scholarly journals Debates with Small Transparent Quantum Verifiers

2016 ◽  
Vol 27 (02) ◽  
pp. 283-300 ◽  
Author(s):  
Abuzer Yakaryilmaz ◽  
A. C. Cem Say ◽  
H. Gökalp Demirci
Keyword(s):  
Polynomial Time ◽  
Regular Languages ◽  
Quantum Model ◽  
Correct Decision ◽  
Classical Counterpart ◽  
Context Free ◽  
Classical Constant

We study a model where two opposing provers debate over the membership status of a given string in a language, trying to convince a weak verifier whose coins are visible to all. We show that the incorporation of just two qubits to an otherwise classical constant-space verifier raises the class of debatable languages from at most NP to the collection of all Turing-decidable languages (recursive languages). When the verifier is further constrained to make the correct decision with probability 1, the corresponding class goes up from the regular languages up to at least E. We also show that the quantum model outperforms its classical counterpart when restricted to run in polynomial time, and demonstrate some non-context-free languages which have such short debates with quantum verifiers.

scholarly journals BOND-FREE LANGUAGES: FORMALIZATIONS, MAXIMALITY AND CONSTRUCTION METHODS

2005 ◽  
Vol 16 (05) ◽  
pp. 1039-1070 ◽  
Author(s):  
LILA KARI ◽  
STAVROS KONSTANTINIDIS ◽  
PETR SOSÍK
Keyword(s):  
Polynomial Time ◽  
Structural Characterization ◽  
Regular Languages ◽  
Important Case ◽  
Similarity Relation ◽  
Construction Methods ◽  
Large Sets ◽  
Code Property ◽  
Context Free

The problem of negative design of DNA languages is addressed, that is, properties and construction methods of large sets of words that prevent undesired bonds when used in DNA computations. We recall a few existing formalizations of the problem and then define the property of sim-bond-freedom, where sim is a similarity relation between words. We show that this property is decidable for context-free languages and polynomial-time decidable for regular languages. The maximality of this property also turns out to be decidable for regular languages and polynomial-time decidable for an important case of the Hamming similarity. Then we consider various construction methods for Hamming bond-free languages, including the recently introduced method of templates, and obtain a complete structural characterization of all maximal Hamming bond-free languages. This result is applicable to the θ-k-code property introduced by Jonoska and Mahalingam.

OVERLAP-FREE LANGUAGES AND SOLID CODES

2011 ◽  
Vol 22 (05) ◽  
pp. 1197-1209 ◽  
Author(s):  
YO-SUB HAN ◽  
KAI SALOMAA
Keyword(s):  
Polynomial Time ◽  
Data Transmission ◽  
Efficient Algorithm ◽  
Formal Language ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Regular Languages ◽  
Nice Property ◽  
Decomposition Problem ◽  
Context Free

Solid codes have a nice property called synchronization property, which is useful in data transmission. The property is derived from infix-freeness and overlap-freeness of solid codes. Since a code is a language, we look at solid codes from formal language viewpoint. In particular, we study regular solid codes (that are solid codes and regular). We first tackle the solid code decidability problem for regular languages and propose a polynomial time algorithm. We, then, investigate the decidability of the overlap-freeness property and show that it is decidable for regular languages but is undecidable for context-free languages. Then, we study the prime solid code decomposition of regular solid codes and propose an efficient algorithm for the prime solid code decomposition problem. We also demonstrate that a solid code does not always have a unique prime solid code decomposition.

