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Dal non-determinismo al determinismo ( nei linguaggi 2dim ): alcune riflessioni

Dal non-determinismo al determinismo ( nei linguaggi 2dim ): alcune riflessioni. Marcella Anselmo, Dora Giammarresi, Maria Madonia, Antonio Restivo. Riunione Prin. Varese, luglio 2006. 2dim Languages.  finite alphabet  ** all 2dim rectangular words ( pictures ) over 

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Dal non-determinismo al determinismo ( nei linguaggi 2dim ): alcune riflessioni

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

  3.     L islocalif 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2sub-picture of is in       p =         p Local 2dim Languages bordered picture p: tile: a square picture of size (2,2)

  4. 0 0 0 1 0 0 0  1 0 1 0 1  0   0 1 0 0  0 0 0    = 0    0         0 0 1  0 0  0 0 1  1 0  1 0     1 0   p = # # # # # 1 0 0 # 1 0 0 # 0 1 0 # 0 1 0 # p = 0 0 1 # 0 0 1 # # # # # # (Usual) Example of local language Ld = the set of square pictures with symbol “1” in all main diagonal positions and symbol “0” in the other positions

  5. Recognizable 2dim Languages L is recognizable by tiling systemif L=(L’) where L’ is a local language and  is a mapping from the alphabet of L’ to the alphabet of L (, , , ) , where L’=L(), is called tiling system 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

  6. a a a a a a a a a (Usual) Example LSq = all squares over  = {a}. LSq is recognizable by tiling system. Set L’=Ld and (1)= (0)= a 1 0 0 0 1 0  Ld 0 0 1

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

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

  10. Scanning strategies (II) Snake-like Free 4 5 6 7 9 4 1 2 3 11 10 9 8 7 12 13 14 36 6 3 8 36 35 34 5 1 2

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

  15. bb bb ab bb aa aa aa aa p’= bb ab bb bb aa ba ba aa New Example Lcol-1n = {p | first col = last col }{a,b}** Lcol-1n  REC Lcol-1n  DREC Local alphabet: = {xy} Projection “erase” subscripts: (xy) = x Lcol-1i = {p | 1<in , first col = col i} DREC

  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) one Unambiguity • Determinism • Possible accepting computations: 1 • Possible backtracking steps at each step of computation: 0

  17. 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). Unambiguous Recognizable Languages (UREC) L ** is unambiguous if it admits an unambiguous tiling system. UREC: all unambiguous recognizable 2dim languages. UREC  REC Generalization in 2dims of unambiguous automata for strings

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

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

  20. one 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) one dimension of p (m or n) “line”-unambiguity • Determinism • Possible accepting computations: 1 • Possible backtracking steps at each step of computation: 0

  21. Column-Unambiguos Languages (Col-UREC) 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-URECif L has a tiling system that is LR- or RL- column unambiguous. Remark: Backtracking at each step of possibly O(m) steps.

  22. Row-Unambiguos Languages (Row-UREC) 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-URECif L has a tiling system that is TD- or DT- column unambiguous. Remark: Backtracking at each step of possibly O(n) steps.

  23. a b a a b a b b a b b a a a a a b a b b b a b b a (A new) Example = {a, b} LSq-cent-a = odd-side squares with a in center position a0 b1 a0 a2 LSq-cent-a  Col-UREC,Row-UREC LSq-cent-a DREC b0 a0 b0 b1 a0 b2 b0 a0 a0 a1 a0 a0 b0 a2 b0 b1 By an “old” proof by Inoue et al. b1 a2 b0 b0 a0

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

  25. A necessary condition for unambiguity 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. MS is the boolean matrix MS=|a|  *,  * where a= 1 iff   L. The number of different rows , Row(MS ), is finite. Theorem Let L  **. There is a k such that, for all m, If L  Col-UREC then Row(ML(m))  km If L  UREC then RankQ(ML(m))  km Use Matz’s Theorem and Hromkovic et al. Theorem Idea of Proof

  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 Am with km states for L(m). Fact If L  UREC, then Am is an unambiguous automaton with km states for L(m). If L  Col-UREC, then Am is a deterministic automaton with km states for L(m).

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

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

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

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

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

  32. REMARK: Diag-Unambiguos Languages 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-URECif L has a tiling system that is TL-, TD-, BL- or BR- diagonal unambiguous. Remark: NO backtracking at each step

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

  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. riflettere a mente fresca…

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