Four heads are better than three

by   Ville Salo, et al.
Turun yliopisto

We construct recursively-presented finitely-generated torsion groups which have bounded torsion and whose word problem is conjunctive equivalent (in particular positive and Turing equivalent) to a given recursively enumerable set. These groups can be interpreted as groups of finite state machines or as subgroups of topological full groups, on effective subshifts over other torsion groups. We define a recursion-theoretic property of a set of natural numbers, called impredictability, which roughly states that a Turing machine can enumerate numbers such that every Turing machine occasionally "correctly guesses" whether they are in the language (by halting on them or not), even if trying not to, and given an oracle for shorter identities. We prove that impredictable recursively enumerable sets exist. Combining these constructions and slightly adapting a result of [Salo and Törmä, 2017], we obtain that four-headed group-walking finite-state automata can define strictly more subshifts than three-headed automata on a group containing a copy of the integers, confirming a conjecture of [Salo and Törmä, 2017]. These are the first examples of groups where four heads are better than one, and they show the maximal height of a finite head hierarchy is indeed four.


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1 Introduction

In this paper, we combine construction techniques from group theory and recursion theory to solve a question of the author and Törmä made in [15] about group-walking automata. The a priori motivation was this conjecture, but the constructions and definitions we give may be of independent interest.

1.1 The results

Theorem 1.

Let be . Then there exists a recursively presented torsion group with bounded torsion, whose word problem is Turing equivalent to .

The exact types of reduction are that many-one reduces to the word problem, and the word problem conjunctively reduces to .

Our recursion-theoretic contributions are of a somewhat technical nature (though not particularly difficult). We define a possibly new notion of -impredictability (Definition 1), which roughly states that a Turing machine can enumerate numbers such that every Turing machine occasionally “correctly guesses” whether they are in the language (by halting on them or not), given an oracle for shorter identities.

Theorem 2.

For every , there exists a -impredictable set.

We now state our main “navigation-theoretic” contribution. In [15], for a f.g. group and , a class of -subshifts is denoted by . This is the class of subshifts defined by group-walking finite-state machines with heads (see [15] or Section 5 for the definitions). The following is a slight adaptation of a result of [15].

Lemma 1.

If has bounded torsion and has -impredictable word problem for fast-enough growing , then .

For example is fast enough. By putting the above results together, we obtain that four heads is better than three, as claimed in the title.

Theorem 3.

There exists a finitely-generated recursively-presented group containing a copy of the integers such that .

See Section 1.3 for some context for this result, and a new conjecture.

1.2 Some relevant existing work

The main group-construction result (recursively presented groups with bounded torsion and with a word problem of a prescribed difficulty) uses the idea from [13, 3] of groups of finite-state machines. It also uses existing torsion groups as a black-box, in particular it is an application the deep theory that arose from the Burnside problem [1].

The idea of hiding information into the word problem is of course not a new idea in combinatorial group theory, but we are not aware of it appearing previously in the context of torsion groups. The Dehn monsters from [11], recursively presented groups where no infinite set of distinct elements can be enumerated, seem strongly related. (Our construction cannot be used to produce such groups due to using another group as a black box, but it is possible that their construction can be adapted to produce our result.)

Topological full groups (already on ) are a well-known source of interesting examples of groups [10, 6, 9], and our groups can also be interpreted as subgroups of topological full groups of subshifts on torsion groups. In a symbolic dynamic context, [7] (independently) uses a similar construction to prove that the automorphism groups of multidimensional SFTs can have undecidable word problem.

Dan Turetsky showed in the MathOverflow answer [12] that the halting problem is not -impredictable for some , and that there exists a set which is simultaneously -impredictable for all . The latter result of course implies Theorem 2.

1.3 Head hierarchies

For context, we state what is now known about the head hierarchies, and state a bold conjecture.

Theorem 4.

For an infinite finitely-generated group , denote by the supremum of such that . Then

Our contribution is the on the left, which is notable because the maximal finite height is now known. We conjecture that four heads are needed if and only if the word problem is undecidable:

Conjecture 1.

Let be a finitely-generated infinite group. Then

  • if and has decidable word problem, then ,

  • if and has undecidable word problem, then ,

  • if is a torsion group, .

The upper bounds are known, and is known for torsion groups [15]. Everything except the second item was conjectured in [15] and [14]. It is known that the sunny-side-up subshift (the subshift of of configurations with at most one ) does not prove , as on these groups two heads suffice for it. We do not know any other non-torsion groups where the sunny-side-up requires fewer heads than the above conjecture would suggest.

Settling this conjecture in the positive would not be the end of the story. A more refined invariant than would be to ask what the precise set of such that is. In particular this is of interest when is a torsion group; [15] only gives an affine function such that .

These results are of course about just one way to associate subshift classes to -headed automata. We believe the results are relatively robust to changes in the definition, but some details are critical. It is in particular open what happens when is a torsion group and the heads are allowed to communicate over distances, i.e. if they have shared state.

