Small Turing universal signal machines

T . Neary , D. W oods, A.K. Seda and N. Murphy (Eds.): The Complexity of Simple Programs 2008 . EPTCS 1, 2009, pp. 70–80, doi:10.4204/EPTCS.1.7 c  J. Durand-Lose Small T uring universal signal machine s J ´ er ˆ ome Durand-Lose Laboratoire d’Informatique Fondam entale d’Orl ´ eans, Univ ersit ´ e d’Orl ´ eans, B.P . 6 759, F-45067 ORL ´ EANS Cede x 2. Jerome.Dur and-Lose@u niv-orleans.fr This article aims at providing signal machines as sm all as p ossible able to perform any computation (in the classical under standing). After presenting signal mach ines, it is shown how to get universal ones from T u ring machines, cellular-automa ta a nd cyclic tag systems. Finally a halting universal signal machine with 13 meta-signals and 21 collision rules is presented. 1 Introd uction Computatio n and uni vers ality hav e been defined in the 1930’ s. In the last five or so decades, it has been un veiled how common they are. T he questio n about the frontier moved from pro ving the univ ersa lity of dynamic al systems to the complexit y of univ ersal machines . This is not only an intellec tual challeng e, b ut also importa nt to find niche s escaping tons of non decidabil ity results or otherwise assert that these niches are too small to be of any inte rest. There hav e already been a lot of in vestig ations on small Tur ing machines (Rogozhin, 198 2, 1996; Mar genste rn , 1995; Kudlek, 1996; Baiocchi, 200 1), re gister mach ines (K orec, 1996), and cel lular au- tomata (Ollinge r , 2002; Cook, 2004). Moreo ver , a s (T uring) uni ver sality (capabili ty to ca rry out an y T urin g/clas sical computa tion) has been de ve loped in very limited systems, some “adaptati ons” were made and variou s noti ons of uni versalit y exi st: • polynomial time univer sal ity w hen polynomial computatio ns (as defined in complexity theo ry) are still done in polynomia l time, as oppos ed to e xpone ntial time univer sali ty provid ed by e.g. 2-coun ter automata, and • semi-univer sality when the computation m ust be started on an infinite configuration –for example the whole tape of a T uring machine is fi lled w ith some ultimately -period ic infinite world– or in many c ases like cellula r automata w hich natur ally work on infinite configu rations . In the prese nt article, minimal uni versal machine i n the c onte xt of Abstr act g eometri cal computation (A G C) is in ve stigate d. A GC has been introduced as a continuou s counterp art of cellular automata. This mov e is inspired by the way dynamics of CA is often designe d or analyzed in an Euclidean space. In A G C, signals are movin g with constan t speed in an Euclidean space. When the y meet, they are replac ed/re writte n; inter acting in a collision based computing way . A signa l machin e (SM) defines exi sting kinds of signa l, meta-si gnals , and their interactions , collision rules . A G C allo ws, since this is a graphi cal model, to understa nd the way the information is m ov ed around and interac ts as sho wn by the v arious illustr ations. Abstract geometric al computation uses continuo us space and time so that the possibilit y Z eno ef fect has to be consid ered. Indeed, it can happen and be used to compute beyo nd T uring comput ability and to J. Durand-Los e 71 climb the arithmetical hierarch y by using accumulati ons of collis ions (Durand-Lose, 2009a). But since only T uring comp utabili ty is addre ssed here, accumulation s are not consi dered. The straightforw ard measure of complexity (or simple/ small-nes s) of an SM is the number of meta- signal s. If there are m meta-sign als defined, then there are at most 2 m − m − 1 po ssible collision rules (at least two signals are needed in a collisi on and the y must be all dif fere nt). Since parallel signals cannot interac t, the number of rules c ould be much lower , b ut ne vertheles s exceed ing by far the numb er of meta- signal s. In many co nstruc tions, only a small part is defined, the rest being either undefined