Freezing, Bounded-Change and Convergent Cellular Automata
DDiscrete Mathematics and Theoretical Computer Science
DMTCS vol.
VOL :ISS, 2015,
Freezing, Bounded-Change and ConvergentCellular Automata ∗ Nicolas Ollinger Guillaume Theyssier Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, Orléans, France Institut de Mathématiques de Marseille (Université Aix Marseille, CNRS, Centrale Marseille), France received September 2019 , This paper studies three classes of cellular automata from a computational point of view: freezing cellular automatawhere the state of a cell can only decrease according to some order on states, cellular automata where each cell onlymakes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges tosome fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristalgrowth started by S. Ulam in the 60s. The central question addressed here is how the computational power and com-putational hardness of basic properties is affected by the constraints of convergence, bounded number of change, orlocal decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long termreachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communi-cation complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overallcomputational complexity of such cellular automata depends on the class considered (among the three above), the di-mension, and the precise problem studied. In particular, we show that all settings can achieve universality in the senseof Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides,the computability of limit configurations starting from computable initial configurations separates bounded-changefrom convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellularautomata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.
Keywords: freezing cellular automata, convergent cellular automata, complexity, computability
Cellular automata (CA for short) are a well-studied model appearing in different research areas underdifferent points of view. It is widely used as a modeling tool of fundamental physical phenomena [8]or high-level phenomena from other disciplines [1, 33, 12]. It is a rich class of symbolic dynamicalsystems[21] extensively studied both from the topological [26] and ergodic point of view [38]. It is acomputational model very close to Turing machines but with a massively parallel feature: this translatesinto a specific algorithmic complexity theory [41], into universal computational power and existence of ∗ This work has been supported by ECOS Sud – CONICYT project C12E05ISSN subm. to DMTCS c (cid:13) a r X i v : . [ c s . D M ] F e b Nicolas Ollinger, Guillaume Theyssier universal objects [35], but also into the ubiquity of undecidability and computational hardness in most ofthe properties and problems concerning them [24, 9, 25, 16]. These points of view can have interactions.For instance, the computational power of several classes of CA defined by dynamical features inspired byphysics has been studied [20, 32, 13].Here we follow that line of research and focus on the overall computational complexity of a class ofCA defined by a very simple dynamical property: point-wise convergence. More precisely, we study threeclasses which form a hierarchy by inclusion: freezing CA ( i.e. CA which are locally decreasing accordingto some order on states), bounded-change CA ( i.e. CA with a global bound on the number of state changesa cell can make in any orbit), and the general class of point-wise convergent CA. One of our inspiration isthe profusion of examples in the literature, from the seminal works of S. Ulam [42] to the recent and fastdevelopment of self-assembly models [44], which all share the convergence property: a larger and largerzone of the configuration get frozen by the dynamics while changes continue outside the zone. On theother hand, some previous works explicitly studied the class of freezing CA [14, 3] or bounded-changeCA [43] and established that universal computation is possible in any dimension but “slowed down” by thebounded-change constraint in dimension 1. This was done by studying the short-term prediction problemon one hand, and by giving an explicit encoding of Minsky machines into such CA on the other hand.The present paper is an extension (i) of [14] with two new ingredients. First, we extend the benchmarkproblems used to evaluate computational complexity and include long-term topological reachability as inthe notion of universality of [10], but also limit fixed points and their dependence on the initial configura-tion. Second, we consider the whole class of point-wise convergent CA through the same approach, whichto our knowledge is completely new. Our main results show that the overall computational complexityvaries both within the hierarchy of the three classes and with the dimension considered (see Section 6for a synthetic view). In particular, convergent CA are generally more powerful than bounded-changeCA, while freezing and bounded-change CA have the same overall computational complexity. However,we show that even the most constrained setting (freezing 1D CA) can achieve universality in the senseof [10] (Theorem 5). This is counterbalanced by the fact that such CA cannot produce uncomputable limitpoints starting from computable initial configurations (Theorem 8), while convergent CA in dimension1 can (Theorem 9). Concerning dimension sensitiveness, we show that various aspects are affected: thecapacity to efficiently compute or the capacity to produce uncomputable limit points from computableinitial configuration for bounded-change CA, and the decidability of nilpotency for all the three classes.
Organization of the paper.
The classical definitions and formal setting for CA in general is recalled inthe next paragraph. In section 2 we introduce the three classes of CA considered in this paper togetherwith several examples. In section 3 we focus on dynamical properties of such CA including nilpotency.Section 4 is devoted to the computational complexity of two canonical problems of CA theory: short-term prediction and long term reachability. In section 5 we study problems specific to convergent CAconcerning limit fixed-point reached from a given configuration. Finally, section 6 gives a brief recap ofour main results.
Formal setting.
For any finite set Q (the alphabet) and positive integer d (the dimension), we considerthe symbolic space Q Z d , i.e. the set of maps called configurations giving a state from Q to each positionin Z d . The state of configuration c ∈ Q Z d at position z ∈ Z d will be denoted either c ( z ) or c z .For any n ∈ N , we define B ( n ) as the set of positions of Z d of norm (maximum of absolute values of (i) In particular, we give complete detailed proofs of all results of [14], some of which were only sketched in this preliminary paper. reezing, Bounded-Change and Convergent Cellular Automata n : B ( n ) = { z ∈ Z d : (cid:107) z (cid:107) ∞ ≤ n } . Then for any u ∈ Q B ( n ) , we define the cylinder set [ u ] centered on cell 0 by: [ u ] = { c ∈ Q Z d : ∀ z ∈ B ( n ) , c z = u z } . These cylinder sets can be chosen as a base of open sets of the space Q Z d endowing it with a compact(and totally disconnected) topology [26]. Equivalently, it can be defined by the following metric: δ ( c, c (cid:48) ) = 2 − min {(cid:107) z (cid:107) ∞ : c z (cid:54) = c (cid:48) z } often called the Cantor metric.A cellular automaton of dimension d and state set Q is a map F acting on the set of configuration Q Z d in a continuous and translation invariant way. Equivalently (Curtis-Lyndon-Heldund theorem [21]), it canbe defined locally by a neighborhood V (a finite subset of Z d ) and a local transition function f : Q V → Q as follows: ∀ z ∈ Z d , F ( c ) z = f (cid:0) c (cid:12)(cid:12) z + V (cid:1) where c (cid:12)(cid:12) z + V is the map z (cid:48) ∈ V (cid:55)→ c z + z (cid:48) . The radius of F is the smallest integer r such that V ⊆ B ( r ) where V is some neighborhood for which there is a local map f V : Q V → Q defining F as above. F induces an action on finite patterns as follows. For any n ∈ N and any u ∈ Q B ( n + r ) , F ( u ) is the finitepattern v ∈ Q B ( n ) obtained by application of f on u at each position from B ( n ) , i.e. such that ∀ c ∈ [ u ] , F ( c ) ∈ [ v ] . We will frequently use the so called von Neumann neighborhood which is the following subset of Z : V = { ( x, y ) : | x | + | y | ≤ } .Finally, we will use the following standard complexity classes: • PTIME is the set of problems which can be solve be a deterministic Turing machine in polynomialtime; • LOGSPACE is the set of problems which can be solve be a deterministic Turing machine in loga-rithmic space; • NLOGSPACE is the set of problems which can be solve be a non-deterministic Turing machine inlogarithmic space.Without explicit mention, and in particular when speaking about P-completeness, we consider LOGSPACEreductions.
Definition 1
A CA F is a freezing CA if, for some (partial) order ≤ on states, the state of any cell canonly decrease, i.e. F ( c ) z ≤ c z for any configuration c and any cell z . Nicolas Ollinger, Guillaume Theyssier (a) (b) (c) (d)
Fig. 1:
Some examples of freezing CA studied in the literature: (a) configuration of Ulam’s rule from Example 1after some steps starting from a single 1 in a sea of 0s; (b) spatio-temporal representation of the standard bootstrappercolation model (Example 2) where shades of gray represent time at which a cell turned to 1 (the whiter the sooner);(c) a configuration of a model of forest fire as in Example 4 (fire is represented in yellow, trees in green, and emptyspaces in black and hashes in shades of gray indictaing the time modulo some larger integer at which the fire occupiedthat position for last time); (d) typical configuration of rule ’life without death’ from Example 3
For any F and states q (cid:54) = q (cid:48) , denote by q → F q (cid:48) the fact that in some context a cell can change fromstate q to q (cid:48) in one step. If F is a freezing CA, then the transitive and reflexive closure of → F is a (partial)order satisfying the condition of Definition 1. Conversely, if → F is acyclic then F is freezing for the ordergiven by the reflexive and transitive closure of → F . This gives a polynomial time algorithm (in the sizeof the transition table) to test whether a given CA is freezing: build → F and test its acyclicity. Fact 1
There is an algorithm to decide whether a given CA is freezing that runs in time O ( n | V | ) where n is its number of states and V is its neighborhood. Our purpose is to study freezing cellular as a class and give general results for it, as started in [14].However, many particular freezing cellular automata have already been considered in the literature. Wegive below several examples showing the variety of possible behaviors within the class (see Figure 1).
Example 1
It turns out that one of the very first defined CA is a freezing CA. Indeed in [42], as an attemptto study models of cristal growth, the following CA over alphabet Q = { , } is defined: F ( c ) z = (cid:40) if c z = 0 and ( c | z + V ) (cid:54) = 11 else,where V is the von Neumann neighborhood and ( P ) denote the number of occurrences of in thefinite pattern P . Example 2
The threshold growth models in 2D [17] are CA with states , where becomes if thenumber of s in the neighborhood is above some threshold, and s stay unchanged forever. They were inparticular considered as theoretical models of bootstrap percolation and a lot of work was dedicated tothe experimental and rigorous analysis of the phase transitions they exhibit (see for instance [22]). Thefact that these examples are monotone (with respect to the order extended to configurations) is to be takeninto account when studying their computational complexity [15, 3].reezing, Bounded-Change and Convergent Cellular Automata Example 3
By taking any CA on alphabet Q and endowing Q with some order ≤ , one can defined afreezing CA as follows: F ≤ ( c ) z = min ≤ ( c z , F ( c ) z ) . There is a priori no relation to expect between F and F ≤ , but this construction is a source of examples.For instance a freezing 2D CA with 2 states called “life without death” received a lot of attention and wasin particular shown P -complete [18]: it is F ≤ where F is the “game of life” CA. Example 4
SIR epidemic propagation models [12], in their simplest form, are states freezing CA: a S usceptible person can become I nfected, and then R ecover and acquire immunity so that it nether goesback to susceptible or infected states. A similar CA was also considered to model (ii) forest fire propagation[2]. These examples are in fact multi-state versions of the threshold rules from Example 2. Example 5
Self-assembly tilings [36, 44], and more precisely the so called abstract tile assembly model(aTAM), consists in a set of Wang tiles where each edge has a color and a bond strength. They areclassically studied as asynchronous non-deterministic systems where tiles are added one by one to thecurrent assembly, starting from a single tile (the seed) and according to the following rule: a tile canbe added if the sum of strengths values of edges with a color matching the corresponding neighbor isabove some threshold. An important property of this model (no longer true for some of its generalizationslike mismatch-free [4] or negative glue [11, 37]), is that if a tile can be added at some position in somecontext, then it can also be added in the context where new tiles are added at empty neighboring positions.Fomally, an aTAM system can be seen as follows: • Q = T ∪ { (cid:15) } is the alphabet, endowed with the order (cid:22) such that t (cid:22) (cid:15) for all t ∈ T ; • s ∈ T is the seed tile; • U = V \ { (0 , } where V is the von Neumann neighborhood V ; • R ⊆ Q U × Q is the local compatibility rule (iii) which is monotone with respect to (cid:22) (extended to Q U ), i.e. if ( m, t ) ∈ R and m (cid:48) (cid:22) m then ( m (cid:48) , t ) ∈ R , and verifies ( m, (cid:15) ) ∈ R for all m ∈ Q U (notadding a tile is always possible).For two configurations c, c (cid:48) ∈ Q Z we write c → c (cid:48) if c and c (cid:48) differ at position z , c z = (cid:15) and ( c | z + U , c (cid:48) z ) ∈ R .We denote by → ∗ R the reflexive and transitive closure of → R . An assembly is a configuration c with c → ∗ R c where c is the configuration everywhere equal to (cid:15) except in position (0 , where it is s . A terminal assembly is a configuration c such that c → ∗ R c and for any c (cid:48) with c → ∗ R c (cid:48) we have c = c (cid:48) .From such a system we can define a freezing cellular automaton F R with ordered alphabet ( Q, (cid:22) ) andneighborhood V as follows: F R ( c ) z = (cid:40) c z if c z (cid:54) = (cid:15), min { t : ( c | z + U , t ) ∈ R } else.In particular, for any finite configuration c , c → ∗ R F R ( c ) . We will come back to this simple translation ofaTAM systems to freezing CAs in subsection 5.3. (ii) Technically, the CA contains a probabilistic part describing a rate of tree growth. Setting this parameter to gives a deterministicfreezing CA. (iii) Here we forget about the details of the aTAM model involving bond strength and threshold, we simply observe that it can berepresented as a local compatibility relation. Of course not all such relations correspond to an aTAM model.
