Random Context and Programmed Grammars of Finite Index Have The Same Generative Power Doc. RNDr. Alexander Meduna, CSc. Ing. Zbyněk Křivka DIFS, FIT, Brno University of Technology, Czech Republic

Contents Introduction, Motivation Preliminaries: –Programmed Grammars, –Random Context Grammars, –Finite Index Families of Languages & Relationships Main Result Conclusion, Discussion

A quadruple G = (V, T, P, S), where: –V … total alphabet (a finite set of symbols) –T … alphabet of terminals –P … set of rules of the form: p: A  x, where A  (V – T), x  V* and p is a unique label of the rule –S … axiom (the starting nonterminal) Derivation step: uAv  uxv [p: A  x], where u,v,x  V*, A  (V – T) Language: L(G) = { w | S  * w, w  T* }. Context-free Grammar

Created in sixtieth of 20th century Modified form of the rules: –p:  A  x, g(p) , where A  (V – T), x  V* –g(p) is a set of rule labels Derivation step: uAv  uxv [p] = wBz  wyz [q], where q  g(p), u,v,w,z,x,y  V*, A,B  (V – T) –For every used rule is given set of next potentially applicable rules Programmed Grammar

Random Context Grammar Created in sixtieth of 20th century Modified form of the rules: –p:  A i  x, f(p) , where A i  (V – T), x  V* –f(p)  (V – T) is a set of nonterminals called permitting context Derivation step: u 0 A 1 u 1 …u i-1 A i u i …u n-1 A n u n  u 0 A 1 …u i-1 xu i …A n u n [p], where u 0,u 1,…, u n  V*, {A 1,…,A n }= f(p) –Rule p is applicable if sentential form contains all nonterminals from f(p).

Grammar of Finite Index For a derivation S  * x, such that w 0  w 1  …  w n, where n  1, w i  V*, 1  i  n, S = w 0, w n = x, x  T* Ind(S  * x, G) = max { occur(w i,V – T) | 1  i  n } G of index k – the smallest positive integer that every word x  L(G) satisfies Ind( S  *x,G)  k. G of finite index – exists some k  1 such that G is of index k.

Families of Languages P finac RC finac P fin P EDT0L RC fin SM LIN ? CF 1989

Main Result P finac RC finac P fin P EDT0L RC fin SM LIN ? CF Our result, but…

Main Result P finac RC finac P fin P EDT0L RC fin SM LIN ? CF K fin Our alternative way of the proof 1996

Main Result - Theorem Theorem: P fin = RC fin 1989 [Dassow, Paun] : RC fin  P fin 1996 [Fernau, Holzer] : K fin = P fin …NOT USED Second direction of inclusion proved by construction.

Basic Idea Nonterminals of form  p q, A, j, h  4 essential atomical steps of the algorithm: 1)Inside of all nonterminals update h to h+m-1 (number after application of p). 2)In nonterminals following rewritten nonterminal, change their positions. 3)Rewrite a nonterminal by chosen rule p. 4)Choose next rule q to be applied as would the programmed grammar do.

Example of Simulation 1 Step in Programmed Grammar of index k: x 0 Ax 1 Bx 2 Cx 3  x 0 Ax 1 yx 2 Cx 3 [p:B  y,{q}] Simulation in Random Context Grammar of index k: x 0  p,A,1,3  x 1  p,B,2,3  x 2  p,C,3,3  x 3  x 0  p q,A,1,2  x 1  p,B,2,3  x 2  p,C,3,3  x 3  x 0  p q,A,1,2  x 1  p q,B‘,2,2  x 2  p,C,3,3  x 3  x 0  p q,A,1,2  x 1  p q,B‘,2,2  x 2  p q,C,2,2  x 3  x 0  p q,A,1,2  x 1 y x 2  p q,C,2,2  x 3  x 0  q,A,1,2  x 1 y x 2  p q,C,2,2  x 3  x 0  q,A,1,2  x 1 y x 2  q,C,2,2  x 3 where x 0,…,x 3,y  T*, A,B,C  (V PG – T)*,  …   (V RC – T)*

Conclusion Alternative way of the proof P fin =RC fin. A Practical usage of this result ? Other open problems in theory of regulated grammars of finite index Thank you for your attention!