Forgetting Automata and Unary Languages

2006 ◽  
pp. 186-197
Author(s):  
Jens Glöckler
Keyword(s):  
Unary Languages

CONJUNCTIVE GRAMMARS GENERATE NON-REGULAR UNARY LANGUAGES

2008 ◽  
Vol 19 (03) ◽  
pp. 597-615 ◽  
Author(s):  
ARTUR JEŻ
Keyword(s):  
Negative Answer ◽  
The Body ◽  
Regular Languages ◽  
Natural Numbers ◽  
Open Problems ◽  
Conjunctive Grammars ◽  
Unary Languages ◽  
Context Free ◽  
Conjunctive Grammar ◽  
Context Free Grammars

Conjunctive grammars, introduced by Okhotin, extend context-free grammars by an additional operation of intersection in the body of any production of the grammar. Several theorems and algorithms for context-free grammars generalize to the conjunctive case. Okhotin posed nine open problems concerning those grammars. One of them was a question, whether a conjunctive grammars over a unary alphabet generate only regular languages. We give a negative answer, contrary to the conjectured positive one, by constructing a conjunctive grammar for the language {a4n : n ∈ ℕ}. We also generalize this result: for every set of natural numbers L we show that {an : n ∈ L} is a conjunctive unary language, whenever the set of representations in base-k system of elements of L is regular, for arbitrary k.

Expressive capacity of subregular expressions

2018 ◽  
Vol 52 (2-3-4) ◽  
pp. 201-218 ◽  
Author(s):  
Martin Kutrib ◽  
Matthias Wendlandt
Keyword(s):  
Regular Languages ◽  
Regular Expressions ◽  
Different Types ◽  
Finite Set ◽  
Unary Languages

Different types of subregular expressions are studied. Each type is obtained by either omitting one of the regular operations or replacing it by complementation or intersection. For uniformity and in order to allow non-trivial languages to be expressed, the set of literals is a finite set of words instead of letters. The power and limitations as well as relations with each other are considered, which is often done in terms of unary languages. Characterizations of some of the language families are obtained. A finite hierarchy is shown that reveals that the operation complementation is generally stronger than intersection. Furthermore, we investigate the closures of language families described by regular expressions with omitted operation under that operation. While it is known that in case of union this closure captures all regular languages, for the cases of concatenation and star incomparability results are obtained with the corresponding language families where the operation is replaced by complementation.

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 DETERMINISTIC PUSHDOWN AUTOMATA AND UNARY LANGUAGES

2009 ◽  
Vol 20 (04) ◽  
pp. 629-645 ◽  
Author(s):  
GIOVANNI PIGHIZZINI
Keyword(s):  
Finite Automata ◽  
Point Of View ◽  
Deterministic Finite Automaton ◽  
Pushdown Automaton ◽  
Nondeterministic Automaton ◽  
Letter Alphabet ◽  
Finite State ◽  
Unary Languages ◽  
Context Free ◽  
Pushdown Automata

The simulation of deterministic pushdown automata defined over a one-letter alphabet by finite state automata is investigated from a descriptional complexity point of view. We show that each unary deterministic pushdown automaton of size s can be simulated by a deterministic finite automaton with a number of states that is exponential in s. We prove that this simulation is tight. Furthermore, its cost cannot be reduced even if it is performed by a two-way nondeterministic automaton. We also prove that there are unary languages for which deterministic pushdown automata cannot be exponentially more succinct than finite automata. In order to state this result, we investigate the conversion of deterministic pushdown automata into context-free grammars. We prove that in the unary case the number of variables in the resulting grammar is strictly smaller than the number of variables needed in the case of nonunary alphabets.

scholarly journals State Complexity of Overlap Assembly

2020 ◽  
pp. 1-20
Author(s):  
Janusz A. Brzozowski ◽  
Lila Kari ◽  
Bai Li ◽  
Marek Szykuła
Keyword(s):  
Self Assembly ◽  
Binary Operation ◽  
The State ◽  
Regular Languages ◽  
Deterministic Finite Automaton ◽  
State Complexity ◽  
Dna Strands ◽  
Unary Languages ◽  
Number Formula ◽  
Dna Strand

The state complexity of a regular language [Formula: see text] is the number [Formula: see text] of states in a minimal deterministic finite automaton (DFA) accepting [Formula: see text]. The state complexity of a regularity-preserving binary operation on regular languages is defined as the maximal state complexity of the result of the operation where the two operands range over all languages of state complexities [Formula: see text] and [Formula: see text], respectively. We determine, for [Formula: see text], [Formula: see text], the exact value of the state complexity of the binary operation overlap assembly on regular languages. This operation was introduced by Csuhaj-Varjú, Petre, and Vaszil to model the process of self-assembly of two linear DNA strands into a longer DNA strand, provided that their ends “overlap”. We prove that the state complexity of the overlap assembly of languages [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text], is at most [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text] there exist languages [Formula: see text] and [Formula: see text] over an alphabet of size [Formula: see text] whose overlap assembly meets the upper bound and this bound cannot be met with smaller alphabets. Finally, we prove that [Formula: see text] is the state complexity of the overlap assembly in the case of unary languages and that there are binary languages whose overlap assembly has exponential state complexity at least [Formula: see text].

scholarly journals Uncountable Realtime Probabilistic Classes

2019 ◽  
Vol 30 (08) ◽  
pp. 1317-1333
Author(s):  
Maksims Dimitrijevs ◽  
Abuzer Yakaryılmaz
Keyword(s):  
Linear Space ◽  
Logarithmic Space ◽  
Unary Languages ◽  
Bounded Error

We investigate the minimal cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On non-unary case, we obtain the same result for double logarithmic space, which is also tight. When replacing the work tape with a few counters, we can still achieve similar results for unary linear-space two-counter automata, unary sublinear-space three-counter automata, and non-unary sublinear-space two-counter automata. We also show how to slightly improve the sublinear-space constructions by using more counters.

Sign in / Sign up

Export Citation Format

Share Document