scholarly journals EQUATIONAL COMPLEXITY OF THE FINITE ALGEBRA MEMBERSHIP PROBLEM

2008 ◽  
Vol 18 (08) ◽  
pp. 1283-1319 ◽  
Author(s):  
GEORGE F. McNULTY ◽  
ZOLTÁN SZÉKELY ◽  
ROSS WILLARD
Keyword(s):  
Finite Automata ◽  
Finite Algebra ◽  
Equational Theory ◽  
Upper And Lower Bounds ◽  
Variety Of Algebras ◽  
Membership Problem ◽  
Finite Algebras ◽  
Shift Automorphism ◽  
Positive Integers ◽  
Nonfinitely Based

We associate to each variety of algebras of finite signature a function on the positive integers called the equational complexity of the variety. This function is a measure of how much of the equational theory of a variety must be tested to determine whether a finite algebra belongs to the variety. We provide general methods for giving upper and lower bounds on the growth of equational complexity functions and provide examples using algebras created from graphs and from finite automata. We also show that finite algebras which are inherently nonfinitely based via the shift automorphism method cannot be used to settle an old problem of Eilenberg and Schützenberger.

Finite axiomatizability for equational theories of computable groupoids

1989 ◽  
Vol 54 (3) ◽  
pp. 1018-1022 ◽  
Author(s):  
Peter Perkins
Keyword(s):  
Binary Operation ◽  
Finite Algebra ◽  
Equational Theory ◽  
Natural Numbers ◽  
Finite Axiomatizability ◽  
Finite Algebras ◽  
Equational Theories ◽  
Primitive Recursive ◽  
Positive Integers ◽  
Recursive Set

A computable groupoid is an algebra ‹N, g› where N is the set of natural numbers and g is a recursive (total) binary operation on N. A set L of natural numbers is a computable list of computable groupoids iff L is recursive, ‹N, ϕe› is a computable groupoid for each e ∈ L, and e ∈ L whenever e codes a primitive recursive description of a binary operation on N.Theorem 1. Let L be any computable list of computable groupoids. The set {e ∈ L: the equational theory of ‹N, ϕe› is finitely axiomatizable} is not recursive.Theorem 2. Let S be any recursive set of positive integers. A computable groupoid GS can be constructed so that S is inifinite iff GS has a finitely axiomatizable equational theory.The problem of deciding which finite algebras have finitely axiomatizable equational theories has remained open since it was first posed by Tarski in the early 1960's. Indeed, it is still not known whether the set of such finite algebras is recursively (or corecursively) enumerable. McKenzie [7] has shown that this question of finite axiomatizability for any (finite) algebra of finite type can be reduced to that for a (finite) groupoid.

scholarly journals Hardness results for the subpower membership problem

2018 ◽  
Vol 28 (05) ◽  
pp. 719-732
Author(s):  
Jeff Shriner
Keyword(s):  
Finite Algebra ◽  
Membership Problem ◽  
Finite Algebras ◽  
Maltsev Condition ◽  
Cube Term ◽  
Maltsev Conditions ◽  
Congruence Distributive

The main result of this paper shows that if [Formula: see text] is a consistent strong linear Maltsev condition which does not imply the existence of a cube term, then for any finite algebra [Formula: see text] there exists a new finite algebra [Formula: see text] which satisfies the Maltsev condition [Formula: see text], and whose subpower membership problem is at least as hard as the subpower membership problem for [Formula: see text]. We characterize consistent strong linear Maltsev conditions which do not imply the existence of a cube term, and show that there are finite algebras in varieties that are congruence distributive and congruence [Formula: see text]-permutable ([Formula: see text]) whose subpower membership problem is EXPTIME-complete.

COMPUTATIONAL COMPLEXITY OF THE FINITE ALGEBRA MEMBERSHIP PROBLEM FOR VARIETIES

2002 ◽  
Vol 12 (06) ◽  
pp. 811-823 ◽  
Author(s):  
ZOLTÁN SZÉKELY
Keyword(s):  
Finite Algebra ◽  
Order Theory ◽  
Finite Basis ◽  
Membership Problem ◽  
Graph Algebras ◽  
Finite Basis Problem ◽  
Finite Algebras ◽  
First Order Theory ◽  
Open Questions ◽  
Graph Algebra

We exhibit finite algebras each generating a variety with NP-complete finite algebra membership problem. The smallest of these algebras is the flat graph algebra belonging to the tetrahedral graph, a graph of 6 vertices obtained by cutting and spreading out the surface of a tetrahedron on the plane. The sequence of graphs we use to build up our flat graph algebras is similar to the sequence exhibited by Wheeler in [36] , 1979, to describe the first order theory of k-colorable graphs. Graph algebras were introduced by Shallon in [34] , 1979, and investigated, among others, by Baker, McNulty and Werner in [2] , 1987. Flat algebras were constructed and used by McKenzie in [27] , 1996, to settle some open questions related to decidability, like Tarski's Finite Basis Problem. Flat graph algebras were also discussed by Willard in [37] , 1996, and Delić in [8] , 1998.

