Recognizable Infinite Triangular Array Languages

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1 IOSR Jounal of Mathematics (IOSR-JM) e-issn: , -ISSN: X. Volume 14, Issue 1 Ve. I (Jan. - Feb. 2018), PP Recognizable Infinite iangula Aay Languages 1 V.Devi Rajaselvi, 2.Kalyani Reseach Schola,Sathyabama Institute of Science and echnology,chennai-119,india. Deatment of Mathematics,St.Joseh s Institute of echnology,chennai-119,india. Coesonding Autho: 1 V.Devi Rajaselvi Abstact: In this ae we extend to infinite tiangula aays the concet of tiangula domino systems and tiangula tiling systems which ecognize infinite tiangula aays and show that the class of -tiangula aay languages ecognized by tiangula domino systems is same as the class of -tiangula aay languages ecognized by tiangula tiling systems. Also we intoduce tiangula Wang systems and ove that the class of -tiangula languages obtained by tiangula Wang systems is the same as the class of ecognizable -tiangula languages Keywods: Infinite tiangula domino systems,wang ecognizable infinite tiangula aay languages, labelled tiangula Wang tile Date of Submission: Date of accetance: I. Intoduction Infinite tiangula ictues ae the digitized images which occu in the two dimensional lane. Infinite tiangula ictue is a tiangula aay of elements of alhabets. It is useful to intoduce the notation Σ fo the set of all infinite tiangula ictues ove the same alhabet. Infinite tiangula ictue has infinite numbe of ows and infinite numbe of ight slanting lines. he size of a ictue is secified by the ai ( ow, sline ) of its numbe of ows and numbe of ight slanting lines. A ixel (i, j), 1 i ow, 1 j sline is the element at osition (i, j) in the tiangula aay P. Conventionally the indices gow fom bottom to to fo the ows and fom left to ight fo ight slanting lines. Fo convenience we usually conside the bodeed vesion of ictue obtained by suounding the ictue with the secial bounday symbol # which is assumed not to be in the alhabet. In [4] domino ecognizability of tiangula ictue languages and hl-domino systems ae defined. Also in [4] we define the ovelaing of iso-tiangula ictues. In this ae we extend to infinite tiangula aays the concet of tiangula domino systems and tiangula tiling systems which ecognize infinite tiangula aays and show that the class of -tiangula aay languages ecognized by tiangula domino systems is same as the class of tiangula aay languages ecognized by tiangula tiling systems. In [1,2] we study about infinite aays.in [5] we study the Wang systems fo ectangula ictues. Hee we intoduce tiangula Wang systems and ove that the class of -tiangula languages obtained by tiangula Wang systems is the same as the class of ecognizable -tiangula languages. II. Peliminaies We will see some definitions of infinite ectangula aays [20]. Definition 2.1 A tiling system = (,,, ) whee and ae two finite alhabets, is a finite set of tiangula tiles ove {#} and : is a ojection fom to. he tiling system ecognizes an - language L ove if L = (L ) whee L = L ( θ) is the local - language ove coesonding to the set of tiles. We denote by L (S), the family of - languages ecognized by tiling systems. his family is also denoted by -REC. Definition 2.2: A labelled Wang tile is a 5-tule, consisting of 4 colous,choosen in a finite set Q of colous and a label chosen in a finite alhabet. Definition 2.3: DOI: / Page

