Computational complexity of some problems involving congruences on algebras (original) (raw)

The complexity of the Chinese Remainder Theorem

arXiv (Cornell University), 2023

The Chinese Remainder Theorem for the integers says that every system of congruence equations is solvable as long as the system satisfies an obvious necessary condition. This statement can be generalized in a natural way to arbitrary algebraic structures using the language of Universal Algebra. In this context, an algebra is a structure of a first-order language with no relation symbols, and a congruence on an algebra is an equivalence relation on its base set compatible with its fundamental operations. A tuple of congruences of an algebra is called a Chinese Remainder tuple if every system involving them is solvable. In this article we study the complexity of deciding whether a tuple of congruences of a finite algebra is a Chinese Remainder tuple. This problem, which we denote CRT, is easily seen to lie in coNP. We prove that it is actually coNP-complete and also show that it is tractable when restricted to several well-known classes of algebras, such as vector spaces and distributive lattices. The polynomial algorithms we exhibit are made possible by purely algebraic characterizations of Chinese Remainder tuples for algebras in these classes, which constitute interesting results in their own right. Among these, an elegant characterization of Chinese Remainder tuples of finite distributive lattices stands out. Finally, we address the restriction of CRT to an arbitrary equational class V generated by a two-element algebra. Here we establish an (almost) dichotomy by showing that, unless V is the class of semilattices, the problem is either coNP-complete or tractable.

Complexity of certain decision problems about congruential languages

Journal of Computer and System Sciences, 1985

Given an arbitrary finite Church-Rosser Thue system S, it is shown that the question of whether a given congruence class is finite is undecidable, and the question of whether every congruence class is finite is not even semidecidable (in fact, I7,-complete). It is shown that the question of whether a given (resp. every) congruence class is a context-free language is at least as hard. Also, given a finite rewriting system over a commutative monoid, the question of whether every congruence class is finite is shown to be tractable. This contrasts with the known result that the question of whether a given congruence class is finite requires space at least exponential in the square root of the input length. 0 1985 Academic press, IN.

The Chinese Remainder Theorem, Associative Algebras, and Multiplicative Complexity

1990 Conference Record Twenty-Fourth Asilomar Conference on Signals, Systems and Computers, 1990.

Polynomial multiplication continues to play a fundamental role in many important algorithms (e.g., convolution, correlation). Many methods have been developed which facilitate highly efficient polynomial multiplication. One such method is baaed on the Chinese Remainder Theorem (CRT), a classic result from ring theory. The CRT is known to reduce the complexity of polynomial multiplication from O (N 2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algebras. This new point of view provides a clearer understanding o€ the CRT.

The Algorithmic Complexity of Modular Decomposition

2001

Modular decomposition is a thoroughly investigated topic in many areas such as switching theory, reliability theory, game theory and graph theory. We propose an O(mn)-algorithm for the recognition of a modular set of a monotone Boolean function f with m prime implicants and n variables. Using this result we show that the computation of the modular closure of a set can be done in time O(mn2). On the other hand, we prove that the recognition problem for general Boolean functions is NP-complete. Moreover, we introduce the so called generalized Shannon decomposition of a Boolean functions as an efficient tool for proving theorems on Boolean function decompositions.

Efficient Algorithms for Finite mathbbZ\mathbb{Z}mathbbZ-Algebras

Groups, complexity, cryptology, 2024

For a finite Z-algebra R, i.e., for a Z-algebra which is a finitely generated Z-module, we assume that R is explicitly given by a system of Z-module generators G, its relation module Syz(G), and the structure constants of the multiplication in R. In this setting we develop and analyze efficient algorithms for computing essential information about R. First we provide polynomial time algorithms for solving linear systems of equations over R and for basic ideal-theoretic operations in R. Then we develop ZPP (zero-error probabilitic polynomial time) algorithms to compute the nilradical and the maximal ideals of 0-dimensional affine algebras K[x1,. .. , xn]/I with K = Q or K = Fp. The task of finding the associated primes of a finite Z-algebra R is reduced to these cases and solved in ZPPIF (ZPP plus one integer factorization). With the same complexity, we calculate the connected components of the set of minimal associated primes minPrimes(R) and then the primitive idempotents of R. Finally, we prove that knowing an explicit representation of R is polynomial time equivalent to knowing a strong Gröbner basis of an ideal I such that R = Z[x1,. .. , xn]/I.

