Combinatorial remarks on two-dimensional Languages

Transcription

1 Combinatorial remarks on two-imensional Languages Franesa De Carli To ite this version: Franesa De Carli. Combinatorial remarks on two-imensional Languages. Mathematis [math]. Université e Savoie English. <tel > HAL I: tel Submitte on Sep 2009 HAL is a multi-isiplinary open aess arhive for the eposit an issemination of sientifi researh ouments whether they are publishe or not. The ouments may ome from teahing an researh institutions in Frane or abroa or from publi or private researh enters. L arhive ouverte pluriisiplinaire HAL est estinée au épôt et à la iffusion e ouments sientifiques e niveau reherhe publiés ou non émanant es établissements enseignement et e reherhe français ou étrangers es laboratoires publis ou privés.

2 Combinatorial remarks on Two-Dimensional Languages Franesa De Carli

3 Contents Introution 3 Two-imensional languages 7. Loal Languages Tiling System reognizable languages Logi Formulas an Loal Languages First Orer Sentenes Logi Formulas for Loal Languages Existential Seon Orer Formulas Regular Expressions On-line Tesselation Automata Grammars Loal languages an algebrai strutures The lattie Lo n The simplest ase: the lattie Lo The general ase Meet-irreuible elements an oatoms in Lo n Join-irreuible elements Some uneiable problems The lattie Lo Chains The lattie Lo h Further Works an Open Problems Two-imensional languages an omputability 48

4 4 Tiling reognizability of various lasses of polyominoes Polyominoes Polyominoes an tiling systems Constrution of L-onvex polyominoes using tiling system Two-imensional languages an DNA-omputation Algorithm to transform Tiling Systems into labele Wang Tiles Convex polyominoes onstrute on labele Wang tiles Example of Parallelogram polyomino on labele Wang tiles Form labele Wang tiles to DNA Wang tiles

5 Introution The stuy of two-imensional languages is a researh topi whih reently aquainte some interest in ifferent fiels of mathematis an omputer siene. Sine 967 there has been an attempt to exten results an tehniques use on string (one-imensional) languages to two-imensional ase efining the two-imensional languages or piture languages. Starting from a finite alphabet Σ we all pitures or two-imensional wors over Σ the elements of Σ = nm (Σn ) m where we ientify an element p = p... p m of (Σ n ) m with the matrix p = (q ij ) of size m n. We all piture language over Σ any subset L Σ. One of the most interesting an simple lass of piture languages in pratie the two-imensional ounterpart of string regular languages is that of tilingsystem reognizable languages. This thesis is eiate to the stuy of some theoretial algebrai an ombinatorial properties of suh languages. There are several reasons to etermine an analogy between tiling-system reognizable languages an regular string languages. As a matter of fat the various known haraterizations of tiling system reognizable languages are very similar to the lassi haraterizations of regular languages: they are the languages reognize by partiular finite state automata the online tesselation automata; or the languages oiniing with ertain regular expressions; or also efinable by existential seon orer formulas. The haraterization that we will use in the thesis is that one propose by Giammarresi an Restivo in [33]. It is base on the fat that reognizable string languages an be haraterize in terms of loal string languages an alphabeti projetions. The same iea is applie to the two imensional languages:. it is given the efinition of loal languages over an alphabet Σ an it is ae the symbol. A language L Σ is sai loal if there exist a 3

6 set Θ of pitures over Σ {} of size 2 2 suh that L onsists exatly of pitures p suh that all the sub-bloks of size 2 2 of p (obtaine surrouning p with ) are in Θ. In this ase we say L = L(Θ). 2. A language L Σ is sai tiling-system reognizable if there exists a loal language L over Γ an a projetion π : Γ Σ suh that π(l ) = L. The stuies have pointe out also some ifferenes between the lasses of regular string languages an tiling system reognizable languages. To begin with there are some losure properties in one-imensional ase whih o not hol in two-imensional ase for instane the family of regular string languages is lose uner omplementation instea the family of tiling system reognizable languages that we enote by REC has not this property. Subsequently there have been some works ([2 6 47]) eiate to the stuy of the lass REC o-rec of reognizable languages or suh that their omplements are reognizable an the lass UREC of piture languages that amits an unambiguous tiling system [2] (that is the languages of pitures whih have a unique ounter-image in the loal language). In those works it was prove a strit inlusion among the lasses REC o-rec REC an UREC preisely we have: U-REC REC REC o-rec. Moreover in [47] Matz esribes a tehnique for showing that a piture language is non-reognizable; in this paper he gives a neessary onition satisfie by tiling reognizable piture languages an he poses the problem of fining a non-reognizable piture language for whih his tehnique for proving the non-reognizability fails. In this thesis we investigate the possibility of using two-imensional languages to reognize partiular lasses of isrete objets espeially polyominoes. The onnetion oul be interesting sine polyominoes are simple an important isrete strutures that appear in several problems relate to theoretial omputer siene an isrete mathematis; moreover the stuy of polyominoes has prove a fertile topi of researh. We reall that in the Cartesian plane Z Z a ell is a unit square; a polyomino is a finite onnete union of ells. Polyominoes are firstly stuie by Golomb [36] an ivulge by Garner [29]. They are relate to many problems as enumeration ([7]) tiling ([37 35]) an isrete tomography. In orer to simplify these problems several lasses of polyominoes have been efine using the notions of onvexity an irete growth. 4