Nicolas Ollinger, Guillaume Theyssier
In a freezing CA, a cell can change at most a finite number of time during its evolutions, precisely atmost n − times for a CA with n states. This property is our second main definition. Definition 2
A CA F is k-change if the number of state change of any cell in any orbit is at most k ,formally: ∀ c ∈ Q Z d , ∀ z ∈ Z d : |{ t ∈ N : F t +1 ( c ) z (cid:54) = F t ( c ) z }| ≤ k. A CA F is bounded-change if it is k -change for some k . Bounded-change CA have been studied previously [43] as language recognizers. CA with boundedcommunications have also been considered [28] (again from the language recognition point of view)and are very close: after a bounded communication a cell do no longer depends on its neighbors andmust enter a temporal cycle bounded by the number of states, which means that some power the CA isbounded-change.A CA F is nilpotent if there is t > such that F t is a constant map. Any nilpotent CA is bounded-change. Example 6
A bounded-change CA is not necessarily freezing. For instance, the following one-dimensionalnilpotent CA over { , , } with neighborhood { , } is not freezing: f ( a, b ) = (cid:40) if b = 02 else.because both → and → are possible state changes in one cell (but after 2 steps all cells are instate ). If F is bounded-change and c is any configuration then for any cell z ∈ Z d there is some time τ ( c, z ) such that the state of cell z does no longer change after time τ ( c, z ) when starting from c . Then for anyfinite set of cells E ⊆ Z d , the states of cells in E does no longer change after time max z ∈ E τ ( c, z ) . Saiddifferently, the sequence ( F t ( c )) t ∈ N is convergent. This property is our third main definition. Definition 3
A CA F is convergent if, for any configuration c , the sequence ( F t ( c )) t ∈ N is convergent. Inthis case, the limit configuration reached is denoted by: F ω ( c ) = lim t →∞ F t ( c ) . Moreover, given an initial configuration c and a cell z ∈ Z d , we call freezing time the first time τ ( c, z ) after which cell z no longer changes its state, formally: τ ( c, z ) = min { t ∈ N : ∀ t (cid:48) ≥ , F t + t (cid:48) ( c ) z = F t ( c ) z } . The next example gives a construction technique to produce convergent CA that are not bounded-change. The idea is to divide the configuration into working zones and ensure the presence of at mostone working head per zone. Each head permanently bounces between the two extremities of its zone andshrinks the zone by one cell at each bounce. With this behavior, each finite working zone gets completelyshrinked in finite time and converges to a fixed point. There is no way to check locally that a working zoneis finite, but there is no problem of convergence with infinite zones because the head will escape towards reezing, Bounded-Change and Convergent Cellular Automata F interpreted as old state and new state respectively.The head sequentially updates these states to simulate a parallel synchronous application of F at eachpass. Example 7 (Shrinking zone construction)
Let F be any 1D CA on alphabet Q with radius and localmap δ : Q → Q . We define Z F on alphabet R = { b, b + , e } ∪ Q (cid:48) with Q (cid:48) = Q × Q × {← , → , l, r } andradius as follows: • e , the error state , is a spreading state: any cell with e in its neighborhood turns into state e ; aconfiguration c is valid if e never appears in its orbit; • b , the blank state, never changes except in presence of the error state; b + becomes b except inpresence of the error state; a maximal connect component of cells in state Q (cid:48) is a working zone ; • in a working zone, patterns of the form ( x, y, r )( x (cid:48) , y (cid:48) , l ) , or ( x (cid:48) , y (cid:48) , l )( x, y, r ) , or ( x, y, z )( x (cid:48) , y (cid:48) , z (cid:48) ) with { z, z (cid:48) } ⊆ {← , →} , or ( x, y, r )( x (cid:48) , y (cid:48) , z ) or ( x, y, z )( x (cid:48) , y (cid:48) , l ) with z ∈ {← , →} , are forbiddenand generate an e state when detected; therefore in a valid configuration and in each working zonethere is at most one occurrence of a state of the form ( x, y, {← , →} ) called the head ; • a cell without forbidden pattern (from previous item) and without head in its neighborhood doesn’tchange its state; • the movements and actions of the heads are as follows: – inside a working zone, the head in state ← moves left, the head in state → moves right; thelocal map δ is only applied the head moves left to right; precisely we have the followingtransitions: ( x, y, l ) , ( x (cid:48) , y (cid:48) , ← ) , ( x (cid:48)(cid:48) , y (cid:48)(cid:48) , r ) (cid:55)→ ( x (cid:48) , y (cid:48) , r )( x, y, l ) , ( x (cid:48) , y (cid:48) , l ) , ( x (cid:48)(cid:48) , y (cid:48)(cid:48) , ← ) (cid:55)→ ( x (cid:48) , y (cid:48) , ← )( x, y, l ) , ( x (cid:48) , y (cid:48) , → ) , ( x (cid:48)(cid:48) , y (cid:48)(cid:48) , r ) (cid:55)→ ( x (cid:48) , y (cid:48) , l )( x, y, → ) , ( x (cid:48) , y (cid:48) , r ) , ( x (cid:48)(cid:48) , y (cid:48)(cid:48) , r ) (cid:55)→ ( δ ( y, x (cid:48) , x (cid:48)(cid:48) ) , x (cid:48) , → ) – when a boundary of the working zone is reached, the head bounces, changes of direction and Nicolas Ollinger, Guillaume Theyssierthe working zone get shrinked by one cell; precisely we have the following transitions: b, ( x, y, l ) , ( x (cid:48) , y (cid:48) , ← ) (cid:55)→ ( x, y, ← ) b, ( x, y, ← ) , ( x (cid:48) , y (cid:48) , r ) (cid:55)→ ( x, y, → ) b, ( x, y, → ) , ( x (cid:48) , y (cid:48) , r ) (cid:55)→ ( y, x, l ) b, ( x, y, l ) , ( x (cid:48) , y (cid:48) , → ) (cid:55)→ b + ( x, y, → ) , ( x (cid:48) , y (cid:48) , r ) , b (cid:55)→ ( x (cid:48) , y (cid:48) , → )( x, y, l ) , ( x (cid:48) , y (cid:48) , → ) , b (cid:55)→ ( x (cid:48) , y (cid:48) , ← )( x, y, l ) , ( x (cid:48) , y (cid:48) , ← ) , b (cid:55)→ ( x (cid:48) , y (cid:48) , r )( x, y, ← ) , ( x (cid:48) , y (cid:48) , r ) , b (cid:55)→ b + (note the swap between x and y in the third transition above to initialize the sequential appli-cation of δ ) – finally the head disappears in a working zone of size , precisely: b (cid:48) , ( x, y, z ) , b (cid:48)(cid:48) (cid:55)→ ( x, y, r ) for any { b (cid:48) , b (cid:48)(cid:48) } ⊆ { b, b + } . Given two configurations c, c (cid:48) ∈ Q Z and any n > we define the valid configuration λ n,c,c (cid:48) ∈ R Z asfollows: λ n,c,c (cid:48) ( z ) = b if z < − n , ( c z , c (cid:48) z , → ) if z = − n, ( c z , c (cid:48) z , r ) if − n < z ≤ n,b if z > n. Lemma 1
For any F , the CA Z F is convergent but not bounded-change. Moreover for any configuration c of F any t > and any n ≥ t and any z with | z | ≤ n − t it holds: Z t n F ( λ n,c,c ) z = λ n − t,F t ( c ) ,F t − ( c ) where t n = n (cid:88) i = n − t +1 i + 1 . Moreover, for any ≤ t (cid:48) ≤ t n and for any | z | < n , Z t (cid:48) F ( λ n,c,c ) z is either b or b + , or there is some ≤ t (cid:48)(cid:48) ≤ t such that it is of the form ( F t ( c ) z , F t ( c ) z , x ) with | t − t | = 1 . Proof:
First Z F is convergent. To see this let’s consider any configuration c and show that (cid:0) F t ( c ) (cid:1) t ∈ N is convergent. We have the following cases: • if c is not valid then F t ( c ) = e for any large enough t ; • if c is valid and c ∈ { b, b + } then F t ( c ) = b for any t ; • finally, if c is valid and c ∈ Q (cid:48) then: reezing, Bounded-Change and Convergent Cellular Automata – either cell belongs to an infinite working zone and the head will no longer be in its neigh-borhood after some finite time its state will no longer change; – or cell belongs to some finite zone with no head and its state will never change; – or cell belongs to some finite zone containing a head and after some finite time the zone hasshrinked to size and doesn’t change anymore, so that cell stays in state b or some state q ∈ Q (cid:48) forever.To see that Z F is not bounded-change it is sufficient to check that, for any n ∈ N , cell changes morethan n times in the orbit of λ n,c,c (cid:48) whatever c, c (cid:48) ∈ Q Z .Finally, it is straightforward to check that Z n +1 F ( λ n,c,c (cid:48) ) = d where d is defined by: d ( z ) = b if z ≤ − n , ( F ( c ) z , c z , l ) if − n < z < n, ( c z , c (cid:48) z , ← ) if z = n,b if z > n. and then Z n +1+2 nF ( λ n,c,c (cid:48) ) = λ n − ,F ( c ) ,c . The formula on Z t n F ( λ n,c,c ) z follows by induction, and thelast point of the lemma follows by definition of Z F in working zones. (cid:50) From the discussion above we have the following strict hierarchy.