2 Recognizable Infinite iangula Aay Languages A Wang system is a tilet W=(,Q,) whee is a finite alhabet,q is a finite set of colous and is a finite set of Wang tiles, Q 4. Definition 2.4: Let W=(,Q,) be a Wang system. An infinite aay M ove is a tiling of W,if it satisfies the following conditions. 1. M(1,1) = M(m,1) = 2. M(m,n) = B B s a B c a Hee,,, s B., M(1,n) =, m = 2, 3,..., m, n = 2, 3,... b B, n = 2, 3,... If M is a tiling of W, the label of M, denoted by M, is an infinite aay ove,defined by M (m,n)=a M(m,n) = s a, fo some,,,s. Definition 2.5: Let W is a Wang system. An infinite aay w is geneated by W if thee exists a tiling M such that M = w. We denote by L (W ),the language of infinite aays geneated by the Wang system W. Definition 2.6: An infinite aay language L is Wang ecognizable if thee exists a Wang system W such that L = L (W). he family of all Wang ecognizable infinite aay languages is denoted by Definition 2.7: L (WS). Let M = (Q,,, I, F) be a two diection online tessellation automata (2DOA). We say that L ecognized by M in (inf, )-mode if L = { L = L (M). Σ is Σ : inf(()) F fo some un ()} and we wite III. Recognizability Of -iangula Languages Definition 3.1: A tiangula tiling system = (,,, ) whee and ae two finite alhabets, is a finite set of tiangula tiles ove {#} and : is a ojection fom to. he tiangula tiling system ecognizes an -tiangula aay language L ove if L = (L ) whee L = L ( θ) is the local - tiangula aay language ove coesonding to the set of tiles. We denote by L (IS), the family of -tiangula aay languages ecognized by tiangula tiling systems.his family is also denoted by - REC. heoem 3.1 L (IDS) L (IS) DOI: / Page

3 Fo oving this theoem we need to ove the following oositions. Poosition 3.1. Clealy L (IDS) L (IS). o ove the convese at we need the following oosition. Poosition 3.2. he family L (IS) is closed unde ojection. Let 1, 2 be two alhabets. Let : 1 2 be a ojection. Let L Σ be ecognized by the tiangula tiling system. 1 = ( 1,,, 1 ) whee L = 1 (L ) and L = L ( θ) Conside the tiangula tiling system 2 = ( 2,,, 2 ) whee 2 = 1 : 2. Now (L) = ( 1 (L )) = ( 1 )(L ) = 2 (L ). heefoe (L) is ecognized by the tiangula tiling system 2. Hence L (IS) is closed unde ojection. Recognizable Infinite iangula Aay Languages Poosition 3.3. If L is a local -tiangula language ove, then thee is an hl-local -tiangula language L ove an alhabet and a maing : such that L = (L ). Let L = L ( θ) whee is a finite set of tiangula tiles ove {#}. Let = and Let L = L ( Δ). hen L is an hl-local -tiangula language ove. Define : by hen L= (L ). a 4 = a a 1 a 4 2 a 3 Combining oositions we have the following esult. DOI: / Page

4 Recognizable Infinite iangula Aay Languages IV. Labeled Wang iles In this section we use a diffeent fomalism to ecognize -tiangula ictue languages. Wang tiles ae intoduced fo the tiling of Euclidean lane. De Pohetics and Vaicchio [5] have intoduced the notion of Wang tile by adding a label taken as a finite alhabet, to a Wang tile.hey have also defined Wang systems and oved that the family of aay languages ecognized by Wang systems coincides with the family of ecognizable aay languages. We extend the concet of -languages to -tiangula aay languages and ove that the family of -tiangula aay languages ecognized by tiangula Wang systems is the same as the family of ecognizable -tiangula aay languages. Definition 4.1: A labelled tiangula Wang tile is a 4-tule consisting of thee colous chosen in a finite set Q of colous and a label chosen in a finite alhabet. Definition 4.2: A tiangula Wang system is a tile W = <, Q, > whee is a finite alhabet, Q is a finite set of colous, is a finite set of tiangula Wang tiles, Q 3. Definition 4.3: Let W = <, Q, > be a tiangula Wang system. An infinite tiangula aay M ove is a tiling of W, if it satisfies the following conditions: ove, defined by Definition 4.5: DOI: / Page

5 Recognizable Infinite iangula Aay Languages Let W be a tiangula Wang system. An infinite tiangula aay w is geneated by W if thee exists a tiling M such that M = w. We denote by L (W), the language of infinite tiangula aays geneated by the tiangula Wang system W. Definition 4.6: An infinite tiangula aay language L is Wang ecognizable if thee exists a tiangula Wang system W such that L = L (W). he family of all Wang ecognizable infinite tiangula aay languages is denoted by L (IWS). heoem REC L (IWS) Fo oving this theoem we need to ove the following oositions. Poosition 4.1. L (IWS) is closed unde ojection. Let W = (,Q,) be a tiangula Wang system and : be a ojection. We have to show that if L = L (W), then L = (L) = L (W ) fo some W. Let W = (, Q, ) whee = (a) : a (b) : b hen L = L (W ). Poosition 4.2. L (IWS) - REC. Let L L (IWS) and L= L (W) whee W = (, Q, ). Let = and = whee DOI: / Page

