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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

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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 Overview

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Two-dimensional (2dim) Languages Problem: generalizing formal language theory from 1dim to 2dim Two-dimensional string (or picture) over a finite alphabet: abaab bcabc acbba finite alphabet ** all 2dim strings (pictures) over L ** 2dim language

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From 1dim to 2dim 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 A unifying point of view (G, Restivo, 1992): Recognizability by tiling system (= local language + projection) REC family

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

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

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REC family is defined in terms of local languages It is necessary to identify the boundary of a picture p using a boundary symbol REC family I p = L is local if 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 2 of is in p

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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# #####

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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 REC family II Example: L Sq = all squares over = {a} is recognizable by tiling system. Set L’=L d and (1)= (0)= a REC is the family of two-dimensional languages recognizable by tiling system ( , , , ), where L’=L( ), is called tiling system

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1dim case: from an automaton to a tiling system 0 2 a a b b 1 b b1b1 b2b2 a1a1 a0a0 # # b0b0 Θ =Θ = b0b0 #b0b0 b1b1 b0b0 a1a1 b1b1 b1b1 a1a1 b2b2 b2b2 b2b2 a1a1 a0a0 b2b2 a0a0 b1b1 a1a1 a0a0 b1b1 a0a0 a1a1 b0b0 # b1b1 # a0a0 # L= strings over ={a,b} starting with b and with even occurrences of a w = b a b a a b a w’= # b 0 a 1 b 2 a 0 a 1 b 2 a 0 # ( a 0 ) = ( a 1 ) = a; ( b 0 )= ( b 1 ) = ( b 2 ) =b ;

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w’= # b 0 a 1 b 2 a 0 a 1 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) # 0 2 a a b b b1b1 b2b2 a1a1 a0a0 # 1 b b0b0 1dim case: “Computing” by a tiling system (from a tiling system to an automaton) ##a0a0 w = b a b2b2 b a b a a Θ =Θ = b0b0 #b0b0 b1b1 b0b0 a1a1 b1b1 b1b1 a1a1 b2b2 b2b2 b2b2 a1a1 a0a0 b2b2 a0a0 b1b1 a1a1 a0a0 b1b1 a0a0 a1a1 b0b0 # b1b1 # a0a0 #

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#### 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 2dim case: “Computing” by a tiling system (from a tiling system to an automaton) First, decide a scanning strategy! Recall last computed local symbols!

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“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.

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####### ## ## ## ####### ####### ## ## ## ####### Scanning strategies (I) Diagonal (“OTA”)By column 2 4 3 5 6 1 12 3 4 5 6 13 1415

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####### ## ## ## ## ####### ####### ## ## ## ## ####### Scanning strategies (II) FreeSnake-like 2435 6 1 1 2 3 4 5

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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)

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

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Supports updating operations extract 3 local symbols 1, 2, 3 needed to compute the next local symbol (ex. ) insert new local symbol 11 22 33 Data Structure Depends on the chosen scanning procedure In 1dim automata the data structure saves the current state (local symbol) and updates it. In 2dim

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Tiling Automata (1) A Definition: A tiling automaton (TA) of type tl2br is A =(T, S, D 0, ) where: T = ( , , , ) is a tiling system S is a tl2br-directed scanning strategy D 0 initial content of data structure : ( 1, 2, 3, a) = 4 if Θ and ( 4 )=a 11 22 33 44

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Example ###### # 1000 # # 0100 # # 0aa # # aaaa # ###### 0100##0 (*) = a A Consider a tiling automaton A=(T Sq, S r, D 0, ) where T sq tiling system for L Sq and S r a scanning strategy that goes row by row (from the left to the right) #00#001 0 a 1 * 0 0

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Tiling Automata (2) 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)

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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)

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Languages of Tiling Automata 1)L(TA) = REC = L(OTA) 2)L(DTA) L(DOTA) 3)L(DTA) is incomparable with L(4DFA) 4)L(UTA) L(DTA) L(4DFA) Proposition: The following properties hold

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Sketch proof item 3) 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)

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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

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Grazie

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Complexity issue 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.

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Baciamo le mani!

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

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Θ =Θ = b0b0 #b0b0 b1b1 b0b0 a1a1 b1b1 b1b1 a1a1 b2b2 b2b2 b2b2 a1a1 a0a0 b2b2 a0a0 b1b1 a1a1 a0a0 b1b1 a0a0 a1a1 b0b0 # b1b1 # a0a0 # 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) # 0 2 a a b b b1b1 b2b2 a1a1 a0a0 # 1 b b0b0 1dim case: “Computing” by a tiling system (from a tiling system to an automaton) w’= # b 0 a 1 b 2 a 0 a 1 b 2 a 0 # w = b a b a a b a ## a0a0

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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) # 0 2 a a b b b1b1 b2b2 a1a1 a0a0 # 1 b b0b0 Θ =Θ = b0b0 #b0b0 b1b1 b0b0 a1a1 b1b1 b1b1 a1a1 b2b2 b2b2 b2b2 a1a1 a0a0 b2b2 a0a0 b1b1 a1a1 a0a0 b1b1 a0a0 a1a1 b0b0 # b1b1 # a0a0 # 1dim case: “Computing” by a tiling system (from a tiling system to an automaton) w = b a b a a b a w’= # b 0 a 1 b 2 a 0 a 1 b 2 a 0 #

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“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???)

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Definition: String language L is local if all substrings of length 2 are in a finite set Θ. (L=L(Θ) ) Local (string) languages…(TOGLIERE w with border # # 0 0011 string w over Γ= {0, 1} w=w=0 0011 finite set of strings of length 2 over Γ # allowed substrings Θ = … 0000 0 11 1 ## Sono equivalenti ai regular

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(Usual) Example1DIM case: “Computing” by a tiling system (from a tiling system to an automaton) 0 1 a a b b w= a b b a a b a Θ corresponds to undirected edges! To use Θ for a computation we need to decide a scanning procedure +variable to keep current local symbol (state) b0b0 b1b1 a1a1 a0a0 # # # w’= # a 1 b 1 b 1 a 0 a 1 b 1 a 0 # Θ = b0b0 b0b0 b0b0 b0b0 b1b1 a0a0 # # a1a1 # b0b0 # a1a1 a1a1 a0a0 a1a1 b1b1 b1b1 a0a0 b1b1 b0b0 a0a0 a1a1 a0a0

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A second example Diagonal scanning strategy (“2OTA”) ####### ## ## ## ####### 12 3 4 5 6 13 1415 Filling-all-the-picture Start in a corner Comp. next-position function: (i, j) (i+1, j-1) Mono-directional (tl2br) Contiguity property

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