Fact 2 F freezing ⇒(cid:54)⇐ F bounded-change ⇒(cid:54)⇐ F convergent. Contrary to the freezing property which is easy to check (Fact 1), being bounded-change or convergentare undecidable properties.
Theorem 1
The following holds in any dimension: • nilpotent CA are recursively inseparable from non-convergent CA; • given F convergent, it is undecidable whether it is bounded-change. Proof:
In the following we do the proof for dimension 1, but the result follows straightforwardly for anydimension. In both cases we prove a reduction from the nilpotency problem for 1D CA of radius with aspreading state. This problem was shown undecidable in [23]. Consider any F of radius on alphabet Q with spreading state s ∈ Q and local map δ : Q → Q , and construct the following CAs: • F on alphabet Q × { , } with transition map: (cid:0) ( q , b ) , ( q , b ) , ( q , b ) (cid:1) (cid:55)→ (cid:40) ( s, if s ∈ { q , q , q } , ( δ ( q , q , q ) , − b ) otherwise . It is straightforward to check that F is bounded-change (and more precisely nilpotent) if F isnilpotent and non-convergent else. The first item of the theorem follows;0 Nicolas Ollinger, Guillaume Theyssier • F is Z F with the following modification (using notation of Example 7): whenever a state ( q, q (cid:48) , x ) with s ∈ { q, q (cid:48) } appears in the neighborhood of cell it turns into state e . Z F is always convergent byLemma 1, and so is F because for any initial configuration c , either F and Z F produce the sameorbit, or c is invalid for F ( i.e. , the orbit converges to ω e ω ). Moreover a straightforward adaptationof Lemma 1 shows that: for any configuration c of F any t > and any n > t it holds: Z t n F ( λ n,c,c ) = (cid:40) e if F t ( c ) = s ( F t ( c ) , F t − ( c ) , h ) else, with h ∈ {→ , r } where t n is the time constant from Lemma 1. First if F is not nilpotent, then there is a configuration c whose orbit does not contain state s . Therefore, for any n , λ n,c,c has an orbit under F with atleast n changes for cell . This shows that F is not bounded-change in this case. If on the contrary F is nilpotent, then there is t such that F t ( c ) = s for any t ≥ t and any c . Therefore any cell inany working zone of F becomes e after t passes of the working head. This implies that a cell’sstate can change at most t times while staying in Q (cid:48) ( t times head entering/leaving the cell).Since the other possible state changes are Q (cid:48) → { b, b + } , Q (cid:48) → e , b + → b and { b, b + } → e , wededuce that F is (2 t + 2) -change and the second item of the theorem follows. (cid:50) The convergence property has obviously strong consequences on the dynamics. This can be seen throughthe trace or column factor which is a well-studied object [26, 19].For any finite W ⊆ Z d and any CA F , the trace of F of base W starting from c is the followingsequence: T W ( c ) = ( F t ( c ) (cid:12)(cid:12) W ) t ∈ N . Lemma 2
Let W ⊆ Z d be any non-empty finite set. A CA F is convergent if and only if T W ( c ) is even-tually constant for any configuration c . Moreover, in that case, there is always a fixed point c (cid:48) (i.e. F ( c (cid:48) ) = c (cid:48) ) such that T W ( c ) and T W ( c (cid:48) ) are eventually equal. This does not imply that the limit set is made of fixed points, however we can prove the followingproposition which is not true for CA in general.
Proposition 1
Let F be a convergent CA which is not nilpotent. Then it possesses two distinct fixedpoints. Proof:
First, any convergent CA must have a fixed-point c which is uniform: c z = q for some q ∈ Q and any z ∈ Z d . Now suppose that c is the unique fixed-point, then by Lemma 2 F is asymptotically- q -nilpotent, meaning that for any z ∈ Z d and any c (cid:48) ∈ Q Z d , the trace T { z } ( c (cid:48) ) is eventually constant equalto q . By Theorem 2 of [40] this implies that F is nilpotent. The proposition follows. (cid:50) As a corollary, we get the following dimension-sensitive decidability result.
Theorem 2
The nilpotency problem is:reezing, Bounded-Change and Convergent Cellular Automata • decidable (in time polynomial in the size of the transition table) for 1D convergent CA; • undecidable for freezing CA in higher dimension. Proof:
First, in dimension and more, the classical proof of undecidability of nilpotency (see [23])works without any modification for freezing CA: given a Wang tile set, we build a freezing CA with aspreading error state, that checks locally if the configuration is a valid tiling and produces the error statein case of local error detection. This CA is nilpotent if and only if the tile set does not tile the plane. Theundecidability follows from [5].In dimension , from Proposition 1, we know that nilpotency is equivalent to the property of having asingle fixed-point. By the classical construction of De Bruijn graph, the existence of at least two fixed-point is easy to decide in polynomial time. Precisely, and for the sake of completeness, suppose F hasradius r and local transition map f : Q r +1 → Q and let G = ( V, E, λ ) be the edge-labeled digraph with V = Q r and • E = { (cid:0) ( q − r , . . . , q r − ) , ( q − r +1 , . . . , q r ) (cid:1) : f ( q − r , . . . , q r ) = q } , • λ (cid:0) ( q − r , . . . , q r − ) , ( q − r +1 , . . . , q r ) (cid:1) = q . Since F necessarily possesses a uniform fixed point ω q ω , testing if F possesses at least two amounts tocheck whether there is a circuit in G with at least one edge not labeled by q . This can be done in timepolynomial in | Q | r . (cid:50) Convergence also implies some kind of topological regularity as shown in the following proposition.
Proposition 2
No convergent CA is sensitive to initial conditions, whatever the dimension.
Proof:
Suppose by contradiction that F on Q Z d is convergent and sensitive to initial conditions: thereis N ∈ N such that for any c ∈ Q Z d and any p ∈ N there exists c (cid:48) ∈ Q Z d such that c (cid:12)(cid:12) B ( p ) = c (cid:48) (cid:12)(cid:12) B ( p ) and F t ( c ) (cid:12)(cid:12) B ( N ) (cid:54) = F t ( c (cid:48) ) (cid:12)(cid:12) B ( N ) for some t ∈ N . Consider any configuration c ∈ Q Z d and p ≥ N . By sen-sitivity there is c (cid:48) ∈ Q Z d and t ∈ N such that either T B ( N ) ( c ) or T B ( N ) ( c (cid:48) ) is non-constant on timeinterval [0 , t ] . Denote by c the one among c and c (cid:48) that corresponds to the non-constant trace. Let p = p + N + r ( t + 1) where r is the radius of F . Applying sensitivity again on c and p we knowthere exist c (cid:48) and t such that: • c and c (cid:48) are identical on B ( p ) ; • therefore, by choice of p , T B ( N ) ( c ) and T B ( N ) ( c (cid:48) ) coincide on interval [0 , t ] ; • T B ( N ) ( c ) and T B ( N ) ( c (cid:48) ) differ at time t .So one of c or c (cid:48) , denoted c , is such that T B ( N ) ( c ) is not constant on interval [ t , t ] . Going on withthe same reasoning we construct a converging sequence ( c n ) n ∈ N of configurations such that T B ( N ) ( c n ) isnot constant on each interval [ t i , t i +1 ] for ≤ i < n . Taking c = lim n c n we get a trace T B ( N ) ( c ) whichis not eventually constant contradicting Lemma 2. (cid:50) The convergence behavior is intrinsically irreversible except in the trivial case of the identity, moreprecisely there is no surjective convergent CA appart from the identity. It was shown in [14] for freezing2
Nicolas Ollinger, Guillaume Theyssier cellular automata by an elementary proof. In fact, as pointed to us by V. Salo, it is an immediate conse-quence of the Poincaré recurrence theorem for all convergent CA: if a CA is not the identity, then their issome local context u in which a cell changes its state, but the Poincaré recurrence theorem implies thatin a surjective CA there must be an orbit where u is recurrent which yields infinitely many states changeand contradicts convergence. Without going into details about measures and cellular automata which areout of the scope of the present paper (see [38]), we give a rather self-contained proof of this fact in thefollowing proposition. Proposition 3
If a convergent CA is surjective, then it is the identity map.
Proof:
Let F be surjective and convergent and let µ denote the uniform product measure on configurationsof F . By the balance theorem [30], µ is preserved under F : µ ( X ) = µ ( F − ( X )) for any measurable set X (see [6] for a more general result). If we suppose that F is not the identity map, then there is a word u ∈ Q B ( n ) such that for all x ∈ [ u ] it holds F ( x ) (cid:54) = x . We claim that there is a configuration x such that F t ( x ) ∈ [ u ] for infinitely many t . From this claim we deduce that F is not convergent because the orbit of x is not convergent. The proposition follows. To prove the claim, let us denote R t = ∪ t (cid:48) ≥ t F − t (cid:48) ([ u ]) and R = ∩ t R t . By definition R is the set of configurations whose orbit visits [ u ] infinitely many times and wewant to show that R (cid:54) = ∅ . We actually show that µ ( R ) > . Since R t +1 = F − ( R t ) for all t ≥ we have µ ( R t ) = µ ( R ) . Moreover [ u ] ⊆ R so we deduce that µ ([ u ] \ R t ) ≤ µ ( R \ R t ) = 0 since R t ⊆ R .Finally, by Boole’s inequality we have µ ([ u ] \ R ) = 0 so µ ( R ) > since µ ([ u ]) > . (cid:50) Computational complexity and universality in cellular automata is a very well-studied topic and the gen-eral ability for cellular automata to do any Turing computation is now considered as an obvious fact. Inthis section, we shall show how the three definitions of section 2 affect this computational power and howit also depends on the dimension.
We start be defining two classical problems associated to any CA which will serve as a canonical measure-ment of the computational complexity of the considered CA. They are very different and complementary.The first one is about short-term predictability and provides a fine-grained complexity measurementwithin class
PTIME . Definition 4
Let F be any CA of radius r and alphabet Q . The prediction problem PRED F is defined asfollows: • input: t > and u ∈ Q B ( rt ) • output: F t ( u ) ∈ Q . The second one is about long-term reachability and provides a coarse-grained complexity measurethat can go (and is expected to go) beyond the decidable. It is inspired by the notion of universality fordynamical symbolic systems from [10].
Definition 5
Let F be any CA of radius r and dimension d . The cylinder reachability problem CYREACH F is defined as follows:reezing, Bounded-Change and Convergent Cellular Automata • input: two cylinders u and v . • output: decide whether there is c ∈ [ u ] and t ∈ N such that F t ( c ) ∈ [ v ] . For CA in general, hard examples of both problems are well-known. Let us fix a 1D CA G of radius such that PRED G is P-complete, for instance rule 110 [34], and a 1D CA H of radius such that CYREACH H is undecidable. More precisely, we suppose that there is a fixed state h ∈ Q H of H suchthat the following sub-problem of CYREACH H , that we will denote CYREACH ∗ H , is also undecidable:given a cylinder [ u ] , decide whether there is t ∈ N and c ∈ [ u ] such that F t ( c ) ∈ [ h ] . Such an H can beobtain by adapting the example of [10, section 6.1]. When restricting to freezing CA, and provided thedimension is at least 2, one can find hard examples for both problems. Note that “life without death” isknown to have a P-complete PRED problem [18] but the hardness of
CYREACH is not clear.
