Membership Problem Research Articles

THREE CONVEX HULL THEOREMS ON TRIANGLES AND CIRCLES

We prove three convex hull theorems on triangles and circles. Given a triangle <TEX>${\triangle}$</TEX> and a point p, let <TEX>${\triangle}^{\prime}$</TEX> be the triangle each of whose vertices is the intersection of the orthogonal line from p to an extended edge of <TEX>${\triangle}$</TEX>. Let <TEX>${\triangle}^{{\prime}{\prime}}$</TEX> be the triangle whose vertices are the centers of three circles, each passing through p and two other vertices of <TEX>${\triangle}$</TEX>. The first theorem characterizes when <TEX>$p{\in}{\triangle}$</TEX> via a distance duality. The triangle algorithm in [1] utilizes a general version of this theorem to solve the convex hull membership problem in any dimension. The second theorem proves <TEX>$p{\in}{\triangle}$</TEX> if and only if <TEX>$p{\in}{\triangle}^{\prime}$</TEX>. These are used to prove the third: Suppose p be does not lie on any extended edge of <TEX>${\triangle}$</TEX>. Then <TEX>$p{\in}{\triangle}$</TEX> if and only if <TEX>$p{\in}{\triangle}^{{\prime{\prime}}$</TEX>.

The IO and OI hierarchies revisited

We study languages of λ-terms generated by IO and OI unsafe grammars. These languages can be used to model meaning representations in the formal semantics of natural languages following the tradition of Montague. Using techniques pertaining to the denotational semantics of the simply typed λ-calculus, we show that the emptiness and membership problems for both types of grammars are decidable. In the course of the proof of the decidability results for OI, we identify a decidable variant of the λ-definability problem, and prove a stronger form of Statman's finite completeness Theorem.

Rational subsets and submonoids of wreath products

It is shown that membership in rational subsets of wreath products H≀V with H a finite group and V a virtually free group is decidable. On the other hand, it is shown that there exists a fixed finitely generated submonoid in the wreath product Z≀Z with an undecidable membership problem.

Two-dimensional Sgraffito automata

We present a new model of a two-dimensional computing device called Sgraffito automaton . In general, the model is quite simple, which allows a clear design of computations. When restricted to one-dimensional inputs, that is, strings, the Sgraffito automaton does not exceed the power of finite-state automata. On the other hand, for two-dimensional inputs, it yields a family of picture languages with good closure properties that strictly includes the class REC of recognizable picture languages. The deterministic Sgraffito automata define a class of picture languages that includes the class of deterministic recognizable picture languages DREC, the class of picture languages that are accepted by four-way alternating automata, those that are accepted by deterministic one-marker automata, and the sudoku-deterministically recognizable picture languages, but the membership problem for the accepted languages is still decidable in polynomial time. In addition, the deterministic Sgraffito automata accept some unary picture languages that are outside of the class REC.

Global effective versions of the Briançon–Skoda–Huneke theorem

We prove global effective versions of the Briancon–Skoda–Huneke theorem. Our results extend, to singular varieties, a result of Hickel on the membership problem in polynomial ideals in Cn, and a related theorem of Ein and Lazarsfeld for smooth projective varieties. The proofs rely on known geometric estimates and new results on multivariable residue calculus.

The complexity of intersecting finite automata having few final states

The problem of determining whether several finite automata accept a word in common is closely related to the well-studied membership problem in transformation monoids. We raise the issue of limiting the number of final states in the automata intersection problem. For automata with two final states, we show the problem to be $${\oplus}$$źL-complete or NP-complete according to whether a nontrivial monoid other than a direct product of cyclic groups of order 2 is allowed in the automata. We further consider idempotent commutative automata and (Abelian, mainly) group automata with one, two, or three final states over a singleton or larger alphabet, elucidating (under the usual hypotheses on complexity classes) the complexity of the intersection nonemptiness and related problems in each case.

