141 research outputs found
Satisfiability in multi-valued circuits
Satisfiability of Boolean circuits is among the most known and important
problems in theoretical computer science. This problem is NP-complete in
general but becomes polynomial time when restricted either to monotone gates or
linear gates. We go outside Boolean realm and consider circuits built of any
fixed set of gates on an arbitrary large finite domain. From the complexity
point of view this is strictly connected with the problems of solving equations
(or systems of equations) over finite algebras.
The research reported in this work was motivated by a desire to know for
which finite algebras there is a polynomial time algorithm that
decides if an equation over has a solution. We are also looking for
polynomial time algorithms that decide if two circuits over a finite algebra
compute the same function. Although we have not managed to solve these problems
in the most general setting we have obtained such a characterization for a very
broad class of algebras from congruence modular varieties. This class includes
most known and well-studied algebras such as groups, rings, modules (and their
generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie
algebras), lattices (and their extensions like Boolean algebras, Heyting
algebras or other algebras connected with multi-valued logics including
MV-algebras).
This paper seems to be the first systematic study of the computational
complexity of satisfiability of non-Boolean circuits and solving equations over
finite algebras. The characterization results provided by the paper is given in
terms of nice structural properties of algebras for which the problems are
solvable in polynomial time.Comment: 50 page
Systems of Diagonal Equations Over pâAdic Fields
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135491/1/jlms0257.pd
Expressive Power, Satisfiability and Equivalence of Circuits over Nilpotent Algebras
Satisfiability of Boolean circuits is NP-complete in general but becomes polynomial time when restricted for example either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is connected with solving equations over finite algebras. This in turn is one of the oldest and well-known mathematical problems which for centuries was the driving force of research in algebra. Let us only mention Galois theory, Gaussian elimination or Diophantine Equations. The last problem has been shown to be undecidable, however in finite realms such problems are obviously decidable in nondeterministic polynomial time.
A project of characterizing finite algebras m A with polynomial time algorithms deciding satisfiability of circuits over m A has been undertaken in [Pawel M. Idziak and Jacek Krzaczkowski, 2018]. Unfortunately that paper leaves a gap for nilpotent but not supernilpotent algebras. In this paper we discuss possible attacks on filling this gap
Criteria of measure-preserving for -adic dynamical systems in terms of the van der Put basis
This paper is devoted to (discrete) -adic dynamical systems, an important
domain of algebraic and arithmetic dynamics. We consider the following open
problem from theory of -adic dynamical systems. Given continuous function
Let us represent it via special convergent series, namely van
der Put series. How can one specify whether this function is measure-preserving
or not for an arbitrary ? In this paper, for any prime we present a
complete description of all compatible measure-preserving functions in the
additive form representation. In addition we prove the criterion in terms of
coefficients with respect to the van der Put basis determining whether a
compatible function preserves the Haar measure
Counterexamples to the Hasse principle
In this article we develop counterexamples to the Hasse principle using only
techniques from undergraduate number theory and algebra. By keeping the
technical prerequisites to a minimum, we hope to provide a path for
nonspecialists to this interesting area of number theory. The counterexamples
considered here extend the classical counterexample of Lind and Reichardt. As
discussed in an appendix, this type of counterexample is important in the
theory of elliptic curves: today they are interpreted as nontrivial elements in
the Tate--Shafarevich group
Classical simulations of Abelian-group normalizer circuits with intermediate measurements
Quantum normalizer circuits were recently introduced as generalizations of
Clifford circuits [arXiv:1201.4867]: a normalizer circuit over a finite Abelian
group is composed of the quantum Fourier transform (QFT) over G, together
with gates which compute quadratic functions and automorphisms. In
[arXiv:1201.4867] it was shown that every normalizer circuit can be simulated
efficiently classically. This result provides a nontrivial example of a family
of quantum circuits that cannot yield exponential speed-ups in spite of usage
of the QFT, the latter being a central quantum algorithmic primitive. Here we
extend the aforementioned result in several ways. Most importantly, we show
that normalizer circuits supplemented with intermediate measurements can also
be simulated efficiently classically, even when the computation proceeds
adaptively. This yields a generalization of the Gottesman-Knill theorem (valid
for n-qubit Clifford operations [quant-ph/9705052, quant-ph/9807006] to quantum
circuits described by arbitrary finite Abelian groups. Moreover, our
simulations are twofold: we present efficient classical algorithms to sample
the measurement probability distribution of any adaptive-normalizer
computation, as well as to compute the amplitudes of the state vector in every
step of it. Finally we develop a generalization of the stabilizer formalism
[quant-ph/9705052, quant-ph/9807006] relative to arbitrary finite Abelian
groups: for example we characterize how to update stabilizers under generalized
Pauli measurements and provide a normal form of the amplitudes of generalized
stabilizer states using quadratic functions and subgroup cosets.Comment: 26 pages+appendices. Title has changed in this second version. To
appear in Quantum Information and Computation, Vol.14 No.3&4, 201
A history of Galois fields
This paper stresses a specific line of development of the notion of finite field, from Ăvariste Galoisâs 1830 âNote sur la thĂ©orie des nombres,â and Camille Jordanâs 1870 TraitĂ© des substitutions et des Ă©quations algĂ©briques, to Leonard Dicksonâs 1901 Linear groups with an exposition of the Galois theory. This line of development highlights the key role played by some specific algebraic procedures. These intrinsically interlaced the indexations provided by Galoisâs number-theoretic imaginaries with decompositions of the analytic representations of linear substitutions. Moreover, these procedures shed light on a key aspect of Galoisâs works that had received little attention until now. The methodology of the present paper is based on investigations of intertextual references for identifying some specific collective dimensions of mathematics. We shall take as a starting point a coherent network of texts that were published mostly in France and in the U.S.A. from 1893 to 1907 (the âGalois fields network,â for short). The main shared references in this corpus were some texts published in France over the course of the 19th century, especially by Galois, Hermite, Mathieu, Serret, and Jordan. The issue of the collective dimensions underlying this network is thus especially intriguing. Indeed, the historiography of algebra has often put to the fore some specific approaches developed in Germany, with little attention to works published in France. Moreover, the âGerman abstract algebraâ has been considered to have strongly influenced the development of the American mathematical community. Actually, this influence has precisely been illustrated by the example of Elliakim Hasting Mooreâs lecture on âabstract Galois fieldsâ at the Chicago congress in 1893. To be sure, this intriguing situation raises some issues of circulations of knowledge from Paris to Chicago. It also calls for reflection on the articulations between the individual and the collective dimensions of mathematics. Such articulations have often been analysed by appealing to categories such as nations, disciplines, or institutions (e.g., the âGerman algebra,â the âChicago algebraic research schoolâ). Yet, we shall see that these categories fail to characterize an important specific approach to Galois fields. The coherence of the Galois fields network had underlying it some collective interest for âlinear groups in Galois fields.â Yet, the latter designation was less pointing to a theory, or a discipline, revolving around a specific object, i.e. Gln(Fpn) (p a prime number), than to some specific procedures. In modern parlance, general linear groups in Galois fields were introduced in this context as the maximal group in which an elementary abelian group (i.e., the multiplicative group of a Galois field) is a normal subgroup. The Galois fields network was actually rooted on a specific algebraic culture that had developed over the course of the 19th century. We shall see that this shared culture resulted from the circulation of some specific algebraic procedures of decompositions of polynomial representations of substitutions
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