7 Sine polyominoes an be naturally represente by pitures languages it is natural to ask if the several lasses of polyominoes are tiling reognizable or not. Reinhart in [54] proves that the piture language whih represents the lass of the polyominoes is tiling reognizable. Moreover as we will see later in etails also the lasses of onvex olumn-onvex parallelogram an irete-onvex polyominoes an be reognize by tiling system reognizable languages [24 25]. Later in orer to stuy the borerline between reognizability an non reognizability (by a tiling system) in the theory of piture language other authors ([60]) fae the problem by stuying the family of L-onvex polyominoes an some lose families stritly relate both to the tomographial haraterization of L-onvexity an to the reognizable family of all polyominoes. They prove that the family of L-onvex polyominoes satisfies the neessary onition given by Matz for the reognizability an they onjeture that the family of L-onvex polyominoes is non-reognizable. Another fiel of researh in whih two-imensional languages reently took relevane is that one of DNA omputation. The DNA omputing using self assembling of DNA oligo-nueoties is a very ative omain an the transfer of onepts from theoretial omputer siene to nano struture is a hallenge in orer to onstrut biomoleular mahines [23 6]. The works in this fiel have shown how information an algorithms an be enoe in biohemial systems an some effiieny problems eserve interest for example what kins of shapes an patterns an be assemble using a small number of tiles or how quikly they an be assemble. Of late several works pointe out the onnetion between partiular lasses of two-imensional languages an the DNA omputation. More preisely it seems possible to onstrut with DNA some pitures or better some two imensional languages. In [27] the authors show an algorithm to translate a tiling system in a set of Wang tiles (Wang tiles are a set of squares in whih eah ege of eah tile is olore; mathing olore eges are aligne to tile the plane). In [6] it is shown how to onstrut with DNA an assembling of Wang tiles. In our thesis we use these results with the aim to show that the isrete objets whih an be reognize by tiling systems reognizable languages an be onstrute with DNA. Another aspet we have treate in the thesis is the stuy of the algebrai struture of loal languages. These languages form a lattie with the inlusion 5

8 relation. To simplify the subjet we ivie the problem in easier problems: first we stuie the algebrai struture limiting the alphabet of the loal languages to one symbol then we passe to the alphabet of two symbols an finally we extene the results to the general ase i.e. to the loal languages over an alphabet of n symbols. Moreover we have eepene some lassi omputability problems whih have been solve for string languages but have not been yet onsiere for the two-imensional ase. As usual in that fiel we are about the eiability/ineiability of some problems regaring loal languages an tiling system reognizable languages. For instane we onsier the emptiness problem. Suh a problem asks to establish if a language is empty; this problem is eiable for regular string languages but it is not for loal languages (an onsequently for tiling system reognizable languages). In the thesis we show that this problem is even Σ 0 -omplete an we plae in the arithmetial hierarhy the other ommon problems. Overview of the thesis Chapter ontains preliminaries on two-imensional languages we give a brief review of the main results an the ifferent haraterizations of tiling system reognizable languages whih play the entral role in the thesis. In Chapter 2 we esribe the algebrai struture of the families of loal languages. We show that this struture is a lattie with respet to the inlusion an we investigate the properties of the lattie. In Chapter 3 we eal with omputational problems stuying their eiability. Moreover we give the position in the arithmetial hierarhy of the lassial problems on string languages now turne to two-imensional languages. In Chapter 4 after some basi efinitions onerning polyominoes we eal with the reognizability of several lasses of polyominoes by tiling system reognizable languages. In partiular we give the tiling systems for languages representing some lasses of onvex polyominoes as the h-onvex or parallelogram. Moreover we investigate the reognizability of L-onvex polyominoes. Finally Chapter 5 is evote to the appliation of tiling system reognizable languages to DNA omputation. We give the iea about the onstrution with DNA of some lasses of polyominoes (i.e. the lass of parallelogram polyominoes) get through to the family of REC. 6

9 Chapter Two-imensional languages This hapter ontains the basi efinitions an notations about two-imensional languages. In partiular we give the efinitions of the most important lasses of two-imensional languages use in our work (that are the loal languages an the tiling system reognizable languages) an their ifferent haraterizations (for all the etails see [ ]). We reall that for all set X it is efine X = n ω Xn where ω is the set of natural numbers. Given an alphabet (that is a finite non empty set) Σ the elements of Σ are wors (or strings) over Σ. If u Σ is a wor then we say length of u the minimum n suh that u Σ n. On Σ it is possible to efine the binary operation onatenation so that Σ with the onatenation is a monoi with ientity λ the string of length 0 or empty string. We say that any subset L Σ is a language over Σ. In [33] the authors exten these notions to the biimensional ase. We apply the same notations to the rest of the thesis. Definition Let Σ be a finite alphabet a two-imensional string or piture over Σ is a retangular array of elements of Σ. p = p... p n p m... p mn. The set of all the pitures over Σ is enote by Σ. language over Σ is a subset of Σ. A two-imensional 7

10 Definition 2 Let p Σ we enote by ˆp the piture obtaine by surrouning p with a speial symbol. (That is ˆp is a wor of (Σ { }) ) ˆp =... p... p n... p m... p mn.... Definition 3 Let p be a piture with m rows an n olumns we say p has size (m n). A blok (or subpiture) of p is a submatrix of p formally it is a piture p suh that if p has size (m n ) then m m n n an there exist h k N suh that an 0 i m 0 j n k n n h m m p (i j) = p(i + h j + k). Let p Σ has size (m n) h m an k nwe enote by B hk (p) the set of all possible bloks of p that have size (h k). Example Given Σ = {0 } onsier the following piture p over Σ: p = ; an p = We give some examples of subpitures of p:. a = b = 0 0 =

11 where a B 32 (p) b B 22 (p) an B 23 (p). But is not a subpiture of p. = Now we an introue two partial operations between pitures. Definition 4 The row onatenation between p an q (p q) is a partial operation whih is efine if p an q have the same number of olumns. In this ase we have: p q = p... p n p m... p mn q... q n q m... q m n. Analogously we an efine the olumn onatenation between p an q if they have the same number of rows (an it is enote by p ɵ q). We an exten the efinitions of onatenation above introue as in the uniimensional ase to pitures languages. Definition 5 Let L L 2 be two-imensional languages over an alphabet Σ. The row onatenation between L an L 2 (enote by L L 2 ) is efine as follows: L L 2 = {p q p L e q L 2 }. Similarly we efine the olumn onatenation between L an L 2 (enote by L ɵ L 2 ). We an also efine the transitive losure of the onatenation operations by iterating the operations (the analogous of the Kleene star operation). 9

12 Definition 6 Let L be a two-imensional language. The row losure of L (enote by L ) is efine as: L = i 0 L i where L 0 = λ L = L L n = L L (n ). Analogously we have the olumn losure of L (enote by L ɵ ).. Loal Languages Now we introue a lass of piture languages alle the loal languages. Definition 7 Let Σ be a finite alphabet a two-imensional language L Σ is loal if there exists a finite set Θ of bloks of size (2 2) whih we all a set of tiles over Σ { } suh that L = {p Γ B 22 (ˆp) Θ}. In this ase we write L = L(Θ) an we say that Θ is a rapresentation by tiles of the loal language L = L(Θ). We give two examples of loal languages whih will be realle later in the thesis. Example 2 Given Σ = {a b } onsier the following set of tiles Θ over Σ: a a a a a b Θ = b b b a a b b a b b b b. 0

