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### One-Nonterminal Conjunctive Grammars over a Unary Alphabet [chapter]

Artur Jeż, Alexander Okhotin
2009 Lecture Notes in Computer Science
This implies undecidability of some properties of one-nonterminal conjunctive grammars over a unary alphabet.  ...  of equations Xi = ϕi(X1, . . . , Xn) with 1 i n, in the sense that the solution of a system is encoded in the solution of an equation.  ...  There exists an algorithm, which, given a one-nonterminal conjunctive grammar G = ({a}, {S}, P, S) over a unary alphabet and a finite automaton M over an alphabet Σ k = {0, 1, . . . , k − 1}, determines  ...

### Parsing Boolean grammars over a one-letter alphabet using online convolution

Alexander Okhotin, Christian Reitwießner
2012 Theoretical Computer Science
Consider context-free grammars generating strings over a one-letter alphabet.  ...  For the membership problem for such grammars, stated as "Given a grammar G and a string a n , determine whether a n is generated by G", only a naïve O(|G| · n 2 )-time algorithm is known.  ...  Subsequent research on conjunctive grammars over a unary alphabet revealed their nontrivial properties, such as the following.  ...

### Unambiguous Conjunctive Grammars over a One-Letter Alphabet [chapter]

Artur Jeż, Alexander Okhotin
2013 Lecture Notes in Computer Science
It is demonstrated that unambiguous conjunctive grammars over a unary alphabet Σ = {a} have non-trivial expressive power, and that their basic properties are undecidable.  ...  The key result is that for every base k 11 and for every one-way real-time cellular automaton operating over the alphabet of base-k digits k+9 4 , . . . , k+1 2 , the language of all strings a n with the  ...  The following grammar becomes the first evidence of non-triviality of unambiguous conjunctive grammars over a unary alphabet. Example 2.  ...

### Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth

Artur Jeż, Alexander Okhotin
2008 Theory of Computing Systems
It has recently been proved (Jeż, DLT 2007) that conjunctive grammars (that is, context-free grammars augmented by conjunction) generate some nonregular languages over a one-letter alphabet.  ...  The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as nonexistence of an r.e. bound on the growth rate of generated languages.  ...  For conjunctive grammars over a unary alphabet there exist no algorithm to decide whether a given grammar generates a finite language (a regular language). Proof (a sketch).  ...

### On the closure properties of linear conjunctive languages

Alexander Okhotin
2003 Theoretical Computer Science
Linear conjunctive grammars are conjunctive grammars in which the body of each conjunct contains no more than a single nonterminal symbol.  ...  They can at the same time be thought of as a special case of conjunctive grammars and as a generalization of linear context-free grammars that provides an explicit intersection operation.  ...  Acknowledgements I am grateful to Kai Salomaa for his valuable advices on di erent versions of this paper. I would also like to thank my anonymous referee for important comments and suggestions.  ...

### Conjunctive and Boolean grammars: The true general case of the context-free grammars

Alexander Okhotin
2013 Computer Science Review
This paper surveys the results on conjunctive and Boolean grammars obtained over the last decade, comparing them to the corresponding results for ordinary context-free grammars and their main subfamilies  ...  Conjunctive grammars extend the definition of a context-free grammar by allowing a conjunction operation in the rules; Boolean grammars are further equipped with an explicit negation.  ...  Theoretical topics Grammars over a one-symbol alphabet Conjunctive grammars over a unary alphabet form a special area of study.  ...

### Conjunctive grammars with restricted disjunction

Alexander Okhotin, Christian Reitwießner
2010 Theoretical Computer Science
The first author also wishes to thank Christos Nomikos and Panos Rondogiannis for discussions about the Greibach normal form for conjunctive grammars.  ...  Acknowledgements The authors are grateful to the anonymous referees for careful reading, and in particular for noticing a flaw in the earlier proof of Theorem 4.  ...  First of all, this stronger restriction on conjunctive grammars still gives a non-trivial family. For instance, the important grammar over a unary alphabet given in Example 3 is of this form.  ...

