Hierrchy o pushdown grphs Didier Cucl CNRS / LIGM University Pris- Est Frnce
The hierrchy o pushdown grphs recursive trnsition grphs Corresponding hierrchies o lnguges, terms, ordinls, ininite words higher order recursive schemes Didier Cucl Pushdown grphs
Grphs lphet L o edge lels lphet C o vertex lels: colours Grph : G V L V C V on n rritrry countle set V o vertices Didier Cucl Pushdown grphs
A hierrchy o grph milies 2002 Two sic grph opertions: unolding nd pth unctions Didier Cucl Pushdown grphs
Unolding i i i Didier Cucl Pushdown grphs
Pth unctions set Exp o pth expressions C L {ε} Exp or ny u,v Exp pth s u, u v, u +, u, u v, u v Exp u G t or u Exp Didier Cucl Pushdown grphs
s s s t or (s,,t) G c t or s = t (c,s) G ε t or s = t s u t or t u s s u v t or r (s s u+ t or s ( ) u + t s u t or (s s u v t or s u r r u t) u t s v t v t) For instnce s ε t mens tht s = t s Didier Cucl Pushdown grphs
Pth unction h : L C Exp pplied y inverse on grph G h (G) = { (s,,t) s h() G h() t } { (c,s) s h(c) s } G c s t s h(c) Didier Cucl Pushdown grphs
Inverse pth unction i i i Didier Cucl Pushdown grphs
Inverse pth unction c i c i c i Didier Cucl Pushdown grphs
Inverse pth unction c (i + not( )) c i i c i Didier Cucl Pushdown grphs
Inverse pth unction c d i (i + not( )) i + not( ) c c i i d i d i Didier Cucl Pushdown grphs
Inverse pth unction c i (i + not( )) c d d c d i i + not( ) Didier Cucl Pushdown grphs
Unolding i c d c d c d c d c d c d c d c d c d c d c d Didier Cucl Pushdown grphs
Inverse pth unction i Didier Cucl Pushdown grphs
A hierrchy o grph milies Tree0 = mily o inite trees Grphn = Pth -1 (Treen) Treen+1 = Un(Grphn) Didier Cucl Pushdown grphs
A hierrchy o grph milies Finite trees Pth Finite grphs Tree 0 Regulr trees Tree 1 Un Pth Un Grph 0 Equtionl grphs Grph 1 Algeric trees Tree 2 Pth Grph 2 Didier Cucl Pushdown grphs
VR-equtionl grphs Courcelle 1989 Sme hierrchy Pth unctions = regulr sustitutions Exp = set o regulr expressions or L ; u v, u +, u v or u,v Exp Pth = mondic interprettions Su-hierrchy o inite degree grphs Pth unctions = inite sustitutions Didier Cucl grph 1 = HR-equtionl grphs Courcelle 89 Pushdown grphs
Tree(g) or some integer mpping g i n g(0) g(n) n g(n) Tree(2 n ) Grph 2 Tree(n!), Tree(2 2n ) Grph 3 Tree(2 n) hierrchy Didier Cucl Pushdown grphs
Genertors For ech level n ind genn Grphn such tht Grphn = Pth (genn) Didier Cucl Pushdown grphs
Grph itertion Shelh, Stupp 1975 G Proposition with Knpik 2011 The itertion opertion preserves Grphn or ech n > 0 Didier Cucl Pushdown grphs
Tree-grph G Proposition Colcomet 04, Cryol, Wöhrle 03 TreeGrph(Grphn) Grphn+1 Didier Cucl Pushdown grphs
Theorem Muchnik 1984, Wlukiewicz 1996 The tree-grph opertion preserves the decidility o the mondic theory Corollry Courcelle Wlukiewicz 1998 The unolding opertion preserves the decidility o the mondic theory Corollry Didier Cucl Any grph o the hierrchy hs decidle mondic theory Pushdown grphs