13 This set Θ is a representation by tiles of the two-imensional language of pitures with three rows over Σ where the first row ontains only the symbol a the seon the symbol b an the thir the symbol. An example of piture of L(Θ) is: a a a a p = b b b b Example 3 Given Σ = {0 } an Θ = This set of tiles is a representation by tiles of the two-imensional language of the pitures with 0 an arrange as the blak an white of a hessboar. Consier: This piture is in L(Θ). We reall that in the uniimensional ase the loal string languages are efine in analogous way through 2 tiles. Informally a (string) language is loal if it ontains strings suh that eah substring of length 2 is ontaine in a ertain set of 2 tiles. The loal (string) languages are a sublass

14 of regular languages as known for the literature an showe in the following lemma (Lemma ) for whih we give a simple iret proof. Notation. Let us use the following notations: - by writing [awe mean the set of two tiles { a a } ; the meaning of a] is similar; - by writing ab we mean the set of two tiles { a b a b Lemma Eah string languages is regular. Proof. Let L = L(Θ). Consier the following noneterministi finite state automaton with three states q q 0 q an state transition funtion δ: q is the initial state; a a transition δ(q i) = q i if an only if [i Θ; q i is an aepting state if an only if i] Θ }. a a transition δ(q i j) = q j if an only if ij Θ. It is easy to see that the automaton M aepts the language L..2 Tiling System reognizable languages In this paragraph we introue a new lass of piture languages the lass of tiling system reognizable languages. Definition 8 Let p Σ be a piture with size (m n) Γ a finite alphabet an π : Σ Γ a surjetive funtion. The projetion of p by π is the piture p Γ suh that p (i j) = π(p(i j)) for all i m j n. Definition 9 Let L Σ be a two-imensional language Γ a finite alphabet. The projetion of L by π : Σ Γ is the language L = {p Γ p = π(p) p L} Γ. 2

15 Definition 0 A tiling system (briefly TS) is a 4-tuple T = (Σ Γ Θ π) where Σ Γ are finite alphabets Θ is a finite set of tiles over Γ { } an π : Γ Σ a projetion. Definition We say that a tiling system T reognizes a language L Σ if there exists a projetion π : Γ Σ an a finite set of tiles Θ over Γ where L = L(Θ) suh that L = π(l ). In this ase we write L = L(T ). We say L(Θ) is an unerlying language for L(T ). A language L Σ is tiling system riognizable (briefly TS-riognizable) if there exists a tiling system T = (Σ Γ Θ π) suh that L = L(T ). Example 4 We give an example of a tiling system reognizable language. Consier the set of tiles Θ of Example 3 an the projetion π : {0 } {a} that maps the symbols 0 in a (π(0) = π() = a). Then the tiling system T = {a} {0 } Θ π reognizes the language of all square pitures over {a}. Theorem Eah loal language is a tiling system reognizable language. Proof. Let L(Θ) be a loal language. That language is a tiling system reognizable language it suffies to onsier Σ = Γ L(Θ) as unerlying language the ientity as projetion: π : Σ Σ. So we have T = (Σ Γ L(Θ) π) with L(Θ) = L(T ). Thus the thesis follows. Corollary The family of loal languages is ontaine stritly into the family of tiling system reognizable languages. In orer to show the valiity of the orollary we give an example of a loal language that is not a tiling system reognizable language. Example 5 We reall the loal language L(Θ) showe in Example 2. Let Γ = {} π : Σ Γ be the projetion given by: π(a) = π(b) = π() = ; onsier T = (Σ Γ L(Θ) π) then we have that the language of only three rows is tiling system reognizable. Observe that L(T ) is not loal. To prove this assertion let us proee by ontraition an suppose that there exists a set of tiles Θ suh that L(T ) = L(Θ ). Then we an obtain the wor p = 3

16 whih belongs to L(Θ ) an the wor p = whih also belongs to L(Θ ) too: B 22 (p ) = B 22 (p) Θ. But p oes not belong to the language L(T ) whih is a ontraition. Later in the thesis we will use the following efinition an lemma: Definition 2 Let Θ be a finite set of tiles we say Θ is irreunant if it ontains tiles whih are use at least one time in a wor of the language L(Θ). Lemma 2 Let Θ an Θ 2 be two finite set of tiles if Θ Θ 2 then L(Θ ) L(Θ 2 ). If Θ an Θ 2 are irreunant then also the vieversa is true. Proof. ( ) Let Θ Θ 2 p L(Θ ) then B 22 ( p) Θ Θ 2 an so p L(Θ 2 ). ( ) Let Θ Θ 2 be irreunant sets an L(Θ ) L(Θ 2 ). If t Θ then there exists p L(Θ ) whih belongs also to L(Θ 2 ) then t Θ 2. Now we give an example of an irreunant set of tiles an of a reunant one. Example 6 Reall the set Θ of the Example 3 Θ is irreunant. Consier the set Θ obtaine aing to Θ the tile: this tile oes not math with any other tile in Θ then it will not use in any wor of L(Θ ). Then Θ is reunant. Observe that the two languages L(Θ) an L(Θ ) oinie. We reall that in the uniimensional ase the family of languages reognize by finite-state automata oinies with the one efine by means of regular expressions (Kleene s theorem) with the one efine in terms of seon orer formulas (Buhi s theorem) an with the one generate by partiular grammars. In the following subsetions we show that there is an equivalent situation for two imensional ase. 4

