Hard Asymptotic Sets for One-Dimensional Cellular Automata
aa r X i v : . [ c s . CC ] J u l Hard Asymptotic Sets for One-DimensionalCellular Automata ⋆ Ville Salo
University of TurkuTUCS – Turku Centre for Computer Science
Abstract.
We prove that the (language of the) asymptotic set (and thenonwandering set) of a one-dimensional cellular automaton can be Σ -hard. We do not go into much detail, since the constructions are relativelystandard. It is well-known that (the language of) the limit set of a cellular automatoncan be Π -hard. Usually, [3] is given as the reference, since this was presumablywhere the result was first claimed, although the proof given is wrong. We givea similar result for the asymptotic set (hopefully with a correct proof). It turnsout that asymptotic sets live in the analytical hierarchy instead of the arithmetichierarchy, and their level is Σ . A similar problem on countable SFTs is solvedin [7].In [2], a two-dimensional cellular automaton with a maximally complicatedasymptotic set in terms of Kolmogorov complexity is constructed. There is noreason why the asymptotic set of this CA should have high computational com-plexity since these two notions are relatively orthogonal. Our best guess is thatit is in Π , since high Kolmogorov complexity can be checked at this level,and since it is clearly computable whether a partial Robinson tiling extends toa tiling of the plane. Similarly, our construction says nothing about the Kol-mogorov complexity of the asymptotic set. We note, however, that compared tothe construction in [2] our construction is completely trivial.While the proof is not interesting, the result is a bit more so, since we arenot aware of many examples of ‘practical’ Σ -complete sets. A well-known Π -complete set is the set of notations for countable ordinals, though. We consider cellular automata, shift-commuting continuous functions, on thefull shift X = S Z . Our reference for the analytical hierarchy is [6]. Our main ⋆ Research supported by the Academy of Finland Grant 131558 nterest is in Σ -predicates P ( w ) with a free variable w ranging over N (usuallybijected with S ∗ ), since these turn out to characterize the asymptotic set. Thedefinition of such predicates is that exactly one existential second-order (set)quantifier, no universal second-order quantifier and any number of first-order(number) quantifiers is used. For these, the natural normal form (see Part A,Chapter 1, Theorem 1.5 in [6]) is P ( w ) = ( ∃ C ⊂ N )( ∀ m ∈ N )( ∃ ℓ ∈ N ) R ( C, m, ℓ, w ) , (1)where R is recursive. A predicate with subsets of N as inputs is of course said tobe recursive if the corresponding Turing machine halts no matter what the setis, after inspecting some (arbitrarily long but finite) prefix of the set. Definition 1.
The asymptotic set of a CA f : X → X is the set A ( f ) = [ x ∈ X \ n ∈ N [ k ≥ n f k ( x )This is the union of sets of limit points of f -orbits of configurations. Here,and in all that follows, the language of a subset Y of S Z is the set of wordsthat occur as y [0 ,k − for y ∈ Y and k ∈ N , even if the set if not closed (it iswell-known, and easy to see, that asymptotic sets need not be closed). Lemma 1.
The (language of the) asymptotic set of a CA f : X → X is always Σ .Proof. Given w ∈ S ∗ , we wish to check whether there exists y ∈ A ( f ) with y [0 , | w |− = w . This is the case if and only if there exists x ∈ X such that y withthis property appears in T n ∈ N S k ≥ n f k ( x ). Thus, whether w is in the asymptoticset is equivalent to ( ∃ C ⊂ N )( ∀ m ∈ N )( ∃ ℓ ∈ N ) R ( C, m, ℓ, w ) , where R checks that ℓ ≥ m , and for the configuration y encoded by C in somereasonable way, we have f ℓ ( y ) [0 , | w |− = w . By form, this is a Σ check. ⊓⊔ Theorem 1.
