Random Context and Programmed Grammars of Finite Index Have The Same Generative Power

1. 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

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

3. Context-free Grammar • 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* }.

4. Programmed 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

5. Random Context Grammar • Created in sixtieth of 20th century • Modified form of the rules: • p: Ai x, f(p), where Ai(V – T), xV* • f(p)  (V – T) is a set of nonterminals called permitting context • Derivation step:u0A1u1…ui-1Aiui…un-1Anun u0A1…ui-1xui…Anun[p], where u0,u1,…, unV*, {A1,…,An}= f(p) • Rule p is applicable if sentential form contains all nonterminals from f(p).

6. Grammar of Finite Index • For a derivation S* x, such that w0  w1  …  wn, where n  1, wiV*, 1  i  n, S = w0, wn = x, xT* • Ind(S* x, G) = max { occur(wi,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.

7. Families of Languages P EDT0L ? CF acPfin Pfin acRCfin 1989 SMLIN RCfin

8. Main Result P EDT0L ? CF acPfin Pfin acRCfin Our result, but… SMLIN RCfin

9. Main Result P EDT0L Kfin ? CF 1996 acPfin Pfin acRCfin Our alternative way of the proof SMLIN RCfin

10. Main Result - Theorem Theorem: Pfin = RCfin 1989[Dassow, Paun]: RCfinPfin 1996[Fernau, Holzer]: Kfin=Pfin…NOT USED Second direction of inclusion proved by construction.

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

12. Example of Simulation • 1 Step in Programmed Grammar of index k: x0Ax1Bx2Cx3 x0Ax1yx2Cx3 [p:B  y,{q}] • Simulation in Random Context Grammar of index k:x0 p,A,1,3 x1 p,B,2,3 x2 p,C,3,3 x3x0 pq,A,1,2 x1 p,B,2,3 x2 p,C,3,3 x3 x0 pq,A,1,2 x1 pq,B‘,2,2 x2 p,C,3,3 x3 x0 pq,A,1,2 x1 pq,B‘,2,2 x2 pq,C,2,2 x3 x0 pq,A,1,2 x1 y x2 pq,C,2,2 x3  x0q,A,1,2x1 y x2pq,C,2,2x3 x0q,A,1,2x1 y x2q,C,2,2x3 • where x0,…,x3,yT*, A,B,C(VPG – T)*, …(VRC – T)*

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