How to simulate Turing machines by invertible one-dimensional cellular automata

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1 How to simulte Turing mchines by invertible one-dimensionl cellulr utomt Jen-Christophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, Lyon Cedex 07, Frnce Jnury 4, 2008 Abstrct The issue of testing invertibility of cellulr utomt hs been often discussed. Constructing invertible utomt is very useful for simulting invertible dynmicl systems, bsed on locl rules. The computtion universlity of cellulr utomt hs long been positively resolved, nd by showing tht ny cellulr utomton could be simulted by n invertible one hving superior dimension, Toffoli proved tht invertible cellulr utomton of dimension d 2 were computtion-universl. Morit proved tht ny invertible Turing Mchine could be simulted by one-dimensionl invertible cellulr utomton, which proved computtion-universlity of invertible cellulr utomt. This rticle shows how to simulte ny Turing Mchine by n invertible cellulr utomton with no loss of time nd gives, s corollry, n esier proof of this result. Keywords: cellulr utomt, universlity, invertibility. 1 Introduction A cellulr utomton cn be seen s simple rry of cells, ech cell intercting with defined set of neighbors, nd chnging of stte ccording to n internl tble in synchronous wy (i.e. ech cell considers the old sttes of the other cells to compute its new stte). One of the best known exmple of cellulr utomton is Conwy s gme of Life. We define nd use subclss of cellulr utomt tht cn be esily used for hndling simultions of smll locl processes. These prtitioned cellulr utomt re convenient tool for describing some processes of cellulr computtions. In this subclss, we consider tht there re severl flows of informtion tht go through ech cell. The set of neighbors defines the direction nd ll the chrcteristics of tht informtion flow, tht is quntified by product of finite sets. All tht cn be done by cell is to melt the different flows, creting new ones or deleting some others, ccording to deterministic tble of 2,rue Lebudy, Rosny/Seine, Frnce ; emil: jcdubcq@ens.ens-lyon.fr

2 trnsition. The description of these utomt llows esier hndling thn the one of rndom cellulr utomton. The purpose of the second prt of this pper is to prove the following result : Theorem 1 Any 1-tpe Turing Mchine cn be simulted without loss of time by n invertible prtitionned cellulr utomton. Morit, in [3], proved tht invertible Turing Mchines could be simulted by invertible cellulr utomt, nd he lso presented proof tht Turing Mchines could be simulted by invertible ones, but the trnsformtion of TM into n invertible one needs qudrtic time to run. Our construction is independnt of Morit s result. The ide of the proof is to hve ech cell simulting one tpe cell of the Turing mchine, while conveying to the left n history of ll the trnsformtions performed by the hed. 2 Definitions nd properties In this section, we shll define subclss of cellulr utomt tht re clled prtitionned cellulr utomt. These utomt re very simple to use in simultion of locl processes, nd esy to define. Their forml definition llows primry results tht re quite interesting. 2.1 Prtitioned cellulr utomt. Prtitioned cellulr utomt (PCA) re formlly defined s qudruplets (d, S, N, f) where: d is the dimension of the PCA; S = S 1 S 2... S r where S 1, S 2,..., S r re finite sets of stte; r N is the number of neighbors; N = {z 1, z 2,..., z r } is finite subset of Z d (the neighbourhood of the PCA); f is the trnsition function defined from S N S with the following conditions: f is the composition of two functions Φ G N ; Φ is function from S into S; G N is the function defined from S N into S by where s (j) z i G N (s 1, s 2,..., s r ) = (s (1) z 1, s (2) z 2,..., s (r) z r ) is the j th component of the i th neighbor.

3 Figure 1: The set of neighbors of A nd the ction of G { 2, 1,+1} Initil configurtion C 0 A single step of computtion over C 0 by G { 2, 1,+1} A single step of computtion over C 0 by A Figure 2: Applying A to configurtion As n exemple we give the ction of the following PCA: A = (1, {0, 1, 2} {0, 1, 2} {0, 1, 2}, { 2, 1, +1}, sorting G { 2, 1,+1} ) where the function sorting is the function tht tkes list of three numbers s input nd gives the sme list in incresing order s output. A PCA is useful for moving round informtion (s quntified by the vrious sets of sttes). Informtion will move in stright line if Φ is the identity function or could be melted loclly by the ction of Φ. The difference between prtitionned cellulr utomt nd cellulr utomt is tht the locl trnsition function is lwys defined s being the composition of G N nd of nother fonction, nd tht the set of sttes is relly defined s being product of sets. 2.2 Invertibility Lemm 1 A prtitioned cellulr utomton A is invertible iff its function Φ is bijection. Proof. First we prove tht if Φ is not bijection, then A is not invertible. We exhibte two distincts configurtions tht hve the sme imge. As Φ is not bijective, it is lso not injective (s it goes from finite set into the sme finite set). Hence, there does exist two sttes of the PCA tht hve the sme imge. Let s denote them by (σ 1, σ 2,..., σ r ) nd (θ 1, θ 2,..., θ r ). Then, let s tke two configurtions tht re identicl except for the neighbors of cell c in which the i th component of the i th neighbor is σ i in the configurtion C 1 nd θ i in the configurtion C 2. If we pply A on these two