or blank (i.e. signal s just cross ea ch ot her). So th at the number o f spec ially defined ru les is also a good comple ment to the size of an SM. In Dura nd-Lose (2 005a,b), T uring u ni ve rsality is prov en b y red uction from 2-counter auto mata. T his result is not interesting here since the uni v ersal SM they provide are expo nentia l time univ ersal and not ver y small. (For an S M compu tation , the time comple xity is the longest chain of collis ions linked by signal s.) In the p resent article, uni versal SM are generated from T uring machines , cellul ar aut omata a nd cyclic tag systems. In each case, specia l care is taken in order to sav e signals and collisio n rules. For Tur ing machines , only results based on the construct ion in D urand -Lose (2009a) are presented; this direct sim- ulatio n is not detailed since it does not provide the best bound. Cel lular automata (CA) are massi ve ly paralle l devic es w here bo unds are also very tight and pro vide a very small semi-un iv ers al SM . Cyclic tag systems (CTS ) work by considering a binary word and a circu lar list of binary words . At each iteration , the fi rst bit of the word is deleted and if it is 1 the fi rst word of the list is added to th e end of the word, then the list is rotated. CTS are polynomia l time T uring univ ersal (Neary and W ood s, 2006a) and hav e been used to provide the best kno wn bounds on T uring machines (W oods and Neary, 2006, 2007; N eary and W oo ds, 2006b) and cellular automata (Cook, 2004). They also provide the best bound s presented here. Signal machines are presen ted in Section 2. Each followin g sectio n deals with a differe nt univ ersal model and present s ways to simulate them with SM: Tu ring machines in Section 3, cellula r automata in Section 4, and cycli c tag systems in Section 5. Section 6 gathers some conclud ing remarks . 2 Definitions In Abstr act ge ometrica l computat ion , dimensionles s objects are m ovi ng on the real axis. When a colli- sion occurs they ar e replaced according to rules. This is defined by the follo wing machines: Definition 1 A signal machine is defined by ( M , S , R ) where M ( meta-si gnals ) is a fi nite set, S ( spee ds ) a mapping from M to R , and R ( collision rules ) a function from the subsets of M of cardinality at least two int o subsets of M (all these sets are composed of m eta-sig nals of distinct speed). Each instance o f a meta -signal i s a signal . The mapping S assigns speeds to signals. T hey corre spond to the in v erse slopes of the l ine segment s i n space-time diagrams. A collision r ule , ρ − → ρ + , defines what emer ges ( ρ + ) from the collision of two or more signals ( ρ − ). Since R is a function, signal m achine s are determin istic. The ex tended valu e set , V , is the union of M and R plus one symbol for void, ⊘ . A config ura tion , c , is a mapping from R to V such that the set { x ∈ R | c ( x ) 6 = ⊘ } is finite. A n infinite config ura tion , is a similar mapping such that the pre vio us set has no accumulatio n point. A signal correspon ding to a meta-sign al µ at a position x , i.e. c ( x ) = µ , is moving uniformly with consta nt speed S ( µ ) . A signal must start (resp . end) in the initial (resp. final) configur ation or in a 72 Small T uri ng uni ve rsal signal machines collisi on. This corresp onds to con dition 2 i n Def. 2 . At a ρ − → ρ + collisi on, signals corre spond ing to t he meta-sig nals in ρ − (resp. ρ + ) must end (resp . start) and no other signal should be present (condition 3). Definition 2 The space-t ime diagr am issued from an initial configura tion c 0 and lasting for T , is a mapping c from [ 0 , T ] to con figuration s (i.e. from R × [ 0 , T ] to V ) such that, ∀ ( x , t ) ∈ R × [ 0 , T ] : 1. each { x ∈ R | c t ( x ) 6 = ⊘ } is finite, 2. if c t ( x )= µ then ∃ t i , t f ∈ [ 0 , T ] with t i < t < t f or 0 = t i ≤ t < t f or t i < t ≤ t f = T s.t.: • ∀ t ′ ∈ ( t i , t f ) , c t ′ ( x + S ( µ )( t ′ − t )) = µ , • t i = 0 or ( c t i ( x + S ( µ )( t i − t )) = ρ − → ρ + and µ ∈ ρ + ), • t f = T or ( c t f ( x + S ( µ )( t f − t )) = ρ − → ρ + and µ ∈ ρ − ); 3. if c t ( x )= ρ − → ρ + then ∃ ε , 0 < ε , ∀ t ′ ∈ [ t − ε , t + ε ] ∩ [ 0 , T ] , ∀ x ′ ∈ [ x − ε , x + ε ] , • ( x ′ , t ′ ) 6 = ( x , t ) ⇒ c t ′ ( x ′ ) ∈ ρ − ∪ ρ + ∪ {⊘} , • ∀ µ ∈ M , c t ′ ( x ′ )= µ ⇔ or  µ ∈ ρ − and t ′ < t and x ′ = x + S ( µ )( t ′ − t ) , µ ∈ ρ + and t < t ′ and x ′ = x + S ( µ )( t ′ − t ) . On space-ti me diagrams, time is increa sing upward. T he traces of signals are line segments whose directi ons are d efined b y ( S ( . ) , 1 ) (1 is the tempora l coordi nate). Collisions c orrespo nd to the extre mities of these seg ments. This definition can easily be exten ded to T = ∞ and to infinite initial configur ation. Although speeds may be any real and thus encode informat ion, in the follo w ing, only a few integer v alues are used. Similarly , the distance between signals m ay be any real but only integer positions are used. 2.1 T ime complexity measure As a computing dev ice, the input is the initial configurati on and the output is the fi nal configuratio n. A SM is T urin g univer sal if there exis ts encodi ngs/re presentations thro ugh which it can go from the code of a T uring machine (or any equiv alent model of computation) and an entry to the output (if any). A GC pro vides dynamical systems with no halting feature. Mainly two appro aches exist to settle this: • an observ er checks that t he end of the computa tion is reached acc ording to so me propert y ove r the configura tion, e.g. the presence of a meta-signal , or • the system reaches a stable state, i.e. no more collisio n is possible. The las t one is prefer red since the hal ting is then a part of the sy stem; althou gh it is somet imes meaning - less: for example, when simul ating a cellular automata, since there is no halting feature in CA, the same discus sion arises again. T o consider polyn omial time uni v ersality , the time complexity of a computation should be defined. Space and time are con tinuou s, by resc aling a finite computation, it can be made as short as wan ted, and e ven w orse: any infinite comp utation startin g from a finite c onfigurat ion can b e auto matically fol ded into a finite portion of the space-ti me dia gram (Durand-Los e , 2009a). So that a correct notion of time comple xity lies elsewhe re. Collision s are cons idered as the discrete steps related by signals : a colli sion is causally befor e ano ther if a signal generate d by the first one ends in the second one. This yields a direct acyc lic graph. Definition 3 The time complex ity of an SM computation is the longest length of a chain in the collision causal ity D A G . For space comple xity one may consi der the longest length of an anti-chain or the maximum number of signals present at the same time (which form a cut). J. Durand-Los e 73 2.2 Generating an infinite periodic signal patter n The aim is to generate a p eriodic infini te seq uence of signa ls on t he sid e of a space-time d iagram. This is useful to generate, starting from a finite number of signals the infinite data for semi-uni v ersalit y (when it is periodic ). For TM, since it happens away from the head, there is no problem. The C A case is not so simpl e as ex plaine d later . In both cases, the increasi ng of the size all o ws to f all back into regular uni v ersalit y . The constru ction works as follo ws: a signal bounce s way and back between two signals , each time it bounces on the upper signal, a signal is emitted. T he boundary signa ls are used to record the locatio n in the pattern. The constructio n is simple and straig htforw ard. It is only presented on an examp le: an infinite sequenc e of period 3, ( µ 1 µ 2 µ 3 ) ω , where each µ i is already defined. The added signals and rules as well as the resulti ng space-time diagram are displayed on Fig. 1. Id Speed boun L 2 bord L 1 , bord L 2 , b o rd L 3 1 Collision rules { boun L , b o rd L 3 } → { µ 1 , bord L 1 } { boun L , b