Proposition 4
There exists a 2D freezing CA F such that PRED F is P-complete and CYREACH F isundecidable. Proof:
Any 1D CA F with states Q and neighborhood V can be simulated by a 2D freezing CA withstates Q ∪ {∗} as follow. Let V (cid:48) = { ( v, −
1) : v ∈ V } . A cell in a state from Q never changes. A cellin states ∗ look at cells in its V (cid:48) neighborhood: if they are all in a state from Q then it updates to thestate given by applying F on them, otherwise it stays ∗ . Starting from a all- ∗ configuration except onone horizontal line where it is a Q -configuration c , this 2D freezing CA will compute step by step thespace-time diagram of F on configuration c . Applying this construction to G , we obtain a 2D freezingCA G (cid:48) such that PRED G reduces to PRED G (cid:48) as follows: given t > and u ∈ Q B ( t ) G (of dimension 1),we compute u (cid:48) ∈ Q B ( t ) G (cid:48) (of dimension 2) defined by u (cid:48) z = (cid:40) u x when z = ( x, − t ) , ∗ else.By definition of G (cid:48) it holds: ( G (cid:48) ) t ( u (cid:48) ) = G t ( u ) . We now apply the same construction to H to obtain H (cid:48) with the following modifications: • H (cid:48) has neighborhood B ( r ) where r ≥ and is large enough so that V (cid:48) ⊆ B ( r ) • H (cid:48) as an additional state b that never changes whatever the context, and that propagates downwardand to the left and to the right; • if a cell in a state from Q has a left or right neighbor not in Q , it becomes b ; • if a cell z in state ∗ has a neighbor which is not in z + v (cid:48) and not in state ∗ then it becomes b ; • if none of the above cases holds and cell z is in a state from Q and cell z + (0 , is in state h thencell z turns into state h .Then CYREACH ∗ H reduces to CYREACH H (cid:48) as follows: given a cylinder u of radius n , we compute u (cid:48) ∈ Q B ( n ) H (cid:48) defined by u (cid:48) z = u x when z = ( x, , b if z = ( x, y ) with y < , ∗ else.4 Nicolas Ollinger, Guillaume Theyssier
Then by definition of H (cid:48) we have that H t ( c ) ∈ [ h ] for some t ∈ N and c ∈ [ u ] if and only ( H (cid:48) ) t (cid:48) ( c (cid:48) ) ∈ [ h ] for t (cid:48) ∈ N and c (cid:48) ∈ [ u (cid:48) ] . Indeed, if H t ( c ) = h then ( H (cid:48) ) t ( c (cid:48) ) (0 ,t ) = h and therefore ( H (cid:48) ) t ( c (cid:48) ) (0 , where c (cid:48) is obtained from c as u (cid:48) is from u . For the other direction, ( H (cid:48) ) t (cid:48) ( c (cid:48) ) (0 , = h implies thatthere is some t ≥ t (cid:48) with ( H (cid:48) ) t ( c (cid:48) ) (0 ,t ) = h and it can only be possible if c (cid:48) ( x, ∈ Q for all | x | ≤ t and c (cid:48) ( x,y ) = ∗ for all ( x, y ) ∈ B ( t ) and y > (because no b can be produced close to (0 , at first step).We deduce that there is a configuration c ∈ [ u ] with c x = c (cid:48) ( x, for all | x | ≤ t it holds H t ( c ) = h . Thereduction from CYREACH ∗ H to CYREACH H (cid:48) follows.To conclude, G (cid:48) × H (cid:48) fulfills the conditions of the proposition because since both G (cid:48) and H (cid:48) have aquiescent state, then PRED G (cid:48) reduces to PRED G (cid:48) × H (cid:48) and CYREACH H (cid:48) reduces to CYREACH H (cid:48) × G (cid:48) . (cid:50) Proposition 5
There exists a 1D convergent CA F such that PRED F is P-complete and CYREACH F is undecidable. Proof:
This proof relies on construction from Example 7 which we apply on automata G and H men-tioned above in this section. We claim that Z G × Z H has the desired property. Indeed, on one hand PRED G reduces to PRED Z G as follows: given t > and u ∈ Q B ( t ) G , we compute t n from Lemma 1with n = t + 1 and u (cid:48) ∈ Q B ( t n ) Z G such that u (cid:48) = λ t n ,c,c (cid:12)(cid:12) B ( t n ) where c (cid:12)(cid:12) B ( t ) = u and c is constant outside B ( t ) . Lemma 1 shows that the first component of state Z t n G ( u (cid:48) ) is precisely G t ( u ) . The reduction follows.On the other hand CYREACH ∗ H truth-table reduces to CYREACH Z H as follows: there is t and c ∈ [ u ] with H t ( c ) ∈ [ h ] if and only if there is t and c (cid:48) ∈ [ u (cid:48) ] with Z tH ( c (cid:48) ) ∈ [ v (cid:48) ] where • u (cid:48) z = (cid:0) u z , u z , r (cid:1) and u and u (cid:48) have same domain; • v (cid:48) is one of the (finite number of) cylinders of the form v (cid:48) z = (cid:0) h, x, y (cid:1) .Indeed, if H t ( c ) ∈ [ h ] for some c ∈ [ u ] then Lemma 1 shows that Z t n H ( λ n,c,c ) ∈ [ v (cid:48) ] for n > t and some v (cid:48) as above and we have λ n,c,c ∈ [ u (cid:48) ] . Conversely if Z tH ( c (cid:48) ) ∈ [ v (cid:48) ] for some v (cid:48) as above and c (cid:48) ∈ [ u (cid:48) ] ,then necessarily c (cid:48) contains a working zone that (possibly) extends u (cid:48) and survives at least t steps. Then,either this zone contains no head and then states don’t change inside the zone which implies c (cid:48) ∈ [ v (cid:48) ] andtherefore [ u ] ∩ [ h ] (cid:54) = ∅ , or the orbit of c (cid:48) coincides during t steps with that of some λ n,c,c with c ∈ [ u ] and Lemma 1 implies that H t (cid:48) ( c ) ∈ [ h ] . To conclude that Z G × Z H has the desired property, it is enoughto remark that both Z G and Z H have a quiescent state so that PRED Z G reduces to PRED Z G ×Z H and CYREACH Z H reduces to CYREACH Z G ×Z H . (cid:50) The following result for was essentially present in [43] using the formalism of bounded-change CAseen as language recognizers and therefore focusing on the complexity of recognizable languages ratherthan prediction problems.
Proposition 6
Let F be any 1D bounded-change CA. Then PRED F ∈ NLOGSPACE . Moreover,if in addition the neighborhood V of F is “one-way”, i.e. if V ⊆ N or V ⊆ − N then PRED F ∈ LOGSPACE . Proof:
Consider a k -change CA F of radius r and alphabet Q . Given some u ∈ Q B ( rt ) , the followingnon-deterministic algorithm allows to compute F t ( u ) : reezing, Bounded-Change and Convergent Cellular Automata i with − rt ≤ i ≤ rt do:(a) guess some column vector of states of height rt − | i | + 1 whose first state is u i : C i ∈ Q rt −| i | +1 with C i (0) = u i ;(b) if − rt + r ≤ i ≤ rt − r then check that C i is compatible with C i − r , . . . , C i + r i.e. • for each τ with < τ ≤ | i | /r check that C i ( τ ) = f ( C i − r ( τ − , . . . , C i + r ( τ − where f is the local map of F .2. return C ( t ) The key observation is that in a k -change CA a column of states of height at most t appearing in avalid space-time diagram can be represented in space O (log( t )) as a list of at most k pairs (state,duration)describing the sequence of state change and the duration between them. Therefore, the above algorithmcan be implemented in non-deterministic logspace by keeping in memory only columns C i − r to C i + r .The compatibility test (loop on τ ) can be done directly on the compact representation of columns since C i ( τ ) can be retrieved by a bounded number of comparisons and sums of small integers. This shows PRED F ∈ NLOGSPACE .If we suppose that V ⊆ − N , then the compatibility test only involves column C i − r to C i . Moreover,there is a unique possible column C i compatible with the given columns C i − r , . . . , C i − which can becomputed by C i ( τ ) = f ( C i − r ( τ − , . . . , C i ( τ − . Again this can be realized using compact repre-sentation using O (log( t )) space. We deduce in this case that PRED F ∈ LOGSPACE . (cid:50) Under the hypothesis
PTIME (cid:54) = NLOGSPACE , bounded change CA have simpler prediction prob-lems than convergent CA. The goal of the next two subsections is to show that the same happens whenconsidering the communication complexity of prediction problems, but that freezing CA can have unde-cidable cylinder reachability. Hence the difference between bounded-change and convergent CA is in theshort-term prediction complexity rather than in the long-term reachability complexity.
Communication complexity was introduced by Yao [45] to study distributed computing. We first brieflyrecall the classical definition of communication complexity for any function (see [27] for general refer-ence), and then apply it to the prediction problem of cellular automata as it was done in [16]. The goalof this section is to show that bounded-change CA have a lower communication complexity than CA ingeneral.Consider a function f : A × B → C where A , B and C are finite sets. The communication complexityof f is the amount of information (number of bits) that need to be exchanged in the worst case betweentwo parties (say Alice and Bob) in order to decide the value f ( x, y ) when one of the parties knows only x and the other only y (both know f and they have unlimited computing power).More precisely, a communication protocol π is a finite rooted binary tree where each leaf holds avalue from C and each internal node n holds either a map α n : A → { , } (Alice speaks) or a map β n : B → { , } (Bob speaks). The run of π on input ( a, b ) is the path ρ ( a, b ) starting from the rootand defined by the following descent in the tree until reaching a leaf: when at node n , go to left child6 Nicolas Ollinger, Guillaume Theyssier if α n ( a ) = 0 (resp. β n ( b ) = 0 ) and to right child otherwise. The output π ( a, b ) ∈ C of the protocol π on input ( a, b ) is the value held by the leaf reached by ρ ( a, b ) . The cost c π ( a, b ) of π on input ( a, b ) isthe length of ρ ( a, b ) . A protocol solves f if f ( a, b ) = π ( a, b ) on all inputs ( a, b ) ∈ A × B . Finally, the communication complexity of f is CC ( f ) = min π solves f max a,b c π ( a, b ) . For any CA F we consider the communication complexity of the prediction problem by dividing thespace into two roughly equal parts through an hyperplane (we extend the definition of [16] to any dimen-sion). Precisely, given any n ∈ N we define A n and B n as follows: A n = B ( n ) ∩ { z ∈ Z d : π ( z ) ≤ } ,B n = B ( n ) ∩ { z ∈ Z d : π ( z ) > } , where π : Z d → Z denotes the projection on the first coordinate. The proof idea in Theorem 3 belowworks for other choices of hyperplane, but we prefer to keep notations simple.For any alphabet Q let ( u, v ) ∈ Q A n × Q B n (cid:55)→ u | v ∈ Q B ( n ) denote the concatenation map defined asthe inverse map of w ∈ Q B ( n ) (cid:55)→ (cid:0) w (cid:12)(cid:12) A n , w (cid:12)(cid:12) B n (cid:1) . Given a CA F of radius r , we then define for each n the communication problem P F,n associated to
PRED F as follows: ( u, v ) ∈ Q A rn × Q B rn (cid:55)→ F n ( u | v ) ∈ Q and we consider the communication complexity of prediction of F through the following map: CC F : n ∈ N (cid:55)→ CC ( P F,n ) . Clearly CC F ∈ O ( n d ) for any F of dimension d since it is always possible to apply the trivial protocolwhere Alice sends all its input to Bob and Bob then knows all the information to answer. Let us first recallthat general CAs can reach maximal communication complexity (this is a straightforward extension of[16] which is written in the one-dimensional setting). The intuition is that a d -dimension CA in time n can test equality between Ω( n d ) pairs of bits, one in the Alice region, one in the Bob region, and have theconjunction of all these tests readable at a given position. Proposition 7
For any d , there is a CA F of dimension d with CC F ∈ Ω( n d ) and it can be chosen ofradius 1. Proof:
Let e = (1 , , . . . , and define F of neighborhood B (1) on alphabet Q = {← , → , T } × { , } as follows: F ( c ) z = c z − e if c z ∈ {→} × { , } ,c z + e if c z ∈ {←} × { , } , ( T, else if ∃ z (cid:48) ∈ B (1) with c z + z (cid:48) = ( T, or π ( c z − e ) (cid:54) = π ( c z + e ) , ( T, in any other case.Consider configurations c satisfying the following property : reezing, Bounded-Change and Convergent Cellular Automata • c z ∈ ( → , { , } ) if π ( z ) < , • c z ∈ ( ← , { , } ) if π ( z ) > , • c z = ( T, else.We call such configuration correct configurations. For any z = ( i , . . . , i d ) ∈ Z d , denote z = ( − i , i , . . . , i d ) .Let X n be the set of correct configurations such that π ( c z ) = π ( c z ) for all z (cid:54)∈ B ( n ) . It is straight-forward to show that for c ∈ X n/ it holds that π ( F n ( c ) ) = 1 if and only if π ( c z ) = π ( c z ) for all z ∈ B ( n/ . If we now consider the problem P F,n , we deduce a fooling set of size n d / , i.e. a set ofinput ( u i , v i ) such that for all i (cid:54) = j it holds F n ( u i , v j ) (cid:54) = F n ( u i , v i ) . This implies CC F ∈ Ω( n d ) (see[27]). (cid:50) Bounded-change CA are intrinsically limited in communication complexity compared to general CAas shown by the following theorem.