17 .3 Logi Formulas an Loal Languages In this setion we propose a ifferent haraterization for the lasses of twoimensional languages just introue; a haraterization through logi formulas. For a start we give some basi efinitions an we show how to represent a loal language by means of logi formulas..3. First Orer Sentenes Definition 3 Let Σ be a finite alphabet an p be a a piture of Σ of size (m n). We an represent p as follows: where p = (om(p) S S 2 (P a ) a Σ ) om(p) = { 2... m} { 2... n} is the set of the positions where the elements of p lay; S an S 2 are the suessor relations for the omponents of the points belonging to om(p) that is: (i j)s (i + j) an (i j)s 2 (i j + ) for i < m j < n; P a = {(i j) p(i j) = a} a Σ is the set of the points of om(p) whih are labele with a. We an mule symbols of preiates with the interpretation relations so that the atomi formulas are of the kin xs y xs 2 y P a (x) an are naturally interprete as follows: xs y xs 2 y x P a respetively with S 2 S 2 (the suessor relations just introue) an P a. The formulas are onstrute starting from the atomi ones an using the Boolean onnetives an the quantifiers an. A sentene is a formula without free variables. If ϕ(x x 2... x n ) is a formula with at maximum x x 2... x n free variables p Σ is a piture an q 2 q 2... q n are elements of om(p) then we write (p q... q n ) = ϕ(x... x n ) if p satisfies ϕ with the natural interpretation above where x i is interprete by q i. If ϕ is a sentene we write p = ϕ. The language L(ϕ) efine by a sentene ϕ is the set of all pitures p Σ suh that p = ϕ. 5

18 Definition 4 Let L a two-imensional language L is first orer efinible if there exists a sentene ϕ whih ontains only first orer quantifiers suh that L = L(ϕ). Now we an esribe partiular positions (that is partiular elements of om(p)) in a piture by first orer formulas. A position into the higher row x = ( j) on j n is esribe by: ϕ t (x) = y ys x. Analogously we an efine the positions into the lower row by ϕ b (x) = y xs y. ϕ r (x) = y xs 2 y efines the positions into the rightmost olumn an ϕ l (x) = y ys 2 x the positions into the leftmost olumn. Now it is lear the meaning of ϕ tr ϕ tl ϕ br ϕ bl. For example: ϕ tr = ϕ t ϕ r. We give some examples of first orer sentenes that efine loal linguages. Example 7 Let L(Θ) be the loal language given by the set of tiles over the alphabet Σ = {0 }: 0 Θ =

19 The pitures p L(Θ) will be of the kin: It is easy to see that we have exatly the language of only one row L(Θ) = {p of only one row an p {0 } }. The first orer sentene ϕ = xϕ t (x) efines L(Θ). By this sentene we say that all the position in eah piture that satisfies ϕ are that ones of the first row onsequently there is only one row thus L(Θ) is: L(Θ) = L(ϕ) = {p Σ p = ϕ}. Analogously we oul use ϕ b instea of ϕ t. Moreover we an use ϕ r or ϕ l to mean the languages of only one olumn. Remark The language of the square pitures over Σ of only one symbol is not loal moreover we an not esribe this language by a first orer sentene; see [33]..3.2 Logi Formulas for Loal Languages In this subsetion we will etermine first orer sentenes in orer to efine loal languages. The same result is obtaine in [32] using a ifferent argumentation. Moreover we will show that it will be suffiient to use only universal quantifiers. Consier B(u v x y) := xs u ys v vs 2 u ys 2 x. Intuitively four positions into a piture satisfy B(u v x y) if an only if they onstitute a blok of size (2 2). Formally: Theorem 2 Let p Σ then (p q q 2 q 3 q 4 ) = B(u v x y) if an only if (q q 2 q 3 q 4 ) onstitute a tile.. 7

20 Proof. (p q q 2 q 3 q 4 ) satisfies B(u v x y) if an only if q 3 S q q 4 S q 2 q 2 S 2 q q 4 S 2 q 3 ; this is true if an only if (q q 2 q 3 q 4 ) onstitutes a tile: q q 2 q 3 q 4 Theorem 3 Let Θ be a finite set of tiles an t a tile: Consier the formula: an the sentene: Then L(Θ) = L(ϕ). t = a b. t ab (u v x y) := P a (u) P b (v) P (x) P (y) ϕ = u v x y[b(u v x y) t Θ t ab (u v x y)]. Proof. p L(Θ) if an only if all tiles whih belong to p are in Θ that is if an only if all tiles in p satisfy the sentene ϕ if an only if p = ϕ this hols if an only if p L(ϕ). Example 8 We show the first orer sentene whih efines the language L(Θ) over Σ = {0 } of square wors whih have in the main iagonal an 0 in the other positions. First of all the set of tiles Θ for the language L(Θ) is the following: Θ =

21 Then the first orer sentene for L(Θ) is: u v x y[b(u v x y) (t (u v x y) t 0 (u v x y) t 00 (u v x y) t 0 (u v x y) t 0 (u v x y) t 0 0 (u v x y) t 0 (u v x y) t 00 (u v x y) t 0 (u v x y) t (u v x y) t 0 0 (u v x y) t 0 (u v x y) t 00 (u v x y) t 000 (u v x y) t 000 (u v x y) t 0000 (u v x y)] We foun a simply way to efine all the loal languages: starting from a set of tiles we foun a first orer sentene of only four universal quantifiers whih allows us to axiomatize the loal language generate..3.3 Existential Seon Orer Formulas In the previous subsetion we observe that is possible ientify a piture p Σ with the struture p = (om(p) S S 2 (P a ) a Σ ). Moreover the properties of the two-imensional pitures an be esribe by first orer logi formulas an existential seon orer formulas using first orer variables x y z x x 2... for the elements of om(p) that is for positions an seon orer variables X Y Z X X 2... for sets of positions. In other wors variables are interprete as subsets of om(p). Definition 5 A two-imensional languages L is seon orer efinable if there exists a seon orer sentene ϕ suh that L = L(ϕ). Definition 6 A two-imensional languages L is existential seon orer efinable if there exists a sentene of the kin: ϕ = X X 2... X n ψ(x X 2... X n ) suh that L = L(ϕ) where ψ ontains only first orer quantifiers. Example 9 Now we verify that the language of square wors over a given alphabet is existential seon orer efinable. It suffies to esribe a set of positions whih () ontains the left-upper orner; (2) is lose for iagonal 9