o rd L 1 } → { µ 2 , bord L 2 } { boun L , b o rd L 2 } → { µ 3 , bord L 3 } { bord L 1 , µ 1 } → { µ 1 , bord L 2 , boun L } { bord L 2 , µ 2 } → { µ 2 , bord L 3 , boun L } { bord L 3 , µ 3 } → { µ 3 , bord L 1 , boun L } boun L bord L 1 bord L 3 bord L 1 bord L 1 µ 1 µ 2 µ 3 µ 1 boun L boun L boun L bord L 2 bord L 3 bord L 2 bord L 3 Figure 1: Generating a periodic pattern. T o sav e a signal, the signal is emitted at th e bottom. T he number of added signals is one plus the period (the outpu t signals are not c ounted ). The number of r ules is twice the p eriod. If the out put signal s are to be set at unequal distances, the generato r uses one plus twice the period meta-signals (one for the two bor ders and differe nt way and back for the whole period). 3 T uring machines Due to the poor results genera ted and the lack of space, the presenta tion of T uring machin es and the full constr uction from Durand-Lose (2009a) are not giv en here. The nu mber of meta-si gnals in the sign al machine simulat ing a TM is: 1 for each tape symbol, 2 for each state, and 4 extra signals for enlar ging the tape. The number of collisio n rules is up boun ded by: 2 for each entry of the transition table of the TM, and plu s 2 for each tr ansitio n on the bla nk sy mbol pl us 2 (fo r e nlar gin g the t able). In the cited pap er , TM haltin g correspond s to the head lea ving the tape. W ith a halting state, the correspo nding colli sion rules produce the disapp earance of the state signal, at no extra cost. Consider ing the curv e of Neary and W oods (2006b ); W oods and Neary (20 06, 200 7), this lead s to the follo wing valu es: 18 meta-signals for polynomial time univ ersal ity . The number of collision rules is bound ed by 62. The exact number s hav e not been compute d since fewer meta -signa ls is possible. Semi-uni v ersal T M with fewer stat es exists. L ike the o ne presented in Subsect . 2.2, add hoc construc- tions to generate the extensio n of the tape –to achiev e full univ ersali ty– would add too many states and 74 Small T uri ng uni ve rsal signal machines has not been con sidere d. Fo r semi-u ni ve rsality , using Smith (200 7), the 2-states 3-sy mbol TM generat es a SM with 7 meta-sign als (the ones for enlar ging are not needed) and 6 collision rules. 4 Cellular automata Cellular aut omata (CA) operate ov er infinite arrays of cells. E ach cell can be in finitely many states. (Infinite array is the only way to ensure unboun ded m emory .) A CA changes the states of all the cells simultan eously according to a local function and the states of the two surr oundin g cells. This is parallel, synch ronous , local and uniform process. Definition 4 A cellular automa ton is defined by ( Q , f ) where: Q is a finite set of states , f : Q 3 → Q , is the local function . The glo bal function , G : Q Z → Q Z , is defined by: ∀ i ∈ Z , G ( c ) i = f ( c i − 1 , c i , c i + 1 ) . Only CA of dimension 1 and radius 1 are consider ed here. Higher dimensions can be cover ed simi- larly by higher dimensio n signal machines. Radius 1 means that a cell only communica tes with its two closes t neighbo rs (one on each side). Broader radii could hav e been consid ered, but more signals are needed to c on vey i nformatio n at g reater dista nces (e.g. 5 meta -signal s per stat e instead of 3 for radiu s 2). Halting is not pro vided by C A, it can be defined by reaching a stable /perio dic configuration, the appari tion of a stat e/patte rn in the configuratio n or on some designated cell. In any case, more meta- signal s are needed. There are two ways to manipulate finite C A-configura tions: use some quiescent state , q # (satisf ying f ( q # , q # , q # ) = q # ), for unde fined cells; or use a period ic spatial e xtensio n on both sid e. This patt ern is also (ultimately ) time-periodi c in the CA ev olution. In the simulatio n, the computation is framed by period ic signals according to oblique discrete lines in the s pace-ti me diagram of the CA as illu strate d by the examp