Theorem 3
For any bound-change CA F of dimension d , we have CC F ∈ O (cid:0) n d − log( n ) (cid:1) . Proof:
Let r be the radius of F and k be such that F is k -change. Fix some n ∈ N . We distinguish twozones of the space that play an important role: Z A = A rn ∩ { z ∈ Z d : − r + 1 ≤ π ( z ) ≤ } ,Z B = B rn ∩ { z ∈ Z d : 1 ≤ π ( z ) ≤ r } . These zones are such that Alice (resp. Bob) only needs to know the configuration on A rn ∪ Z B (resp. B rn ∪ Z A ) to know the configuration after one step of F on A r ( n − (resp. B r ( n − ). The protocol is asfollows: first, as an initialization step, Alice and Bob exchange the content of the Z A and Z B and initializetheir time counter at . Then, they repeat as much as necessary the following loop:1. Alice (resp. Bob) separately compute the evolution step by step on A rn (resp. B rn ) supposing thatnothing changes in Z B (resp. Z A ) and until a state change is observed in Z A (resp. Z B );2. Alice (resp. Bob) send the time counter at which her change was observed; let t m be the minimumof the time value between Alice and Bob;3. if Alice (resp. Bob) has a time counter equal to t m then she sends a “diff report” which containsthe list of cell changes that occurred inside Z A (resp. Z B ) between the previous time counter valueand t m ;4. Alice (and symmetrically for Bob) possibly receives the other “diff report” and updates its knowl-edge as follows: • updates her knowledge of Z B according to the diff report (if received); • if its time counter value is strictly larger than t m , then revert its knowledge of her half-spaceto what it was at time t m (which she can do without any further information from Bob). • sets its time counter as t m ;8 Nicolas Ollinger, Guillaume Theyssier
This protocol allows Alice and Bob to correctly compute the evolution of F during n steps (without evensupposing that it is bounded-change) and thus solves P F,n . Indeed, the fact that they retrospectively jumpback in time to the step of the first change in Z A ∪ Z B zones ensures that the hypothesis of no change in Z A ∪ Z B is correct for all the retained computation steps.Let us now upper bound the cost of this protocol: • the initialization costs O ( n d − ) (the sizes of Z A and Z B are rn d − ); • the total communication for time counters is O ( n log( n ) since the main loop of the protocol isexecuted at most n times; • each diff report with i cell changes costs at most i (log( | Q | ) + ( d + 1) log( n )) : log( n ) for the timestep, d log( n ) for communicating the position of each cell, log( | Q | ) to describe the new state;The protocol is such that each cell change of each diff report sent corresponds to a real state change in Z A ∪ Z B during the first n steps of the evolution of F . Since F is k -change, the total number of changesin Z A ∪ Z B is less than rkn d − . Therefore the total cost of the protocol is at most O (cid:0) n d − log( n ) (cid:1) . (cid:50) Convergent CA in general require higher communication complexity as shown in the following propo-sition.
Proposition 8
There exists a 1D convergent CA F with CC F ∈ Ω( √ n ) . Proof:
Let G be any 1D CA of radius 1 such that CC G ∈ Ω( n ) (it exists by Proposition 7). We claim that F = Z G is such that CC F ∈ Ω( √ n ) . To see this consider for any n the communication problem P G,n associated to G on some input ( u, v ) ∈ Q A n G × Q B n G and let ( u (cid:48) , v (cid:48) ) = ( c (cid:48) (cid:12)(cid:12) A τn , c (cid:48) (cid:12)(cid:12) B τn ) where c (cid:48) = λ n,c,c and τ n is the constant from Lemma 1 that guaranties Z τ n G ( λ n,c,c ) = ( G n ( c ) , G n − ( c ) , r ) . This transformation from ( u, v ) to ( u (cid:48) , v (cid:48) ) is a reduction of the communication problem P G,n to P Z G ,τ n since u (cid:48) (resp. v (cid:48) ) can be computed from u only (resp. v only), so we have CC ( P G,n ) ≤ CC ( P Z G ,τ n ) .Since τ n ≤ kn for some k > we deduce that CC Z G ( n ) ≥ CC G (cid:0)(cid:112) n/k (cid:1) . The proposition followsfrom the hypothesis on CC G . (cid:50) In this section we show that any k -counter Minsky machine [31] can be simulated by a 1D freezingCA where the evolution of both the state and counter values are encoded by signals moving through thespace-time diagram, one transition per space unit. The instant configuration (state + counters) of thecounter machine at time t is represented by the temporal column (trace) of cell t of the cellular automata,where the value of counters is encoded in unary as the (temporal) distance between occurrences of twospecial states of the CA.More precisely, the control of the machine is represented as a signal of equation (cid:54) y − x (cid:54) whereboth cells in column x carry the state of the counter machine and an emptiness flag for each counter at time x . A counter can easily be tested for equality to zero. Incrementation corresponds to a local deceleration reezing, Bounded-Change and Convergent Cellular Automata CYREACH and the maximal number of state change per cell in a freezing CA, sowe present a construction with additional technical details useful for our goals.
Definition 6
A (deterministic) k - counter Minsky machine is a -tuple M = ( Q M , q , h, τ ) where q , h ∈ Q M are the initial and halting states and τ : Q M × { , } k → Q M × {− , , } k is its transition map, whichverifies τ ( h, · ) = ( h, (0 , . . . , . A configuration of M is an element of Q M × N k . M changes any con-figuration c = ( q, ( χ i ) ≤ i ≤ k ) in one step into configuration M ( c ) = ( q (cid:48) , (max(0 , χ i + δ i )) ≤ i ≤ k ) where ( q (cid:48) , ( δ i ) ≤ i ≤ k ) = τ ( q, (min(1 , χ i )) ≤ i ≤ k ) . M halts on input ( χ i ) ≤ i ≤ k ∈ N k if there is a time t suchthat M t ( q , ( χ i ) ≤ i ≤ k ) ∈ ( h, N k ) . Minsky machine might be slow, but they have the features of an acceptable computational model andparticular have an undecidable halting problem.