Genertors Grphn+1 = Pth -1 (Un(Grphn Det CoDet)) = Pth -1 (genn+1) genn+1 Treen+1 Det Didier Cucl Pushdown grphs
Tree genertor t level 1 Didier Cucl Pushdown grphs
Grph genertor t level 2 1 1 1 Didier Cucl Pushdown grphs
Tree genertor t level 2 1 1 1 Didier Cucl Pushdown grphs
Higher order pushdown utomt Theorem Cryol 2006 Grphn = ε-closure((pdn) Reg) = i Wi(Ui i Vi) pusdown hierrchy Didier Cucl Pushdown grphs
Hierrchies o lnguges ordinls ininite words terms Didier Cucl Pushdown grphs
A hierrchy o lnguges Finite grphs Trce Regulr lnguges Grph 0 Index 0 Equtionl grphs Trce Context ree lnguges Grph1 Index 1 Grph 2 Trce Indexed lnguges Index 2 Trce Level 2 indexed lnguges Grph 3 Index 3 Indexed lnguges Aho 1968 The hierrchy o indexed lnguges Mslov 1974 Didier Cucl Pushdown grphs
Proposition Mslov 1976 For ech n 0, the lnguge mily Indexn is closed under intersection y ny regulr lnguge inverse regulr sustitution Indexn sustitution Didier Cucl Pushdown grphs
The hierrchy on ordinls Ordinl ω +1 : trnsitive closure o Theorem Brud 2009 For ech n, the ordinls in Grphn re the ordinls < ω (n+1) Grphω? ǫ 0? MSO? Didier Cucl Pushdown grphs
The hierrchy on ininite words Equtionl grphs Ult. periodic words Grph 1 In 1 Grph2 Morphic words In 2 Grph 3 Morphic words t level 2 In 3 Level 2 morphic words: Chmpernowne numer 0 1 10 11 100 101 110 111... The Liouville numer: 10 i! =,110001000000000000000001... i 1 Didier Cucl Pushdown grphs
Questions chrcterize morphic words t level 2 or α = 0,u with u In n nd n > 0 is α rtionl or trnscendentl numer? is 2 in the hierrchy? Didier Cucl Pushdown grphs
The hierrchy on terms By irst order sustitutions Theorem 2002 Treen+1 Term = Sustn Courcelle, Knpik 2002 Sust0 = regulr terms Sustn+1 = Se,u (Sustn) Didier Cucl u word o distinct constnts e symol o rity u +1 Pushdown grphs
First order sustitution (evlution) unction e constnt word x y e y x y Didier Cucl Pushdown grphs
Finite terms S e,u (e t 0 t 1... t n ) = S e,u (t 0 ) [S e,u (t 1 )/u(1),...,s e,u (t n )/u(n)] S e,u ( t 1... t m ) = S e,u (t 1 )...S e,u (t m ) Ininite terms S e,u (t) = sup n S e,u (t n Ω ) without Ω Didier Cucl Pushdown grphs
The hierrchy on terms By higher order schemes Dmm 1982 Scheme t level 0 S R ; F F g S Solution R ω g g g Didier Cucl Pushdown grphs
Scheme t level 1 Nivt 1975, Guessrin 1987, Courcelle 1990 R S F ; F x x F g Solution x R ω g g g Didier Cucl Pushdown grphs
Scheme t level 2 F F S ; g R x G x F F : ( 0 0 ) 0 0 G Solution R ω x g g g Didier Cucl Pushdown grphs
Sety condition: scope o vriles Dmm 1982 x 1 F Unse scheme x n No level( x i ) < level( s ) x i s R F S ; g G F x x F G Solution R ω g t n+2 = t t n n+1 x x t 2 Didier Cucl t 3 Pushdown grphs
Theorem 2002 Treen+1 Term = SeSchemen Knpik, Niwiński, Urzyczyn 2002 Any se scheme R t level n+1 is trnsormed into se scheme S t level n such tht R ω = Un(h (S ω ),r) or some pth unction h Didier Cucl Pushdown grphs
Conjecture The non rtionl lgeric numers re not in the pushdown hierrchy Didier Cucl Pushdown grphs