On the Period of a Periodic-Finite-Type Shift

On the Period of a Periodic-Finite-T ype Sh ift Akiko Manada and Navin Kashy ap Dept. Mathematics and Statistics Queen’ s University Kingston, ON, K7L 3 N6, Canad a. Email: { ak iko,nkashyap } @mast.queen su.ca Abstract — Periodic-finite-type shifts (PFT’s) f orm a class of sofic shi fts that strictly contains the class of shifts of fi nite type (SFT’s). In this paper , we inv estigate how the notion of “period” inherent in the defin ition of a PFT causes i t to differ from an SFT , and h ow the period influ ences the properties of a PFT . I . I N T R O D U C T I O N Shifts of finite type (SFT’ s) are objects of fundamen tal im- portance in symbo lic dyna mics and the theory o f con strained coding [2]. A well-known example of an SFT would be th e ( d, k ) run- length limited ( ( d, k ) -RLL) shift, w here the nu mber of 0’ s b etween suc cessi ve 1’ s is at least d a nd a t most k . Constrained codes based on these ( d, k ) -RLL shifts are used in most storag e media such a s magnetic tape s, CD’ s and D VD’ s. A genera lization of SFT’ s was introd uced b y Moision and Siegel [4] w ho were in terested in examining th e properties of distance-enh ancing constrained codes, i n which the app earance of certain words is fo rbidden in a per iodic manner . This new class of shifts, called per iodic-finite-ty pe shifts (PFT’ s), contains th e class of SFT’ s and some other inter esting classes of shifts, such as constrained systems with unconstrain ed positions [ 1],[7], and sh ifts arising from the time-v ary ing maximum transition ru n constraint [6]. T he class o f PFT’ s is in turn proper ly co ntained with in the class of so fic sh ifts [3], a fact we discuss in more detail in Section II. The prop erties of SFT’ s are now q uite we ll under stood (cf. [2]), but the same can not be said fo r PFT’ s. The study of PFT’ s h as prim arily focused on finding efficient algorithm s for con structing their p resentations [1], [3], [5]. Th e difference between the definitions of SFT’ s and PFT’ s is quite small. An SFT is defined as a set of bi- infinite seque nces (over som e alphabet) that do n ot co ntain as subword s any word fro m a certain finite set. Thus, an SFT is defined by f orbidd ing the appearan ce of finitely m any word s at any p osition of a bi- infinite seq uence. A PFT is a lso defined b y fo rbiddin g th e appearan ce of finitely many words, except that these words are on ly for bidden to a ppear at positions of a b i-infinite sequence that are ind exed by certain pr e-defined perio dic integer sequences; see Section II f or a form al definition. This paper aims to initiate a study of how the “per iod” inherent in the definition of a PFT in fluences its properties. After a review of relev ant d efinitions and backgro und in Section II, we will see in Sectio n III that giv en an SFT Y , ∗ This work was supported in part by a Disco very Grant from the Natural Science s and Engineering Research Council (NSERC) of Canada. we can associate with it a PFT X in such a way that it is only the period that differentiates X fro m Y . W e then seek to understand how the pe riod d etermines th e p roperties o f the PFT X by means of a compar ati ve stu dy of X and Y . W e in vestigate a different aspect of periods in Sectio n IV, wh ere we study the influ ence of the perio d of a PFT X on th e period s of periodic sequences in X , and on the period s of gra phical presentation s of X . I I . B A S I C B AC K G R O U N D O N S F T ’ S A N D P F T ’ S W e begin with a revie w of basic backg round , ba sed o n material from [2] an d [3]. Let Σ be a finite set of symb ols; w e call Σ an alphab et . W e alw ays assume that | Σ | = q ≥ 2 since q = 1 giv es us a trivial case. Let w = . . . w − 1 w 0 w 1 . . . be a bi-infinite sequenc e over Σ . A word (fin ite-length sequen ce) u ∈ Σ n (for som e integer n ) is said to be