6 Recognizable Infinite iangula Aay Languages Let L 1 = L (θ). Define : by = a. hen L = (L 1 ) and theefoe L -REC. Poosition REC (WS). L Let L -REC. hen thee exists a local -tiangula aay language L 1 ove and a ojection : such that L = (L 1 ). Let L 1 = L (θ) whee is a finite set of tiangula tiles ove {#}. Conside the tiangula Wang system W = <, Q, > whee Q = ( {#} 2 ) {B} whee B {#} = whee DOI: / Page

7 hen L = L (W). hus L L (IWS) and theefoe - REC L (IWS). Combining oositions 5.3.1,5.3.2 and we have the esult. Recognizable Infinite iangula Aay Languages V. Automata Chaacteization Of Recognizable Infinite iangula Aay Languages In this section we give automata chaacteization of ecognizable infinite tiangula aay languages. Definition 5.1: Let M = (Q,,, I, F) be a two diection online tessellation automata (2DOA). We say that L Σ is ecognized by M in (inf, )-mode if L = { L = L (M). Σ : inf(()) F fo some un ()} and we wite heoem 5.1. L -REC if and only if L is ecognized by a 2DOA in (inf, )-mode in which evey state is a final state. Let L -LOC. hen thee exists a finite set of tiangula tiles such that L = L (θ). Conside the 2DOA, M = (Q,,, 0, F) whee Q = a a 0 = : # # # # # and : Q Q 2 Q such that # a 22 a 12 a 13 a 14, a 21 a 13 a 11 a 12, x = x a 21 a 22 a : 23 x a 21 a 22 a 23 hen L = L (M). Since the languages ecognized by 2DOA is closed unde mohism, evey L - REC is ecognized by a 2DOA with F = Q. Convesely let L be ecognized by a 2DOA M = (Q,,, 0, F). Let = Q ( {#}), 1 = (, a) ( 0, #) ( 0, #) ( 0, #) ( 0, 0, a) 2 = 3 = ( 0, #) ( 0, #) ( 0, #) (, a) 0 ( 0, 0, #) (, b) (, a) (, c) (t, d) (,, a) and = Let L 1 = L (θ). Define : by (a, ) = a. hen L = (L 1 ). heefoe L -REC. Remak 5.1. If L is ecognized by a 2DOA M = (Q,,, 0, F), then we can show that L is mohic image of an hllocal -language. Using the notation of theoem let 1 = DOI: / Page

8 Recognizable Infinite iangula Aay Languages VI. Leaning Of Recognizable Infinite iangula Aay Languages In this section we give leaning algoithm fo ecognizable infinite tiangula aay languages. In [3], we have given an algoithm to lean hl-local -tiangula aay languages in the limit fom ositive data that ae ultimately eiodic aays. We have oved that an -tiangula aay language is ecognizable if and only if it is a ojection of a hl-local -tiangula aay language. We now show how to deive a leaning algoithm, fo ecognizable -tiangula aay languages, fom one that lean hl-local -tiangula aay languages. Let L -REC. hen L is ecognized by some 2DOA, M = (Q,,, 0, F). Let = Q ( {#}) and let 1 and 2 be ojections on defined by 1 (, a) = and 2 (, a) = a. A tiangula aay ove is called a comutation descition aay if 1 () is an acceting un of M on 2 (). We note that 1. he alhabet contains n(m+1) elements, whee n is the numbe of states of minimum 2DOA fo L and m =. 2. Fo any ositive examle of L, let C() the set of all comutation descition tiangula aays fo. hen C() has atmost n A() tiangula aays. 3. If U is an hl-local -tiangula language ove such that (U)=L and E is a chaacteistic samle fo U, then thee is a finite set S L of ositive data of L such that E -1 (S L ). We obtain a leaning algoithm fo the class -REC. Algoithm REC Inut: A ositive esentation of an unknown ecognizable -tiangula aay language L, n = Q fo the minimal 2DOA which accets L. Outut: A finite set fo dominoes such that L= ( L ( ) ) DOI: / Page