Theorem 4 [31] There is a -counter machine M u such that it is undecidable to know whether M u haltson a given input. It is also undecidable whether some given -counter machine halts on the empty input(all counters equal to zero). We now detail our construction that transform any k -counter machine M = ( Q M , q , h, τ ) into a 1Dfreezing CA F M that simulates it. Its neighborhood is V = {− , , } and its state set is Q = { b, w } ∪ Q kc ∪ Q s ∪ I where: • b is a “blank” symbol used to fill a configuration and leave room for a computation to expandspatially, • w is a “wall” state to protect computation zones and avoid back propagation of wrong haltinginformation, • Q c = { , − , , } × {− , , } is the counter alphabet used to encode the value of onecounter and the elementary operation to be applied on it, • Q s = Q M × {− , , } k ∪ Q M is the control alphabet used to encode the state of M (an elementof Q M ) and (possibly) an elementary operation per counter, • I = { i , . . . , i K } is a “large enough” ordered set serving as a bounded countdown before initial-izing a computation of M on an empty input in such a way that it maximizes the number of statechanges in a cell; in practice we fix K = 3 k + 3 .Before giving the transition rule of F , let us give the (pre-)order (cid:22) on Q ensuring that it is a freezingCA together with some useful intuitions about the behavior of F : • h (cid:22) q c for any q c ∈ Q c ( h is used to propagate the halting information of an halting computationof M back to the cell where it started ),0 Nicolas Ollinger, Guillaume Theyssier q q q h +1 − (cid:54) = 0 : −
1= 0 : 0 time 0 1 2 3state q q q hcounter 0 1 0 0 t = 0 t = 1 t = 2 t = 3 · · · b i i b · · · ... iK iK b · · · w q , b · · · w - q b · · · w q , b · · · w , q b · · · w - q , - b · · · w - q b · · · w h, b · · · w - h b · · · w h ... w hw h ... w h ... w h ... w h ... Fig. 2:
Example of space-time diagram of F M (on the right) simulating the Minsky machine M started on an emptycounter (on the left). To simplify, we supposed K = 1 and any state ( α , δ ) is represented as α . • w (cid:22) q for any q ∈ Q ( w can be triggered from any state ), • ( c i , δ i ) ≤ i ≤ k (cid:22) ( c (cid:48) i , δ i ) ≤ i ≤ k if c i ≺ c (cid:48) i for all i where ≺ ≺ − ≺ ( a counter is repre-sented as a sequence of 1s terminated by the sequence − , used to adjust the offset of thenext counter value on the next column, and carrying an invariable operation ), • q c (cid:22) q (cid:48) s (cid:22) q s (cid:22) b for any q c ∈ Q c , q (cid:48) s ∈ Q M × {− , , } k , q s ∈ Q M ( the “normal sequence” ofstate changes is: blank, then control state, then control state plus instruction, then counter ), • ( − , k (cid:22) i K (cid:22) · · · (cid:22) i and ( q , k ) (cid:22) i K ( the special sequences with K state changes to ini-tialize a computation on an empty input ).The transition rule f M : Q V → Q of F M is defined as follows (Figure 4.3 illustrate almost all transi- reezing, Bounded-Change and Convergent Cellular Automata f M ( x, y, z ) = h if y = h or ( z = h and y ∈ ( { } × {− , , } ) k , (1) q else if y = b and x = ( q, · ) ∈ Q M × {− , , } k , (2) b else if y = b and x (cid:54)∈ Q M × {− , , } k , (3) w else if y (cid:54) = b and x = ( q, · ) ∈ Q M × {− , , } k , (4) ( min(1 ,α +1) , δ i ) ≤ i ≤ k else if y = ( α , δ i ) i and x = w , (5) (cid:0) α (( c i , δ i ) , d i ) (cid:1) ≤ i ≤ k else if y = ( c i , δ i ) i ∈ Q kc and x = ( d i , δ (cid:48) i ) i ∈ Q kc , (6) τ (cid:0) q, ( v ( c i )) ≤ i ≤ k (cid:1) else if y = q and x = ( c i , δ i ) ≤ i ≤ k , (7) (cid:0) β ( c i , δ i ) (cid:1) ≤ i ≤ k else if y = ( q, ( δ i ) ≤ i ≤ k ) and x = ( c i , δ (cid:48) i ) ≤ i ≤ k , (8) i n +1 else if x = y = i n , ≤ n < K , (9) ( q , k ) else if y = i K and z = b , (10) ( − , k else if x = w and y ∈ Q M × {− , , } k , (11) w else, (12)where α : Q c × { , − , , } → Q c is the counter update map defined by: α (( − , δ ) , x ) = ( , δ ) ,α (( , δ ) , x ) = ( , δ ) ,α (( , δ ) , x ) = ( , δ ) ,α ((1 , δ ) , x ) = (cid:40) ( − , δ ) if x = δ , (1 , δ ) elseand v : { , − , , } → { , } is a counter positivity test map defined by: v ( x ) = (cid:40) if x = 1 , elseand β : Q c → Q c is a counter copy map defined by: β ( x, δ ) = ( − , δ ) if x = − and δ = − , ( − , δ ) if x = and δ < , (1 , δ ) else.A simple verification on the transition rule gives the following. Fact 3 F M is freezing for the order (cid:22) . As said above, the encoding of the counter values of M in F M is in the time interval between twoparticular state changes, more precisely: for any initial configuration c of F , the (possibly infinite) valueof counter i ( ≤ i ≤ k ) at position z ∈ Z is defined by V i ( c, z ) = (cid:12)(cid:12) { t ∈ N : F t ( c ) z = ( c j , δ j ) j ∈ Q kc and c i = 1 } (cid:12)(cid:12) . Nicolas Ollinger, Guillaume Theyssier
For this definition to be useful in F M we need some hypothesis on the sequence of state at the con-sidered position: we say that the trace at z ∈ Z is valid if it contains for each counter i the subsequence − , , and does not contain state w .The M -state at position z starting from c is Q ( c, z ) = q where ( q, ( δ i ) ≤ i ≤ k ) is the unique occurrenceat cell z of a state from Q M × {− , , } k (uniqueness comes from the freezing order (cid:22) and case 8 of thetransition rule), and it is undefined if there is no such occurrence.The next lemma shows that on well-formed columns, F M correctly compute a transition of M fromone column to the next, via the above encoding. Lemma 3 (Correct simulation)
Let c be a configuration of F M and z ∈ Z such that c z ∈ Q M × {− , , } k and c z +1 = b and the trace at z is valid. Let ( q, ( δ i ) ≤ i ≤ k ) = τ (cid:0) Q ( c, z ) , (min(1 , V i ( c, z ))) ≤ i ≤ k (cid:1) . Thenthe trace at z + 1 is valid, Q ( c, z + 1) is well-defined and equal to q and we have V i ( c, z + 1) = max(0 , V i ( c, z ) + δ i ) for all ≤ i ≤ k . Proof: F M ( c ) z ∈ Q c and F M ( c ) z +1 = Q ( c, z ) (by case 2 of f M ). Then F M ( c ) z +1 = ( q, ( δ i ) i ) by case7 of f M and F M ( c ) z +1 = ( c i , δ i ) ≤ i ≤ k where for each i : • either c i = − and thus V i ( c, z + 1) = 0 , but the definition of β ensures that either V i ( c, z ) = 0 and δ i < , or V i ( c, z ) = 1 and δ i = − ; • or c i = 1 and then case 6 of the definition of f M and the validity of the trace at z ensures thatcounter i at position z + 1 will turn into state ( − , δ i ) at times t i + 2 + δ i where t i is the time ofoccurrence of − in counter i at position z (by definition of the map α ).In any case we have a valid trace at z + 1 and V i ( c, z + 1) = max(0 , V i ( c, z ) + δ i ) . (cid:50) Lemma 4 (Correct halting information)
Let c be a configuration of F M and z ∈ Z a position such that c z = ( q , (0 , . . . , and c z +1 = b and the trace at z is valid. Then F tM ( c ) z = h for some t ∈ N impliesthat the counter machine M halts on input ( V i ( c, z )) ≤ i ≤ n . Proof:
Suppose that M doesn’t halt on input ( V i ( c, z )) ≤ i ≤ n . Using Lemma 3, we prove by induction on n ∈ N that if traces at position z, z + 1 , . . . , z + n are valid then Q ( c, z + n ) (cid:54) = h . Indeed it is true for n = 0 and by induction we prove that if z + n has a valid trace then c z + n = b (otherwise case 4 wouldproduce state w contradicting validity). It implies that case 2 cannot produce state h at position z + n , andtherefore h appears at position z + n only if it appears at position z + n + 1 (case 1 is the only remainingcase of f M to produce h ). So if all traces z + n for n ∈ N are valid then there is no t such that F t ( c ) z = h .Suppose now that some trace is not valid and take n ≥ minimum so that the trace at z + n is not valid.Since the trace at z + n − is valid and has a well-defined Q ( c, z + n − it means that there is some t such that F tM ( c ) z + n − ∈ Q s . Then there are two cases: • either F tM ( c ) z + n = b but then Lemma 3 would apply to configuration F tM ( c ) and position z + n − showing that the trace is valid at position z + n which contradicts the hypothesis on n ; reezing, Bounded-Change and Convergent Cellular Automata • or F tM ( c ) z + n (cid:54) = b so by case 4 of f M we would have F t + iM ( c ) z + n = w for all i ≥ showing that h does not appear at position z + n .The lemma follows. (cid:50) Theorem 5
There exists a 1D freezing CA F such that CYREACH F is undecidable. Proof:
Let M u be the machine from Theorem 4. We claim that CYREACH F Mu is undecidable becausethe halting problem of M u on a given input reduces to it. Indeed, given an input ( χ i ) ≤ i ≤ k , let l = max i χ and define v ∈ Q B ( l +2) by: v j = w if j = − l − , ( α − j,i ) ≤ i ≤ k if − l − ≤ j < , ( q , ( − , . . . , − if j = 0 ,b if < j ≤ l + 2 ,where α j,i ∈ Q c is (1 , − if j − < χ i and ( − , else. On can check that for any c ∈ [ v ] and anyposition z with − l − ≤ z ≤ l + 2 the trace at z in c is valid: it follows from the definition of f M for − l − ≤ z ≤ (case 5 at z = − l − and 6 elsewhere) and for z > it follows from Lemma 3. More-over, by choice of v and for any i ( ≤ i ≤ k ), we have V i ( c, − χ i −
1) = 0 and V i ( c, − χ i − p ) = p for all ≤ p ≤ χ i by a straightforward induction. Therefore we have V i ( c, −
1) = χ i and V i ( c,
0) = χ i by choice of v . We deduce from Lemma 4 that cylinder [ h ] can be reached from cylinder [ v ] only if M u halts on input ( χ i ) i . Finally, by choosing c ∈ [ v ] such that c z = b for z > l + 2 , it follows from Lemma 3that if M u halts on ( χ i ) i then F tM ( c ) = h for some t . This proves the reduction of the halting problemof M u to CYREACH F Mu and the theorem follows. (cid:50) The same construction allows to prove the following result which might seem surprising at first: evenif it is easy to check that a CA is freezing (Fact 1), and if an immediate bound on the number of changesin a freezing CA is given by the size of the alphabet, it is still undecidable to determine what is the actualmaximal number of changes.
Theorem 6
There exists k ∈ N such that it is undecidable to determine whether a given freezing CA F is k -change. Proof:
Let M be any counter machine and F M the associated freezing CA. It follows from the freezingorder (cid:22) that the unique sequence of state changes giving K + 5 changes is (repetition of a same stateremoved): i → · · · i K → ( q , k ) → ( − , k → ( , k → ( , k → h, and any other sequence compatible with (cid:22) has strictly less state changes. Indeed, it is the longest amongthe sequences containing i K and the choice of K = 3 k + 3 ensures that all (cid:22) -admissible sequenceswithout occurrence of i K have strictly fewer than K changes.Consider a configuration c of F M such that the maximal sequence of state changes above occurs at cell . By case 9, c must be such that c z = i for any − K + 1 ≤ z ≤ otherwise w would appear (by case 12)at position . Moreover, we must have c = b and otherwise we would have F K +1 ( c ) = w . Thereforewe have F K +1 ( c ) = ( q , k ) and F K +1 ( c ) − = w (by cases 10 and 12). Moreover cases 11 and 3 force4 Nicolas Ollinger, Guillaume Theyssier F K +1 ( c ) = b . Finally, Lemma 4 can be applied at position of configuration F K +1 ( c ) showing that h appears at position only if M halts on empty input (because the state change sequence at position is such that V i ( c,
0) = 0 for any i ). Conversely, if M halts on the empty input and if we considerconfiguration c such that c z = i for z ≤ and c z = b for z > , it is easy to check (using Lemma 3) thatthe sequence of state changes at position in c is exactly the above sequence. We conclude that M haltson the empty input if and only if F M is not K + 4 -change. The theorem follows. (cid:50) In the previous section we studied canonical problems of prediction and reachability defined for anycellular automaton. Here we consider problem related to the convergence property. We can first considerthe limit value of a cell, and the time its takes to reach this limit given an initial configuration.