a subword o f w , denoted by u ≺ w , if u = w i w i +1 . . . w i + n − 1 for some inte ger i . If we want to emp hasize the fact that u is a subword o f w starting at the index i , ( i.e. , u = w i w i +1 . . . w i + n − 1 ), we write u ≺ i w . By co n vention, we assume that the empty word ǫ ∈ Σ 0 is a sub word of any bi-infin ite sequence . Also, we define σ to be th e shift map, th at is, σ ( w ) = . . . w ∗ − 1 w ∗ 0 w ∗ 1 . . . is the bi-infinite sequ ence satisfyin g w ∗ i = w i +1 for all i . Giv en a labeled directed gr aph G , whe re labels come from Σ , let S ( G ) be the set o f bi-in finite sequen ces which are generated by reading off labels along bi-infinite paths in G . A sofi c shift S is a set of b i-infinite seq uences such th at S = S ( G ) for some labeled directed graph G . In this ca se, we say th at S is p r esented by G , or that G is a pr esentation of S . It is well known that every sofic shift has a deterministic presentation , i.e. , a presentation such that outgo ing edges from the same state (vertex) are labeled distinctly . For a sofic sh ift S , B n ( S ) d enotes th e set of words u ∈ Σ n satisfying u ≺ w for som e bi- infinite sequen ce w in S , and B ( S ) = ∪ n ≥ 0 B n ( S ) . A sofic shift S is irr edu cible if there is an irreducib le ( i. e. , stron gly connected) presentation of S , or equiv alently , for e very o rdered pair o f word s u and v in B ( S ) , there exists a word z ∈ B ( S ) such that u z v ∈ B ( S ) . A shift of fi nite type (SFT) Y F ′ , with a fin ite set of forbid den words (a forbid den set) F ′ , is the set of all bi- infinite sequences w = · · · w − 1 w 0 w 1 · · · over Σ such that w contains no word f ′ ∈ F ′ as a subword . Tha t is, the finite number of word s f ′ in F ′ are not in B ( Y F ′ ) . A periodic- finite-type shift , which we abbreviate as PFT , is characterized by an ord ered list of finite sets F = ( F (0) , F (1) , . . . , F ( T − 1) ) and a period T . The PFT X {F ,T } is d efined as the set of all b i-infinite sequ ences w over Σ such that f or som e in teger r ∈ { 0 , 1 , . . . , T − 1 } , th e r - shifted sequ ence σ r ( w ) o f w satisfies u ≺ i σ r ( w ) = ⇒ u 6∈ F ( i mod T ) for every integer i . For simplicity , we say that a word f is in F ( symbolically , f ∈ F ) if f ∈ F ( j ) for som e j . Since the ap pearance of words f ∈ F is forb idden in a periodic manner, note that f can be in B ( X {F ,T } ) . Also, ob serve that a PFT X {F ,T } satisfying F (0) = F (1) = · · · = F ( T − 1) is simply the SFT Y F ′ with F ′ = F (0) . T hus, SFT’ s are sp ecial ca ses o f PFT’ s. W e call a PFT p r ope r when it cannot be represented as an SFT . Any SFT can b e c onsidered to be an SFT in which ev ery for bidden word has the same length. Mor e precisely , giv en an SFT Y = Y F ∗ , find the lo ngest forb idden word in F ∗ and say it has length ℓ . Set F ′ = { f ′ ∈ Σ ℓ : f ′ has so me f ∗ ∈ F ∗ as a prefix } . Then , Y F ∗ = Y F ′ , and each word in F ′ has the same length, ℓ . Furtherm ore, we can also assume that B ℓ ( Y ) = Σ ℓ \ F ′ since if not (that is, if B ℓ ( Y ) ( Σ ℓ \ F ′ ), every word in (Σ ℓ \ F ′ ) \ B ℓ ( Y ) can be added to F ′ , with out affecting Y in any way . Correspon dingly , ev ery PFT X has a r epresentation of the form X {F ,T } such that F ( j ) = ∅ for 1 ≤ j ≤ T − 1 , a nd e very word in F (0) has the same leng th. An ar bitrary repr esentation X {F ,T } can be co n verted to one in th e above f orm as fo llows . If f ∈ F ( j ) for some 1 ≤ j ≤ T − 1 , list out all words with length j + | f | whose suf fix is f , add them to F (0) , a nd delete f fro m F ( j ) . Co ntinue this p rocess until F (1) = · · · = F ( T − 1) = ∅ . Then, apply the metho d d escribed ab ove fo r SFT’ s to make every word in F (0) have th e same length. It is known that PFT’ s b elong to the class of sofic shifts. Theorem II.1 (Moision and Siegel, [3]) All periodic -finite- type shifts X are sofic shifts. T hat is, f or any PFT X , ther e is a presentation G of X . Moision an d Siegel p roved the theore m by gi ving an algo - rithm that, giv en a PFT X , generates a p resentation, G X , of X . W e call the p resentation G X the MS pres enta tion of X . The MS algorithm , g i ven a PFT X as input, run s as f ollows. 