4 configurtions, it is cler tht G N lets ll the cells identicl except the cell c, nd tht Φ pplied to c lets ll the cells identicl (s both configurtions get the sme stte for cell c). Hence, we get configurtion tht hs two ncestors, nd A is not invertible. Now, we shll prove tht if Φ is bijection, the PCA A is invertible. We simply construct A 1. Let s denote the stte of cell c by s c. Let A 1 = (d, S, N, g) with g function tht cn be written s G N Φ 1 where Φ 1 is the extension of Φ 1 to S N : Φ 1 (s c1, s c2,..., s cr ) = (Φ 1 (s c1 ), Φ 1 (s c2 ),..., Φ 1 (s cr ))) Hence, the i th component of cell c is (Φ 1 (s c z i )) (i), where s c is the stte of the cell c fter hving pplied A to the initil configurtion. We cn expnd it into: (Φ 1 (Φ(G N (s c zi +z 1, s c zi +z 2,..., s c zi +z r ))) (i) i.e. (G N (s c zi +z 1, s c zi +z 2,..., s c zi +z r )) (i), i.e. s (i) c z i +z i, i.e. s (i) c. As tht computtion stnds for ny cell c nd ny i, 1 i r, the utomt A is invertible nd dmits A 1 for inverse. Let us note tht the inverse for the PCA we built is not exctly PCA since the melting is done before the trnsltion, but it is not fundmentlly different (nd it is still CA). 2.3 Consequences We hve just built subclss of CA tht is very esily described, nd for which invertibility is esily decidble. This section tries to explin why the trnsformtion doesn t preserve invertibility. As Kri showed in [1] tht in generl cse, invertibility of 2D cellulr utomt is undecidble, there is difference between the two clsses. CA cn obviously simulte PCA, s the definition of these cn be seen s just definition of subset. PCA cn simulte CA s follows: just mke ech cell send the whole informtion to ll the neighbors, nd then let ech cell pply the trnsition function of the CA to cell (s every cell will know the sttes of the neighbors). But the simultion does not preserve bijectivity (becuse just few sttes of the cells correspond to the simultion). Thus it is not possible to prove tht invertibility of the PCA is equivlent to the invertibility of the corresponding CA. In fct, we build mpping from one set of configurtions into nother set of configurtions tht is bijection in the cse of the simultion of PCA by CA (hence decidbility of invertibility is kept) nd tht is not surjective in the cse of the simultion of generic CA by PCA. 3 Simultion of Turing Mchines by one-dimensionl invertible prtitioned cellulr utomt We re interested in the possibility of simulting processes with prtitioned cellulr utomt, especilly the simultion of Turing Mchines. This resolves the problem of

5 universlity of invertible cellulr utomt, s invertible prtitioned cellulr utomt re subset of invertible cellulr utomt nd simultion of Turing Mchine llows the computtion of ny computble function. Morit lredy nlysed the reltions between universlity nd cellulr utomt in [3], [4] nd [2], but this study brings new pproch to the problem. 3.1 Deterministic Turing Mchine The Turing Mchine tht we shll consider in this section cn be defined s triplet T = (F, Q, τ) where: F is finite set of symbols (the set of chrcters of the Turing Mchine) in which we cn distinguish blnk symbol, denoted by 0, Q is finite set of sttes of the R/W hed, in which we cn distinguish n cceptnce hlt-stte q Y, τ is function from F Q F Q {left, right}, the trnsition function of the Turing Mchine T. A configurtion of tpe is given by mpping from Z into F. A configurtion of Turing Mchine is the configurtion of tpe nd the position nd internl stte of the hed. For ech computtion step, the R/W hed tht is initited to strting position mkes trnsition ccording to the function τ depending on the vlue of the cell nd the vlue of the stte of the hed, rewrites new vlue in the cell, nd then moves whether to the right or to the left (ccording to τ). A vlid configurtion is configurtion tht cn be deduced from n initil mpping from Z into F with finite number of symbols different from 0 nd the hed t the left end of the tpe in finite number of steps. 3.2 Definition of A τ Being given Turing Mchine T, we shll consider the following PCA A τ : N = {0, 1, 1, 2} Σ = Q {} A τ = (1, F Σ Σ (F Σ {left, right}), N, Φ G N } We shll denote the stte of cell by (, b, c, d) nd prtilly define Φ in the following wy: If b, c =, d = (0,, left), then, if we suppose (, b, δ) = τ(, b): Φ(, b,, d) = (, b,, (, b, left)) if δ = left = (,, b, (, b, left)) if δ = right