The lattice of congruence lattices of algebras on a finite set

Algebra universalis

The congruence lattices of all algebras defined on a fixed finite set A ordered by inclusion form a finite atomistic lattice E. We describe the atoms and coatoms. Each meet-irreducible element of E being determined by a single unary mapping on A, we characterize completely those which are determined by a permutation or by an acyclic mapping on the set A. Using these characterisations we deduce several properties of the lattice E; in particular, we prove that E is tolerance-simple whenever |A| ≥ 4. * supported by Slovak VEGA grant 1/0063/14 * * This research started as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project, supported by the European Union, co-financed by the European Social Fund 113/173/0-2.

Some Complexity Results for Polynomial Ideals

Journal of Complexity, 1997

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)-tuple P = ( f, g 1 , g 2 , . . . , g w ) where f and the g i are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases. © 1997 Academic Press

Semi-algebraic Complexity of Quotients and Sign Determination of Remainders

Journal of Complexity, 1996

For two nonzero polynomials p 0 ϭ ͚ m iϭ0 x 0,i T i , p 1 ϭ ͚ mϪ1 iϭ0 x 1,i T i ʦ R[T ] the successive signed Euclidean division yields algorithms, that is, semialgebraic computation trees, for tasks such as computing the sequence of successive quotients, deciding the signs of the sequence of remainders in a point a ʦ R, deciding the number of remainders, or deciding the degree pattern of the sequence of remainders. In this paper we show lower bounds of order m log 2 (m) for these tasks, within the computational framework of semi-algebraic computation trees. The inevitably long paths in semi-algebraic computation trees can be specified as those followed by certain prime cones in the real spectrum of a polynomial ring. We give in the paper a positive answer to the question posed in T. Lickteig (J. Pure Appl. Algebra 110(2), 131-184 (1996)) whether the degree of the zero-set of the support of a prime cone provides a lower bound on the complexity of isolating the prime cone. The applications are based on the degree inequalities given by Strassen and Schuster and extend their work to the real situation in various directions.

Computational Recognition of Properties of Finite Algebras

2008

In 1993 Ralph McKenzie resolved several of the most intriguing and challenging problems concerning varieties generated by finite algebras with only finitely many basic operations. These accomplishments can be summarized as follows. Accomplishment 1. There is a finite algebra generating a residually countable inherently nonfinitely based variety. This refutes the R-S Conjecture that every finitely generated residually small variety should be residually very finite. It also refutes an old and provocative speculation that the finite algebras of finite type which generate residually small varieties might all be finitely based. Accomplishment 2. There is no algorithm for deciding which finite algebras generate residually finite varieties. Accomplishment 3. There is no algorithm for deciding which finite algebras are finitely based. So Ralph McKenzie settled Tarski's celebrated Finite Basis Problem. Until McKenzie's breakthrough, no algbraically reasonable property of finite algebras was known to be undecidable. Indeed, a number of properties (generating a minimal variety, generating a congruence modular variety, etc.) were long known to be decidable. Ralph McKenzie invented a robust technique for interpreting an arbitrary Turing machine T into a finite algebra A(T) so that the machine computations would be available in the variety generated by A(T). It seems likely that it can be used to demonstrate the undecidability of a wide assortment properties of varieties generated by finite algebras. Indeed, further undecidability results have already to obtained by Charles Latting, Ralph McKenzie, and Ross Willard. These lectures are intended to provide a path to these accomplishments. There are only two main differences between McKenzie's exposition and the one found here. First, I organized the material into lectures which are each, more or less, amenable to presentation in fifty-minutes. The second and more significant deviation is that I followed the work of Ross Willard to prove that A(T) is finitely based when T halts. I added no new results (and I hope no errors either). This presentation is intended to be concrete, and to reveal how the ideas develop toward the ultimate results. See Ross Willard's work for a more abstract perspective. These notes arose from five occasions on which I gave series of lectures on this material. My colleagues at LaTrobe University (1994), at the University of South Carolina (1994-95,1996-97), at the University of Hawaii (1995-96), at the Beijing Workshop in Logic, Universal Algebra, and Computer Science (1998), and in the Ulam Seminar at the University of Colorado (1998) all endured my struggles to talk reasonably about these results. Their criticisms and ideas have become part of these notes. Ralph McKenzie and Ross Willard both shared early drafts of their work with me. Of i PREAMBLE ii course, these lectures owe a large debt to Ralph McKenzie (say about 98%) who originated these spectacular results.

