ICALP 1979: Automata, Languages and Programming pp 104-117

# Strict deterministic languages and controlled rewriting systems

• Laurent Chottin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 71)

## Abstract

A controlled rewriting system over an alphabet X is a finite set of rules vi → wi (l⩽i⩽n) with vi , wi in X* such that |vi|<|wi| , each rule being associated with a regular language Ri ... X*. Given such a system, f ⇒ g means that f=αviβ and g=αwiβ for some i, α in Ri , β in X*. The system is said to be injective if and only if f ⇒ g ⇐ f′ implies f=f′. Controlled rewriting systems are a special case of finite relations with computable left context (P. Butzbach [5], 1973), which can be defined as above, with the Ri's recursive instead of regular. P. Butzbach proved [5] that every simple deterministic language [11] is generated by some finite relation with computable left context iterating from a finite set of words. Here we improve this result with our THEOREM 1 : "Every strict deterministic language is generated by some injective controlled rewriting system iterating from a finite set of words". Moreover, let A be a deterministic pushdown automaton and ⇒ be the rewriting relation associated with A by the above theorem. Let θ : X* ⇒ X* defined by θ(u)=v if and ∃ w, w ⇒ v (v is unique, for ⇒ is injective); in some sense, θ generalizes the semi-Dyck simplification. We state :

THEOREM 2 : "If R ... X* is regular then so is θ(R)". This extends a result of M. Benois [1], also obtained by M. Fliess [10] (using quite different methods).

THEOREM 3 : "Let L be a language accepted by A by some accepting states. Then θ(L) is regular and"

This is a reformulation of a result recently obtained by J. Sakarovitch [18] with a different method.

## Keywords

Normal Form Regular Language Finite Automaton Deterministic Finite Automaton Pushdown Automaton
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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