1 / 35

Tiling Automata: a computational model for recognizable two-dimensional languages

Tiling Automata: a computational model for recognizable two-dimensional languages. Marcella Anselmo Dora Giammarresi Maria Madonia Univ. of Salerno Univ. Roma Tor Vergata Univ. of Catania ITALY. Overview.

yetty
Download Presentation

Tiling Automata: a computational model for recognizable two-dimensional languages

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Tiling Automata: a computational model for recognizable two-dimensional languages Marcella Anselmo Dora Giammarresi Maria Madonia Univ. of Salerno Univ. Roma Tor Vergata Univ. of Catania ITALY

  2. Overview • Topic:recognizability of 2dim languages (picture languages) by finite devices • Motivation:define a computational device for 2dim languages based on tiling systems • Results:definition of tiling automata, comparison of deterministic and non-deterministic models, with OTA, 4NFA

  3. Two-dimensional string (or picture)over a finite alphabet: a b b c a c b a c b b a a b a • finite alphabet •  ** all 2dim strings (pictures) over  • L **2dim language Two-dimensional (2dim) Languages Problem: generalizing formal language theory from 1dim to 2dim

  4. From 1dim to 2dim A unifying point of view (G, Restivo, 1992): Recognizability by tiling system (= local language + projection) REC family Several attempts since 60s: • Automata (4NFA and OTA, AFA, … ) • Logics (monadic second-order, first-order, existential monadic second-order) • Grammars (matrix, image, array, TRG,… grammars) • Operations (column-, row- concatenation, stars, …) Definition of different classes of picture languages

  5. First model by Blum & Hewitt (1967) • Generalization of classical 2-way automata: They can move: Left, Right, Up, Down 4-way automata (4NFA) Transition function (p,a)= (q,d )d{,,,} - The deterministic model is denoted by 4DFA - L(4DFA)L(4NFA) - L(4DFA), L(4NFA) not closed under concatenations and *

  6. OTA: a restricted type of 2dim cellular automata ….computing by diagonal waves q i-1,j-1 q i-1,j q i,j-1 • L(DOTA)L(OTA) / On-line tesselation automata (OTA) : Q  Q  Q   2Q • DOTA if : Q  Q  Q   Q

  7.          p = p =        • L islocalif there exists a set  of tiles (i. e. square pictures of size 22) such that, for any p in L, any sub-picture 22of is in   p REC family I • REC family is defined in terms of local languages • It is necessary to identify the boundary of a picture p • using a boundary symbol

  8. 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

  9. REC family II • 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 Example: LSq = all squares over  = {a} is recognizable by tiling system. Set L’=Ld and (1)= (0)= a • REC is the family of two-dimensional languages recognizable by tiling system

  10. b b0 a0 b1 b0 b0 a0 b2 a1 b1 b2 a1 b1 a1 b1 a0 a1 b2 b2 a1 a0 b1 # b b1 b2 a b a1 b0 # Θ= 1 2 0 # # # # a0 b1 b0 a0 a (a0)=(a1) =a; (b0)= (b1) = (b2)=b; w = b a b a a b a w’= # b0 a1 b2 a0 a1 b2 a0# 1dim case: from an automaton to a tiling system L= strings over ={a,b}starting with b and with evenoccurrences of a

  11. b b0 b0 b1 b0 b2 a0 a1 a0 b2 b1 a1 a1 a1 b2 b1 b1 b2 a1 a0 b1 a0 # b b1 b2 a b a1 # b0 Θ= 1 2 0 # # # # a0 a0 b0 b1 a # # 1dim case: “Computing” by a tiling system (from a tiling systemto an automaton) b w = w’= # b0 a1 b2 a0 a1 b a b a a a a0 b2 Θ corresponds to undirected edges! To use Θ for a computation we need to decide a scanning strategy + variable to keep current local symbol (state)

  12. 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 2dim case: “Computing” by a tiling system (from a tiling systemto an automaton) LSq = squares over {a}. Use L’=Ld(1)= (0)= a # # # # # First, decide a scanning strategy! Recall last computed local symbols! 1 a a 0 a # # 0 a a a # # 0 a a a # # # # # # #

  13. “Computing” by a tiling system (from a tiling system to an automaton) Remark : Tiling system = “undirectional” transitions For a 2dim finite tiling automaton we need Tiling system+scanning strategy+data structure Local picture is the run of the automaton.

  14. Scanning strategies (I) By column Diagonal (“OTA”) 1 2 4 1 4 5 3 13 2 5 3 6 6 14 15

  15. Scanning strategies (II) Snake-like Free 1 2 3 4 5 4 6 3 5 1 2

  16. Scanning strategy We need a “good” one! • Start in a corner • Filling-all-the-picture property • Computable next-position function • Contiguity property • Mono-directional (tl2br or tr2bl or br2tl or bl2tr)

  17. By column scanning strategy Start in a corner # # # # # # # # # Filling-all-the-picture 1 4 # # 2 5 Comp. next-position function: (i, j) (i+1, j) # # 3 6 # # # # # # # Contiguity property Mono-directional (tl2br) An example