le on Fig. 3. The idea is to locat e the cells at inte ger positions and each time the local function is used, three signal s are emitted, one for the cell and one for each of the closest cell on each side. For each state, s , there are three meta-signals: s L , s and s R or speed s − 1, 0 and 1 respecti v ely . A transi tion is performed when a cell recei ve d simultaneousl y the valu es from th is two neighbo rs (spe cial care has to be pu t on locatio ns to ensure exact meetings). T he local function is encod ed in the collision rules : if f ( s , t , u ) = v then the follo wing rule is defined { s R , t , u L } → { v L , v , v R } . Rule 110 is presented Fig. 2 as w ell as a genera ted rule. Output 0 1 1 0 1 1 1 0 Input 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 zero R one zero L one L one one R Figure 2: L ocal funct ion and the rule implementing f ( 0 , 1 , 0 ) = 1. Evo lution and simulati on on the configurati on 11, framed on the left by ω ( 10 ) and on the right by ( 011 ) ω is presen ted on Fig. 3. As it can be seen, the way the frames are position ed (in boldf ace) in the e v olutio n is not trivi al and neither is the generated periodic pattern on both side of the simulation. W ith the references in Durand-Lose (200 9b) and Ollinger (20 08), it is hard to go belo w 30 meta- signal s for full univ ersa lity . Ollinger (2002); Cook (2004) and Richard and Ollinger (2008) prov ide uni v ersal C A with ve ry few states b ut they use period ic extensio ns that encode a boolean function or a cyc lic tag system which leads to too many meta-signa ls unless a cle v er way to generate them is found. J. Durand-Los e 75 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 0 1 Figure 3: E v olutio n and simulation of rule 110 on 11 framed by ω ( 10 ) and ( 011 ) ω . Nev erth eless, with rule 110 (prov ed univ ersal in Cook (2004)), a 6 meta-sign als 8 rules semi-uni v ersal SM is generated. This SM is no t h alting . T o get a halting SM from a CA , one has to kno w precise ly what corres ponds to halt and add signals and rules for the SM to halt. 5 Cyclic tag systems simulation Cyclic tag systems (CTS) are defined by a word and a circ ular list of appenda nts. Both th e word and the appen dants are binary words. The system is updated in the follo wing way: the first bit of the word is remov ed. If it is 1 the n the first a ppenda nt is appended at the end of the word (otherwise nothing is done). Then, the list is rotate d circularly . The list repres ents the code and the word the inpu t. Not only are CTS able to compute (Cook, 2004) b ut also the y can do it with polynomia l slo wdo w n (Neary and W oods, 2006a). The simulation is done with two object s: parallel signals encodin g the word, and a cycle list: each time it cycl es, depe nding on the signals that started it can deliv er a cop y of the first appendan t or not. Signals encoding the word are placed so that deliv ered copies automatical ly enlar ge the wor d on the right. The initial configuration is presente d before the dynamics of the vari ous elements . The signals are defined on F igure 4. T he follo w ing naming con ventio n is used : meta-signals with no subscript hav e speed 0 and the ones with subscrip t L L , R and RR ha v e speed − 2, 1 and 2 respecti v ely . Speed Meta-Signals -2 go LL 0 zero , one , first , sep , last 1 zero R , one R , false R , true R 2 zero RR , one RR , go RR Figure 4: L ist of all the meta-si gnals. As illustra ted on Figure 5, the initial configur ation is composed, left to right, of: last , go LL that starts the dynamics , then one ’ s and z ero ’ s to encode the word, then first to indic ate the beginn ing of the cyclic list then alte rnati v ely one ’ s and zero ’ s to encode each appen dant and sep to separa te them and finally last . The iteration starts when the go LL signal erases last and bounces as go RR . The later erases the fi rst bit and send the correspondi ng signal, zero R or one R to cyclic list. Once the rotation is initiated and the