An infinite hierarchy of intersections of context-free languages

Liu, Leonard Y.; Weiner, Peter

doi:10.1007/bf01762237

Cited by 52 publications

(29 citation statements)

References 4 publications

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“…Let 𝒞ℱℒ 0 =𝒟𝒞ℱℒ 0 be the class of regular languages. Liu and Weiner (1973), (see also Kintala 1978) proved that for each d > 0, the language

\begin{matrix} left \\ L_{d} = false{a_{1}^{k_{1}} a_{2}^{k_{2}} \dots a_{d}^{k_{d}} b_{1}^{k_{1}} b_{2}^{k_{2}} \dots b_{d}^{k_{d}} : k_{1}, k_{2}, \dots, k_{d} \geq 0 false} \\ \subseteq {a_{1}, a_{2}, \dots, a_{d}, b_{1}, b_{2}, \dots, b_{d}}^{*} \end{matrix}

satisfies L d +1 ∉ 𝒞ℱℒ d . But as

L_{d} = true \cap_{i = 1}^{d} L_{d, i}

where

L_{d, i} = {a_{1}^{k_{1}} a_{2}^{k_{2}} \dots a_{d}^{k_{d}} b_{1}^{l_{1}} b_{2}^{l_{2}} \dots b_{d}^{l_{d}} : k_{1}, k_{2}, \dots, k_{d}, l_{1}, l_{2}, \dots, l_{d} \geq 0 & k_{i} = l_{i}},

and L d , i is a DCFL for each i , we have L d +1 ∈ 𝒟𝒞ℱℒ d +1 − 𝒞ℱℒ d for each d .…”

Section: Counter Machinesmentioning

confidence: 97%

“…It is known that context-free languages (CFLs) lie at the base of an intersection hierarchy of languages (see Liu and Weiner 1973 or Kintala 1978). If we let 𝒞ℱℒ n (𝒟𝒞ℱℒ n ) be the class of languages that are intersections of n CFLs ( n deterministic CFLs, respectively) then for each n , 𝒞ℱℒ n ⊊𝒞ℱℒ n +1 and 𝒟𝒞ℱℒ n ⊊𝒟𝒞ℱℒ n +1 Liu and Weiner (1973) showed that for each n , there is a language L n +1 ∈ 𝒟𝒞ℱℒ n +1 −𝒞ℱℒ n , which, together with the fact that 𝒟𝒞ℱℒ n ⊆ 𝒞ℱℒ n for each n (to our knowledge, proper containment has not been established for the entire hierarchy), results in two hierarchies, one embedded in the other.…”

Section: Introductionmentioning

confidence: 99%

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Counter machines and crystallographic structures

2015

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One way to depict a crystallographic structure is by a periodic (di)graph, i.e., a graph whose group of automorphisms has a translational subgroup of finite index acting freely on the structure. We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages 𝒟𝒞ℒd, d = 0, 1, 2, …, within the intersection classes of deterministic context-free languages. We introduce a class of counter machines that accept these languages, where the machines with d counters recognize the class 𝒟𝒞ℒd An intersection of d languages in 𝒟𝒞ℒ1 defines 𝒟𝒞ℒd. We prove that there is a one-to-one correspondence between sets of walks starting and ending in the same unit of a d-dimensional periodic (di)graph and the class of languages in 𝒟𝒞ℒd. The proof uses the following result: given a digraph Δ and a group G, there is a unique digraph Γ such that G ≤ Aut Γ, G acts freely on the structure, and Γ/G ≅ Δ.

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“…Let 𝒞ℱℒ 0 =𝒟𝒞ℱℒ 0 be the class of regular languages. Liu and Weiner (1973), (see also Kintala 1978) proved that for each d > 0, the language

\begin{matrix} left \\ L_{d} = false{a_{1}^{k_{1}} a_{2}^{k_{2}} \dots a_{d}^{k_{d}} b_{1}^{k_{1}} b_{2}^{k_{2}} \dots b_{d}^{k_{d}} : k_{1}, k_{2}, \dots, k_{d} \geq 0 false} \\ \subseteq {a_{1}, a_{2}, \dots, a_{d}, b_{1}, b_{2}, \dots, b_{d}}^{*} \end{matrix}

satisfies L d +1 ∉ 𝒞ℱℒ d . But as

L_{d} = true \cap_{i = 1}^{d} L_{d, i}

where

L_{d, i} = {a_{1}^{k_{1}} a_{2}^{k_{2}} \dots a_{d}^{k_{d}} b_{1}^{l_{1}} b_{2}^{l_{2}} \dots b_{d}^{l_{d}} : k_{1}, k_{2}, \dots, k_{d}, l_{1}, l_{2}, \dots, l_{d} \geq 0 & k_{i} = l_{i}},

and L d , i is a DCFL for each i , we have L d +1 ∈ 𝒟𝒞ℱℒ d +1 − 𝒞ℱℒ d for each d .…”

Section: Counter Machinesmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Counter machines and crystallographic structures

2015

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“…A conjunctive grammar is linear context-free, if it is at the same time linear conjunctive and context-free. We shall also consider finite intersections of context-free and of linear context-free languages, which were studied by Liu and Weiner [10], by Wotschke [17] and by Kutrib, Malcher and Wotschke [9]; these families are properly contained in conjunctive and linear conjunctive languages, respectively. Finally, regular languages are those recognized by finite automata, while deterministic context-free languages are recognized by deterministic pushdown automata, see Harrison [7].…”

Section: Example 3 [13]mentioning

confidence: 99%

“…Consider finite intersections of context-free [10,17] or linear context-free languages [9]: both families have the closure properties required by Theorem 9, and they are therefore closed under dual concatenation with co-finite languages.…”

Section: ) Union (2) Intersection and (3) Concatenation With Singletmentioning

confidence: 99%

“…Deterministic context-free languages are closed under dual concatenation with regular languages from the right, while dually concatenating finite or co-finite languages from the left leads out of this class. Finite intersections of context-free languages [10,17,9] are closed under dual concatenation with co-finite languages, but not with finite languages. Finally, conjunctive languages [13] are closed under dual concatenation with any regular languages.…”

Section: Introductionmentioning

confidence: 99%

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Notes on Dual Concatenation

Okhotin

2007

Int. J. Found. Comput. Sci.

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It was recently found that concatenation of formal languages has a logical dual (A. Okhotin, The dual of concatenation, Theoret. Corn-put. Sci., 345 (2005), 425-447). In this paper, the closure or nonclosure of common language families under dual concatenation with finite, co-finite and regular languages is determined. In addition, language equations with union, linear concatenation and dual concatenation with co-finite constants are shown to be almost equal in power to linear conjunctive grammars.

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The Descriptive Complexity Approach to LOGCFL

Lautemann

McKenzie

Schwentick

et al. 2001

Journal of Computer and System Sciences

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Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers in first-order logic with linear order. Our work extends the elaborate theory relating monoidal quantifiers to NC 1 and its subclasses. In the absence of arithmetical predicates for plus and times (equivalently, in the absence of the BIT predicate), we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's hardest context-free language is LOGCFL-complete under quantifier-free reductions without arithmetic. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with predicates for plus and times than without. Considering a particular groupoidal quantifier, we prove that first-order logic with the``majority of pairs'' quantifier is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define

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An infinite hierarchy of intersections of context-free languages

Cited by 52 publications

References 4 publications

Counter machines and crystallographic structures

Counter machines and crystallographic structures

Notes on Dual Concatenation

The Descriptive Complexity Approach to LOGCFL

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