1. 1 Dal non-determinismo al determinismo ( nei linguaggi 2dim ): alcune riflessioni Marcella Anselmo, Dora Giammarresi, Maria Madonia, Antonio Restivo Riunione Prin. Varese, luglio 2006

2. 2  finite alphabet  ** all 2dim rectangular words (pictures) over  L   ** 2dim language p  L has size (m,n) Column concatenation Row concatenation Column/Row star 2dim Languages pq p q =  p  q = p q

3. 3 Local 2dim Languages L is local if there exists a finite set  of tiles that contains all allowed subpictures of size (2,2,), i.e. p  L if and only if any 2  2 sub-picture of is in  p  tile: a square picture of size (2,2) bordered picture p:         p = 

4. 4 L d = the set of square pictures with symbol “1” in all main diagonal positions and symbol “0” in the other positions (Usual) Example of local language 10 01  10 00 10 00 00 0  1  1    1  0  0  0 00  01   0   00 0  0   0    1  = 01 00 100 010 001 p = ##### #100# #010# #001# ##### 

5. 5 L is recognizable by tiling system if L=  (L’) where L’ is a local language and  is a mapping from the alphabet  of L’ to the alphabet  of L Recognizable 2dim Languages REC is the family of two-dimensional languages recognizable by tiling system REC is closed almost under all operations but it is not closed under complement ( , , ,  ), where L’=L(  ), is called tiling system

6. 6 (Usual) Example L Sq = all squares over  = {a}. L Sq is recognizable by tiling system. Set L’=L d and  (1)=  (0)= a 100 010 001  L d aaa aaa aaa 

7. 7 #### (Usual) Example L Sq = squares over {a}. Use L’=L d  (1)=  (0)= a aaa aaa aaa 10 01  10 00 10 00 00 0  1  1    1  0  0  0 00  01   0   00 0  0   0    1  = 01 00 ##### ## ## ## # p =  10 0 0 0 1 1 0 0 “Computing” by a tiling system (from a tiling system to an automaton) First, decide a scanning strategy!

8. 8 “Computing” by a tiling system (from a tiling system to an automaton) Remark :Tiling system = “undirectional” transitions Definition: A 2dim finite automaton is Tiling system + scanning procedure Local picture is the run of the automaton. Remark : All 2dim finite automata “correspond” to family REC (i.e. scanning procedure does not matter!) Ex: 2OTA (2dim on-line tesselation automata)

9. 9 Scanning strategies (I) ######## ## ## ## ## ## ## ######## 12 3 4 5 6 7 8 9 10 34 3536 Diagonal (“2OTA”) ######## ## ## ## ## ## ## ######## By column 1 2 3 4 5 6 7 8 9 10 11 12 31 32 33 34 35 36

10. 10 Scanning strategies (II) ######## ## ## ## ## ## ## ######## Snake-like 123 456 7891011 12 1314 343536 Free ######## ## ## ## ## ## ## ######## 1 2 3 4 5 6 7 8 9 36

11. 11 A remark about REC A tiling system (= local language + projection) “generalizes” to 2 dim a non-deterministic finite automaton. Family REC is not closed under complement Definition of REC is intrinsically non-deterministic and it is not possible to eliminate non-determinism without getting a smaller class!

12. 12 From non-determinism to determinism.... Non-determinism Possible accepting computations: “several” Possible backtracking steps at each step of computation: linear in the size of input [if pictures: =O(m  n)] Determinism Possible accepting computations: 1 Possible backtracking steps at each step of computation: 0

13. 13 Remark on 1DIM case : In string languages 2 definitions possible: -Determinism from left to right - Co-determinism from right to left Correspond to same class!....choose one definition… ?? Remark: Languages recognized by automata that are both deterministic and co-deterministic are smaller class!

14. 14 A tiling system is Top-Left-deterministic if  a,b,c   and s    unique tile such that  (s)=d. ab cd (Analogously TR-,BL-,BR-deterministic tiling system) ?? There is an unique way to fill this position with a symbol of  L is deterministic if it has a TL- or TR- or BL- or BR- deterministic tiling system Deterministic Recognizable Languages (DREC) Classical definition (only a bit extended)

15. 15 L col-1n  REC a baba babaa bb ababb aaaab ababbb p’= L col-1n = {p | first col = last col }  {a,b} ** New Example Local alphabet:  = {x y } Projection “erase” subscripts:  ( x y ) = x L col-1n  DREC L col-  1i = {p |  1<i  n, first col = col i}  DREC

16. 16 From non-determinism to determinism: what can we define in beetween? Non-determinism Possible accepting computations: “several” Possible backtracking steps at each step of computation: linear in the size of p (m  n) Determinism Possible accepting computations: 1 Possible backtracking steps at each step of computation: 0 Unambiguity one

17. 17 Unambiguous Recognizable Languages (UREC) Def [GR92] A tiling system ( , , ,  ) is unambiguous for L   ** if the projection π is injective on L(  ) (i.e. for any p  L there is a unique p’  L’ such that  (p’)=p). UREC: all unambiguous recognizable 2dim languages. L   ** is unambiguous if it admits an unambiguous tiling system. UREC  REC Generalization in 2dims of unambiguous automata for strings

18. 18 UREC and REC L col-  ij  REC L col-  ij  UREC L col-  ij =  ** L col-1n  ** and REC is closed with respect to   UREC REC  L col-  ij = i j  i,j: col i = col j Necess. Cond. for UREC

19. 19 Properties of UREC Proposition UREC is not closed under row/column concatenation/closure. Proposition UREC is closed under intersection and rotation operations.