### Linear grammars with one-sided contexts and their automaton representation

Mikhail Barash, Alexander Okhotin
2015 RAIRO - Theoretical Informatics and Applications
The main result is the undecidability of the emptiness problem for grammars restricted to a one-symbol alphabet, which is proved by simulating a Turing machine by a cellular automaton with feedback.  ...  The paper considers a family of formal grammars that extends linear context-free grammars with an operator for referring to the left context of a substring being defined, as well as with a conjunction  ...  Defining a non-regular unary language Ordinary context-free grammars over a unary alphabet Σ = {a} define only regular languages.  ...

### Linear Grammars with One-Sided Contexts and Their Automaton Representation [chapter]

Mikhail Barash, Alexander Okhotin
2014 Lecture Notes in Computer Science
The main result is the undecidability of the emptiness problem for grammars restricted to a one-symbol alphabet, which is proved by simulating a Turing machine by a cellular automaton with feedback.  ...  The paper considers a family of formal grammars that extends linear context-free grammars with an operator for referring to the left context of a substring being defined, as well as with a conjunction  ...  Defining a non-regular unary language Ordinary context-free grammars over a unary alphabet Σ = {a} define only regular languages.  ...

### On the expressive power of univariate equations over sets of natural numbers

Alexander Okhotin, Panos Rondogiannis
2012 Information and Computation
The results have direct implications on the power of unary conjunctive grammars with one nonterminal symbol.  ...  Equations of the form X = ϕ(X) are considered, where the unknown X is a set of natural numbers.  ...  Acknowledgments We are grateful to an anonymous referee for suggesting to improve the equation in Lemma 2, which resulted in a shorter and clearer proof.  ...

### AN OVERVIEW OF CONJUNCTIVE GRAMMARS [chapter]

ALEXANDER OKHOTIN
2004 Current Trends in Theoretical Computer Science
Conjunctive grammars were introduced in 2000 as a generalization of context-free grammars that allows the use of an explicit intersection operation in rules.  ...  Several theoretical results on their properties have been obtained since then, and a number of efficient parsing algorithms that justify the practical value of the concept have been developed.  ...  In [20] , every linear conjunctive language over unary alphabet was shown to be regular.  ...

### Complexity of solutions of equations over sets of natural numbers

Alexander Okhotin, Artur Jez, Marc Herbstritt
2008 Symposium on Theoretical Aspects of Computer Science
Systems of equations over sets of natural numbers (or, equivalently, language equations over a one-letter alphabet) of the form Xi = ϕi(X1, . . . , Xn) (1 i n) are considered.  ...  Expressions ϕi may contain the operations of union, intersection and pairwise sum A+B = {x+y | x ∈ A, y ∈ B}.  ...  The algorithm begins with representing the given system as a conjunctive grammar over a unary alphabet, with a linearly bounded blow-up.  ...

### Expressive power of LL(k) Boolean grammars

Alexander Okhotin
2011 Theoretical Computer Science
It is demonstrated that over a one-letter alphabet, these languages are always regular, while Boolean LL subsets of Σ * a * obey a certain periodicity property, which, in particular, makes the language  ...  It is also shown that linear conjunctive LL grammars cannot generate any language of the form L · {a, b}, with L non-regular, and that no languages of the form L · c * , with non-regular L, can be generated  ...  Acknowledgements The author is grateful to the anonymous reviewers for careful reading and numerous valuable comments, and particularly for noticing a serious error in the earlier ad hoc proof of Example  ...

### Unambiguous Boolean grammars

Alexander Okhotin
2008 Information and Computation
One could conjecture that unambiguous Boolean grammars over a unary alphabet generate only regular languages.  ...  An important contribution to the study of conjunctive grammars has recently been made by Jeż [10] , who constructed a conjunctive grammar over a unary alphabet generating the nonregular language {a 4  ...

### A simple P-complete problem and its language-theoretic representations

Alexander Okhotin
2011 Theoretical Computer Science
The known linear conjunctive grammar for a P -complete problem simulates a 45-state trellis automaton over a 9-letter alphabet [18] , and thus contains as many as 164035 = 45 2 · 9 2 + 9 + 1 rules according  ...  The problem, which remains P-complete, is encoded as a simple formal language over a two-letter alphabet, which can be succinctly represented by language equations of several types.  ...  There exists a 20-rule linear conjunctive grammar generating a P -complete language over a 2-letter alphabet.  ...
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