1) Represent X in the form X {F ,T } , such that every word in F has the same length ℓ and belon gs to F (0) . 2) Prepare T copies of Σ ℓ and name the m V (0) , V (1) , . . . , V ( T − 1) . 3) Consider the words in V (0) , V (1) , . . . , V ( T − 1) as states. Draw an edg e labeled a ∈ Σ fr om u = u 1 u 2 · · · u ℓ ∈ V ( j ) to v = v 1 v 2 · · · v ℓ ∈ V ( j +1 m od T ) if and only if u 2 · · · u ℓ = v 1 · · · v ℓ − 1 and v ℓ = a . 4) Remove states cor respondin g to words in F (0) from V (0) , together with th eir incoming and outgoing edg es. Call this labeled d irected g raph G ′ . 5) If there is a state in G ′ having o nly incoming edges or only outg oing edges, remove the state from G ′ as well as its incoming or outgoing ed ges. Continu e this p rocess until we canno t fi nd such a state. Th e resulting graph G X is a presentation o f X . Remark II.2 It is evident that the MS presentatio n of a PFT is always d eterministic. Also, for a path α in G X with length | α | ≥ ℓ , α terminates at so me state th at is a co py of u = u 1 u 2 . . . u ℓ iff the length- ℓ suffix of the w ord gene rated by α is equal to u . I I I . I N FL U E N C E O F T H E P E R I O D T O N A P F T From this p oint on, whenever we consid er an SFT Y F ′ in this paper , we will implicitly assume that e very forbidden word in F ′ has the same length ℓ , and that B ℓ ( Y ) = Σ ℓ \ F ′ . As we observed in the pr e vio us section , there is no loss of generality in doing so. Gi ven an SFT Y F ′ , consider the PFT X = X {F ,T } in which F = ( F (0) , F (1) , . . . , F ( T − 1) ) = ( F ′ , ∅ , . . . , ∅ ) . While Y F ′ ⊆ X {F ,T } , eq uality does not hold in general. No te that it is only the influen ce of the p eriod T tha t ca uses th e shifts X = X {F ,T } and Y = Y F ′ to dif fer . So, a comp arativ e study of X and Y is a usef ul means of under standing h ow the period T d etermines th e p roperties of the PFT X . In this section, we p resent a samp ling of resu lts that illustrate how proper ties of the SFT Y can affect those o f the PFT X . The following result, which shows that the irredu cibility of Y has a significant effect on th e irreduc ibility of X , may b e considered typica l of the comp arative study p roposed ab ove. Theorem III.1 Suppose th at Y = Y F ′ is an irre ducible SFT . Let X = X {F ,T } be the PFT satisfying F = ( F (0) , F (1) , . . . , F ( T − 1) ) = ( F ′ , ∅ , . . . , ∅ ) . If there exists a periodic bi- infinite seque nce y in Y with a period p satisfying p ≡ 1 (mod T ) , then th e MS p resentation, G X , of X is irreducib le as a g raph. That is, X is irreducib le. Pr oo f : Th rougho ut this pro of, fo r a path η in a g raph, let s ( η ) an d t ( η ) b e the starting state a nd the term inal state, respectively , of η in the g raph. Also, f or a state v = v 1 v 2 . . . v ℓ in G X , v ∈ V ( j ) is denoted by v ( j ) for 0 ≤ j ≤ T − 1 . Let G ′ be the gr aph d efined in Step 4 of the M S algo rithm. Consider the subgrap h H of G ′ that is induce d by the states in Σ ℓ \ F ′ . Since Σ ℓ \ F ′ = B ℓ ( Y ) , all states in H have incom ing edges an d ou tgoing edges. Hence, H is a subgraph of G X . Ke y p oints of the pr oof are the following. Claim 1 : H is a presentatio n of Y . Claim 2 : H is ir reducible as a graph if there exists a periodic bi-infinite sequen ce y in Y with a perio d p satisfying p ≡ 1 (mo d T ) . Once these claim s are proved, it is straightfo rward to ch eck that the MS pr esentation G X of X is irr educible. Note that the graph G ′ is obtained from H by ad ding word s in F (0) to V (1) , V (2) , . . . , V ( T − 1) and corr esponding inco ming an d outgoin g ed ges. Ob serve that (b y Step 5 of the MS algor ithm) a word f ′ ∈ F (0) is a state in G X if and only if there exist paths ρ 1 , ρ 2 in G ′ satisfying s ( ρ 1 ) = f ′ , t ( ρ 1 ) ∈ Σ ℓ \ F ′ and s ( ρ 2 ) ∈ Σ ℓ \ F ′ , t ( ρ 2 ) = f ′ . Since H is irreducib le, G X is irreducib le as well. Pr oo f of Claim 1 : W e need to show th at S ( H ) ⊆ Y and Y ⊆ S ( H ) . It is clear that S ( H ) ⊆ Y since, by Remar k II.2, there is no path in H which generates word s in F ′ . Con versely , take an arbitrary bi-infinite sequence x = . . . x − 1 x 0 x 1 . . . ∈ Y . Since f ′ 6≺ x for ev ery fo rbidden word f ′ ∈ F ′ , we see that for any in teger i , the states correspo nding to x i − ℓ +1 x i − ℓ +2 . . . x i are in H . Th erefore , there exists an edge labeled x i +1 from x i − ℓ +1 x i − ℓ +2 . . . x i ∈ V ( j ) to x i − ℓ +2 . . . x i x i +1 ∈ V ( j +1 m od T ) for all integers i and 0 ≤ j ≤ T − 1 . Hence, x ∈ S ( H ) , that is, Y ⊆ S ( H ) . Pr oo f of Claim 2 : A periodic bi-infinite seq uence y ∈ Y with period p ≡ 1 (mo d T ) can be written as y = ( y 1 y 2 . . . y n ) ∞ , for some y 1 y 2 . . . y n ∈ Σ n , where n is some mu ltiple of p satisfying n ≡ 1 (mo d T ) a nd n ≥ ℓ . As y ∈ Y , y n − ℓ +1 . . . y n y 1 y 2 . . . y n ∈ B ( Y ) . Thus, for ev ery i ∈ { 0 , 1 , . . . , T − 1 } , th ere exists a path α in H satis- fying s ( α ) = z ( i ) = y n − ℓ +1 . . . y n and gener ating y 1 y 2 . . . y n . Observe that t ( α ) is also z ( i ′ ) = y n − ℓ +1 . . . y n for som e i ′ ∈ { 0 , 1 , . . . , T − 1 } . Howev er, since | y 1 y 2 . . . y n | = n ≡ 1 (mo d T ) , we h av e i ′ = i + 1 mo d T . This automatica lly implies that fo r the word z = y n − ℓ +1 . . . y n in B ( Y ) , there is a path β j k in H such that s ( β j k ) = z ( j ) and t ( β j k ) = z ( k ) for any ordered pair ( j, k ) , whe re 0 ≤ j, k ≤ T − 1 . Now take an arbitr ary pair of states u ( r ) and v ( s ) in H . Since Y is irred ucible, th ere exist words w ′ and w ∗ in B ( Y ) so that u w ′ z and z w ∗ v are in B ( Y ) . Thu s, the re exists a path γ genera ting w ′ z such th at s ( γ ) = u ( r ) and t ( γ ) = z ( j ) for some 0 ≤ j ≤ T − 1 , a nd a p ath δ gen erating w ∗ v such that s ( δ ) = z ( k ) for som e 0 ≤ k ≤ T − 1 and t ( δ ) = v ( s ) . As there is a p ath β j k from z ( j ) to z ( k ) from the argument above, we have a path γ β j k δ starting f rom u ( r ) and ter minating at v ( s ) . Hence, th e presen tation H is ir reducible as a grap h. From Th eorem III.1, we can obtain the f ollowing c orollary . Corollary III.2 Let Y = Y F ′ be an irreducible SFT suc h tha t |F ′ | < | Σ | . Then for all T ≥ 1 , the PFT X = X {F ,T } with F = ( F (0) , F (1) , . . . , F ( T − 1) ) = ( F ′ , ∅ , . . . , ∅ ) is irreducib le. Pr oo f : Since |F ′ | < | Σ | , there is a sym bol a ∈ Σ which is not u sed as the first sym bol of any word in F ′ . Hence, the bi-infinite seq uence a = a ∞ is in Y . As a has p eriod 1 , we have f rom Theo rem III.1 that X is irredu cible. The pro of of Theor em III.1 shows that the SFT Y = Y F ′ has a presentatio n H th at is a subg raph of th e MS p resentation G X of X = X {F ,T } , where F = ( F ′ , ∅ , . . . , ∅ ) . Th is fact may allow us to compare som e of th e inv ariants associated with the two sh ifts Y and X , for examp le, the ir entropies and their zeta f unctions (see [2, Chap ters 4 an d 6 ]). The entropy (or the Sha nnon capacity) h ( S ) of a so fic shift S can be co mputed from a deterministic presenta tion G of S as follows: h ( S ) = log 2 λ , where λ is the largest eigenv alue of the ad jacency matrix A G of G . Equ iv a lently , λ is the largest root o f the char acteristic poly nomial χ A G ( t ) = det( tI − A G ) of A G (see, e.g. , [2, Chapter 4 ]). Returning to the shifts X and Y as a bove, since H is a subgrap h of G X , it ma y be possible to express the character istic polyno