INTERPRETING GRAPH COLORABILITY IN FINITE SEMIGROUPS

2006 ◽  
Vol 16 (01) ◽  
pp. 119-140 ◽  
Author(s):  
MARCEL JACKSON ◽  
RALPH McKENZIE
Keyword(s):  
Membership Problem ◽  
Finite Semigroups ◽  
Variety Of Semigroups ◽  
Membership Problems

We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55-element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3-colorability problem.

UNDECIDABILITY, AUTOMATA, AND PSEUDOVARITIES OF FINITE SEMIGROUPS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 455-473 ◽  
Author(s):  
JOHN RHODES
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Membership Problem ◽  
Joint Paper ◽  
Finite Semigroups ◽  
Membership Problems

The author proves for each of the operations # equalling *, °, **, □ or m, there exist pseudovarieties of finite semigroups [Formula: see text] and [Formula: see text] with decidable membership problems, such that [Formula: see text] has an undecidable membership problem. In addition, if [Formula: see text] denotes the pseudovariety of all finite aperiodic semigroups, [Formula: see text] denotes the pseudovariety of all finite groups, and [Formula: see text](E) denotes the pseudovariety of all finite aperiodic semigroups satisfying the finite number of equations E, then it is proved that there exists E such that [Formula: see text](E) has an undecidable membership problem. Note [Formula: see text] equals all semigroups of complexity ≤1. Section 6 is expanded into a joint paper with B. Steinberg, following this paper.

Undecidability of the identity problem for finite semigroups

1992 ◽  
Vol 57 (1) ◽  
pp. 179-192 ◽  
Author(s):  
Douglas Albert ◽  
Robert Baldinger ◽  
John Rhodes
Keyword(s):  
Word Problem ◽  
Finite Semigroup ◽  
Identity Problem ◽  
Finite Semigroups ◽  
Or Groups ◽  
Finite Set ◽  
Variety Of Semigroups ◽  
Finite Products

In 1947 E. Post [28] and A. A. Markov [18] independently proved the undecidability of the word problem (or the problem of deducibility of relations) for semigroups. In 1968 V. L. Murskil [23] proved the undecidability of the identity problem (or the problem of deducibility of identities) in semigroups.If we slightly generalize the statement of these results we can state many related results in the literature and state our new results proved here. Let V denote either a (Birkhoff) variety of semigroups or groups or a pseudovariety of finite semigroups. By a very well-known theorem a (Birkhoff) variety is defined by equations or equivalently closed under substructure, surmorphisms and all products; see [7]. It is also well known that V is a pseudovariety of finite semigroups iff V is closed under substructure, surmorphism and finite products, or, equivalently, determined eventually by equations w1 = w1′, w2 = w2′, w3 = w3′,… (where the finite semigroup S eventually satisfies these equations iff there exists an n, depending on S, such that S satisfies Wj = Wj′ for j ≥ n). See [8] and [29]. All semigroups form a variety while all finite semigroups form a pseudovariety.We now consider a table (see the next page). In it, for example, the box denoting the “word” (identity) problem for the psuedovariety V” means, given a finite set of relations (identities) E and a relation (identity) u = ν, the problem of whether it is decidable that E implies u = ν inside V.

Membership Problem for Two-Dimensional General Row Jumping Finite Automata

2020 ◽  
Vol 31 (04) ◽  
pp. 527-538
Author(s):  
Grzegorz Madejski ◽  
Andrzej Szepietowski
Keyword(s):  
Computational Model ◽  
Turing Machine ◽  
Finite Automaton ◽  
Finite Automata ◽  
The Other ◽  
Membership Problem ◽  
Two Dimensional ◽  
Np Complete ◽  
Nondeterministic Turing Machine ◽  
Membership Problems

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

COMPUTATIONALLY AND ALGEBRAICALLY COMPLEX FINITE ALGEBRA MEMBERSHIP PROBLEMS

2007 ◽  
Vol 17 (08) ◽  
pp. 1635-1666 ◽  
Author(s):  
MARCIN KOZIK
Keyword(s):  
Finite Algebra ◽  
Membership Problem ◽  
Membership Problems

In this paper we produce a finite algebra which generates a variety with a PSPACE-complete membership problem. We produce another finite algebra with a γ function that grows exponentially. The results are obtained via a modification of a construction of the algebra A(T) that was introduced by McKenzie in 1996.

