Chapter 7 On Erasing Rules and Their Elimination

Abstract

The present three-section chapter studies how to eliminate erasing rules, having the empty string on their right-hand sides, from context-free grammars and their regulated versions. The chapter points out that this important topic still represents a largely open problem area in the theory of regulated grammars. Section 7.1 gives two methods of eliminating erasing rules from ordinary context-free grammars. One method is based upon a well-known technique, but the other represents a completely new algorithm that performs this elimination. Section 7.2 establishes workspace theorems for regular-controlled grammars. In essence, these theorems give derivation conditions under which erasing rules can be removed from these grammars. Section 7.3 discusses the topic of this chapter in terms of scattered context grammars. First, it points out that scattered context grammars with erasing rules characterize the family of recursively enumerable languages while their propagating versions do not. In fact, propagating scattered context grammars cannot generate any non-context-sensitive language, so some scattered context grammars with erasing rules are necessarily unconvertible to equivalent propagating scattered context grammars. This section establishes a sufficient condition under which this conversion is always possible.