Definition 7
Let F be any convergent CA. The limit value problem LIMIT F is defined as follows: • input: a computable configuration c (given as a Turing machine); • output: F ω ( c ) .The freezing time problem TIME F is defined as follows: • input: a computable configuration c (given as a Turing machine); • output: the freezing time τ ( c, of cell starting from c . Remark 1
Problems
LIMIT F and TIME F are Turing-equivalent since we have F ω ( c ) = F τ ( c, ( c ) and τ ( c,
0) = min { t : F t ( c ) = F ω ( c ) } . Theorem 7
There is a freezing CA F such that both TIME F and LIMIT F are uncomputable. Proof:
From the remark above it is sufficient to prove uncomputability of
LIMIT F . The example F M ofTheorem 5 has the desired property and the proof of the theorem actually shows it because: • it shows undecidabilty of reachability of cylinder [ h ] where h is an invariable state of F M , there-fore it shows that given some cylinder [ u ] it is undecidable whether there is c ∈ [ u ] such that F ω ( c ) = h ; • there is in fact a canonical computable (actually ultimately constant) configuration c u such that [ h ] is reachable from [ u ] if and only if F ω ( c u ) = h . (cid:50) In the following subsection we are going to consider a non-uniform problem on the limit fixed-pointfor which a difference between bounded-change and convergent CA will appear. Let us first give somecomputability upper bounds for such limit fixed-points.Given a configuration c ∈ Z d and a state q , denote by χ q ( c ) the set of positions in Z d which are in state q in c : χ q ( c ) = (cid:8) z ∈ Z d : c z = q (cid:9) . The following proposition gives immediate upper-bounds on such characteristic sets for limit fixed pointsin terms of the arithmetical hierarchy [39]. reezing, Bounded-Change and Convergent Cellular Automata Proposition 9
Let F be a convergent CA and c ∈ Z d be any computable configuration. For any state q the set χ q ( F ω ( c )) is a ∆ arithmetical set. Moreover, if F is freezing with order (cid:22) then: • χ q ( F ω ( c )) is recursively enumerable if q is (cid:22) -minimal; • χ q ( F ω ( c )) is co-recursively enumerable if q is (cid:22) -maximal. Proof:
First for a convergent CA F we have χ q ( F ω ( c )) = { z : ∃ t , ∀ t, t ≥ t ⇒ F t ( c ) z = q } and also χ q ( F ω ( c )) = { z : ∀ t, ∃ t , F t + t ( c ) z = q } which shows that χ q ( F ω ( c )) ∈ ∆ . When F is freezing with order (cid:22) and q is (cid:22) -minimal we have χ q ( F ω ( c )) = { z : ∃ t, F t ( c ) z = q } and if q is (cid:22) -maximal then χ q ( F ω ( c )) = { z : ∀ t, F t ( c ) z = q } . (cid:50) In dimension 1, bounded-change CA are too much restricted to produce uncomputable limits from com-putable initial configurations. The argument in the following theorem is due to G. Richard.
Theorem 8
For any 1D bounded-change F and any computable configuration c , F ω ( c ) is computable. Proof:
First we can suppose without loss of generality that F has radius (if F has radius r and alphabet Q one can consider F (cid:48) of radius and alphabet Q r obtained from F by grouping cells by blocks of size r ,computability of configuration is not affected by this grouping operation). Now let c be a fixed computableconfiguration and denote by λ ( z ) the number of changes occurring at position z in the orbit of c : λ ( z ) = (cid:12)(cid:12) { t : F t +1 ( c ) z (cid:54) = F t ( c ) z } (cid:12)(cid:12) . We claim that there is an algorithm that, given z ≤ z (cid:48) and λ ( z ) and λ ( z (cid:48) ) , correctly computes F ω ( c ) [ z,z (cid:48) ] .Indeed, knowing λ ( z ) and λ ( z (cid:48) ) , we can compute F t ( c ) [ z,z (cid:48) ] with t the first time such that F t ( c ) z = F ω ( c ) z and F t ( c ) z (cid:48) = F ω ( c ) z (cid:48) (it is sufficient to compute more and more time steps until having observed λ ( z ) changes at z and λ ( z (cid:48) ) changes at z (cid:48) ). Then, since the states of cells z and z (cid:48) do not change after time t , andsince F has radius , F ω ( c ) [ z,z (cid:48) ] only depends on F t ( c ) [ z,z (cid:48) ] and more precisely F ω ( c ) [ z,z (cid:48) ] = F t + k ( z (cid:48) − z ) ( c ) [ z,z (cid:48) ] if F is k -change.Let us define the following constants, depending (iv) on c , which are the maximal number of changesoccurring infinitely often to the left/right respectively, and the positions beyond which these maxima are (iv) They can depend on c in a non-recursive way. In fact the remainder of the proof shows that they cannot depend recursively on c otherwise it would contradict Theorem 7. Nicolas Ollinger, Guillaume Theyssier bounds on the number of changes: L = max { i : ∀ z < , ∃ z (cid:48) < z, λ ( z (cid:48) ) = i } z L = max { z : ∀ z (cid:48) < z, λ ( z (cid:48) ) ≤ L } R = max { i : ∀ z > , ∃ z (cid:48) > z, λ ( z (cid:48) ) = i } z R = min { z : ∀ z (cid:48) > z, λ ( z (cid:48) ) ≤ R } The algorithm to compute F ω ( c ) z given z is the following: compute larger and larger portions of thespace-time diagram around position z until finding z ≤ z ≤ z and t such that:1. z ≤ z L and z ≥ z R ,2. the state of z has changed L times before time t and the state of z has changed R times beforetime t .By definition of L, z L , R, z R , such values z , z and t can always be found for any z and we have λ ( z ) = L and λ ( z ) = R . Then it is sufficient to apply the algorithm of the above claim to compute F ω ( c ) [ z ,z ] and therefore obtain F ω ( c ) z . (cid:50) Contrary to bounded-change CA, convergent CA can have arbitrarily many changes at a given cell de-pending on the context. This is enough to obtain uncomputable limit fixed point from computable initialconfigurations as shown by the following CA F . Intuitively, F is a 1D CA that simulates progressivelyall Turing machines for more and more time steps and, each time a machine i halts, the head travels to apredetermined computable position p ( i ) to write a mark, and then goes back to the simulation zone andgoes on simulating other Turing machines. Any position p ( i ) would ultimately contain the informationof whether machine i halts or not. The Turing simulation is done in a finite zone that grows as neededbut also that moves regularly to the right, therefore any given cell is out of this active computation zoneafter a finite time. The position map p ( i ) is increasing so that less than i positions lie to the left of p ( i ) ,which means that p ( i ) will be crossed at most i times by the marking mechanism. This intuitive descrip-tion is probably enough to be convinced that there is a CA F and a computable configuration c such that (cid:0) F t ( c ) (cid:1) t converges to an uncomputable configuration. However, we want F to be a convergent CA, i.e. we want the convergence of (cid:0) F t ( c ) (cid:1) t for any inital configuration c , which requires a much more carefuldesign of F . In what follows it is important to keep in mind that the construction is not sensitive to detailsin the implementation of the Turing machines: in fact the property of F being convergent is independentof the actual Turing computation. That’s why we focus on the marking process.The general idea is to implement an addressing mechanism working in unary: positions to be markedare special zones of a certain length, and the marking process consists in a ’snake’ of some length movingto the left until it reaches a special zone of matching length. The convergence is guaranteed by a counteron each special zone that allow only a bounded number of successive ’snakes’ to cross the zone. A keyaspect of the construction to help the analysis and the proof of convergence is the presence of a uniqueglobal head that can either work on the Turing computation or be the ’head of the snake’ in the markingprocess. Essentially all state changes occur in the neighborhood of the head, so convergence is guaranteedby the fact that this global head never visits infinitely often the same cell. reezing, Bounded-Change and Convergent Cellular Automata { , } { , } { , } { , } { , } { , } { , } { , } { , } Fig. 3:
Conveyor belt and zigzags in the addressing component.
Global structure. F has alphabet Q = Q T × Q M × Q A × Q H ∪ { e } where e is a special error statethat spreads over the entire configuration as soon as it appears, and all normal states have four components: • the Turing component Q T = { L, R } ∪ E T which holds the simulation of Turing machines; • the marker component Q M = { , } which is used to mark predetermined positions p ( i ) for haltingmachines i ; • the addressing component Q A = { , } ∪ { , } which is used to handle the low level mechanismfor the marking head to stop at the correct position; • the global head component Q H = { L, R }∪ S H which ensures unicity of a global head and controlsthe succession of phases of Turing simulation and marking process.The global head is not the Turing head which is encoded by states in E T . Addressing component.
This component is a kind of “conveyor belt” with zigzags that globally shiftinformation as follows: • A zigzag is any maximal connected zone in state { , } . More precisely, states from { , } corre-spond to a simple belt and states from { , } correspond to three layers of belt forming a zigzag(plus a forth layer used as a passage counter for the zigzag). Each position in this virtual belt has awell-defined predecessor and successor as depicted in Figure 3. Note that in some degenerate cases,this virtual belt as several connected components but always finite ( in the case where all cells withpossibly finite exceptions are in a state from { , } in their addressing component, in any othercase). • The default dynamics of F on this addressing component is simply that each bit b at any positionin the virtual belt is shifted to its successor’s position. This default dynamics can be changed onlywhen the global head is near (details below). Validity checks and computation cycle.