22 suessor (that is we an move from the position (i j) to the position (i + j + )); an (3) we on t reah the lower borer or the righter borer exept the right-lower orner. An existential seon orer whih efines the three onitions is: X( x(ϕ tl (x) X(x)) x y z((x(x) xs y ys 2 z) X(z)) x((ϕ b (x) ϕ r (x)) ( X(x) ϕ br (x))..4 Regular Expressions The basi operations between two-imensional wors that we have efine in the first paragraph an be use to obtain larger family of two-imensional languages starting from elementary languages. Using those operations we are then able to give a new haraterization for loal languages an tiling system reognizable languages. Definition 7 Given an alphabet Σ we say that the empty language an eah language { a } where a Σ are atomi languages over Σ. Remark 2 We enote by R the following set of operations: R = { ɵ ɵ C }. The elements of R are alle regular operations. Definition 8 A language over Σ is regular if it is obtaine from atomi languages by applying a finite sequenes of regular operations. Informally a regular expression is a formula whih speifies how a ertain language is obtaine from atomi languages by regular operations. Definition 9 A regular expression RE over an alphabet Σ is reursively efine as: an all symbols a Σ are regular expressions; if α β are regular expressions then 20

23 (α) (β) (α) (β) C (α) (α) (β) (α) ɵ (β) (α) (α) ɵ are regular expressions. Eah regular expression over Σ enotes a two-imensional language. Using stanar notations: enotes the empty language a enotes the language with the only wor {a} (α) (β) enotes the union of the languages whih are enote by α an β (α) (β) enotes their intersetion (α) (β) an (α) ɵ (β) enote their row an olumn onatenation respetively (α) (α ɵ ) enote the row an olumn losure of the language enote by α respetively finally (α) C enotes its omplement. Definition 20 A two-imensional language L Σ is regular if it an be represente by a regular expression over Σ. Example 0 Consier Σ = {0 } the regular expression: E = (((0 ) ) ɵ (( 0) )) ɵ enotes the language of the hess wors with an o number of rows. For example the following wor belong sto E:

24 .5 On-line Tesselation Automata In literature there have been epite ifferent kins of automata to reognize two-imensional languages [ ]. In this setion we esribe a partiular moel of ellular automata introue in [6] whih reognizes the tiling system reognizable languages. Informally an on-line tesselation automata is a finite sequene of ells or better an array where a wave of transition passes in iagonal over the array. Eah ell hanges its state epening on the state of the two neighbor ells the upper an the leftmost respetively. Formally: Definition 2 A two-imensional non eterministi (eterministi) on-line tesselation automata enote by 2OTA (2DOTA) is efine by A = (Σ Q I F δ) where: Σ is the alphabet in input; Q is a finite set of states; I Q (I = {i} Q) is the set of initial states; F Q is the set of final states (or aepting states); δ : Q Q Σ 2 Q (δ : Q Q Σ Q) is the transition funtion (see also [33]). The automata A runs over a piture p Σ assoiating a state (in Q) to eah position (i j) in p. This state is given by the transition funtion δ an it epens to the other states alreay assoiate with the positions (i j) an (i j ) an to the symbol p(i j). At the zero step an initial state q 0 is assoiate with eah position in the first row an olumn of p. The omputation onsists of l (p) + l 2 (p) steps (where l (p) l 2 (p) mean the number of rows an olumn of p respetively). At the following step the automata reas p( ) an the state δ(q 0 q 0 p( )) is assoiate with the position ( ). At the seon step the states are simultaneously assoiate to the positions ( 2) e (2 ) an so on the automata passes to the next iagonal. At the k step the states are simultaneously assoiate with the position (i j) suh that i + j = k. 22

25 Definition 22 A 2OTA A reognizes a piture p Σ if there exists a omputation of A over p suh that the state assoiate with the position (l (p) l 2 (p)) is a final state. We notie that a 2OTA is reue to a stanar automata over strings when we restrit the omputation to pitures of one row. To summarize the results epite in the previous subsetions we have the following theorem of haraterization: Theorem 4 For a language L the following are equivalent:. L is tiling system reognizable; 2. L an be obtaine from atomi languages by applying a finite number of times the regular operations (exluing the omplementation) an the projetion; 3. L = L(ϕ) for an existential seon orer formula; 4. L is reognize by a 2OTA. Proof. See [33 Theorem 8.7]..5. Grammars In literature ifferent systems for generating pitures using grammars have been presente ([ ]) with the attempt of generalize the grammars of the uniimensional ase for formal string languages. Among the most reently efine grammars we reall the tile rewriting grammar of ([2]). Informally in tile rewriting grammars a rewriting rule hanges a homogeneous retangular piture into an isometri one tile with speifie tiles. Derivation an language generation with tiling rewriting grammar rules are similar to ontext-free grammars. Anyway there is not a grammar that exatly generalizes the uniimensional ontext free grammars. The moel propose by Crespi Reghizzi et al. has greater generative apaity than the tiling systems at yet any moels of grammars have the same generative power as tiling systems. Thus ifferently to regular languages in one imension there is not a haraterization for tiling system reognizable languages. 23

26 Chapter 2 Loal languages an algebrai strutures In this hapter we stuy the algebrai struture of loal languages an tiling system reognizable languages. We observe that it is possible to efine over these two families of languages an orer relation (an onsequently the two operations meet an join) so that we have a lattie of languages. Our prinipal purpose is to stuy the main features of this lattie in partiular the istributivity the moularity et. The problem appears rather omplex so we start to stuy the simplest ase i.e. we onsier the lattie of the loal languages efine over an alphabet of only one symbol (Lo ). Easily we observe that in this ase the lattie is istributive. Then we onsier the lattie of loal languages efine over an alphabet of two symbols (Lo 2 ) an we exten the results to the languages over alphabets of more than two symbols. We also onsier the haraterization of atoms an o-atoms of the lattie an the join-irreuible elements. Finally we restrit the researh to the uniimensional ase i.e. we examine the lattie of the horizontal ominoes whih is a proper sub-lattie of Lo 2. In this ase we give a omplete haraterization of atoms an o-atoms. Notation We will use two ouble brakets p to enote the set of tiles we an extrat from the piture p within brakets (of ourse surroune by ). For instane if for a b Σ we write a b b 24