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.

Diving into the queue

2018 ◽  
Vol 52 (2-3-4) ◽  
pp. 89-110
Author(s):  
Simon Beier ◽  
Martin Kutrib ◽  
Andreas Malcher ◽  
Matthias Wendlandt
Keyword(s):  
Finite Number ◽  
Finite Automata ◽  
Time Constraints ◽  
Membership Problem ◽  
Storage Medium ◽  
Logarithmic Space ◽  
Better Than

We introduce and study the model of diving queue automata which are basically finite automata equipped with a storage medium that is organized as a queue. Additionally, two queue heads are provided at both ends of the queue that can move in a read-only mode inside the queue. In particular, we consider suitable time constraints and the case where only a finite number of turns on the queue is allowed. As one main result we obtain a proper queue head hierarchy, that is, two heads are better than one head, and one head is better than no head. Moreover, it is shown that the model with one queue head, finitely many turns, and no time constraints as well as the model with two queue heads, possibly infinitely many turns, and time constraints is captured by P and has a P-complete membership problem. We obtain also that a subclass of the model with two queue heads is already captured by logarithmic space. Finally, we consider decidability questions and it turns out that almost nothing is decidable for the model with two queue heads, whereas we obtain that at least emptiness and finiteness are decidable for subclasses of the model with one queue head.

LIMITED AUTOMATA AND REGULAR LANGUAGES

2014 ◽  
Vol 25 (07) ◽  
pp. 897-916 ◽  
Author(s):  
GIOVANNI PIGHIZZINI ◽  
ANDREA PISONI
Keyword(s):  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Finite State Automata ◽  
Turing Machines ◽  
Double Exponential ◽  
Finite State ◽  
A Cell ◽  
Fixed Constant ◽  
The One ◽  
Context Free

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. In the case d = 1, namely, when a rewriting is possible only during the first visit to a cell, these models have the same power of finite state automata. We prove state upper and lower bounds for the conversion of 1-limited automata into finite state automata. In particular, we prove a double exponential state gap between nondeterministic 1-limited automata and one-way deterministic finite automata. The gap reduces to a single exponential in the case of deterministic 1-limited automata. This also implies an exponential state gap between nondeterministic and deterministic 1-limited automata. Another consequence is that 1-limited automata can have less states than equivalent two-way nondeterministic finite automata. We show that this is true even if we restrict to the case of the one-letter input alphabet. For each d ≥ 2, d-limited automata are known to characterize the class of context-free languages. Using the Chomsky-Schützenberger representation for contextfree languages, we present a new conversion from context-free languages into 2-limited automata.

SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

2014 ◽  
Vol 25 (07) ◽  
pp. 877-896 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER ◽  
MATTHIAS WENDLANDT
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Unary Languages ◽  
Order Of Magnitude ◽  
Simulation Results ◽  
Special Case ◽  
Nondeterministic Finite Automaton

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

scholarly journals On numbers n dividing the nth term of a linear recurrence

2012 ◽  
Vol 55 (2) ◽  
pp. 271-289 ◽  
Author(s):  
Juan José Alba González ◽  
Florian Luca ◽  
Carl Pomerance ◽  
Igor E. Shparlinski
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Linear Recurrence ◽  
Recurrent Sequence ◽  
Positive Integers

AbstractWe give upper and lower bounds on the count of positive integers n ≤ x dividing the nth term of a non-degenerate linearly recurrent sequence with simple roots.

Upper and Lower Bounds for a Function Related to Brown's Lemma

Integers ◽  
2010 ◽  
Vol 10 (6) ◽  
Author(s):  
Hayri Ardal
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fixed Positive Integer ◽  
Positive Integers

AbstractThe well-known Brown's lemma says that for every finite coloring of the positive integers, there exist a fixed positive integer

Sign in / Sign up

Export Citation Format

Share Document