2 Preliminaries

For two functions write (resp. ) if for some choice of , (resp. ) for large enough . Write if and . Write for .

We assume some familiarity with computability/recursion theory, but we state some (not necessarily standard) conventions. We identify partial computable functions with Turing machines, and also their Gödel numbers. Let be the set of total recursive functions, or codings of Turing machines which halt on every input. Let the set of all partial recursive functions (we could take simply ). “Recursive” means the same as “computable” and refers to the existence of a Turing machine, which always halts unless the function is explicitly stated to be partial. If is a Turing machine, we write if halts on input . A partial (not necessarily computable) function from to is denoted .

Let us recall some basic definitions of reductions. A set many-one reduces to a set if there is such that . For the following three reductions, we give quantitative versions, so we have a handle on the rate at which reduction happens. The definitions are stated in terms of characteristic sequences, but correspond to their usual meaning.

We say weakly truth table reduces, or wtt-reduces, to if there exists and nondecreasing with such that when applied to words as , if

is the characteristic function of

and that of , we have that is defined on prefixes of , and

We call the rate of the reduction.

We say positively reduces to if it wtt-reduces to , for some , with the following additional properties for : , for all , and is monotone in the sense that , where is letterwise comparison. If further only depends on whether for some computable from , then conjunctively reduces to .

We say Turing reduces to if there is an -oracle machine that can determine membership in . Clearly many-one reducibility implies conjunctive reducibility implies positive reducibility implies weak truth table reducibility implies Turing reducibility.

We assume some familiarity with group theory, but state some conventions. Our groups are discrete and mostly finitely-generated. Finitely-generated groups come with a finite generating set, which we usually do not mention. The identity of a group is denoted by (or just ).

The word problem of a group is the following subset : Let be the fixed symmetric generating set, and order elements of (finite words over ) first by length and then lexicographically. Include if the th word evaluates to the identity of . Slight inconvenience is caused by linearizing the word problem this way, but on the other hand sticking to subsets of slightly simplifies the discussion in Section 3. We denote the word problem of as .

If a countable group acts on a compact zero-dimensional space , the corresponding topological full group is smallest group of homeomorphisms which contains every homeomorphism with the following property: there exists a clopen partition of and such that for . It turns out that the group contains precisely such homeomorphisms, i.e. they are closed under composition. One may think of as a local rule for .

A group is torsion if all elements have finite order, i.e.  does not contain a copy of the integers. The torsion function of a torsion group is defined by

where and is the word norm with respect to the implicit generating set. A group is of bounded torsion if .

In [8] Ivanov shows that the free Burnside group

is infinite and has decidable word problem ([8, Theorem A]), and we obtain

Lemma 2.

There exists a finitely-generated torsion group with bounded torsion and decidable word problem.

The decision algorithm is given in [8, Lemma 21.1]. For a survey on the Burnside problem see [1]. Theorem 3 could be proved using any torsion group with recursive torsion function (e.g. the Grigorchuk group), with minor modifications, as explained after the proof of Theorem 3. Our technical construction results (in particular Theorem 6) are stated and proved without assuming the existence of infinite f.g. groups of bounded torsion, though obviously to obtain Theorem 1 the existence of one is necessary (since it states the existence of one).

We assume some familiarity with symbolic dynamics on groups (see [2] for more information), but state some conventions. If is a group, and a finite set, with the product topology and the -action is called the full shift, and the actions are called shifts. A subshift is a topologically closed subset satisfying . A particularly important subshift is the sunny-side-up subshift . Equivalently a subshift is defined by a family of forbidden patterns, i.e. a (possibly infinite) family of clopen sets that the orbit of may not intersect. If has decidable word problem then is effective if there exists a Turing machine that enumerates a family of forbidden patterns which define the subshift. A cellular automaton on a subshift is a continuous shift-commuting self-map of it.

3 Impredictability

Definition 1.

For a function , a set is -impredictable if

To unravel this definition a bit, we want to have a Turing machine which always halts and gives us positions on the number line so that, for any Turing machine , is just the halting information about on input , even if is allowed oracle access to the first bits of . Since is quantified universally, this definition is one way to formalize the idea that it is hard to predict whether even given access to the first bits of .

Lemma 3.

Let and for all large enough . Then every -impredictable set is -impredictable, with the same choice of .


Let and suppose is -impredictable. We need to show -impredictability where for all large enough . Clearly changing finitely many initial values does not change -impredictability, so we may assume . Let be given by -impredictability, so

In particular, we can restrict to which given first cut off all but at most the first symbols of , and then apply . This restriction is

which is just the definition of -impredictability. ∎

Theorem 5.

For every , there exists a -impredictable set.


We construct an increasing total computable function , and will be contained in its image. We want to have

infinitely many times for all . The way we construct is we go through and set either , or set up the rule for some Turing machine whose behavior we describe (informally). We refer to those already considered as determined.

