end only I X h I R 1 X t t t F t X 1 h I R l R t F I F t X t in V ones The but the 3 n V a be a k lfunction V 1 h is e il R is a is i hiRi l i t DEFINITION 2 V I t E1T V k V 1 1 w q1 u2 V ID: 868541
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1 GRAMHATICAL TYPES end only I X = h I (R
GRAMHATICAL TYPES end only I X = h I (R 1 X t = t t~ F t (X 1 h I R l R t F I ..... F t .... X t in V ones. The ( ) but the 3 n / V a be a k l]'-function V 1 h is e il) R is a ~ is i = (hi,Ri), l~ i~ t. DEFINITION 2 V I t E-
1T, V k V 1 1 w ~q1" u2' V--UlW'U2 V = {Yl ..... yq} finite substitution is defined rules �yi--- zij, lI. let .... ... distribution (i.e. zij is U ~ cs-parentheses ~ ={~l ..... Tn}' ~i • u ' generated the empty (
4 sets, The study of .... PART equations i
sets, The study of .... PART equations is interesting of genera- theory for following families ~ , giving its Let~ be., s T be a u { ~-polynomials, i.e. ~ = x o initial; if An ~ t X , V s infinite collection arbitrary finite
5 ={Xo,X I X m TT, is ell pairs VU NuPar
={Xo,X I X m TT, is ell pairs VU NuPar finite substitution R a function lr : 2 ~ = ~ 2 s . T~lthe closure ) U for it follows: if f'E p), function f. the ad- x o to the of the given by: {f if'e , t X Y Y Y . t~. Each O(-syst
6 em possesses unique minimal A language L
em possesses unique minimal A language L 0 • • 2 1 ), where ho~ is given by ho(C_l) =~ r-_l Xo~ end EII~ gives fixed-point characterization ~-free context-sensitive l, (equivalence their first solution) is U X t U x
7 o X i = h i i ~ U X t U x o i) V i = =
o X i = h i i ~ U X t U x o i) V i = = V V V N a f Y = = ~ X I X i, = U x = in order = { s I, V u N N : V N, VTf = V N ) ; s , R = (3) our bility relation ~ , ~={ ~l .... ~ ' = ( ~ ~ ~i }' and v I = ( U ]~i I ~ . PROPOSI
8 TION 1 t = = t X, , .... k ' ' i s T ~Ti
TION 1 t = = t X, , .... k ' ' i s T ~Ti, e T = TU{[ITT, ~T~T}. We shall o° the eat v G V c = l.l~.G ~T u j = 3-1} &#xki~U;&#xV' 0;'~-l=~l~ h o x ° % = R = i P U ~o, C-Z, no, ~-l~ EQUAT~ , s met ~ , V, such L G EQUAT~ R L
9 c = finite substitution this point, (uni
c = finite substitution this point, (universal) reducibility relation ~ = u v u l • v . ~ , a union of ~v~ iff u ~ v ~ , type ~ over is the class ¢~-reduoibiltty relation ~ . It is 1 o Code (E 1 U E central notion PROP
10 OSITION 2 context-sensitive set, is cont
OSITION 2 context-sensitive set, is context- sensitive too. R L defining it) ::X-t~, Y-t? t x = ~ , 3]Ti. j l~i~n, l~J~k and~1[ T T . . ^ Code(f Let /~be the regular set Left = ~J shall define R L " Eo Eo l~i~n l~J(kt~) ~ ~ ~
11 A ' ~ ~(a)=e, for aEV, for zEk. Now we
A ' ~ ~(a)=e, for aEV, for zEk. Now we = k n second order ~ . (i) Z~=~U - M~UM , u g = , Z } A ~ x'=tx., Y'=ty,. z = and ~ i It lsa : 2 X 2 ~ 2 (t x , t , ty, , t ) is a a ~ as: ~ , can be the cor- L ~ EQUAT~ on ~ . result
12 S an for a ~ . ~ is such that Any gramma
S an for a ~ . ~ is such that Any grammatical possesses a machine M V T ~ Q F a use two symbols x h is a set of rules only for the ) = s I a X - I (ho(X) n v T slav T ) (2) Move-riqht: X= t ---)c} (ho(X)~ s i - ~ ~ ) Similar
13 equations constructed for * ~ IF~O ) U
equations constructed for * ~ IF~O ) U x Y =3Ffin(X U x final state its "protected" initial content). It clear that only {Sfin~U{Sinit +} APPENDIX 2 N(D)TIHE(f(n)), f(n)=n k V T C V "terminal" alphabet. , u U { 1...cp. V-dl.
14 ..dp, di,cieVU {B} , we o 1 of the ! . "
..dp, di,cieVU {B} , we o 1 of the ! . " u u terminal rule x a Y - ( ~ it= I(XUXo). h 3 W N ~ = ..,a t~. ell pairs exist k~ ~o~such = a first-order type~ difference is a p EQUAT~ = It ls w I ..... the CF the set C C hr(~r)
15 ={Ur}, hr(rule)=r N U V T and x o" ~ ) U
={Ur}, hr(rule)=r N U V T and x o" ~ ) U 5) q For ETOL the first of s TYPE POLY POLY U (SSo+)4r ss O Reg-contr. ~ST~ u es 0 U (SSo } U Sinit+(s+)* s u u {sT~ J,of Math. programs" 6th grammatical characterization R ROZENBERG,