Patterns with bounded treewidth

A pattern is a string consisting of variables and terminal symbols, and its language is the set of all words that can be obtained by substituting arbitrary words for the variables. The membership problem for pattern languages, i.e., deciding on whether or not a given word is in the pattern language of a given pattern is NP-complete. We show that any parameter of patterns that is an upper bound for the treewidth of appropriate encodings of patterns as relational structures, if restricted, allows the membership problem for pattern languages to be solved in polynomial time. Furthermore, we identify new such parameters.

Knapsack problems in groups

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem is NP-complete.

Quantum computation of discrete logarithms in semigroups

Abstract We describe an efficient quantum algorithm for computing discrete logarithms in semigroups using Shor's algorithms for period finding and the discrete logarithm problem as subroutines. Thus proposed cryptosystems based on the presumed hardness of discrete logarithms in semigroups are insecure against quantum attacks. In contrast, we show that some generalizations of the discrete logarithm problem are hard in semigroups despite being easy in groups. We relate a shifted version of the discrete logarithm problem in semigroups to the dihedral hidden subgroup problem, and we show that the constructive membership problem with respect to k ≥ 2 generators in a black-box abelian semigroup of order N requires Θ ˜ ( N 1 2 - 1 2 k ) $\tilde{\Theta }(N^{\frac{1}{2}-\frac{1}{2k}})$ quantum queries.

Solution of the membership problem of the prefix monoid in certain one-relator groups

In this work we examine to what extent small cancellation conditions imply the solution of the membership problem of the prefix monoid in one-relator groups. The main results show that with additional combinatorial conditions on the defining relator (prefix pureness and strongly cyclically reduceness), which synchronize the prefixes and an analogue of the Greendlinger Lemma for the set of prefixes (Property P(\(n\))), the membership problem for the prefix monoid can be solved in cases not covered by previous results.

On simple generators of recursively enumerable languages

For each language L, let Fˆ∩(L) (Cˆ∩(L), resp.) be the intersection-closed full AFL (full trio, resp.) generated by L. Furthermore, for each natural number k≥2 let Pk={ank|n∈N}. By applying certain classical and recent results on Diophantine equations we show that LRE=Fˆ∩(Pk), where LRE is the family of all recursively enumerable languages. This allows us to answer to an open problem of S. Ginsburg and J. Goldstine in [3]. A general method to reduce certain restricted membership problems in language families to number theory (in particular to systems of Diophantine equations) is then considered. In the third part of the paper, we show that the Cˆ∩(REP)=Cˆ∩(BREP)=LRE where REP={(amb)n|m,n∈N} and BREP={(anb)n|n∈N}. The results give an answer to two problems left open by P. Turakainen in [11].

Subalgebras of FA-presentable algebras

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. However, it is proven that a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furthermore, it is proven that the class of unary FA-presentable algebras is closed under forming finitely generated subalgebras, and that the membership problem for such subalgebras is decidable.

Approximating amoebas and coamoebas by sums of squares

Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming. Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstellensatz, this allows us to tackle the problem by sums of squares techniques and semidefinite programming. Our method yields polynomial identities as certificates of non-containment of a point in an amoeba or coamoeba. As the main theoretical result, we establish some degree bounds on the polynomial certificates. Moreover, we provide some actual computations of amoebas based on the sums of squares approach.

PSO based Multidimensional Data Clustering: A Survey

Data clustering is considered as one of the most promising data analysis methods in data mining and on the other side KMeans is the well known partitional clustering technique. Nevertheless, K-Means and other partitional clustering techniques struggle with some challenges where dimension is the core concern. The different challenges associated with clustering techniques are preknowledge of initial centers of clusters, problem of stagnation, multiple cluster membership problem, dead unit problem, and slow or premature convergence to local search space. So as to resolve these clustering limitations, an eminent choice is to adapt the Swarm Intelligence (SI) inspired optimization algorithms. This paper presents an overview of the research on an applicability of different Particle Swarm Optimization (PSO) variants for clustering multidimensional data along with the basic concepts of PSO as well as data clustering. It also puts forward an idea of new and advance PSO variant in order to deal with multidimensional data clustering.