mial of A G X in terms of the characteristic polyn omial of A H . Th is would allow us to compar e the entro pies of X and Y . Howev er, this seems to be h ard to do in gen eral. W e have a partial resu lt in the special case whe n Y = Y F ′ with |F ′ | = 1 , a nd X = X {F , 2 } , a s we describe next. Recall tha t | Σ | = q . No w su ppose that Y = Y F ′ is an SFT with the set F ′ consisting of a single fo rbidden word f ′ , and X = X {F , 2 } is the PFT with pe riod 2 and F = ( F (0) , F (1) ) = ( { f ′ } , ∅ ) . Also, let A G X be the ad jacency matr ix o f the MS presentation G X of X , and let A H be that of the subg raph H of G X induced by the states in Σ ℓ \ { f ′ } . Observe th at the matrix A G X is a (2 q ℓ − 1) × (2 q ℓ − 1) 0-1 matrix . W ithout loss of generality , for A G X , we can a ssume the following. • The first q ℓ − 1 rows an d co lumns correspo nd to states in V (0) , a nd the last q ℓ rows and colu mns correspo nd to those in V (1) . • Assign f ′ ∈ V (1) to the (2 q ℓ − 1 ) -th r ow an d co lumn, and arrange the fir st r ow so that the (1 , 2 q ℓ − 1 ) - th en try of A G X is 1. • Let u ∈ V (1) be such that the longest proper suffix of u is equal to that of f ′ . Assign this u to the q ℓ -th row a nd column so that the q ℓ -th row and the (2 q ℓ − 1 ) -th r ow are the same. For a matrix M , set M ( i,j ) to b e the submatr ix of M o btained by deletin g its i - th row and j -th column . Then, observe that A (2 q ℓ − 1 , 2 q ℓ − 1) G X = A H . In this case, by a pplying elementar y row operations to the matr ix N = tI − A G X , we hav e χ A G X ( t ) = det( N ) =     B c d t     , (1) where B is a (2 q ℓ − 2) × (2 q ℓ − 2) matrix satisfying det( B ) = χ A H ( t ) , c is the (2 q ℓ − 2) × 1 co lumn v ector [ − 1 0 . . . 0 ] T , and d ∈ {− 1 , 0 } 2 q ℓ − 2 . Using the for m given in (1 ) for det( N ) , we can d erive the following theo rem. Th e co mplete proof w ill be pu blished in the full version of th is pap er . Theorem III.3 Let Y = Y F ′ and X = X {F , 2 } be the SFT and PFT described above, respectively . Then, the characteristic polyno mial χ A G X ( t ) of the adjacency matrix A G X is giv en by χ A G X ( t ) = t ( χ A H ( t ) + ( − 1 ) q ℓ det( B (1 ,q ℓ ) )) . I V . P E R I O D S I N P F T ’ S The perio d T inv olved in the descr iption of a PFT is no t the on ly notion of “period ” that can b e associated with the shift. For any shift X , we can always defin e its sequen tial period , T ( X ) seq , to be the smallest perio d o f a ny per iodic bi- infinite sequence in X . Furth ermore, if X is an irredu cible sofic shift, we can define a “graphical period” for it as follows. Let G b e a presentation of X with state set V ( G ) = { V 1 , . . . , V r } . For eac h V i ∈ V ( G ) , define p er ( V i ) to be the greatest comm on divisor (gcd ) of the length s of p aths (cycles) in G that begin and end a t V i , and furth er define per ( G ) = g cd( per ( V 1 ) , . . . , p er ( V r )) . It is w ell known that when G is irreducible, per ( V i ) = per ( V j ) for each pair of states V i , V j ∈ V ( G ) , and hence per ( G ) = per ( V ) for any V ∈ V ( G ) . The graphical period , T ( X ) gr aph , o f an irreducible sofic shift X is defined to be the least p er ( G ) of any irredu cible presentation G of X . Giv en a PFT X , define its descriptive period , T ( X ) desc , to be the smallest integer among all T ∗ such th at X = X {F ∗ ,T ∗ } for some F ∗ . In this section, we determin e wh at influen ce, if any , the descriptive period of a PFT h as on its sequ ential and graphica l periods. Let X = X {F ,T } be an ir reducible PFT , and let G be an irreducib le pr esentation of X . Pro position 1 of [3] says that if X is proper, then gcd( per ( G ) , T ) 6 = 1 . Using that p roposition, we ca n obtain the following result, wh ich shows that a prope r PFT X can ha ve T ( X ) desc arbitrarily larger th an T ( X ) seq . Proposition IV .1 Suppose that Y = Y F ′ is an irred ucible SFT , such that the b i-infinite sequence a ∞ ∈ Y for some a ∈ Σ . Let X = X {F ,T } be the PFT satisfying F = ( F (0) , F (1) , . . . , F ( T − 1) ) = ( F ′ , ∅ , . . . , ∅ ) . Then, a ∞ ∈ X , so T ( X ) seq = 1 . Furtherm ore, if X is a p roper PFT and T is prime, we have T ( X ) desc = T . Pr oo f : Since Y ⊆ X , it is clear that a ∞ ∈ X , an d hen ce, T ( X ) seq = 1 . Now , let X = X {F ,T } be a proper PFT with T prime. First o bserve th at the MS p resentation G X of X is irreducib le since the bi-infin ite sequence a = a ∞ is in Y and a has per iod 1. Also, note th at p er ( G X ) mu st be k T f or some k ≥ 1 fr om the con struction of G X . Howe ver, if we consider the period of the states a ℓ in G X , it is T . Thus, per ( G X ) = T by th e irredu cibility of G X . Sin ce X is proper, we ha ve fr om Proposition 1 of [3] that gcd( per ( G X ) , T ∗ ) 6 = 1 f or all T ∗ satisfying X = X {F ∗ ,T ∗ } . As T is prime, gcd( per ( G X ) , T ′ ) = gcd( T , T ′ ) = 1 for all T ′ < T . Theref ore, T is the descriptive period o f X . For example, con sider an SFT Y = Y F ′ with a f orbidd en set F ′ = { b 2 } for so me b ∈ Σ . Th en, Y is ir reducible, and a ∞ ∈ Y fo r any a ∈ Σ \ { b } . In this case, for a PFT X = X {F ,T } with T pr ime, such that F = ( { b 2 } , ∅ , . . . , ∅ ) , it may be verified that X is proper, a nd hence, T = T ( X ) desc . Con versely , T ( X ) seq can be ar bitrarily larger than T ( X ) desc for proper PFT’ s X . W e present such an exam ple next. Set Σ = { 0 , 1 } . W e d efine a sliding-b lock map ψ as follows: fo r a non-e mpty word u = u 1 u 2 . . . u n ∈ Σ n , (resp. a bi-in finite sequ ence w = . . . w − 1 w 0 w 1 . . . over Σ ), defin e ψ ( u ) = u ∗ 1 u ∗ 2 . . . u ∗ n − 1 , where u ∗ i = u i + u i +1 (mo d 2) for 1 ≤ i ≤ n − 1 ( resp. ψ ( w ) = . . . w ∗ − 1 w ∗ 0 w ∗ 1 . . . , where w ∗ i = w i + w i +1 (mo d 2) for each i ). By convention, ψ ( u ) = ǫ when u ∈ Σ 1 . For k ≥ 1 , c onsider the PFT X k = X {F k , 2 } with F k = ( F (0) k , F (1) k ) , d efined as follows. • F (1) k = ∅ for all k ≥ 1 . • F (0) 1 = { 0 } , and for k ≥ 2 , we set F (0) k = ψ − 1 ( F (0) k − 1 ) . That is, F (0) k is the in verse imag e of F (0) k − 1 under ψ . It is easy to see that for each k ≥ 1 , every word f ∈ F (0) k has len gth | f | = k , and in particu lar , we have 0 k ∈ F (0) k . Moreover , as ψ is a two-to- one map ping, we hav e |F (0) k | = 2 k − 1 . Th e fo llowing prop osition contains an other useful o bservation concernin g ψ . W e omit the straig htforward proof b y ind uction. Proposition IV .2 For a binary word u = u 1 u 2 . . . u r of length r > m , let u ∗ 1 u ∗ 2 . . . u ∗ r − m = ψ m ( u ) . If m = 2 j for so me j ≥ 0 , then u ∗ i = u i + u i +2 j (mo d 2) for 1 ≤ i ≤ r − m . Furthermo re, if m = 2 j − 1 for some j ≥ 0 , th en u ∗ i = u i + u i +1 + · · · + u i +2 j − 1 (mo d 2) for 1 ≤ i ≤ r − m . The corollary below simply follows fro m th e fact that fo r any f ∈ F (0) k , we must h av e ψ k − 1 ( f ) = 0 . Corollary IV .3 If z ∈ Σ 2 j (for som e j ≥ 0 ) has an odd number of 1’ s, then z / ∈ F (0) 2 j . W e next record some importan t facts abou t the PFT’ s X k . Proposition IV .4 For k ≥ 1 , the following statements hold: (a) X k +1 = ψ − 1 ( X k ) ; (b) X k is ir reducible iff 1 ≤ k ≤ 6 ; and (c) X k is a proper PFT . Pr oo f : Statement (a) fo llows straightforwardly from the defi- nition o f the PFT’ s X k . For (b), first n ote th at X k is irred ucible fo r 1 ≤ k ≤ 6 since its MS p resentation may be verified to be ir reducible as a graph. When k = 7 , it can b e shown that X k is not irreducible, which im plies that X k is no t irr educible when k ≥ 7 by (a). T o p rove (c), supp ose to the co ntrary that X k is n ot a proper PFT for some k ≥ 1 . T hen, X k = Y for some SFT Y = Y F ′ , where every fo rbidden word in F ′ has the same length, ℓ . Pick a j ≥ 0 such