27 then the orresponing set of tiles is a b b a Θ = b b a b b a b b. We also often use the notation L(p) = L( p ). Example An example of loal language is given by the set of hessboar pitures (i.e. pitures with alternating 0 s an s in eah row an olumn) over the alphabet Σ = {0 } with 0 on eah orner heneforth name L s. Consier the following set of tiles Θ s : Θ s = It is easy to see that L s = L(Θ s ) We now efine a lattie over the set of loal languages on an alphabet of fixe size. Therefore we also nee to reall some efinitions an theorems from lattie theory. For more etails on lattie theory we refer the reaer to [] an [57]. Definition 23 A lattie is a partially orere set L = L suh that every pair of elements x y P has greatest lower boun (or inf or meet) enote by x y an lowest upper boun (or sup or join) enote by x y. It is well known that a lattie on a universe L an be equivalently given either by speifying its partial orer relation or by speifying its binary operations an. Inee for every pair of elements a b L one has. a b a = a b b = a b. It is therefore just a matter of onveniene whether one sees a lattie as an orere struture or an algebrai struture or both. 25

28 Definition 24 In a lattie L = L an element b L is meet-irreuible if b = x y implies b = x or b = y for all x y L. Dually an element b L is join-irreuible if b = x y implies b = x or b = y for all x y L. Definition 25 In a lattie L = L an element b L is a (minimal) over of a if a < b an there is no suh that a < < b. We write a b to enote that b is a over of a. If L has least element 0 then an element a L is sai to be an atom if 0 a. Dually in a lattie with greatest element a oatom is an element a suh that a. Definition 26 Let L = L be a lattie. Then. L is istributive if for all a b L it satisfies: a (b ) = (a b) (a ); 2. L is moular if for all a b L it satisfies: a a (b ) = (a b) ; 3. L is semimoular if for all a b L it satisfies: Theorem 5 Let L be a lattie. Then a b a b a b. L is nonistributive if an only if L has a sublattie isomorphi to N 5 or M 5 (see Figure 2.); L is nonmoular if an only if L has a sublattie isomorphi to N 5. It is known that a moular lattie is also semimoular see e.g. []. Thus the following strit impliations hol: istributive moular semimoular. Definition 27 A partially orere set P satisfies the Joran-Deekin onition if it has only finite hains an for all pair of elements x y P suh that x y all maximal hains between x an y have the same length. 26

29 Figure 2.: The two nonistributive latties N 5 an M 5 rispettivamente. The following result is almost immeiate see for instane [ Proposition 2.]. Theorem 6 Let P be a partially orere set with only finite hains an with least element 0. P satisfies the Joran-Deekin onition if an only if one an efine a funtion (alle the rank funtion) ρ : P N (mapping eah element x P into the rank of x) suh that - ρ(0) = 0; - if y overs x then ρ(y) = ρ(x) +. It is known that a semimoular lattie without infinite hains satisfies the Joran-Deekin onition see for instane [ Lemma 2.26]. Thus if a lattie without infinite hains is semimoular then all maximal hains between the same two elements have the same length. 2. The lattie Lo n Without loss of generality for every n we may fix the alphabet Σ = {0... n } an onfine our investigation to the poset Lo n = Lo n of loal languages on Σ. Definition 28 If Θ is a set of tiles suh that for eah t Θ there exists a piture p L(Θ) suh that t B 22 (ˆp) then we say that Θ is an irreunant set of tiles. A set of tiles is reunant if it is not irreunant. For instane p is irreunant for every piture p. Unless otherwise speifie from this moment on when we are given a set of tiles Θ we will always assume that Θ is irreunant. The problem of reognizing whether a set of tiles is irreunant is in general uneiable [?]. 27

30 Example 2 The following set Θ of tiles is reunant sine the last tile in the list an not be ombine with any other tile in Θ Θ = However the set Θ \ { 0 0 } is irreunant. Theorem 7 For every n Lo n = Lo n is a finite lattie with operations of meet an join given by respetively L(Θ ) L(Θ 2 ) = L(Θ ) L(Θ 2 )(= L(Θ Θ 2 )) L(Θ ) L(Θ 2 ) = L(Θ Θ 2 ). (We remark one more that Θ an Θ 2 are assume to be irreunant.) Proof. It is a simple alulation to hek that L(Θ ) L(Θ 2 ) = L(Θ Θ 2 ). To show the statement about first notie that L(Θ Θ 2 ) is an upper boun of L(Θ ) an L(Θ 2 ). On the other han suppose that L(Θ ) L(Θ 2 ) L(Θ) an let t Θ Θ 2. Whether t Θ or t Θ 2 by irreunany we have that t is a 2 2 subtile of a piture v in L(Θ ) or in L(Θ 2 ); hene v L(Θ) an thus t Θ. This shows that Θ Θ 2 Θ hene L(Θ Θ 2 ) L(Θ). Remark 3 If Θ Θ 2 are sets of tiles then L(Θ ) L(Θ 2 ) L(Θ ) L(Θ 2 ) but equality nee not hol as shown by the following ounterexample. Consier the following two sets of tiles: Θ = 0 Θ2 = 0. It is lear that the piture p = 0 0 belongs to L(Θ Θ 2 ) but not to L(Θ ) L(Θ 2 ). 28

31 Remark 4 In the rest of the paper we will only onsier sets of tiles whih o not ontain the tile hene restriting ourselves to loal languages that o not ontain the empty piture. We use the symbol Θ T ot to enote the set of all possible tiles over {0 } minus the tile. Notie that Θ T ot = { p : p {0 } { } }. Let Lo n enote the loal languages on Σ = {0... n } that o not ontain the empty piture i.e. Lo n = L(Θ T ot ). Clearly Lo n = Lo n is a sublattie of Lo n. Sine we are intereste in the lattie theoreti struture of loal languages restrition to Lo n amounts to no loss of generality by the following theorem: Theorem 8 Lo n Lo n 2 where 2 is the two-element boune lattie an the symbol enotes lattie theoreti isomorphism. Finally we point out that for any m < n the lattie Lo m an be viewe (uner inlusion) as an ieal of Lo n. 2.. The simplest ase: the lattie Lo We exemplify the notions so far introue by faing the simple ase of the lattie Lo relative to the lass of loal languages restrite to the alphabet Σ of only one symbol say Σ = {0}. The loal languages we an onstrut over the alphabet {0} are only six inluing the empty language: See Figure 2.2. More preisely the five nonempty languages are: L 0 = L( 0 ): the language onstitute by the only piture 0 ; L r0 = L( 0 0 ): the language onstitute by the pitures of only one row of 0 s; L 0 = L( 0 0 ): the language onstitute by the pitures of only one olumn of 0 s; L r0 L 0 the language that is the join of the two previous languages i.e. the language onsisting only of rows or olumns of 0 s; 0 : the language onstitute by all possible pitures of 0 s. 29