Quotient complexity of ideal languages

Theoretical Computer Science, 2013

We study the state complexity of regular operations in the class of ideal languages. A language L ⊆ Σ * is a right (left) ideal if it satisfies L = LΣ * (L = Σ * L). It is a two-sided ideal if L = Σ * LΣ * , and an all-sided ideal if L = Σ * L, the shuffle of Σ * with L. We prefer the term "quotient complexity" instead of "state complexity", and we use derivatives to calculate upper bounds on quotient complexity, whenever it is convenient. We find tight upper bounds on the quotient complexity of each type of ideal language in terms of the complexity of an arbitrary generator and of its minimal generator, the complexity of the minimal generator, and also on the operations union, intersection, set difference, symmetric difference, concatenation, star and reversal of ideal languages.

On the intrinsic complexity of the arithmetic Nullstellensatz

Journal of Pure and Applied Algebra, 2000

We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a BÃ ezout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the ÿrst time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function S (x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters. : S 0 0 2 2 -4 0 4 9 ( 9 8 ) 0 0 1 4 8 -0 S := {X d 1 ; X 1 − X d 2 ; : : : ; X n−2 − X d n−1 ; 1 − X n−1 X d−1 n }:

Weak Congruences of Algebras with CONSTANTS1

The paper deals with weak congruences of algebras having at least two constants in the similarity type. The presence of constants is a necessary condition for complementedness of the weak congruence lattices of non-trivial algebras. Some su-cient conditions for the same property are also given. In particular, so called 0,1-algebras have complemented weak congruence lattices if and only if their subalgebra lattices are com- plemented. In this context we also investigate relations among algebras with balanced congruences, balanced weak congruences, consistent and strongly consistent algebras. We prove that an algebra has balanced weak congruences if and only if it is strongly consistent and has balanced con- gruences on all subalgebras. For a variety, strong consistency of algebras is equivalent with having balanced weak congruences. Finally, we prove that for a class of algebras which additionally are Hamiltonian, there is a homomorphism from the congruence lattice onto the subalgebr...

On the Complexity of Generalized Discrete Logarithm Problem

arXiv (Cornell University), 2022

Generalized Discrete Logarithm Problem (GDLP) is an extension of the Discrete Logarithm Problem. The goal is to find x ∈ Z s such g x mod s = y for a given g, y ∈ Z s. Generalized discrete logarithm is similar but instead of a single base element, uses a number of base elements which does not necessarily commute with each other. In this paper, we prove that GDLP is NP-hard for symmetric groups. Furthermore, we prove that GDLP remains NP-hard even when the base elements are permutations of at most 3 elements. Lastly, we discuss the implications and possible implications of our proofs in classical and quantum complexity theory.

The membership problem for unmixed polynomial ideals is solvable in single exponential time

1991

Dickenstein, A., N. Fitchas, M. Giusti and C. Sessa, The membership problem for unmixed polynomial ideals is solvable in single exponential time, Discrete Applied Mathematics 33 (1991) 73-94. Deciding membership for polynomial ideals represents a classical problem of computational commutative algebra which is exponential space hard. This means that the usual al membership problem which are based on linear algebra techniques have doubly exponential sequential worst case complexity. We show that the membership problem has single exponential sequential and polynomial parallel complexity for unmixed ideals. More specific complexity results are given for the special cases of zero-dimensional and complete intersection ideals.

Reduced sub-powers and the decision problem for finite algebras in arithmetical varieties

Algebra Universalis, 1988

The aim of this paper is to prove that every finitely generated, arithmetical variety of finite type, in which every subdirectly irreducible algebra has linearly ordered congruences has a decidable first order theory of its finite members. The proof is based on a representation of finite algebras from such varieties by some quotients of special subdirect products in which sets of indices are partially ordered into dual trees. Then the result of M. O. Rabin about decidability of the monadie second order theory of two successors is applied. 365 366 PAWEL M. IDZIAK ALGEBRA UNIV.

Definable principal congruences and solvability

Annals of Pure and Applied Logic, 2009

We prove that in a locally finite variety that has definable principal congruences (DPC), solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size.

Algorithms for Commutative Algebras Over the Rational Numbers

Foundations of Computational Mathematics, 2016

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel-Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.