A Tale of Conjunctive Grammars

Okhotin, Alexander

doi:10.1007/978-3-319-98654-8_4

Cited by 15 publications

(7 citation statements)

References 62 publications

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“…For instance, describing sequences of declarations and calls, with the declaration before use requirement, is much easier than with conjunctive grammars [2]. Also grammars with left context operators can describe several interesting abstract languages, such as { ww | w ∈ {a, b} * } [16] and { a n 2 | n 0 } [3].…”

Section: Introductionmentioning

confidence: 99%

The hardest language for grammars with context operators

Mrykhin¹,

Okhotin²

2020

Preprint

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In 1973, Greibach ("The hardest context-free language", SIAM J. Comp., 1973) constructed a context-free language L 0 with the property that every context-free language can be reduced to L 0 by a homomorphism, thus representing it as an inverse homomorphic image h −1 (L 0 ). In this paper, a similar characterization is established for a family of grammars equipped with operators for referring to the left context of any substring, recently defined by Barash and Okhotin ("An extension of context-free grammars with one-sided context specifications", Inform. Comput., 2014). An essential step of the argument is a new normal form for grammars with context operators, in which every nonterminal symbol defines only strings of odd length in left contexts of even length: the even-odd normal form.

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Section: Introductionmentioning

confidence: 99%

The hardest language for grammars with context operators

Mrykhin¹,

Okhotin²

2020

Preprint

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“…On the other hand, by the assumption of the Lemma j∈I S j = C I , hence there exists i = i such that n ∈ S i . But by (12): n / ∈ S i , as S i ∩ S i = ∅. Hence n is a missing number for i , a contradiction, as we were supposed to choose a missing number if there was any.…”

Section: Lemmamentioning

confidence: 97%

“…Equations with both union and intersection are equivalent to an extension of context-free grammars, the conjunctive grammars [12], and the question whether any non-periodic set can be specified by such a system of equations has been open for some years, until answered by the following example:…”

Section: Resolved Systems With {∪ ∩ +}mentioning

confidence: 99%

On the Computational Completeness of Equations over Sets of Natural Numbers

Jeż

Okhotin

Automata, Languages and Programming

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Systems of equations of the form ϕ j (X 1 , . . . , X n ) = ψ j (X 1 , . . . , X n ) with 1 j m are considered, in which the unknowns X i are sets of natural numbers, while the expressions ϕ j , ψ j may contain singleton constants and the operations of union and pairwise addition S + T = { m + n | m ∈ S, n ∈ T }. It is shown that the family of sets representable by unique (least, greatest) solutions of such systems is exactly the family of recursive (r.e., co-r.e., respectively) sets of numbers. Basic decision problems for these systems are located in the arithmetical hierarchy. The same results are established for equations with addition and intersection.

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“…Note that if m = 1 and n = 0 in every such rule, then a context-free grammar is obtained. An intermediate family of conjunctive grammars [4] has m 1 and n = 0 in every rule. Linear subclasses of Boolean, conjunctive and context-free grammars are defined by the additional requirement that…”

Section: Boolean Grammarsmentioning

confidence: 99%