Search results for "Universal Algebra"

K4-free Graphs as a Free Algebra

2017

International audience; Graphs of treewidth at most two are the ones excluding the clique with four vertices (K4) as a minor, or equivalently, the graphs whose biconnected components are series-parallel. We turn those graphs into a finitely presented free algebra, answering positively a question by Courcelle and Engelfriet, in the case of treewidth two. First we propose a syntax for denoting these graphs: in addition to parallel composition and series composition, it suffices to consider the neutral elements of those operations and a unary transpose operation. Then we give a finite equational presentation and we prove it complete: two terms from the syntax are congruent if and only if they …

Deduction of the oxidation degree of the group A15 elements at the phthalocyanines on the basis of the Q band

2017

Adjacent vertices can be hard to find by quantum walks

2018

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs Ω(N) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjace…

Varieties of Codes and Kraft Inequality

2007

Decipherability conditions for codes are investigated by using the approach of Guzman, who introduced in [7] the notion of variety of codes and established a connection between classes of codes and varieties of monoids. The class of Uniquely Decipherable (UD) codes is a special case of variety of codes, corresponding to the variety of all monoids. It is well known that the Kraft inequality is a necessary condition for UD codes, but it is not sufficient, in the sense that there exist codes that are not UD and that satisfy the Kraft inequality. The main result of the present paper states that, given a variety V of codes, if all the elements of V satisfy the Kraft inequality, then V is the var…

Varieties of Codes and Kraft Inequality

2005

Decipherability conditions for codes are investigated by using the approach of Guzman, who introduced in [7] the notion of variety of codes and established a connection between classes of codes and varieties of monoids. The class of Uniquely Decipherable (UD) codes is a special case of variety of codes, corresponding to the variety of all monoids. It is well known that the Kraft inequality is a necessary condition for UD codes, but it is not sufficient, in the sense that there exist codes that are not UD and that satisfy the Kraft inequality. The main result of the present paper states that, given a variety $\mathcal{V}$ of codes, if all the elements of $\mathcal{V}$ satisfy the Kraft inequ…

Thin bases of order h

2003

Abstract A subset A⊆ N 0 is called a basis of order h if every positive integer can be represented as a sum of h members of A . Thin bases of order h will be constructed in this paper, for each h ⩾2, where the value of lim sup A(n)/ n h is smaller than that of thin bases known so far. In the most important case h =2 it is shown that for the considered class of bases (which generalizes an ansatz of Stohr) the result is best possible up to an e >0.

On positive P

2002

Continuing a line of research opened up by Grigni and Sipser (1992) and further pursued by Stewart (1994), we show that a wide variety of equivalent characterizations of P still remain equivalent when restricted to be positive. All these restrictions thus define the same class posP, a proper subclass of monP, the class of monotone problems in P. We also exhibit complete problems for posP under very weak reductions.

Subvarieties of the Varieties Generated by the SuperalgebraM1, 1(E) orM2(𝒦)

2003

Abstract Let 𝒦 be a field of characteristic zero, and let us consider the matrix algebra M 2(𝒦) endowed with the ℤ2-grading (𝒦e 11 ⊕ 𝒦e 22) ⊕ (𝒦e 12 ⊕ 𝒦e 21). We define two superalgebras, ℛ p and 𝒮 q , where p and q are positive integers. We show that if 𝒰 is a proper subvariety of the variety generated by the superalgebra M 2(𝒦), then the even-proper part of the T 2-ideal of graded polynomial identities of 𝒰 asymptotically coincides with the even-proper part of the graded polynomial identities of the variety generated by the superalgebra ℛ p  ⊕ 𝒮 q . This description also affords an even-asymptotic desc…

Nondeterministic operations on finite relational structures

1998

Abstract This article builds on a tutorial introduction to universal algebra for language theory (Courcelle, Theoret. Comput. Sci. 163 (1996) 1–54) and extends it in two directions. First, nondeterministic operations are considered, i.e., operations which give a set of results instead of a single one. Most of their properties concerning recognizability and equational definability carry over from the ordinary case with minor modifications. Second, inductive sets of evaluations are studied in greater detail. It seems that they are handled most naturally in the framework presented here. We consider the analogues of top-down and bottom-up tree transducers. Again, most of their closure propertie…

Varieties and Covarieties of Languages (Extended Abstract)

2013

AbstractBecause of the isomorphism (X×A)→X≅X→(A→X), the transition structure of a deterministic automaton with state set X and with inputs from an alphabet A can be viewed both as an algebra and as a coalgebra. This algebra-coalgebra duality goes back to Arbib and Manes, who formulated it as a duality between reachability and observability, and is ultimately based on Kalmanʼs duality in systems theory between controllability and observability. Recently, it was used to give a new proof of Brzozowskiʼs minimization algorithm for deterministic automata. Here we will use the algebra-coalgebra duality of automata as a common perspective for the study of both varieties and covarieties, which are …