The constructive membership problem for discrete free subgroups of rank 2 of

AbstractWe exhibit a practical algorithm for solving the constructive membership problem for discrete free subgroups of rank $2$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{R})$. This algorithm, together with methods for checking whether a two-generator subgroup of $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{SL}_2(\mathbb{R})$ is discrete and free, have been implemented in Magma for groups defined over real algebraic number fields.Supplementary materials are available with this article.

The Canonical Decomposition of $\mathcal{C}^n_d$ and Numerical Gröbner and Border Bases

This article introduces the canonical decomposition of the vector space of multivariate polynomials for a given monomial ordering. Its importance lies in solving multivariate polynomial systems, computing Gröbner bases, and solving the ideal membership problem. An SVD-based algorithm is presented that numerically computes the canonical decomposition. It is then shown how, by introducing the notion of divisibility into this algorithm, a numerical Gröbner basis can also be computed. In addition, we demonstrate how the canonical decomposition can be used to decide whether the affine solution set of a multivariate polynomial system is zero-dimensional and to solve the ideal membership problem numerically. The SVD-based canonical decomposition algorithm is also extended to numerically compute border bases. A tolerance for each of the algorithms is derived using perturbation theory of principal angles. This derivation shows that the condition number of computing the canonical decomposition and numerical Gröbner basis is essentially the condition number of the Macaulay matrix. Numerical experiments with both exact and noisy coefficients are presented and discussed.

(k,l)-Unambiguity and Quasi-Deterministic Structures

We focus on the family of $(k,l)$-unambiguous automata that encompasses the one of deterministic $k$-lookahead automata introduced by Han and Wood. We show that this family presents nice theoretical properties that allow us to compute quasi-deterministic structures. These structures are smaller than DFAs and can be used to solve the membership problem faster than NFAs.

Satisfiability-based Set Membership Filters

Introduced here is a novel application of Satisfiability (SAT) to the set membership problem with specific focus on efficiently testing whether large sets contain a given element. Such tests can be greatly enhanced via the use of filters, probabilistic algorithms that can quickly decide whether or not a given element is in a given set. This article proposes SAT filters (i.e., filters based on SAT) and their use in the set membership problem. Both the theoretical advantages of SAT filters and experimental results show that this technique yields significant performance improvements over previous techniques. Specifically, a SAT filter is a filter construction that is simple yet efficient in terms of both query time and filter size; i.e., SAT filters asymptotically achieve the information-theoretic limit while providing fast querying. As well, this is the first application that makes use of the random k-SAT phase transition results and may drive research into efficient solvers for this and similar applications.

Generalized (identity‐based) hash proof system and its applications

AbstractIn this work, we generalize the paradigm of the hash proof system (HPS) proposed by Cramer and Shoup (EUROCRYPT 2002). In the center of our generalization, we lift a subset membership problem to a distribution‐distinguishing problem. Our generalized HPS clarifies and encompasses all the known public‐key encryption (PKE) schemes that essentially implement the idea of an HPS. Moreover, besides the existing smoothness property, we introduce an additional property named anonymity for HPS. As a natural application, we consider anonymity for PKE in the presence of key leakage and provide a generic construction of leakage‐resilient anonymous PKE from an anonymous HPS. We then extend our generalization to the identity‐based setting. Concretely, we generalize the paradigm of the identity‐based HPS (IB‐HPS) proposed by Boneh et al. (FOCS 2007) and Alwen et al. (EUROCRYPT 2010) and introduce anonymity for it. As an interesting application of the anonymous IB‐HPS, we consider security for PKE with keyword search (PEKS) in the presence of token leakage and provide a generic construction of leakage‐resilient secure PEKS from leakage‐resilient anonymous identity‐based encryption, which in turn is based on anonymous IB‐HPS. Copyright © 2013 John Wiley &amp; Sons, Ltd.

INSIDE THE CLASS OF REGEX LANGUAGES

We study different possibilities of combining the concept of homomorphic replacement with regular expressions in order to investigate the class of languages given by extended regular expressions with backreferences (REGEX). It is shown in which regard existing and natural ways to do this fail to reach the expressive power of REGEX. Furthermore, the complexity of the membership problem for REGEX with a bounded number of backreferences is considered.