  18. 1 2 3 • Supports updating operations • extract 3 local symbols 1,2,3needed to compute • the next local symbol (ex. ) • insert new local symbol Data Structure In 1dim automata the data structure saves the current state (local symbol) and updates it. In 2dim • Depends on the chosen scanning procedure

  19. Definition: A tiling automaton (TA) of type tl2bris A=(T, S, D0, ) where: • T = (, , , ) is a tiling system • S is a tl2br-directed scanning strategy • D0 initial content of data structure • :   (1, 2, 3, a) = 4if  Θ and (4)=a 1 2 3 4 Tiling Automata (1)

  20. 1 0 0 # # 0 0 Example Consider a tiling automatonA=(TSq, Sr, D0, ) where Tsq tiling system for LSq and Sr a scanning strategy that goes row by row (from the left to the right) 0 1 0 (*) = a a 0 *

  21. Similarly define tiling automata of types tr2bl, br2tl,bl2tr L(TA-tl2br)= L(TA-tr2bl)= L(TA-br2tl)= L(TA-bl2tr)= = L(TA) = REC Acceptance defined as usual L(TA- tl2br) Tiling Automata (2)

  22. Tiling Automata (3) • Use standard definitions from string case and define: • Unambiguos Tiling Automata (UTA) • Deterministic Tiling Automata (DTA) L(DTA) =L(DTA-tl2br)  L(DTA-tr2bl)  L(DTA-br2tl) L(DTA-bl2tr)

  23. L(TA) = REC = L(OTA) • L(DTA)  L(DOTA) Languages of Tiling Automata Proposition: The following properties hold • L(DTA) is incomparable with L(4DFA) • L(UTA)  L(DTA)L(4DFA)

  24. Sketch proof item 3) • L(DTA) is incomparable with L(4DFA) Remark : TA are conceptually different from 4NFA in L(4DFA) but not in L(DTA) in L(DTA) but not in L(4DFA) (K. Inoue, A. Nakamura 77)

  25. Conclusions • Tiling Automata necessary: • - to use tiling system as computational devices • - to introduce a “more computational” notion of determinism • Tiling Automata reduce to classical string automata in the • case of one-row pictures

  26. Grazie

  27. Proposition:Deterministic TA with - next-position function in O(1) time, - data structure occupies space O(m+n) and supports the operations in time O(1) parsing in time O(mn) and O(m+n) extra-space. Complexity issue

  28. Baciamo le mani!

  29. Why Tiling Automata? A Tiling System does not correspond to an effective procedure of recognition.

  30. b b0 b2 a1 b2 a1 b1 b0 a0 b0 a0 b1 b1 a0 b1 a1 b2 a1 a0 a1 b1 b2 # b b1 b2 a b a1 # b0 Θ= 1 2 0 # # # # a0 b1 a0 b0 a w = b a b a a b a # # 1dim case: “Computing” by a tiling system (from a tiling systemto an automaton) a0 Θ corresponds to undirected edges! w’= # b0 a1 b2 a0 a1 b2 a0 # To use Θ for a computation we need to decide a scanning strategy + variable to keep current local symbol (state)

  31. b b0 b0 a1 b2 a1 b0 b1 a0 a0 b1 b2 a1 b1 a1 b1 a1 b2 a0 a0 b1 b2 # b b1 b2 a b a1 # b0 Θ= 1 2 0 # # # # a0 b1 a0 b0 a w = b a b a a b a w’= # b0 a1 b2 a0 a1 b2 a0# 1dim case: “Computing” by a tiling system (from a tiling systemto an automaton) Θ corresponds to undirected edges! To use Θ for a computation we need to decide a scanning strategy + variable to keep current local symbol (state)

  32. “Computing” by a tiling system (from a tiling system to an automaton) Remark : Tiling system = “undirectional” transitions For a 2dim finite tiling automaton we need Tiling system+scanning strategy+data structure Local picture is the run of the automaton. Remark : All 2dim tiling automata “correspond” to family REC (i.e. scanning procedure does not matter!) BUT it is necessary to define determinism (= backtracking 0, where???)

  33. w with border # 0 1 0 0 1 # 0 0 1 0 0 1 Θ = … 0 1 # # Local (string) languages…(TOGLIERE string w over Γ= {0, 1} w= 0 1 0 0 1 finite set of strings of length 2 over Γ # allowed substrings Definition: String language L is local if all substrings of length 2 are in a finite set Θ. (L=L(Θ) ) Sono equivalenti ai regular

  34. b0 b1 b0 a1 # # a1 a1 a1 b0 b0 b0 b1 Θ = # b1 b1 a1 b1 a0 a0 a0 a0 b0 a1 # a0 b0 a0 # # 1DIM case: “Computing” by a tiling system (from a tiling system to an automaton) (Usual) Example a b b # 1 0 a w= a b b a a b a Θ corresponds to undirected edges! w’= # a1 b1 b1 a0 a1 b1 a0# To use Θ for a computation we need to decide a scanning procedure +variable to keep current local symbol (state)

  35. Diagonal scanning strategy (“2OTA”) Start in a corner # # # # # # # # # 1 2 4 Filling-all-the-picture # # 5 3 13 Comp. next-position function: (i, j) (i+1, j-1) # # 6 14 15 # # # # # # # Contiguity property Mono-directional (tl2br) A second example

More Related