Every Σ subset of N can be many-one reduced to the language ofthe asymptotic set of some cellular automaton on a full shift. In particular, thereexists an asymptotic set with a Σ -complete language.Proof. First, note that we can restrict to any SFT we like by adding a spreadingstate and having the CA introduce it when a forbidden pattern is seen. Thiscannot decrease the complexity of the asymptotic set. The alphabet S was leftunspecified in the statement of the theorem, since we can also use any (non-trivial) alphabet we like. Namely, any SFT can be recoded to one over { , } through a constant-length substitution, and we can use 0 as the spreading stateif the substitution was chosen so that . . . . . . does not appear in the image.e use the SFT Y with configurations of the form( . . . a a . . . a j | b c b c b c . . . ) × Z × A Z , where | are special symbols, a i , b i ∈ { , } , ( c i ) i is of the form 1 ∗ . . . , A is a finite set of helper states we leave unspecified, and Z is composed ofconfigurations of the form . . . →→ q ←← . . . , where q ∈ Q , and Q is the stateset of a Turing machine M discussed later.Our cellular automaton will simulate the machine M on configurations ofthis form. The head , marked by the q ∈ Q on the middle track, moves around,reading values from the first track and possibly changing values on the A Z -track.On the first track, a i and b i cannot change their values, but the bits c i may beflipped from 0 to 1. The values a i compose the input , the bits of b i representthe guessed set (the part ( ∃ C ⊂ N ) of (1)) and the guesses needed for theuniversal quantification (the part ( ∃ ℓ ∈ N ) of (1)), and the values c i are usedfor the universal quantification itself (the part ( ∀ m ∈ N ) of (1)). Thus, theconfiguration ( b i ) i encodes both an infinite subset C of N and a number ℓ i ∈ N for each i ∈ N . We refer to the latter numbers as a Skolemization of the universalquantification. From now on, we leave the values of the helper states implicit,and discuss instead the projection of Y where they are removed.A signaling configuration is a configuration of Y of the form( . . . .w | x ) × ( . . . →→ .q ←← . . . ) , where . denotes the origin, q is a dedicated state of the Turing machine, w ∈{ , } ∗ , and x ∈ ( { , } × { , } ) N . Note that the set of signaling configurations isopen, since up to shifting, this is just the union of cylinders [ w | ] × [ → q ← | w | ],where w ranges over { , } ∗ .Given a Σ predicate P ( w ) = ( ∃ C ⊂ N )( ∀ m ∈ N )( ∃ ℓ ∈ N ) R ( C, m, ℓ, w ), wechoose the Turing machine and thus the cellular automaton so that { w | P ( w ) } reduces to the asymptotic set via the reduction φ mapping w ( w | ) × ( → q ← | w | )(the A -track containing, say, only unary data).We explain what happens on a signaling configuration. First, the Turingmachine exits the state q ; it will not be re-entered until we explicitly state so.It then looks for the smallest p such that c p = 0. If one is found, the machinedecodes the values ℓ , . . . , ℓ p from ( b i ) i , and checks R ( C, i, ℓ i , w ) for 1 ≤ i ≤ p ,decoding C from ( b i ) i as needed. If these checks are accepting, then c i is flippedto a 1. The Turing machine then returns back to its original position and entersthe state q (and, say, empties the A -track in the progress). If something out ofthe ordinary happens (say, the encoding of C is incorrect), a spreading state isintroduced.A configuration with u = ( w | ) × ( → q ← | w | ) at the origin is in the asymp-totic set if P ( w ) holds, since( . . . .w | x ) × ( → .q ← | w | )as such a configuration as a limit point if x = b b b . . . , if the C encoded in( b i ) i is the correct quess for w , and the values ℓ i are a corresponding Skolemiza-tion of the part ( ∀ m ∈ N )( ∃ ℓ ∈ N ) of (1).If the spreading state ever occurs in the orbit of a point, then only wordsover the spreading state are added to the asymptotic set. Also, if a configurationwith u at the origin appears in the asymptotic set, then in particular a config-uration with u at the origin eventually appears. From such a configuration, thecomputation goes as outlined above assuming that a spreading state is not in-troduced. By compactness, we cannot hope for an encoding of the Skolemization ℓ i where the values cannot be infinite, so that it is necessarily possible that oursimulation of M runs forever without finding the next ℓ i . In such a case, q is ofcourse not re-entered infinitely many times, so