possib le addition of signals is done, a go LL signal is sent back to the word. The emitted one R or zero R signal crosses the one and z ero encoding the word doing nothing until it reache s first where the rotation starts. (For technical reasons, first remains and is remov ed by go LL .) If 76 Small T uri ng uni ve rsal signal machines last go LL go RR last one R one zero one one first zero one one sep one sep zero one one sep zero one last Figure 5: Initial configuration and first collision s for 1011 and list [ 011 , 1 , 011 , 01 ] . the word is empty then go RR meets first . Reaching the empty word is a halting condition so that, is such a case, go RR is just discarde d. Figure 6 lists all the colli sion rules. All non-b lank rules are collision of only two signa ls so that the rules can be pre sented in a two -dimensio nal array . In bla nk collis ion rules, the output is equa l to the input: the sign als just cross each other unaf fecte d. zero one first sep last go LL go RR last , zero R last , one R first one RR zero , zero R , one RR one , one R , one RR go LL , last , false R zero RR zero R , zero RR one R , zero RR go LL , last , false R true R — — first , false R — false R — — sep , go RR — first , one R go LL , true R one R — — first , true R , one RR — — — zero R — — first , false R , zero RR — — — go LL — — go LL go RR zero R one R false R true R go RR last z ero , go RR one , go RR one RR — — — zero RR — — — “—” means blank Other blank rules zero R , zero , go LL zero R , one , go LL one R , zero , go LL one R , one , go LL Figure 6: L ist of all the colli sion rules. The rotati on is handled i n t hree steps: signals are set on move ment –copies are left only if i t i s started by one R – then the signals are mov ing freely to the righ t end of the list, and finally after reaching last the y are positioned . The first part is present ed on Figure 7 . A signal of sp eed 2 ( zero RR or one RR ) is emitted to genera te the speed 1 vers ions of zero ’ s an d one ’ s encoding the first appendan t. A signal of speed 1 is also emitte d as well as anot her one on reach ing the first sep ; these two signals are used to delimit the appenda nt during the translatio n. T he first speed 1 signal is false R for zero R and true R for one R ; so th at the car ried bit is p reserv ed which is u seful to finis h the rota tion. The sig nal to start th e nex t iterati on, go LL , is emitted very quickly (it is not a problem since the next iteration cannot catch up with the rotatio n). This is used to turn false R to true R and to provide a simple halting scheme as explaine d later . From then, the rest of the transla tion is same in both cases. The middle part is straightfo rward: the paralle l f a lse R , zero R and one R are crossin g sep , zero and one until l ast is met. The last part of the rotation could hav e been the symmetric of the first part; in some ways, it is as presen ted on Figure 8 . T o sav e meta-signals , fal se R ha ve been used both to mark the beginnin g and the end of the appe ndant. But the y hav e dif ferent meanings, so the fi rst one changes last to first and the J. Durand-Los e 77 first first zero one one sep zero R one R one R last first go LL false R false R zero R false R true R zero RR (a) Without cop y first first zero one one sep zero R one R one R last first go LL false R false R one R one RR true R zero one one (b) With cop y Figure 7: S tarting the rota tion and restarting the dynamics. second to sep and go RR . The problem is that first intera cts with zero R and one R : zero R (resp. one R ) is chang ed to fal se R (resp. true R ) and one RR (resp. zero RR ). This generate s the lattice in the triangle (the signal s do no t interac t insid e it). The bits of the appendant are now encoded with false R ’ s and true R ’ s and foll o w the pat hs zero R ’ s and one R ’ s wou ld ha ve. It remai ns to g o RR to tu rn the m back i nto zero ’ s a nd one ’ s and to turn the final o ne R to last . go RR , zero RR and o ne RR are p