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| Conjunctive grammars are a class of formal grammars
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| studied in [[formal language]] theory.
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| They extend the basic type of grammars,
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| the [[context-free grammars]],
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| with a [[Logical_conjunction|conjunction]] operation.
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| Besides explicit conjunction,
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| conjunctive grammars allow implicit [[Logical_disjunction|disjunction]]
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| represented by multiple rules for a single nonterminal symbol,
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| which is the only logical connective expressible in context-free grammars.
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| Conjunction can be used, in particular,
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| to specify intersection of languages. | |
| A further extension of conjunctive grammars
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| known as [[Boolean grammar]]s
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| additionally allows explicit [[negation]].
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| The rules of a conjunctive grammar are of the form
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| :<math>A \to \alpha_1 \And \ldots \And \alpha_m</math>
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| where <math>A</math> is a nonterminal and
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| <math>\alpha_1</math>, ..., <math>\alpha_m</math>
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| are strings formed of symbols in <math>\Sigma</math> and <math>N</math> (finite sets of terminal and nonterminal symbols respectively).
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| Informally, such a rule asserts that
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| every string <math>w</math> over <math>\Sigma</math> | |
| that satisfies each of the syntactical conditions represented
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| by <math>\alpha_1</math>, ..., <math>\alpha_m</math>
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| therefore satisfies the condition defined by <math>A</math>.
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| Two equivalent formal definitions
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| of the language specified by a conjunctive grammar exist.
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| One definition is based upon representing the grammar
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| as a system of [[language equation]]s with union, intersection and concatenation
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| and considering its least solution.
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| The other definition generalizes
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| [[Noam Chomsky|Chomsky's]] generative definition of the context-free grammars
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| using rewriting of terms over conjunction and concatenation.
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| Though the expressive means of conjunctive grammars
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| are greater than those of context-free grammars,
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| conjunctive grammars retain some practically useful properties of the latter.
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| Most importantly, there are generalizations of the main context-free parsing algorithms,
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| including the linear-time [[Recursive descent parser|recursive descent]],
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| the cubic-time [[GLR parser|generalized LR]],
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| the cubic-time [[CYK algorithm|Cocke-Kasami-Younger]],
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| as well as [[Leslie Valiant|Valiant's]] algorithm running as fast as matrix multiplication.
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| A number of theoretical properties of conjunctive grammars have been researched,
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| including the expressive power of grammars over a one-letter alphabet
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| and numerous [[Undecidable problem|undecidable decision problems]].
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| This work provided a basis
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| for the study [[language equation]]s of a more general form.
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| ==References ==
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| * Alexander Okhotin, ''Conjunctive grammars.'' [[Journal of Automata, Languages and Combinatorics]], 6:4 (2001), 519-535. [http://users.utu.fi/aleokh/papers/conjunctive.pdf (pdf)]
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| * Alexander Okhotin, ''An overview of conjunctive grammars.'' In: Gheorghe Paun, Grzegorz Rozenberg, Arto Salomaa (Eds.), Current Trends in Theoretical Computer Science: The Challenge of the New Century, Vol. 2, World Scientific, 2004, 545--566. [http://users.utu.fi/aleokh/papers/conjunctive_overview_ct.pdf (pdf)]
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| * Artur Jeż. ''Conjunctive grammars can generate non-regular unary languages.'' International Journal of Foundations of Computer Science 19(3): 597-615 (2008) [http://www.ii.uni.wroc.pl/cms/files/TR012007.pdf Technical report version (pdf)]
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| ==External links==
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| *Artur Jeż. [http://www.ii.uni.wroc.pl/~aje/presentation/PresentationDLT.pdf Conjunctive grammars can generate non-regular unary languages]. Slides of talk held at the [[International Conference on Developments in Language Theory]] 2007.
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| *[http://users.utu.fi/aleokh/conjunctive/ Alexander Okhotin's page on conjunctive grammars].
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| *Alexander Okhotin. [http://www.tucs.fi/publications/insight.php?id=tOkhotin06a Nine open problems for conjunctive and Boolean grammars].
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| [[Category:Formal languages]]
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