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On Some Complexity Results for Even Linear Languages

Liliana Cojocaru

TL;DR

The paper introduces Dyck normal form, a restricted variant of CNF that enforces paired nonterminals on the right-hand side, yielding derivation trees that map to Dyck words. It proves a strong equivalence with CNF and uses this framework to show every CFL L can be represented as L = φ(D'_K) for some K and homomorphism φ, via a trace-language connection to Dyck languages. This leads to an alternate proof that even linear languages are contained in $AC^1$, by constructing an indexing alternating Turing machine that decides membership in $O( abla \,log^2 n)$ time and $O( abla \,log n)$ space with a logarithmic number of alternations. Overall, the work links CFG normal forms, Dyck-language theory, and complexity classifications to provide both syntactic characterizations of CFLs and a complexity-theoretic placement for ELIN.

Abstract

We deal with a normal form for context-free grammars, called Dyck normal form. This normal form is a syntactical restriction of the Chomsky normal form, in which the two nonterminals occurring on the right-hand side of a rule are paired nonterminals. This pairwise property, along with several other terminal rewriting conditions, makes it possible to define a homomorphism from Dyck words to words generated by a grammar in Dyck normal form. We prove that for each context-free language L, there exist an integer K and a homomorphism phi such that L=phi(D'_K), where D'_K is a subset of D_K and D_K is the one-sided Dyck language over K letters. As an application we give an alternative proof of the inclusion of the class of even linear languages in AC1.

On Some Complexity Results for Even Linear Languages

TL;DR

The paper introduces Dyck normal form, a restricted variant of CNF that enforces paired nonterminals on the right-hand side, yielding derivation trees that map to Dyck words. It proves a strong equivalence with CNF and uses this framework to show every CFL L can be represented as L = φ(D'_K) for some K and homomorphism φ, via a trace-language connection to Dyck languages. This leads to an alternate proof that even linear languages are contained in , by constructing an indexing alternating Turing machine that decides membership in time and space with a logarithmic number of alternations. Overall, the work links CFG normal forms, Dyck-language theory, and complexity classifications to provide both syntactic characterizations of CFLs and a complexity-theoretic placement for ELIN.

Abstract

We deal with a normal form for context-free grammars, called Dyck normal form. This normal form is a syntactical restriction of the Chomsky normal form, in which the two nonterminals occurring on the right-hand side of a rule are paired nonterminals. This pairwise property, along with several other terminal rewriting conditions, makes it possible to define a homomorphism from Dyck words to words generated by a grammar in Dyck normal form. We prove that for each context-free language L, there exist an integer K and a homomorphism phi such that L=phi(D'_K), where D'_K is a subset of D_K and D_K is the one-sided Dyck language over K letters. As an application we give an alternative proof of the inclusion of the class of even linear languages in AC1.
Paper Structure (4 sections, 10 theorems)

This paper contains 4 sections, 10 theorems.

Table of Contents

  1. Introduction
  2. Dyck Normal Form
  3. Characterizations of CFLs by Dyck Languages
  4. Even Linear Languages are Contained in $\cal A$$\cal C$$^1$ - An Alternative Proof

Key Result

Theorem 2.2

For each CFG $G=(N, T, P, S)$ there exists a grammar $G'=(N', T, P', S)$ such that $L(G)=L(G')$ and $G'$ is in Dyck normal form.

Theorems & Definitions (18)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Example 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 8 more