u is not added to the asymptoticset. This means that infinitely many appearances of u at the origin in fact provethat P ( w ) holds. This concludes the proof that φ many-one reduces solutions of P to the asymptotic set.Since there exists a Σ -hard subset of N and we can many-one reduce anyΣ subset of N to the asymptotic set of such a cellular automaton, there existsa cellular automaton with a Σ -hard asymptotic set. ⊓⊔ We have shown that the language of an asymptotic set can be Σ -complete,in analogy with the limit set. Since asymptotic sets live far higher in the com-putability hierarchy, it seems natural to also encode configurations into subsetsof N and consider the complexity of the corresponding set of subsets. We do notdiscuss this here.Using Lemma 4.1 in [5], Theorem 1 seems to be extendable to any positiveentropy SFT X . We give a rough outline of this construction: The lemma givesus, inside any positive entropy SFT (even sofic), a subshift Y which is the imageof a full shift in a constant-length substitution. On this subshift, we can simulatethe cellular automaton constructed in Theorem 1, using a cellular automaton f . Of course, there is some leftover to consider, and standard methods suchas the Extension Lemma [1] cannot really be used. However, as we only careabout computational complexity, we can use forbidden patterns of Y as spreadingstates.First, in portions of the configuration containing only patterns of Y , f is ap-plied. Borders of such areas are moved toward the Y -patterns using a pigeonholdargument such as the Pumping Lemma, and by using the Marker Lemma [4] toensure consistency of the process. In the asymptotic set of the cellular automa-ton g obtained, we are left with only configurations where forbidden patterns of Y occur with bounded gaps, and configurations over Y which correspond to theΣ -hard asymptotic set of f . Clearly, the asymptotic set of g is then Σ -complete,since Lemma 1 naturally holds on all SFTs. Having dealt with the positive entropy case, it makes sense to ask what thesituation is on zero-entropy SFTs. Interestingly, things are very different. Now,he natural level where asymptotic sets live is Σ . In particular, these sets arein the arithmetic hierarchy instead of the proper analytical level Σ . Of course,this is intuitive when one compares the normal form P ( w ) = ( ∃ C ⊂ N )( ∀ m ∈ N )( ∃ ℓ ∈ N ) R ( C, m, ℓ, w )of a Σ predicate to the normal form P ( w ) = ( ∃ c ∈ N )( ∀ m ∈ N )( ∃ ℓ ∈ N ) R ( c, m, ℓ, w )of a Σ predicate. Lemma 2.
The asymptotic set of a CA f on a countable SFT X is Σ .Proof. Given a word w , and again leaving encodings implicit, it is in Σ to checkthat ( ∃ x ∈ X )( ∀ n )( ∃ m > n ) f m ( x ) [1 , | w | ] = w. Namely, X is countable, so a single number can encode the contents of a config-uration in X . ⊓⊔ Not all countable SFTs support a cellular automaton with a Σ -completeasymptotic set, but some do. Theorem 2.
There exists a countable SFT, and a CA on it, with a Σ -completeasymptotic set. We refer to [7] for a proof.It is an interesting question what the asymptotic sets of very simple SFTslook like.
Question 1.
Is there a natural characterization of countable SFTs that supportcellular automata with Σ -complete asymptotic sets? Definition 2.
The nonwandering set of a CA f : X → X is the set N ( f ) = { x ∈ X | x ∈ \ n ∈ N [ k ≥ n f k ( x )While y ∈ X is in the asymptotic set of f if it is a limit point of some x ∈ X ,it is in the nonwandering set if it is its own limit point. Again, the languages ofsuch sets live in Σ . The upper bound is proved as Lemma 1. Lemma 3.
The language of the nonwandering set of a CA f : X → X is always Σ . So is the lower bound: heorem 3.
Every Σ subset of N can be many-one reduced to the language ofthe nonwandering set of some cellular automaton on a full shift. In particular,there exists a nonwandering set with a Σ -complete language.Proof. The proof goes as that of Theorem 1, except that the CA does not flipthe values c i to 1 one by one, but instead increments it as a binary counter (sothat ( c i ) i can now be any binary configuration in the SFT). If P ( w ) does nothold, then u = ( w | ) × ( → q ← | w | ) is not even in the asymptotic set, seen asin the proof of Theorem 1. If P ( w ) does hold, then u is in the nonwandering set,as the point ( . . . .w | x ) × ( → .q ← | w | )in the proof of Theorem 1 now has itself as a limit point. ⊓⊔ References
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