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

2009
*
Lecture Notes in Computer Science
*

This implies undecidability of some properties of

doi:10.1007/978-3-642-03351-3_19
fatcat:ha323eea4rb2xm2ggirs5fvlvu
*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 ...##
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Parsing Boolean grammars over a one-letter alphabet using online convolution

2012
*
Theoretical Computer Science
*

Consider context-free

doi:10.1016/j.tcs.2012.06.032
fatcat:2dbfaxle2jfq7ic4qazu6ouxgq
*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. ...##
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Unambiguous Conjunctive Grammars over a One-Letter Alphabet
[chapter]

2013
*
Lecture Notes in Computer Science
*

It is demonstrated that unambiguous

doi:10.1007/978-3-642-38771-5_25
fatcat:kjmtjzyb25ekdjcf27me7pvmgu
*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. ...##
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Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth

2008
*
Theory of Computing Systems
*

It has recently been proved (Jeż, DLT 2007) that

doi:10.1007/s00224-008-9139-5
fatcat:q4r5wpxdljfl7bdpp7kyof4goi
*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

2003
*
Theoretical Computer Science
*

Linear

doi:10.1016/s0304-3975(02)00543-1
fatcat:j52rv3xahrg4rpqrs3xl3amwri
*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. ...##
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Conjunctive and Boolean grammars: The true general case of the context-free grammars

2013
*
Computer Science Review
*

This paper surveys the results

doi:10.1016/j.cosrev.2013.06.001
fatcat:vjrnokycqres5mfpdupksfaumm
*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. ...##
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Conjunctive grammars with restricted disjunction

2010
*
Theoretical Computer Science
*

The first author also wishes to thank Christos Nomikos and Panos Rondogiannis for discussions about the Greibach normal form for

doi:10.1016/j.tcs.2010.03.015
fatcat:k3wwhmpevveapjoe2uigrqxp2u
*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. ...##
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Linear grammars with one-sided contexts and their automaton representation

2015
*
RAIRO - Theoretical Informatics and Applications
*

The main result is the undecidability of the emptiness problem for

doi:10.1051/ita/2015004
fatcat:nhymxnzwq5bvzmx5jemgl2zwp4
*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. ...##
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Linear Grammars with One-Sided Contexts and Their Automaton Representation
[chapter]

2014
*
Lecture Notes in Computer Science
*

The main result is the undecidability of the emptiness problem for

doi:10.1007/978-3-642-54423-1_17
fatcat:e63ob2llvrdh5dx5uf7yhenpyu
*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. ...##
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On the expressive power of univariate equations over sets of natural numbers

2012
*
Information and Computation
*

The results have direct implications

doi:10.1016/j.ic.2012.01.004
fatcat:ml3hmrkkivhqjmqgqy2doprdlm
*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. ...##
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AN OVERVIEW OF CONJUNCTIVE GRAMMARS
[chapter]

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. ...

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

2008
*
Symposium on Theoretical Aspects of Computer Science
*

Systems of equations

doi:10.4230/lipics.stacs.2008.1319
dblp:conf/stacs/OkhotinJ08
fatcat:5jztqkryo5fgnnund22fva2jni
*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. ...##
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Expressive power of LL(k) Boolean grammars

2011
*
Theoretical Computer Science
*

It is demonstrated that

doi:10.1016/j.tcs.2011.05.013
fatcat:mw3g3rip4zferecbwacbfrijou
*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

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 ...

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

2011
*
Theoretical Computer Science
*

The known linear

doi:10.1016/j.tcs.2010.09.015
fatcat:sbpjiaqdwrf2bkkvaac7o44ivu
*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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