6 If b =, c, d = (0,, left), then, if we suppose (, c, δ) = τ(, c): Φ(,, c, d) = (, c,, (, c, right)) if δ = left = (,, c, (, c, right)) if δ = right If b = c =, then Φ(,,, d) = (,,, d) We shll prove tht Φ restricted to the lredy-defined sttes is injective. Hence, we will be ble to extend Φ to bijective function by choosing ny imge not lredy used for the remining undefined sttes. Remrk tht the number of sttes of the A is 2f 2 (q + 1) 3, where f is the size of the set of chrcters of T nd q is the number of sttes of T. Lemm 2 The restriction of Φ to the previous configurtions is injective. Proof. Injectivity of Φ in sme group of sttes is trivil, s d is lwys (0,, left). We shll just check tht there cn t be ny confusion between two different groups. The two first groups (b or c different from ) re obviously distinct becuse of the lst component of the imges. Moreover, no element of these two groups cn be seen s n element of the third one, s the second element of the lst component is necessrily different from in the first two groups nd is in the third. Corollry 1 Φ cn be extended into bijective function. Hence, A τ one-dimensionl cellulr utomton. is n invertible 3.3 Simultion of T by A τ Vlid configurtions of A τ re defined s configurtions of the form (0,,, (0,, left)) (blnk cells) everywhere but in finite number of cells, where it cn be of the form (γ,,, (0,, left)), with γ F (clen cells) or (γ,,, κ), with γ F nd κ F Σ {left, right} (used cells). Moreover, there must be exctly one cell in the line tht hs the form (γ,,, (0,, left)) with nd γ F, for which ll cells on the right re either blnk or clen. The set of these configurtions defines the Turing-vlid configurtions. Lemm 3 There exists n injection from the set of configurtions of T into the set of Turing-vlid configurtions of A τ Proof. Let C be configurtion of T. We build Turing-vlid configurtion of A τ s follows: j 1, C C (j) = (C(j),,, (0,, left)); C C (1) = (C(1), q 0,, (0,, left)), where q 0 is the initil stte of the Turing Mchine T.

7 b 0 left τ(,b)=(,b,left) b b left b 0 left τ(,b)=(,b,right) b b left 0 b left τ(,b)=(,b,left) b b right 0 b left τ(,b)=(,b,right) b b right d e f d e f Figure 3: Trnsition rules for A τ

8 The configurtion we get for A τ is clerly Turing-vlid. The injectivity results from the mpping of C into C C. Now, we shll prove the min theorem: Theoreme 1 The utomton A τ simultes the computtion of T on ny configurtion C of T with no loss of time. Proof. We just hve to show function tht turns the configurtion of A τ obtined from Turing-vlid configurtions by itertions of A τ into configurtions of T obtined by the sme number of computtion steps. We do this by writing tht the line of cells is turned into the tpe of T by keeping just the first field of it (i.e. the chrcter of the cell). The R/W hed is lso in the only stte tht is different from in the second or third fields. We ll just write tht the position of the R/W hed is the bciss minus one of cell from which the second field is different from, or the bciss plus one of cell from which the third field is different from. In fct, we hve to gurntee tht there s only one cell for which the second or the third field is different from to prove the vlidity of tht trnsformtion. We cn prove by induction the following property: there exists one nd only one cell (with bciss x 0 ) for which the second or third field is different from, nd x > x 0, the fourth field is (0,, left). It s true for the Turing-vlid configurtions. Let s suppose tht it is true for the n th itertion. The R/W hed is on the cell x 0 (symbol different from ). After the itertion of G N, the R/W hed is either one cell on left, or one cell on right. The shift on left of the fourth fields tht re different from (0,, left) implies tht they re t most in cell x 0 2; the R/W hed being in position x 0 1 or x 0 + 1, the fourth component is (0,, left) nd the function Φ ensures tht there is only one R/W hed fter the ppliction of Φ. Hence, the condition is true for the (n + 1) th ppliction of A τ becuse the configurtion is still Turing-vlid. The lst check is for the correctness of the simultion of the Turing Mchine T. The move of the R/W hed is correct, s given by the function τ. The tpe is correctly simulted, becuse the cells chnge only if the R/W hed is operting on them, nd, if it opertes, the trnsition is the one tht would be done by the Turing Mchine (ccording to the definition of Φ). Hence, for ny instnt t, there is n exct correspondnce between the tpe of T nd the configurtion of the cellulr utomton. From the preceding, we cn deduce the following corollry: Corollry 2 There exist universl invertible one-dimensionl cellulr utomt. Acknowledgements I would like to thnk the people tht hve directed my work, Jcques Mzoyer nd Bruno Durnd, of the Lbortoire d informtique du Prllélisme of the École Normle Supérieure de Lyon.

9 References [1] J. Kri. Reversbility of 2D cellulr utomt is undecidble. Physic, D 45: , [2] K. Morit. Any irreversible cellulr utomton cn be simulted by reversible. Technicl report, IEICE, [3] K. Morit nd M. Hro. Computtion universlity of one-dimensionl reversible (injective). Trnsctions of the IEICE, E-72(6): , June [4] K. Morit nd S. Ueno. Computtion-universl models of two-dimensionl 16-stte reversible utomt. IEICE Trns. Inf. nd Syst., E75 D(1): , Jnury 1992.