9 Recognizable Infinite iangula Aay Languages Pocedue: o ove the leaning algoithm REC teminates we need to ove the following esult. Poosition 6.1 Let n be the numbe of states fo the 2DOA ecognizing the unknown ecognizable -tiangula aay language L. Afte atmost t(n) numbe of ueies, Algoithm REC oduces a conjectue i such that E i includes a chaacteistic samle fo an hl-local -tiangula language U with the oety that L = (U) whee t(n) is a olynomial in n, deending on L. Let L be an unknown ecognizable -tiangula aay language. Let M = (Q,,, 0,F) be a 2DOA which accets L.Let Q = n and = m.hen thee is an hl-local -tiangula aay language U ove =Q ( {#}) and a ojection such that (Lʹ) = L. Let E be a chaacteistic samle fo U. We have aleady oved that U=U 1 U 2 whee U 1 and U 2 be two local hl -tiangula languages and E = E 1 E 2 ae chaacteistic samles fo U 1 and U 2 esectively,saoudi and Yokomoni [1] have mentioned that the lengths of all stings in E 1 and E 2 ae not moe than 3m 2 1 and 3m 2 2 esectively,whee m 1 = Q 1 2 and m 2 = Q 2 2 (efe 2 lemma 15 in [1] ).heefoe the sizes of the aays in E =E 1 E 2 ae (i,j) whee 1 i 3m 2 and 1 j 3m 2 1.Now we can find a finite set of ositive data S L of L such that E -1 (S L ).Since is aea eseving,the aeas of all the aays in S L ae not moe that 9m 2 1 m 2 2.Let S L ={w 1,w 2,...w } and l=max{a(w 1 ),...A(w )}.hen the numbe of comutation descitions in -1 (S L ) is atmost 1 n... n A( w ) A( w2 ) m l =t(n).note with at most t(n) numbe of ueies,algoithm REC finds a finite set E i of ositive data of U with the oety that E i of ositive data of U with the oety that E i includes a chaacteistic samle fo U and (U)=L. Summaizing we obtain the following esult. heoem 6.1. Given an unknown ecognizable -tiangula aay language L, Algoithm REC leans fom ositive data and sueset ueies, a finite set of dominoes such that L= (L ( )) DOI: / Page

10 Recognizable Infinite iangula Aay Languages Refeences [1]. Ahmed Saoudi,akashi Yokomoni,Leaning Local and Recognizable Omega languages and Monadic Logic Pogams, oceedings of the fist Euoean confeence on comutational leaning theoy,ages ,Octobe 1994, Royal Holloay Univ of Londan,United Kingdom. [2]. Dae V.R., K.G.Subamanian, D.G.homas and R.Siomoney, Infinite aays and ecognizability, Int.J.Patten Recognition and Atificial intelligence 14(2000), [3]. V.Devi Rajaselvi,.Kalyani, D.G.homas HRL-Local infinite tiangula ictue languages" Published in Bazillian Achives of Biology and echnology vol. 59, n.se. 2 e ,januay-decembe 2016,ISSN [4]. V.Devi Rajaselvi,.Kalyani, D.G.homas, Domino Recognizability of tiangula ictue languages, Intenational Jounal of Comute Alications Vol.57, No.15, (ISSN ), Nov [5]. Pohetis L.De and S.Vaicchio.Recognizability of ectangula ictues by Wang systems.jounal of Automata,Languages and Combinatoics,4: ,1997. [6]. homas W., Automta on infinite objects, in: J. VunLeeuauthosn (Ed.), Handbook of heoetical Comute Science, vol. B, Elsevie, Amstedam, , V.Devi Rajaselvi. Recognizable Infinite iangula Aay Languages. IOSR Jounal of Mathematics (IOSR-JM), vol. 14, no. 1, 2018, DOI: / Page