RESTRICTED SETS OF TRAJECTORIES AND DECIDABILITY OF SHUFFLE DECOMPOSITIONS

2005 ◽  
Vol 16 (05) ◽  
pp. 897-912 ◽  
Author(s):  
MICHAEL DOMARATZKI ◽  
KAI SALOMAA
Keyword(s):  
Regular Languages ◽  
Decomposition Problem ◽  
Open Question ◽  
Free Set ◽  
Context Free ◽  
Undecidability Result

The decidability of the shuffle decomposition problem for regular languages is a long standing open question. We consider decompositions of regular languages with respect to shuffle along a regular set of trajectories and obtain positive decidability results for restricted classes of trajectories. Also we consider decompositions of unary regular languages. Finally, we establish in the spirit of the Dassow-Hinz undecidability result an undecidability result for regular languages shuffled along a fixed linear context-free set of trajectories.

scholarly journals IT IS NL-COMPLETE TO DECIDE WHETHER A HAIRPIN COMPLETION OF REGULAR LANGUAGES IS REGULAR

2011 ◽  
Vol 22 (08) ◽  
pp. 1813-1828 ◽  
Author(s):  
VOLKER DIEKERT ◽  
STEFFEN KOPECKI
Keyword(s):  
Polynomial Time ◽  
Dna Computing ◽  
Regular Language ◽  
Decision Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complexity Bound ◽  
The One ◽  
Context Free ◽  
Hairpin Formation

The hairpin completion is an operation on formal languages which is inspired by the hairpin formation in biochemistry. Hairpin formations occur naturally within DNA-computing. It has been known that the hairpin completion of a regular language is linear context-free, but not regular, in general. However, for some time it is was open whether the regularity of the hairpin completion of a regular language is decidable. In 2009 this decidability problem has been solved positively in [5] by providing a polynomial time algorithm. In this paper we improve the complexity bound by showing that the decision problem is actually NL-complete. This complexity bound holds for both, the one-sided and the two-sided hairpin completions.

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.

Regular patterns, regular languages and context-free languages

2010 ◽  
Vol 110 (24) ◽  
pp. 1114-1119 ◽  
Author(s):  
Sanjay Jain ◽  
Yuh Shin Ong ◽  
Frank Stephan
Keyword(s):  
Regular Languages ◽  
Context Free ◽  
Regular Patterns

scholarly journals QUANTUM COUNTER AUTOMATA

2012 ◽  
Vol 23 (05) ◽  
pp. 1099-1116 ◽  
Author(s):  
A. C. CEM SAY ◽  
ABUZER YAKARYILMAZ
Keyword(s):  
Real Time ◽  
Finite Automata ◽  
Affirmative Answer ◽  
Quantum Model ◽  
Open Problems ◽  
Modern Approach ◽  
Context Free Language ◽  
Definition Of ◽  
Context Free ◽  
Free Language

The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.

scholarly journals REPRESENTATIONS AND CHARACTERIZATIONS OF LANGUAGES IN CHOMSKY HIERARCHY BY MEANS OF INSERTION-DELETION SYSTEMS

2008 ◽  
Vol 19 (04) ◽  
pp. 859-871 ◽  
Author(s):  
GHEORGHE PĂUN ◽  
MARIO J. PÉREZ-JIMÉNEZ ◽  
TAKASHI YOKOMORI
Keyword(s):  
Dna Computing ◽  
Research Direction ◽  
Regular Languages ◽  
Turing Computability ◽  
Chomsky Hierarchy ◽  
Recursively Enumerable ◽  
Enumerable Language ◽  
Language L ◽  
Context Free

Insertion-deletion operations are much investigated in linguistics and in DNA computing and several characterizations of Turing computability and characterizations or representations of languages in Chomsky hierarchy were obtained in this framework. In this note we contribute to this research direction with a new characterization of this type, as well as with representations of regular and context-free languages, mainly starting from context-free insertion systems of as small as possible complexity. For instance, each recursively enumerable language L can be represented in a way similar to the celebrated Chomsky-Schützenberger representation of context-free languages, i.e., in the form L = h(L(γ) ∩ D), where γ is an insertion system of weight (3, 0) (at most three symbols are inserted in a context of length zero), h is a projection, and D is a Dyck language. A similar representation can be obtained for regular languages, involving insertion systems of weight (2,0) and star languages, as well as for context-free languages – this time using insertion systems of weight (3, 0) and star languages.

Sign in / Sign up

Export Citation Format

Share Document