32 0 L 0 L r0 L 0 L r0 L 0 Figure 2.2: The lattie Lo. We observe that the lattie Lo is isomorphi to the free istributive lattie with 0 on two generators The general ase We have seen that Lo is istributive. The situation hanges raially if one onsiers latties of loal languages on alphabets with at least two symbols. Theorem 9 If n 2 then Lo n ontains the following sublatties: (i) a sublattie isomorphi to N 5 ; (ii) a sublattie isomorphi to M 5. Hene Lo n is not semimoular Proof. The proof is for Lo 2 an an be extene to Lo n sine Lo 2 is a sublattie of Lo n. (i) Consier the following languages: L 0 = L( 0 ) = { 0 } L 0 = { 0 } L 0 = { 0 }. It is easy to see that L 0 an L 0 are atoms. Obviously L 0 is a over of = L 0 L 0 but L 0 L 0 is not a over of L 0. For example the language L 0 L 0 is suh that L 0 L 0 L 0 L 0 L 0 where inlusions are strit. See also Figure 2.3 (a). 30

33 L 0 L 0 L Θ4 L 0 L 0 L 0 L Θ L Θ2 L Θ3 L 0 Figure 2.3: (a) Sublattie isomorphi to N 5 ; (b) Sublattie isomorphi to M 5. (ii) Consier the sets of tiles: { 0 0 Θ = Θ T ot \ 0 0 } ; Θ = ; Θ 2 = an the loal language L(Θ) (L(Θ ) L(Θ 2 )) = L(Θ 3 ). Diret inspetion shows: L(Θ ) L(Θ 2 ) = L(Θ ) L(Θ 3 ) = L(Θ 2 ) L(Θ 3 ) =. L(Θ ) L(Θ 2 ) = L(Θ ) L(Θ 3 ) = L(Θ 2 ) L(Θ 3 ) Let Θ 4 be suh that L(Θ 4 ) = L(Θ ) L(Θ 2 ). Then Lo 2 ontains the sublattie epite in Figure 2.3 (b). Notie that the embeings of N 5 an M 5 given above preserve 0 as well. ; 2..3 Meet-irreuible elements an oatoms in Lo n We now onsier the set of meet-irreuible elements of the lattie Lo n. In Lo all elements exept for L 0 are meet-irreuible as an be easily seen from Figure 2.2. So in the rest of this setion we onfine ourselves to the ase n 2. Notation. Extening the efinition given in Remark 4 let Θ n T ot = { p : p {0... n } { } }. We will usually write Θ T ot instea of Θ n T ot when the alphabet is learly unerstoo from the ontext. We have: 3

34 Theorem 0 Let n 2 an L = L(Θ) Lo n. The following are equivalent: () L is a oatom; (2) Θ = Θ T ot \ { t } with t Θ T ot ; (3) L is meet-irreuible. Proof. Let us start proving that () an (2) are equivalent. If Θ = Θ T ot \ { t } then learly Θ is irreunant an L(Θ) is a oatom. The onverse is trivial. Clearly () implies (3). So we only have to prove that (3) implies (). Suppose now that L = L(Θ) with at least two ifferent tiles t t 2 Θ T ot \Θ. Let us prove the following useful statement: Claim There are two pitures u(t ) an u(t 2 ) suh that t u(t ) t 2 u(t 2 ) an: u(t ) u(t 2 ) =. One Claim has been prove we have that: L L(Θ) L(u(t )) = L (Θ u(t ) ) L (Θ T ot ) L L(Θ) L(u(t 2 )) = L (Θ u(t 2 ) ) L (Θ T ot ). Notie that all the inlusions are strit. Moreover sine u(t ) an u(t 2 ) are isjoint sets it follows that: (L(Θ) L (u(t )) ( L(Θ) L (u(t 2 )) ) = L(Θ) whih onlues our proof sine this shows that L(Θ) is meet-reuible. The proof of Claim is inee a mere (long) exerise base on onsiering all possible ases of the two tiles t an t 2. Let (resp. 2 ) be the number of ourrenes of the symbol in t (resp. t 2 ). Without loss of generality we examine the six ases where 2 :. = 0 2 = 0; 2. = 2 2 = 0 2; 3. = 3 2 =

35 Sine the stuy of eah of these ases is quite simple we only onsier the first one of them whih is also the most omplex leaving the others to the reaer. Let us assume that t = x y v z an t 2 = x y v z with x x y y v v z z {0... n }. We set u(t ) = x y v z while to etermine a suitable u(t 2 ) we must stuy separately the following ases: (a) x x y y z z an v v. In this ase we easily set: u(t 2 ) = x y v z. (b) x = x y y z z an v v (an similarly we an treat all the ases where t an t 2 have only one ommon element). In this ase we have that t 2 = x y v z where Here we an set: w = u(t 2 ) = { if w = 0 0 if w 0. x y x y v z The reaer an hek that u(t ) an u(t 2 ) have no tiles in ommon.. 33

36 () x = x y = y z z an v v (an similarly we an treat all the ases where t an t 2 have an equal row or olumn) i.e. t 2 = x y v z. Setting x y u(t 2 ) = x y v z we an easily see that u(t ) an u(t 2 ) have no tiles in ommon. () x = x z = z y y an v v (an similarly y = y v = v x x an z z ) i.e. t 2 = x y v z. Setting u(t 2 ) = x y x y v z v z we see that u(t ) an u(t 2 ) have no tiles in ommon. (e) x = x y = y v = v an z z (an similarly we an treat all the ases where t an t 2 iffer in only one element). Here we have t 2 = x y. We set: v z u(t 2 ) = x y y y x x y z v v z z. The piture u(t 2 ) is mae of 20 tiles; it is possible to hek that they are all ifferent from the 9 tiles of u(t ). 34

