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A Uniform Framework for Problems on Context-Free Grammars

Javier Esparza, Peter Rossmanith, Stefan Schwoon

TL;DR

A new algorithm is presented which not only has a better space complexity but is also (in this opinion) easier to read and understand and is competitive to well-known solutions for most (but not all) standard problems.

Abstract

In a paper published in Information Processing Letters in 2000, Bouajjani et al. presented an automata-based approach to a number of elementary problems on context-free grammars. This approach is of pedagogical interest since it provides a uniform solution to decision procedures usually solved by independent algorithms in textbooks. This paper improves upon the work by Bouajjani et al. in a number of ways. We present a new algorithm which not only has a better space complexity but is also (in our opinion) easier to read and understand. Moreover, a closer inspection reveals that the new algorithm is competitive to well-known solutions for most (but not all) standard problems.

A Uniform Framework for Problems on Context-Free Grammars

TL;DR

A new algorithm is presented which not only has a better space complexity but is also (in this opinion) easier to read and understand and is competitive to well-known solutions for most (but not all) standard problems.

Abstract

In a paper published in Information Processing Letters in 2000, Bouajjani et al. presented an automata-based approach to a number of elementary problems on context-free grammars. This approach is of pedagogical interest since it provides a uniform solution to decision procedures usually solved by independent algorithms in textbooks. This paper improves upon the work by Bouajjani et al. in a number of ways. We present a new algorithm which not only has a better space complexity but is also (in our opinion) easier to read and understand. Moreover, a closer inspection reveals that the new algorithm is competitive to well-known solutions for most (but not all) standard problems.

Paper Structure

This paper contains 17 sections, 2 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Notations
  3. Computing ${{\it pre}^*}(L)$
  4. Termination:
  5. Basis.
  6. Step.
  7. Basis.
  8. Step.
  9. The Algorithm
  10. Termination:
  11. Applications
  12. Membership:
  13. Identifying useless variables:
  14. Emptiness:
  15. Finiteness:
  16. ...and 2 more sections

Key Result

lemma thmcounterlemma

$L(A_{{\it pre}^*})\subseteq{{\it pre}^*}(L(A))$

Figures (1)

  • Figure 1: Example automaton before (left) and after the algorithm (right).

Theorems & Definitions (4)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof