Unmbiguous Recognizble Two-dimensionl Lnguges Mrcell Anselmo, Dor Gimmrresi, Mri Mdoni, Antonio Restivo (Univ. of Slerno, Univ. Rom Tor Vergt, Univ. of Ctni, Univ. of Plermo) W2DL, My 26
REC fmily I REC fmily is defined in terms of locl lnguges It is necessry to identify the boundry of picture p using boundry symbol Σ p = M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 2 p = L is locl if there exists set Θ of tiles (i.e. squre pictures of size 2 2) such tht, p in L if nd only if ny sub-picture 2 2 of p is in Θ
M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 3 L d = the set of squre pictures with symbol in ll min digonl positions nd symbol in the other positions Exmple of locl lnguge Θ = p = p =
REC fmily II L is recognizble by tiling system if L= π(l ) where L is locl lnguge nd π is mpping from the lphbet of L to the lphbet of L Exmple: The set of ll squres over Σ = {} is recognizble by tiling system. Set L =L d nd π()= π()= L d REC is the fmily of two-dimensionl lnguges recognizble by tiling system π M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 4
About Unmbiguity Definition of REC is implicitly non-deterministic The determinism nd non-determinism re no more equivlent in REC: the deterministic models (4DFA, 2DOTA, ) don t recognize the whole REC REC is not closed under complement, so it is not possible to eliminte non-determinism from the model (without losing in power of recognition) An intermedite notion between determinism nd non-determinism is the notion of unmbiguity M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 5
Unmbiguous Recognizble Lnguges Def [GR92] A tiling system (Σ, Γ, θ, π) is unmbiguous for L Σ ** if the projection π is injective on L(θ) (i.e. for ny p L there is unique p L such tht π(p )=p). L Σ** is unmbiguous if it dmits n unmbiguous tiling system. UREC denotes the fmily of ll unmbiguous recognizble 2dim lnguges. UREC REC Generliztion in 2dims of unmbiguous utomt for strings M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 6
Exmple: L col-n = {p first col = lst col } {,b} ** L col-n REC Ide: Use Γ = {x y } where the subscript y sves the symbol of the first column nd π(x y ) = x b b b p = b b b b b L col-n UREC p = b b b b b b b b b b b b b b b b M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 7
UREC nd REC UREC REC? Yes i j L col-ij = col i = col j L col-ij REC L col-ij UREC L col-ij = Σ ** L col-n Σ ** nd REC is closed with respect to WHY? M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 8
Towrds necessry condition for unmbiguity Reduce two dimensionl lnguges to string lnguges over the lphbet of the columns (i.e. define L(m)) Use the Theorem of Hromkovic et l. for lower bound on the sttes of n unmbiguous utomton for string lnguge M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 9
From 2dim to dim Let L Σ **. For ny m consider the subset L(m) L of ll pictures with exctly m rows. L(m) cn be viewed s string lnguge over the lphbet of the columns Exmple: b b b b p = L the string w = L(4) b b b b b b b b M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26
An utomton for L(m) Theorem [Mtz 97] Let L Σ **. If L REC, then there is k such tht, for ll m, there is finite string utomton A m with k m sttes for L(m). Ide of Proof: Let (Σ, Γ, θ, π) tiling system for L. The sttes of A m re ll the possible columns (of height m) in the locl lphbet Γ, plus n initil stte. M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26
Ide of Proof (continued) There is n edge from column p to column q if nd only if ny sub-picture 2 2 of p q is in θ. The lbel for this edge is π(q) Exmple: In L col-n we hve b b b b b b b b b b M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 2
Theorem of Hromkovic et l. Def Let S Σ * be regulr string lnguge. Define the infinite boolen mtrix M S = αβ α Σ*, β Σ* where αβ = if nd only if αβ L. Since S is regulr, the number of different rows of M S is finite. Let S Σ * be regulr string lnguge. Denote by uns(s) the size of miniml unmbiguous non-deterministic utomton ccepting S. Theorem (Hromkovic et l.) For every regulr string lnguge S Σ *, uns(s) Rnk Q (M S ). M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 3
A necessry condition for unmbiguity Theorem Let L Σ **. If L UREC, then there is k such tht, for ll m, Rnk Q (M L(m) ) k m. Proof: Note tht if L UREC then the utomton A m for L(m) is unmbiguous Use the inequlity uns(l(m)) Rnk Q (M L(m) ) M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 4
UREC Consider L = L col-ij REC For every m, L(m) is lnguge of strings with t lest two occurrences of the sme symbol. L L(m) It is possible to show tht M L(m) hs Rnk equl to 2 Σ m + ginst the necessry condition for UREC. Theorem (restted) There exist recognizble 2dim lnguges tht re inherently mbiguous. M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 5
Properties of UREC Proposition UREC is closed under intersection nd rottion opertions. Proposition UREC is not closed under row/column conctention/closure. Proof: L col-n UREC. But L col-ij = Σ ** L col-n Σ ** UREC. M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 6
Using utomt chrcteriztion Def A 2UOTA is 2OTA such tht it hs t most one ccepting run on picture p. Theorem L(2DOTA) L(2UOTA) L(2OTA). Proof: Note tht L(2UOTA)=UREC (see lso Mäurer2) nd L(2OTA)=REC. For the first inclusion, consider the lnguge L = {p p is squre lst row = lst col } {,b} ** L L(2DOTA) but L L(2UOTA) The second inclusion follows from L(2UOTA) = UREC REC = L(2OTA) M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 7 b c b c
An undecidbility result Theorem Given tiling system (Σ, Γ, θ, π) for L Σ **, it is undecidble whether it is unmbiguous. Proof: By reduction from the undecidble 2dimensionl Unique Decipherbility Problem. M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 8
Further work Questions relted to UREC Questions relted to (?) DREC (deterministic version of REC) M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 9
Open Problems Is UREC closed under complement? Is UREC lrgest subset in REC closed under complement? Conjecture: If L REC\UREC then L REC M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 2
About Deterministic Recognizble 2dim Lnguges Mny deterministic models : 4DFA, 2DOTA, They don t recognize the whole REC In string lnguges the notion of determinism is, in some sense, oriented : - Determinism from left to right - Co-determinism from right to left In picture four different directions. Two proposls for the definition of 2dimensionl determinism. M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 2
First pproch Ide: A tiling system (Σ, Γ, θ, π) is Top-Left-deterministic if,b,c Γ nd s Σ unique tile c b d such tht π(s)=d.?? There is n unique wy to fill this position with symbol of Γ (Anlogously TR-,BL-,BR-deterministic tiling system) L is deterministic if L hs tiling system tht is deterministic with respect to some direction (TL or TR or BL or BR) M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 22
Second pproch Ide: A tiling system is left-to-right columndeterministic if, fter hving computed the locl symbols in n entire column of picture, the locl symbols on the next one re univoclly determined.???????? L is deterministic if L hs tiling system tht is deterministic with respect to one direction by column nd tiling system tht is deterministic with respect to one direction by row M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 23
Working proposl for these dys Find n pproprite definition for determinism in terms of tiling system tht is not oriented s the recognition by tiling systems M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 24
The end M. Anselmo, D.Gimmrresi, M. Mdoni, A. Restivo W2DL, 3-5 Mggio 26 25