37 2..4 Join-irreuible elements We now turn our attention to join-irreuible elements of Lo n. Let v be a piture over Σ = {0... n }. We an onsier the set of tiles v an the respetive loal language L(v). Theorem A loal language L = L(Θ) Lo n is join-irreuible if an only if there exists v Σ suh that L = L(v) an there are no pitures u... u n suh that the u i are pairwise ifferent; u... u n = v. Proof. (= ) Let L be join-irreuible. Suppose that suh a v oes not exist an let L = L(Θ) for some irreunant Θ. Then for eah tile in Θ there is a piture that ontains it. Then we an write: Θ = u u n an we obtain L(Θ) = L(u ) L(u n ). Thus L(Θ) is not join-irreuible. The right-to-left impliation follows easily from the efinition of a joinirreuible element. Let us remark that it is not suffiient to require that L = L(v) to state that L is join-irreuible as shown in the following example. Example 3 (In Lo 2 ) Consier the sets of tiles: Θ = ; Θ = ; Θ 2 = We have: Θ = Θ Θ 2 an then L(Θ) = L(Θ ) L(Θ 2 ) so L(Θ) is not join-irreuible. Aoring to Theorem the following pitures witness join-reuibility of L(Θ): u = ; u 2 = As a neat onsequene of Theorem we have that if L Lo n is an atom then there exists v Σ suh that L = L(v). On the other han if L is a oatom then L is not join-irreuible: in fat we an not write a oatom L(Θ) as L(v) sine in Θ there are at least 7 orner tiles whereas in every piture there are exatly 4 orner tiles. 35

38 2.2 Some uneiable problems The previous setion shows that the meet-irreuible elements (equivalently the oatoms) of Lo n have an easy haraterization from whih one an immeiately eue that the property of being meet-irreuible (or equivalently a oatom) is eiable. Corollary 2 Given n an a set Θ of tiles on {0... n } one an eie whether L(Θ) is meet-irreuible (equivalently a oatom). Proof. Trivial by Theorem 0. Quite surprisingly we show in this setion that this is true neither of the join-irreuible elements nor of the atoms: in fat that the property of being join-irreuible an the property of being an atom are uneiable. We eal here with sets of tiles that nee not be irreunant. In [24] several uneiable problems onerning loal languages are stuie. For instane we reall that the following problems are uneiable: the equality problem: Is L(Θ) = L(Θ )? where Θ Θ are given sets of tiles. (One again we remark that here Θ an Θ are not suppose to be irreunant to begin with otherwise the problem is eiable sine in the ase of irreunant sets of tiles we have that L(Θ) = L(Θ ) if an only if Θ = Θ ); the irreunany problem: Is Θ irreunant? where Θ is a given set of tiles; the infinity problem: Is L(Θ) infinite? where Θ is a given set of tiles. We are now going to point out some aitional uneiable problems whih relate more iretly to the lattie theoreti struture of loal languages. Theorem 2 Suppose that Θ Θ are given sets of tiles on a ommon alphabet say of n symbols. The problems of asertaining given Θ Θ an m n whether:. L(Θ) is an atom in Lo m 2. L(Θ) is a over of L(Θ ) in Lo m 36

39 3. L(Θ) is join-irreuible in Lo m are uneiable. Proof. We will show below that for every Turing mahine M one an effetively fin a set of tiles Θ M on some alphabet suh that. L(M) if an only if L(Θ M ) ; 2. L(Θ M ) is either empty or a singleton. This is enough to show the laim. Inee onsier any atom L(Θ 0 ) over the alphabet {0}. Without loss of generality we may assume that the harater 0 oes not appear in any tile of Θ M. Then. L(M) if an only if L(Θ M ) is an atom; 2. L(M) if an only if L(Θ 0 Θ M ) is a over of L(Θ 0 ); 3. L(M) if an only if L(Θ 0 Θ M ) is join-reuible. Therefore the laim follows by observing that the problem L(M) is uneiable see for instane [45 Theorem 6.3.()]. It is now left to show how we an buil Θ M starting from M. Let us onsier eterministi Turing mahines working on an alphabet Σ. Our moel of Turing mahines follows losely that of [45] with some minor notational variants. Instrutions are quaruples qaxr where q r are states a Σ {B} an X Σ {B L R} with the stanar meaning. M aepts a string u Σ with a unique halting state h. The initial state is enote by q 0. We may assume that q 0 h. As in [45] we also assume that the tape use by the mahine has a leftmost ell. A onfiguration is a quaruple (v q a w) whih represents that the mahine is in state q; a is the ontent of the ell urrently sanne; v (Σ {B}) is the ontent of the tape to the left of this ell; w (Σ {B}) is the ontent of the tape to the right of this ell with the unerstaning that all remaining ells ontains B. A representative onfiguration (v q a w) is one in whih w oes not en with B. We often write below (v q a w) as vqaw; if a = B an w {B} then we may simply write vq. Following [33 Theorem 9.] a halting omputation of a Turing mahine M an be oe by a two-imensional piture in the following way. Suppose that a halting omputation onsists of a sequene of onfigurations 2... n 37

40 where = q 0 u an n = vhw. We an imagine that all i s have the same length: if not let l = max{l i : i n} where l i is the length of i an let ĉ i = i B l l i. Thus the ĉ i s have the same length. Further let Σ = {a : a Σ} be a opy of the alphabet Σ (with Σ Σ = ) let B be a opy of B an for a wor v (Σ {B}) let v be the orresponing opy of u over the alphabet Σ. For every i if ĉ i = vqw then let γ i = vqw. The omputation an thus be oe by p where p = γ.. γ n p an be viewe as a piture of the loal language given by a suitable set of tiles Θ(M) whose tiles mirror the instrutions of M use to bring the mahine from one onfiguration to the next one. For instane the instrution palq is oe among others by the tiles s p q s p a s a where s Σ {B}. The etails of the onstrution of Θ(M) an be foun in [?]. One has: L(M) L(Θ(M)). We now show how given a Turing mahine M one an effetively onstrut the esire Turing mahine M. First of all note that starting from M one an effetively fin a Turing mahine M (with say initial state q 0 an halting state h ) on the same alphabet suh that L(M ) {e} (where e enotes the empty string) an L(M) M halts on e. For this simply onsier a Turing mahine M suh that on input u M oes not halt if u e; an M on e halts if an only if there is a string v suh that M halts on v. One woul be tempte to take Θ M = Θ(M ) but unfortunately although M an perform at most one halting omputation 38