20. 20 From non-determinism to determinism: what can we define in beetween? (2) Non-determinism Possible accepting computations: “several” Possible backtracking steps at each step of computation: linear in the size of p (m  n) Determinism Possible accepting computations: 1 Possible backtracking steps at each step of computation: 0 one dimension of p (m or n) “line”-unambiguity one

21. 21 A tiling system is Left-Right Column-Unambiguous if, after having computed the local symbols in an entire column, the local symbols on the next column are univocally determined. ?? L is Col-UREC if L has a tiling system that is LR- or RL- column unambiguous. Column-Unambiguos Languages (Col-UREC) Remark: Backtracking at each step of possibly O(m) steps.

22. 22 A tiling system is Top-Down Row-Unambiguous if, after having computed the local symbols in an entire row of a picture, the local symbols on the next row are univocally determined. ?? L is Row-UREC if L has a tiling system that is TD- or DT- column unambiguous. Row-Unambiguos Languages (Row-UREC) Remark: Backtracking at each step of possibly O(n) steps.

23. 23 (A new) Example  = {a, b} L Sq-cent-a = odd-side squares with a in center position abba bbab aaaa babb b b b b a a aa a a0a0 b0b0 b0b0 a2a2 b1b1 b0b0 a2a2 b0b0 a0a0 a1a1 a0a0 a0a0 b2b2 a0a0 b1b1 b0b0 b1b1 b0b0 b0b0 b1b1 a0a0 a0a0 a0a0 a2a2 a0a0 L Sq-cent-a  Col-UREC, Row-UREC L Sq-cent-a  DREC By an “old” proof by Inoue et al.

24. 24 Col-UREC and UREC L col-  1ij n  UREC L col-  1ijn  Col-UREC L col-  1ijn = L col-  1j  L col-  in and UREC is closed with respect to  Col- UREC UREC   i,j: col 1 = col j col i = col n L col-  1ij n = 1ijn  Necess. Cond. for Col-UREC

25. 25 A necessary condition for unambiguity Theorem Let L   **. There is a k such that, for all m, 1.If L  Col-UREC then Row(M L(m) )  k m 2.If L  UREC then Rank Q (M L(m) )  k m L(m)  L is the subset of all pictures with m rows. It can be viewed as a string language over the columns alphabet. S  *, regular string language. M S is the boolean matrix M S =|a  |   *,   * where a  = 1 iff   L. The number of different rows, Row(M S ), is finite. Idea of Proof Use Matz’s Theorem and Hromkovic et al. Theorem

26. 26 From 2dim to 1dim Theorem [ Matz 97 ] Let L   **. If L  REC, then there is a k such that, for all m, there is a finite string automaton A m with k m states for L(m). Fact If L  UREC, then A m is an unambiguous automaton with k m states for L(m). If L  Col-UREC, then A m is a deterministic automaton with k m states for L(m).

27. 27 Theorem of Hromkovic et al. Theorem (Hromkovic et al.) For every regular string language S  *, d(S) = Row(M S ) uns(S)  Rank Q (M S ). d(S) the size of the minimal deterministic automaton accepting S uns(S) the size of a minimal unambiguous non- deterministic automaton accepting S.

28. 28 The following inclusions are all strict: DREC  Col-UREC  UREC  REC Collecting all classes… a   1ijn   i j  

29. 29 A separation result Theorem Whatever we choose a definition of deterministic 2dim finite automaton, the family of corresponding languages is strictly included in UREC. Proof : By previous strict inclusions results (Col-UREC  UREC ) Det-REC is strictly included in UREC, for any definition of Det-REC we choose.

30. 30 An undecidability result for UREC Theorem Given a tiling system ( , , ,  ) for L   **, it is undecidable whether it is unambiguous. Proof : By reduction from the undecidable 2dimensional Unique Decipherability Problem.

31. 31 A decidability result for Col-UREC (Row-UREC) Theorem Given a tiling system T = ( , , ,  ) for L   **, it is decidable whether it is col-unambiguous. Proof : Let M=Card {( ,  ) : ,    }. T col-unambiguous No pair of pictures sp with p, s, t   n,1 s  t  (s) =  (t) Any 2  2 sub-picture of p s, p t in  n  M   tp

32. 32 A tiling system is Top-Left Diagonal-Unambiguous if, after having computed the local symbols in an entire diagonal of a picture, the local symbols on the next diagonal are univocally determined. ?? L is Diag-UREC if L has a tiling system that is TL-, TD-, BL- or BR- diagonal unambiguous. REMARK: Diag-Unambiguos Languages Remark: NO backtracking at each step

33. 33 Conjecture: If L  REC\UREC then  L  REC Is UREC largest subset in REC closed under complement? Is UREC (Col-UREC) closed under complement? Open Problems Is L(4NFA)  UREC?

34. 34 Conclusioni alle riflessioni… Tiling systems are a “compact” way to represent classes of finite state automata on 2 dims. Unambiguos languages are a strict intermediate class between non-deterministic and deterministic families.

35. 35 riflettere a mente fresca…