We call valid a configuration c whose orbit contains no oc-currence of e . By the spreading behavior of e we have that any orbit which is not valid is convergent. F constantly checks and maintains the following conditions and produces the error state e in case ofviolation: • in Turing component the following patterns are forbidden: LR , RL , RE T , E T L , so that a validconfiguration contains at most one connected zone in alphabet E T , called Turing zone ; • in head component the following patterns are forbidden: LR , RL , RS H , S H L , S H S H , so that avalid configuration contains at most one state in S H ;8 Nicolas Ollinger, Guillaume Theyssier • in the addressing component the following patterns are forbidden: ( a, b, c, d, e, f, , so that ina valid configuration the fourth layer of any zigzag holds the unary representation of a counter(distance of the first 0 to the rightmost position of the zigzag) between 0 and the length of thezigzag, called zigzag counter ; • in the addressing component, when a position in the virtual belt is in state then its predecessormust also be in state , except if the head is present on the predecessor’s position (precise definitionbelow); therefore each connected component of the virtual belt seen as a bi-infinite binary wordcontains a single connected component of s; • in the addressing component, no zigzag can accept a too long connected component of enteringthe zigzag, concretely: if there is a 1 in the third layer of the rightmost position of a zigzag, thenthere must be a 0 on the first layer at the same position, otherwise error state e is generated; thisprevents any connected component of s of length ≥ n initially to the right of a zigzag of length n to cross it.The state of the unique global head (if present) is one of the following: S H = {← , → , Preparing , Preparing , Marking , Marking , Marking , Return } . Each one corresponds to a a specific phase of the behavior of F with additional validity checks thatgenerate e in case of failure. Intuitively, ← and → correspond to back and forth movements inside theTuring zone producing one step of Turing computation at each back and forth. Then states Preparing x , Marking x and Return correspond to an interruption of the Turing computation to do a marking cycle:preparation of marking address, movement and marking at the correct position, return to Turing computa-tion. The precise position of the head on the virtual belt is as follows: if the head is in state
Preparing x or Marking x then it is on layer x of the belt in the cell, otherwise it is on layer . In a valid configurationthe sequence of states taken by the global head is always a subword of the language (cid:0) → + ← + (( Preparing +1 + Preparing +3 )( Marking + Marking + Marking ) + Return + )? (cid:1) + • ← : the global head moves back to the leftmost position of the Turing zone; the head must beinside the Turing zone; when this leftmost position is reached, the global head turns into state → ;moreover, if during the crossing of the Turing zone a special interruption state of the Turing head isencountered (in E T ), then the global head turns into state Preparing if the position is inside azigzag and Preparing else; • → : the global head runs through the entire Turing zone from left to right while shifting it one cellto the right, and applying one step of the Turing computation, and adding one more cell to the zoneat the right end; the global head must be inside the Turing zone; when the right end is reached andthe new cell added to the Turing zone, the global head turns into state ← ; • Preparing x : the global head starts to write s on the addressing component while moving twocells left at each step and until it meets the left boundary of the Turing zone; more precisely, whenin state Preparing it writes s onto the first layer if inside a zigzag and onto the third when in reezing, Bounded-Change and Convergent Cellular Automata Preparing ; outside zigzags it always write on the unique layer present (v) ; the global headmust be inside the Turing zone and when it reaches the left boundary of the Turing zone, it turnsinto state Marking ; the purpose of this stage is to write a connected component of s of length n/ onto the virtual belt where n is the distance between the global head and the leftmost positionof the Turing zone when the global state changes from ← to Preparing x ; • Marking x : the global heads moves along the virtual belt from a position to its successor; wheninside a zigzag the x indicates which of the layers of the belt the global head is currently on;the position on the belt preceding that of the global head must contain a and the successor mustcontain a , this forces a connected component of 1s that moves along the belt behind the headwhich can be seen as a unary value called the address value ; when inside a zigzag the global headdoes the following: – when entering a zigzag it checks that the zigzag counter is not otherwise the error state e isgenerated; – when reaching the rightmost position of the second layer (state Marking ), it checks whetherthe address value is exactly twice the width of the zigzag, precisely: if the first layer containsa 1 at its position and the cell immediately to its right contains value , then it enters into state Return and a 1 is written on the marker component; – when in state Marking on the first layer of a zigzag and at the position of the leftmost 1of the zigzag counter, the zigzag counter is decremented by 1 by changing the state 1 at thecurrent position into a 0; • Return : the global head moves along the virtual belt backward (from a position to its predecessor)and transforms all 1s into 0s until it reaches the left boundary of the Turing zone at which point itturns into state → .Given a valid configuration c , we say that a component is convergent if (cid:0) π ◦ F t ( c ) (cid:1) t ∈ N is convergentwhere π is the projection of states onto the considered component. c is convergent when all componentsare convergent. Lemma 5 If c is a valid configuration such that the global head state changes only a finite number oftimes in its orbit, then c is convergent. Proof:
We can suppose without loss of generality that the global state s ∈ S H never change (otherwiseconsider configuration F t ( c ) where t is the time when the last state change occurs). Given the hypothesiswe have the following possible cases: • s = → and the Turing zone extends infinitely to the right: in this case the head moves to the rightforever so the Turing component and the global head component are clearly convergent. The markercomponent is also convergent since it can only change when the global head state changes from Marking x to Return . Finally, the addressing component contains a single connected component (v)
Note that due to the checks done on the addressing component (see above), this
Preparing x stage can generate an e state: forinstance if the head in state Preparing exits a zigzag and later enters into a new one, the s written on the virtual belt will nolonger be connected. This kind of degenerate behavior can easily be avoided by a sufficient spacing between zigzags. Nicolas Ollinger, Guillaume Theyssier of 1s per connected component of the virtual belt (because c is valid). Each cell either stays foreverinside such a connected component of 1s, or stays forever outside after a finite time. In any case,the addressing component is convergent. • s = ← and the Turing zone extends infinitely to the left: this case is symmetric to the previous one. • s = Preparing x and the Turing zone extends infinitely to the left: the Turing component and themarker component don’t change so they are convergent. Moreover the global head moves to the leftforever, so the global head component is also convergent. The global head in this case constantlyextends to the left a connected component of 1s while moving to the left. Therefore, for any cell,after a finite time the global head is on the left and will never come back leaving a similar situationas above: the addressing component is again convergent. • s = Marking x : the global head moves to the left ( x = 1 or ) or to the right ( x = 2 ) foreverfollowed by a (possibly infinite) snake of 1s. The situation is similar to the above cases and theconvergence follows for the same reasons: the global heads moves towards infinity, nothing ischanged on the Turing, head and marker components, and the addressing component convergesbecause of the uniqueness of the component of 1s (per connected component of the virtual belt). • s = Return : the global head moves backward on the virtual belt, so it visits each cell a finitenumber of times (at most in the case of a cell inside a finite zigzag). After this finite numberof visits the state of a cell converges because nothing is changed on the Turing, head and markercomponents, and the addressing component converges as in the previous case. (cid:50) Lemma 6 If c is a configuration such that the global head state changes infinitely many times, then c isconvergent. Proof:
Consider any position z ∈ Z , we will show that (cid:0) F t ( c ) z (cid:1) t ∈ N is convergent. First remark that sucha c must be valid. Moreover the Turing component is convergent because: • either states ← and → do not occur after some finite time in which case the Turing componentbecomes constant; • or there are infinitely many state changes from ← to → or from Return to → which impliesinfinitely many shifts to the right of the computation zone; • other cases are impossible by the hypothesis on c .Considering the sequence of global head states, we must be in one of the following situations: • finitely many s ∈ { Marking x , Return , Preparing x } : in this case after some finite time T ,the global head is restricted to the Turing zone which moves constantly to the right, therefore themarker and global head components are convergent. Moreover, the addressing component evolvesindependently of the global head after time T and, since c is valid, there is only one connectedcomponent of 1s per connected component of the virtual belt. The convergence follows. reezing, Bounded-Change and Convergent Cellular Automata • the state of the global head eventually stays in Marking , Marking , Marking : in this case,after some finite time the global head moves along the virtual belt forever. It cannot be inside orenter an infinite zigzag (otherwise it would eventually stay in a fixed state forever), so it escapesto the left, precisely after some time it stays forever to the left of position z . This shows that theTuring, marking and global head components are convergent at z . By validity of c and the samereasoning as above about connected components of 1s, the addressing component converges at z . • infinitely many changes from Marking x to Return : the global head can only do such statechanges inside finite zigzags, because on one hand this state change can only occur at the rightboundary of a zigzag, and, on the other hand, if the head turns into state
Return inside a zigzagwhich extends infinitely to the left then it could never come back to the leftmost position of theTuring zone, contradicting the infinite state changes hypothesis. The global head can visit onlyfinitely many times each such finite zigzag, therefore after some time it will stay forever to the rightof position z . With the same reasoning as in the previous case, all components are convergent atposition z . (cid:50) Theorem 9
There is a convergent CA F and a computable configuration c such that F ω ( c ) is uncom-putable. Proof:
Lemma 5 and 6 show that the construction technique above always produces convergent CAwhatever the behavior of the Turing component. Let’s define F as the above construction applied to aTuring machine that does the following when started on the empty tape: • Machines ← ∅• n ← • loop forever: – n ← n + 1 – wait n steps – Machines ← Machines ∪ { n } – for each i ∈ Machines do: ∗ simulate n steps of machine i ∗ if i has halted during the simulation then1. Machines ← Machines \ { i }
2. place the Turing head at distance i from the left boundary of thecomputation zone3. turn into special interruption state to launch a marking process This Turing machine has the following properties: • it launches a marking process with address value (vi) i if and only if machine i halts on empty input,and it launches it at most once; (vi) The Turing head placement is i cells away from the left boundary of the computation zone when the interruption launches themarking process; then the preparing stage produce a sequence of 1s of length i . Nicolas Ollinger, Guillaume Theyssier • the simulation of machine i starts after strictly more than i ( i − steps, therefore the markingprocess with address value i cannot be launched before time step i ( i −
1) + 1 .Now consider the following computable configuration c : • the Turing component corresponds to an empty tape with machine head in initial state at position ; • the marker component is uniformly ; • the global head component is in state → with head at position ; • the addressing component is the concatenation for all i of a zigzag of length i with counter at value i followed by i cells of simple belt, concretely: ω , , ,
1) 0 (0 , , , · · · (0 , , , i i · · · where the first zigzag (of length 1) is at position .We claim that the marker component of F ω ( c ) contains a at position p ( i ) = i + (cid:80)
Let us first remark that in dimension 2 and more, it is very easy to produce uncomputable limit fixed pointfrom computable initial configurations.
Proposition 10
Let F be the freezing CA defined over alphabet Q = { , } by: F ( c ) z = min( c z , c z +(0 , ) . There is a computable configuration c such that F ω ( c ) is uncomputable. Proof:
Consider the configuration c defined by: c ( i, j ) = (cid:40) if j > and machine i halts on the empty input in less than j steps else. reezing, Bounded-Change and Convergent Cellular Automata F ω ( c ) ( i, = 0 if and only if machine i halts on the empty input. (cid:50) We can actually obtain much stronger and meaningful results on the uncomputability of limit fixed-points for freezing CA by requiring a finite initial configuration. Let us recall that an aTAM system is directed if there is a unique terminal assembly. Taking the notations of example 5, it means in particularthat this unique terminal assembly is F ωR ( c ) . Then the following result on the computational power ofdirected aTAM systems is directly related to our concern (we use again notations of example 5 to state thetheorem). Theorem 10 (Main construction of [29])
There exists a computable function f : N → N such that forany recursively enumerable set A there exists a directed aTAM whose unique terminal assembly c is suchthat for any n ∈ N : c ( f ( n ) ,
0) = (cid:40) t if n ∈ A(cid:15) if n (cid:54)∈ A where t is some fixed tile of the system. From Proposition 9, the above result is optimal since, in directed aTAM systems and the correspondingfreezing CA, each state is either minimal or maximal with respect to the order, so sets χ q ( F ωR ( c )) are atmost at level one of the arithmetical hierarchy. However for freezing CA in general we can do more asshown below. Corollary 1
There exists a 2D freezing CA F , a finite configuration c , and a state q such that χ q ( F ω ( c )) is neither recursively enumerable nor co-recursively enumerable. Proof:
Fix some computable bijection φ : N → N and consider the two following recursively enumer-able sets: A = { n : π ( φ ( n )) ∈ H } ,A = { n : π ( φ ( n )) ∈ H } where π and π are the projections on first and second components and H is the halting set. FromTheorem10 we get two directed aTAM to which correspond two freezing CA F and F whose limitfixed-point starting from the respective seed configurations c and c verify: F ωi ( c i ) ( f ( n ) , = (cid:40) t i if n ∈ A i (cid:15) i if n (cid:54)∈ A i for i = 1 or . Consider now the freezing CA F = F × F (with the product order), the configuration c whose first component is c and second component is c , and the state q = ( t , (cid:15) ) . One can check that ( f ( n ) , ∈ χ q ( F ω ( c )) if and only if φ ( n ) = ( i, j ) where i ∈ H and j (cid:54)∈ H . The corollary follows. (cid:50) Nicolas Ollinger, Guillaume Theyssier
We give in the following table a synthesis of results showing differences between freezing, bounded-change and convergent CA together with dimension sensitiveness. The table mixes different kinds ofresults, among which: lower bounds L which must be read as “there exists F such that property L holds”,and upper bounds U to be read as “for all F property U holds”.Freezing Bounded change ConvergentMembership PTIME (Fact 1)
Undecidable (Theorem 1)
Undecidable (Theorem 1)
PRED F in 1D NLOGSPACE (Proposition 6)
NLOGSPACE (Proposition 6)
P-complete (Proposition 5)
PRED F in 2D P-complete (Proposition 4) P-complete (Proposition 4)
P-complete (Proposition 4) CC F in 1D O (log( n )) (Theorem 3) O (log( n )) (Theorem 3) Ω( (cid:112) ( n )) (Prop 8) F ω ( c ) in 1D c computable Computable (Theorem 8) Computable (Theorem 8)
Uncomputable (Theorem 9) χ q ( F ω ( c )) in 2D c finite Neither r.e. nor co-r.e. (Corollary 1) Neither r.e. nor co-r.e. (Corollary 1)
Neither r.e. nor co-r.e. (Corollary 1)
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