On the Membership Problem of Permutation Grammars — A Direct Proof of NP-Completeness

2020 ◽  
Vol 31 (04) ◽  
pp. 515-525
Author(s):  
Benedek Nagy
Keyword(s):  
Direct Proof ◽  
Input Word ◽  
Membership Problem ◽  
Context Sensitive ◽  
Language Class ◽  
Language Classes ◽  
Sentential Form ◽  
Np Complete ◽  
Membership Problems ◽  
Context Free

One of the most essential classes of problems related to formal languages is the membership problem (also called word problem), i.e., to decide whether a given input word belongs to the language specified, e.g., by a generative grammar. For context-free languages the problem is solved efficiently by various well-known parsing algorithms. However, there are several important languages that are not context-free. The membership problem of the context-sensitive language class is PSPACE-complete, thus, it is believed that it is generally not solvable in an efficient way. There are various language classes between the above mentioned two classes having membership problems with various complexity. One of these classes, the class of permutation languages, is generated by permutation grammars, i.e., context-free grammars extended with permutation rules, where a permutation rule allows to interchange the position of two consecutive nonterminals in the sentential form. In this paper, the membership problem for permutation languages is studied. A proof is presented to show that this problem is NP-complete.

POINTLIKE SETS, HYPERDECIDABILITY AND THE IDENTITY PROBLEM FOR FINITE SEMIGROUPS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 475-481 ◽  
Author(s):  
JOHN RHODE ◽  
BENJAMIN STEINBERG
Keyword(s):  
Membership Problem ◽  
Identity Problem ◽  
Nilpotent Semigroup ◽  
Finite Semigroups

In this paper, we give a relationship between the identity problem and the problem of deciding whether certain subsets of nilpotent semigroups are pointlike. We then use this to give an example of a pseudovariety which has a decidable membership problem, but for which one cannot decide pointlike sets. Then, by modifying the equations, we show that no graph is fundamentally hyperdecidable by constructing, for each graph, a labeling over a nilpotent semigroup for which we cannot decide inevitability with respect to the pseudovariety defined by these equations.

scholarly journals THE SEMIDIRECTLY CLOSED PSEUDOVARIETY GENERATED BY APERIODIC BRANDT SEMIGROUPS

2001 ◽  
Vol 11 (02) ◽  
pp. 247-267 ◽  
Author(s):  
M. LURDES TEIXEIRA
Keyword(s):  
Universal Algebra ◽  
Brandt Semigroup ◽  
Membership Problem ◽  
Finite Semigroups ◽  
Brandt Semigroups

This paper presents a study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B2, denoted V*(B2). We construct a basis of pseudoidentities for the semidirect powers of the pseudovariety generated by B2 which leads to the main result, which states that V*(B2) is decidable. Independently, using some suggestions given by J. Almeida in his book "Finite Semigroups and Universal Algebra", we constructed an algorithm to solve the membership problem in V* (B2).

scholarly journals COMPRESSED MEMBERSHIP PROBLEMS FOR REGULAR EXPRESSIONS AND HIERARCHICAL AUTOMATA

2010 ◽  
Vol 21 (05) ◽  
pp. 817-841 ◽  
Author(s):  
MARKUS LOHREY
Keyword(s):  
Open Problem ◽  
Regular Languages ◽  
Regular Expressions ◽  
Membership Problem ◽  
Complexity Bounds ◽  
Straight Line ◽  
Membership Problems ◽  
Context Free ◽  
Straight Line Programs ◽  
Extended Regular Expressions

Membership problems for compressed strings in regular languages are investigated. Strings are represented by straight-line programs, i.e., context-free grammars that generate exactly one string. For the representation of regular languages, various formalisms with different degrees of succinctness (e.g., suitably extended regular expressions, hierarchical automata) are considered. Precise complexity bounds are derived. Among other results, it is shown that the compressed membership problem for regular expressions with intersection is PSPACE-complete. This solves an open problem of Plandowski and Rytter.

scholarly journals COMPUTATIONAL COMPLEXITY OF GENERATORS AND NONGENERATORS IN ALGEBRA

2002 ◽  
Vol 12 (05) ◽  
pp. 719-735 ◽  
Author(s):  
CLIFFORD BERGMAN ◽  
GIORA SLUTZKI
Keyword(s):  
Computational Complexity ◽  
Algebraic Structure ◽  
Membership Problem ◽  
Generating Set ◽  
Np Complete ◽  
Membership Problems

We discuss the computational complexity of several problems concerning subsets of an algebraic structure that generate the structure. We show that the problem of determining whether a given subset X generates an algebra A is P-complete, while determining the size of the smallest generating set is NP-complete. We also consider several questions related to the Frattini subuniverse, Φ(A), of an algebra A. We show that the membership problem for Φ(A) is co-NP-complete, while the membership problems for Φ(Φ(A)), Φ(Φ(Φ(A))),… all lie in the class P‖(NP).

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