of sets, ordinal equivalence classes of totally ordered sets. Since programs do not work on abstract entities and require formal representations of objects, we effectivize these abstract notions in order to allow them to be computed by programs. To any such effectivized representation is then associated a notion of Kolmogorov complexity. We prove that these Kolmogorov complexities form a strict hierarchy which coincides with that obtained by relativization to jump oracles and/or allowance of infinite computations.

math.LO↗
arXiv2008

Refinment of the "up to a constant" ordering using contructive co-immunity and alike. Application to the Min/Max hierarchy of Kolmogorov complexities

We introduce orderings between total functions f,g: N -> N which refine the pointwise "up to a constant" ordering <=cte and also insure that f(x) is often much less thang(x). With such orderings, we prove a strong hierarchy theorem for Kolmogorov complexities obtained with jump oracles and/or Max or Min of partial recursive functions. We introduce a notion of second order conditional Kolmogorov complexity which yields a uniform bound for the "up to a constant" comparisons involved in the hierarchy theorem.

math.LO↗
arXiv2008

Kolmogorov complexities Kmax, Kmin on computable partially ordered sets

We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes Max^{X\to D}_{PR} and Max^{X\to D}_{Rec} of functions X \to D which are pointwise maximum of partial or total computable sequences of functions where D = (D,<) is some computable partially ordered set. The enumeration theorem and the invariance theorem always hold for Max^{X\to D}_{PR}, leading to a variant KD;max of Kolmogorov complexity. We characterize the orders D such that the enumeration theorem (resp. the invariance theorem) also holds for Max^{X\to D}_{Rec} . It turns out that Max^{X\to D}_{Rec} may satisfy the invariance theorem but not the enumeration theorem. Also, when Max^{X\to D}_{Rec} satisfies the invariance theorem then the Kolmogorov complexities associated to Max^{X\to D}_{Rec} and Max^{X\to D}_{PR} are equal (up to a constant). Letting K^D_{min} = K^{D^{rev}}_{max}, where D^{rev} is the reverse order, we prove that either K^D_{min} =_{ct} K^D_{max} =_{ct} K^D (=_{ct} is equality up to a constant) or K^D_{min}, K^D_{max} are <=_{ct} incomparable and <_{ct} K^D and >_{ct} K^{0',D}. We characterize the orders leading to each case. We also show that K^D_{min}, K^D_{max} cannot be both much smaller than K^D at any point. These results are proved in a more general setting with two orders on D, one extending the other.

math.LO↗
arXiv2008

Kolmogorov complexity in perspective

We survey the diverse approaches to the notion of information content: from Shannon entropy to Kolmogorov complexity. The main applications of Kolmogorov complexity are presented namely, the mathematical notion of randomness (which goes back to the 60's with the work of Martin-Lof, Schnorr, Chaitin, Levin), and classification, which is a recent idea with provocative implementation by Vitanyi and Cilibrasi.

math.LO↗