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A New Normal Form for Context-Sensitive Grammars

This presentation by Peter Varga and co-authored by Benedek Nagy delves into context-free and context-sensitive grammars, focusing on derivation trees and their depth. It discusses Chomsky normal form, Penttonen normal form, and the implications of rules that equate the lengths of the left and right sides in derivation. The concept of iteration-free normal form is introduced, outlining how any grammar in Penttonen normal form can be transformed into an iteration-free equivalent, preserving language generation. This work enhances our understanding of finite derivation trees in grammar theory.

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A New Normal Form for Context-Sensitive Grammars

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  1. A New Normal Form for Context-Sensitive Grammars Presenter: Peter Varga Co-author: Benedek Nagy

  2. Context-Free derivation trees An upper bound can be given for the derivation tree’s depth in case of context-free grammars, if we use Chomsky normal form:

  3. Derivation tree for context-sensitive grammars In the case of context-sensitive grammars a derivation tree can be given as well with the use of Penttonen normal form. In this tree there will be context branches: Penttonen normal form: Aa ABC ABAC

  4. The problem is the following: A derivation of a finite word can be arbitrarily long because of those rules where the length of the left side equals with the length of the right side. Derivation tree for context-sensitive grammars

  5. In this example the depth of the tree for a finite word has to be exponentially large.

  6. Iteration-free normal form • Definition (Iteration-free normal form): A rule set called H is in iteration-free normal form, if H is in Penttonen normal form, and does not contain any context-sensitive iteration with the same context. • Theorem: Any grammar in Penttonen normal form can be derived to a grammar which is in iteration-free normal form, and the original and the derived grammar generates the same language.

  7. The derivation tree with the use of the iteration-free normal form

  8. Summary Context-free grammars Context-sensitive grammars Chomsky normal form Penttonen normal form Iteration-free normal form Finite derivation tree

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