tha t 2 j ≥ k , an d set r = 2 j − k . By ( a) above, X 2 j = ψ − r ( X k ) = ψ − r ( Y ) . Note that ψ − r ( Y ) is also an SFT , with fo rbidden set ψ − r ( F ′ ) . All words in ψ − r ( F ′ ) have len gth ℓ ′ = ℓ + r . For the PFT X 2 j , o bserve th at the b i-infinite sequenc e w = (0 2 j − 1 1) ∞ 0 2 j (10 2 j − 1 ) ∞ is in X 2 j as w contain s a word in F (0) 2 j ( i.e. , 0 2 j ) o nly once, by Cor ollary IV .3. Theref ore, ev ery subword of w is in B ( X 2 j ) = B ( ψ − r ( Y )) . Now , consider the bi- infinite sequ ence w ′ = (0 2 j − 1 1) ∞ 0 2 j (10 2 j − 1 ) 2 ℓ ′ +1 10 2 j (10 2 j − 1 ) ∞ . Note that every length- ℓ ′ subword o f w ′ is also a subword of w , and hence , is in B ( ψ − r ( Y )) . Th is implies that w ′ ∈ ψ − r ( Y ) . For the two d istinct indices m, n ( m < n ) such th at 0 2 j ≺ m w ′ and 0 2 j ≺ n w ′ , we h av e n − m = 2 j (2 ℓ ′ + 2) + 1 , so that m 6≡ n (mo d 2) . But, since 0 2 j ∈ F (0) 2 j , this imp lies that w ′ 6∈ X 2 j , wh ich is a con tradiction. Statement ( c) of Propo sition IV .4 im plies that T ( X k ) desc = 2 for all k ≥ 1 . In con trast, the f ollowing theorem shows that T ( X k ) seq grows arbitrarily large as k → ∞ . Theorem IV .5 For any j ≥ 0 and 2 j + 1 ≤ k ≤ 2 j +1 , the periods of periodic sequences in X k must be multiples of 2 j +1 . T o prove Theore m IV .5, we n eed the next th ree lemmas. W e omit the simple pr oof of the first lemma. Lemma IV .6 If x ∈ { 0 , 1 } Z is a perio dic sequence, the n so is ψ ( x ) . Fur thermore , any period of x is also a perio d of ψ ( x ) . Lemma IV .7 For any j ≥ 0 , F (0) 2 j +1 = { f ∗ f ∗ 1 : f ∗ = f ∗ 1 f ∗ 2 . . . f ∗ 2 j ∈ Σ 2 j } . Pr oo f : Recall that for a w ord f ∈ F (0) 2 j +1 , ψ 2 j ( f ) = 0 . Since Proposition IV .2 sh ows that ψ 2 j ( f ) = f 1 + f 2 j +1 (mo d 2) , we have f 1 = f 2 j +1 . Noting that |F (0) 2 j +1 | = 2 2 j = | Σ 2 j | , we thus have F (0) 2 j +1 = { f ∗ f ∗ 1 : f ∗ = f ∗ 1 f ∗ 2 . . . f ∗ 2 j ∈ Σ 2 j } . Lemma IV .8 For j ≥ 0 , there is no p eriodic sequence x in X 2 j +1 whose period is (2 t + 1)2 j for some t ≥ 0 . Pr oo f : W e d eal with j = 0 first. Note that F (0) 2 = { 00 , 11 } . So, if X 2 has a p eriodic bi-in finite sequ ence w = ( w 1 w 2 . . . w m ) ∞ with an od d pe riod m , then 00 6≺ w 1 w 2 . . . w m , 11 6≺ w 1 w 2 . . . w m , and w 1 6 = w m . But there is no word w 1 w 2 . . . w m ∈ Σ m that satisfies these co nditions. Now , consider j ≥ 1 . Assume, to the c ontrary , th at there exists a periodic sequenc e x = . . . x − 1 x 0 x 1 . . . ∈ X 2 j +1 whose p eriod is (2 t + 1)2 j for some t ≥ 0 . Then , x is of the form ( x 0 x 1 . . . x (2 t +1)2 j − 1 ) ∞ . Without loss of gen erality , we may assum e tha t fo r every ev en integer i , u ≺ i x imp lies u 6∈ F (0) 2 j +1 . Then , fo r each integer m , x m 2 j x m 2 j +1 . . . x ( m +1)2 j / ∈ F (0) 2 j +1 . So, by Lemma IV .7, we h av e x m 2 j 6 = x ( m +1)2 j . This implies that x 0 = x (2 t )2 j as | Σ | = 2 . But then , x (2 t )2 j . . . x (2 t +1)2 j − 1 x 0 ∈ F (0) 2 j +1 , which is a contradictio n. W e are now in a po sition to prove Th eorem IV .5. Pr oo f of Theorem IV .5 : T o p rove the theor em, it is enou gh to show th at f or j ≥ 0 , the p eriods of per iodic sequen ces in X 2 j +1 must b e multiples of 2 j +1 . It then follows, by Lemma I V .6, that the sam e also ap plies to perio dic seque nces in X k , f or 2 j + 1 < k ≤ 2 j +1 . When j = 0 , the re quired statement cle arly ho lds by Lemma IV .8. So, suppose that th e statement is tr ue for som e j ≥ 0 , so that periodic sequen ces in X 2 j +1 have only multiples of 2 j +1 as periods. There fore, by Lemma IV .6, perio dic sequences in X 2 j +1 +1 also can only have m ultiples of 2 j +1 as period s. Howe ver, by Lemma IV .8, no perio dic sequ ence in X 2 j +1 +1 can have an odd multip le of 2 j +1 as a period. Hence, all perio dic seque nces in X 2 j +1 +1 have per iods that are mu ltiples