Ogden s Lemma for ET0L Languages

Transcription

1 March 2012

2 Overview Introduction to ET0Ls 1 Introduction to ET0Ls Definition Motivation 2 The original lemma Generalising to ET0Ls 3 Direct application Corollary 4

3 What are ET0Ls? Introduction to ET0Ls Definition Motivation An ET0L system (V, Σ, T, S) generates a language by a Lindenmayer system which uses zero context A 1 A 2... A n u 1 u 2... u n A u is table-driven { } {Q QS, S Q, a λ}, {Q a, S a, a a} and extended L = {w : S w} Σ

4 Why ET0Ls? Introduction to ET0Ls Definition Motivation L-systems: natural parallelism biological inspiration

5 Why ET0Ls? Introduction to ET0Ls Definition Motivation L-systems: natural parallelism biological inspiration ET0Ls: generate all the context-free languages have nice closure properties (AFL)

6 Definition Motivation

7 for CFLs The original lemma Generalising to ET0Ls Recall Theorem If L is a context-free language, then there exists an integer l such that for any u L with at least l positions marked, u can be written as u = vwxyz such that 1 x and at least one of w or y both contain a marked position; 2 wxy contains at most l marked positions; and, 3 vw m xy m z L for all m N.

8 Direction of generalisation The original lemma Generalising to ET0Ls Features to lose: arithmetic progressions no moving

9 Direction of generalisation The original lemma Generalising to ET0Ls Features to lose: arithmetic progressions no moving and to keep: infinite sequence of new words duplication of marked symbols

10 The original lemma Generalising to ET0Ls Theorem If L is an ET0L language, then there exists an l N such that for any word w L with at least l marked positions, w can be written as w = u 1 u 2 u n and each u i can be written u i = v (i,1) v (i,2) v (i,ni ) (we will denote the set of subscripts of v, i.e. {(i, j) : i [n], j [n i ]}, by I ); there is a map φ : I [n] such that if each v (i,j) is replaced with u φ(i,j), then the resulting word is still in L, and this process can be applied iteratively to always yield a word in L ; if v (i,j) contains a marked position then so does u φ(i,j) ; there is an (i, j) I such that φ(i, j) = i, and there are at least two marked positions in v (i,j) and at least one in u i but outside of v (i,j).

11 Example Introduction to ET0Ls The original lemma Generalising to ET0Ls L = {a n b m : n N, m 2 n } { {S Ab, A aa, b bb, a a}, } {S S, A λ, b b, b bb, a a}

12 Proof idea Introduction to ET0Ls The original lemma Generalising to ET0Ls A S b a A b b a a A b b b b a a b b b b b

13 Proof idea Introduction to ET0Ls The original lemma Generalising to ET0Ls A S b a A b b a a A b b b b a a a A b b b b b b b b a a a b b b b b b b b b b b b

14 Lemma is not sufficient The original lemma Generalising to ET0Ls L(K) = { } a 2n (2k+1) : n N, k K K uncomputable implies L(K) uncomputable uncountably many examples

15 Proving a language is not ET0L Direct application Corollary L = { a n b m : n N, m 2 2n } Need facts about pumping: unbounded growth no super-exponential growth

16 Rare and nonfrequent symbols Direct application Corollary A symbol is called nonfrequent in a language if there is an upper bound on how many appear in words A symbol is called rare if instances of it appear far apart in long enough words

17 Direct application Corollary Ehrenfeucht and Rozenberg s rare-nonfrequent theorem rare nonfrequent Can prove frequent non-rare using our theorem: Mark all the symbol It s frequent, so above the threshold Pumping makes long words with a fixed subword So it s not rare Need fact about pumping: subword with two marked symbols is preserved

18 What about a shrinking version? And the shrinking lemma for rfcls?