arallel, all the translating signals are paralle l; so that the appendan t is recrea ted with exactl y the same distanc es. last first sep zero one one last false R zero R one R one R false R go RR false R true R true R one R zero RR one RR one RR Figure 8: E nding the rota tion. Figure 9(a) sho ws one full iterat ion of a CTS includ ing a whole rota tion. Figure 9(b) sho ws the haltin g by reaching an empty word. Figure 9(c) shows the effe ct of the haltin g appendan t as expl ained belo w . Figure 9(d) sho w s one entire simulati on with a halting a ppenda nt and cl eaning add ed: blank rules ha ve been modified in order to destro y the garbage signals escaping on the right that nev erthel ess would ne ve r interact with the rest of the configuratio n nor pro v ok e any collisio ns since they are parall el. 5.1 Adding halt at the cost of one rule Three h alting conditions exis t: on emp ty wo rd, on c ycli ng o r o n s pecial h alting a ppenda nt. The first does not yield any result but ne vert heless has to be implemented and already is. T he second needs an extra layer of constru ction to detect cycling and is not considere d. In the third case, when the halt appendant 78 Small T uri ng uni ve rsal signal machines (a) One CTS iteration. (b) Halt by empty word. (c) Use of halt appendant. (d) A full simulation (clean & halt). Figure 9: S imulatio n of a cyclic tag system. is acti vated, go LL has to meet true R . T he posi tion of this collision is determined by the distan ce between first and the nex t sep . It app ends exa ctly at 2 / 3 of the distance. If a one signal is set exac tly at this positi on, it gets into the collision. The extra rule { true R , one , go LL } → { true R } J. Durand-Los e 79 is used to destro y both one and go LL . T hen the rotation fi nished and no m ore collision is possib le. The result of the computa tion is the sequence left of first . The rotating proce ss ensur es that the distances between the signals remain constant, so that a one at 2 / 3 remains there (and a one not at 2 / 3 cannot get to 2 / 3). It should also be ensured that, in the initial positi on, there is no one at 2 / 3 not standin g for hal t which is straight forwar d to reach by , e.g., using locatio ns 1 / 2, 3 / 4, 7 / 8. . . 6 Conclusion Theor em 5 Ther e is a univer sal ha lting signal machine with 13 meta-signals and 21 non-blank rules. Ther e is a non-halti ng semi-un iver sal signa l m ach ine with 6 meta-signa ls and 8 non-b lank rules. The first construc tion uses a collisio n with 3 signals. A 3-signal collision needs perfec t synchron y while 2-signa l collisi ons are more robus t to small perturbatio ns. If only 2 signals collisions are allowed, then a halting meta-sign als can be added and proces sed in the circular list as the va lue 1 (exce pt that it does not generate other signals in the lattice at the end of the rotation). This leads to a signal machine with 15 meta-sig nals (the same ones plus ha l t and hal t R ) and 24 non blan k rules. The race for small uni v ersal devic es also runs for restriction s, e.g. to re vers ible machines. U ni ve r - sality has also been provi ded for rev ersible and conserv ativ e SM Durand-Lose (2006). W e belie ve that the CTS simulation could be turned re ve rsible with less than two extra meta-s ignals . One ke y feat ure of A G C is that the contin uous space and time can be u sed to pr oduce accumulat ions and on top of it to do black hole computations (an accumulatio n contain ing a w hole, poten tially infinite, T uri ng computa tion). One may wond er about the minimal number of meta-signals for an accumula tion and for the black hole ef fect. Let us note that 4 signals is enoug h to make an accumulati on and 3 might to be enoug h, whereas 2 is not. It also seems that the black hole ef fect can be added to the cycl ic tag system simulation with a few ex tra meta-sign als. Refer ences Baiocchi, C. (2001). Th ree small un iv ersal Tur ing machines. 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