o f 2 j +2 . T he theo rem follows by induction. Theorem IV .5 shows th at for 2 j + 1 ≤ k ≤ 2 j +1 , we ha ve T ( X k ) seq ≥ 2 j +1 . I n fact, this hold s with eq uality . Corollary IV .9 T ( X 1 ) seq = 1 , and for k ≥ 2 , if j ≥ 0 is such that 2 j + 1 ≤ k ≤ 2 j +1 , then T ( X k ) seq = 2 j +1 . Pr oo f : When k = 1 , T ( X 1 ) seq = 1 as 1 ∞ ∈ X 1 . So le t k ≥ 2 , and let j ≥ 0 be such that 2 j + 1 ≤ k ≤ 2 j +1 . W e only need to show th at T ( X k ) seq ≤ 2 j +1 . Th e bi- infinite sequ ence w = (0 2 j +1 − 1 1) ∞ is in X 2 j +1 since, by Corollary IV .3, w contains no word in F (0) 2 j +1 as a sub word. Sin ce w has period 2 j +1 , by Lem ma IV .6, w ′ = ψ 2 j +1 − k ( w ) ∈ X k has perio d 2 j +1 as well. Thus, T ( X k ) seq ≤ 2 j +1 . Theorem I V .5 also implies th e f ollowing cor ollary . Corollary IV .10 T ( X k ) gr aph ≥ T ( X k ) seq holds when 1 ≤ k ≤ 6 . Pr oo f : Since X 1 is prope r , T ( X 1 ) gr aph ≥ 2 by Propo sition 1 in [3]. Thu s, T ( X 1 ) gr aph > T ( X 1 ) seq = 1 . So, let k ≥ 2 and suppose 2 j + 1 ≤ k ≤ 2 j +1 for some j ≥ 0 . By Corollary I V .9, we have T ( X k ) seq = 2 j +1 . On the other h and, fo r any irr educible pre sentation G o f X k , we have per ( G ) ≥ 2 j +1 . Indeed, for each vertex V in G , we have per ( V ) b eing a multiple of 2 j +1 ; otherwise we would have a co ntradiction of Theorem I V .5. Hence, T ( X k ) gr aph ≥ 2 j +1 = T ( X k ) seq as req uired. Corollary IV .9 sho ws that T ( X k ) seq grows ar bitrarily large as k → ∞ , while T ( X k ) desc = 2 for all k . It also follows from Corollary IV .10 that T ( X k ) gr aph is strictly larger than T ( X k ) desc when 3 ≤ k ≤ 6 . Eq uality can hold in Corollary I V .10 — for example, wh en k = 2 . In deed, X 2 is pr oper, and its MS presentation , G X 2 , is irred ucible, with per ( G X 2 ) = 2 , so that T ( X 2 ) gr aph = 2 . Fro m Corollary IV .9, we also hav e T ( X 2 ) seq = 2 . Thus, X 2 is an example of a prop er PFT X in wh ich T ( X ) seq = T ( X ) gr aph = T ( X ) desc holds. Thus, to summ arize, there appears to b e no relationship between the descrip ti ve perio d of a PFT an d its sequential period, as we h av e examp les wh ere each of these can be arbitrarily larger than the oth er . W e have also fo und th at, for a PFT X , T ( X ) gr aph can be larger than T ( X ) desc . Howe ver , we b eliev e that the reverse cannot hold; in fact, we conjecture that T ( X ) desc divides T ( X ) gr aph for any PFT X . Finally , we no te that we also have e xamp les of proper PFT’ s X where T ( X ) seq is arbitrarily larger than T ( X ) gr aph . W e omit the proof d ue to space con straints. Theorem IV .1 1 Set Σ = { 0 , 1 } and k ≥ 2 , and let P denote the set of all periodic b i-infinite sequences over Σ with period k ! . Consider the PFT X = X {F , 2 } with F = ( F (0) , ∅ ) , such that F (0) = { w ∈ Σ 2 k ! : ∃ x ∈ P such that w ≺ x } . The following statements h old: (a) X is proper ; (b) X is irreducib le; and (c) T ( X ) seq ≥ k + 1 and T ( X ) gr aph = 2 . R E F E R E N C E S [1] M.-P . B ´ eal, M. Crochemore and G. Fici , “Presentat ions of constraine d systems with unconstrained positi ons, ” IE EE T rans. Inf. Theory , vol. 51, pp. 1891–1900, May 2005. [2] D. Lind and B.H. Marcus, An Intr oduction to Symbolic Dynamics and Coding , Cambridge Uni versity Press, 1995. [3] B.E. Moision and P .H. Siegel , “Periodic-finit e-type shift spaces, ” preprint. [4] B.E. Moision and P .H. Siegel, “Periodi c-finite-type shift spaces, ” Pr oc. ISIT 2001 , W ashington DC, June 24–29, 2001, p. 65. [5] D.P .B. Chav es and C. Pimentel , “ An algori thm for finding the Shannon cov er of a periodic shift of finite type, ” preprint. [6] T .L. Poo and B.H. Marcus, “Time-v arying maximum transition run constrai nts, ” IEEE Tr ans. Inf.. Theory , vol. 52, pp. 4464–4480, O ct. 2006. [7] J.C. de Souza, B. H. Marcus, R. 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