List all partial functions in an infinite-to-one way, i.e. we consider functions successively, so that each appears infinitely many times. When considering , we want to determine new values in so as to make sure that

for at least one new . Suppose we have already determined the first values of , i.e. the word such that the characteristic sequence of will begin with the word is already determined, and that we have not determined whether for any . The idea is that while we do not know what the word actually is, there are only possible choices, and we simply try all of them to get the equivalence above to hold for one new . To achieve this we will determine whether for some interval of choices .

Enumerate the words of as . Now, let be minimal such that for all . Let Set for all . As for pick any distinct values greater than , and determine . If , then for we have

because the characteristic sequence of indeed begins with the word . Thus, we have obtained a new value at which the statement is satisfied for .

For all that are yet undetermined, but some larger number is determined, we determine , so that we determine the values in some new interval . We can then inductively continue to the next value of . This process determines all values of , and by construction is a recursively enumerable set which is -impredictable. ∎

Lemma 4.

Let and suppose many-one reduces to , and wtt-reduces to with rate . If is -impredictable, then is -impredictable.


Let . Let be the many-one reduction from to , and the wtt-reduction from to with rate . Since is -impredictable there exists such that

Setting , we have, by the definition of , that

For define as follows: Given , compute and then evaluate , so that

where we observe that (in particular indeed halts with an output so the formula makes sense).

Specializing the first quantifier, we have


so proves that is -impredictable, as desired. ∎

We show that impredictability implies uncomputability.

Proposition 1.

If is , then is not -impredictable for any .


Suppose for a contradiction that is and is -impredictable with . Let be such that

In particular, this applies to the following : given , we ignore and if , then , and otherwise . This well-defines since is , and for all we have

a contradiction. ∎

4 Impredictable torsion groups

Definition 2.

We define a group which depends on a choice of a finitely-generated groups (and choices of generating sets for them, kept implicit), and a set . First define a subshift on by

where is the (left-invariant) word metric. To each associate the bijection

i.e. the usual shift action in the first component, and to each and associate the bijection

where evaluates to if , and to otherwise. Define

Obviously is finitely-generated, and the implicit generators we use for are the ones from the definition, with taken only for in the generating set of , and only for in the generating set of .

Remark 1.

This group can be interpreted as a group of finite-state machines in the sense of [3] (with obvious nonabelian generalization) when is finite, by simulating the actions of by translations of the head, having states, and changing the state by the left-regular action of (if in the correct clopen set) when is applied. Again if is finite, can also be interpreted as a subgroup of the topological full group of the -subshift under the action , by having act by and having act by either or depending on .

We now prove some important technical properties of these groups, leading up to the proof that they give examples of bounded torsion groups with impredictable word problem.

Lemma 5.

If , the defining action of can be seen as a restriction of the action of in a natural way, thus is a quotient group of .


Map generators to generators in the obvious way. The group acts on and the restriction of the action to is precisely that of . Thus, all identities of are identities also in . ∎

In particular, the defining action of is always a restriction of .

Lemma 6.

For any , there exists a split epimorphism .


We have , so the -translations act nontrivially on the component as the action on is the one-point compactification of the left-regular action of on itself. Observe also that the -component is not modified by any of the maps , so the action of on this component factors is just the shift action . This gives a homomorphism , and the map is a section for it. ∎

We refer to as the natural epimorphism.

Lemma 7.

If is decidable and is , then is an effective subshift.


Since we can list elements of , and can compute the distance between given group elements, we can forbid all finite patterns where two s appear at with . ∎

Lemma 8.

If has decidable word problem and is recursively presented, then the following statements hold.

  • If has decidable word problem, then the word problem of conjunctively reduces to with exponential rate.

  • if is then is recursively presented.


Let and let be the finite generating set of . We begin with the proof of the latter item. We need to find a semi-algorithm that, given , halts if and only if represents the identity. For this, first consider the natural epimorphism image . Because is in particular recursively presented, we can first verify that (if not, then also , and the computation diverges as desired).

Assuming , we next check that acts trivially on all elements of . We define an action of the free group on generators on pairs by

(where and ), and for and we map

when , and otherwise.

If and , clearly if and only if the action of fixes all where . Naturally, if and the action fixes for all , then a fortiori . We can verify this for a particular by using the fact is recursively presented. By the previous lemma, is effective, so we can enumerate upper approximations to which eventually converge. In other words, we eventually obtain the set , and it , then at this point (at the latest) we can conclude that indeed acts trivially and halt.

The proof of the first item is similar. To see that there is a wtt-reduction with exponential rate, observe that if we know the first values of , then we can determine the legal contents of all -patterns with domain in , and using this, and the decidable word problem of , we can determine whether for any word with . Since we list elements of groups in lexicographic order, the resulting rate is exponential, as there are exponentially many words with .

To see that this is a conjunctive reduction, we observe that the reduction function (computing initial segments of the word problem from initial segments of ) of the previous paragraph can be written uniformly for all sets so it is total computable, and that we should set if and only if the th group element acts trivially on the set of patterns not containing elements of . It can be checked by a terminating computation whether the group element acts nontrivially on some pattern containing only one . Our query on should check that whenever the element acts nontrivially on a pattern with two s, then contains the distance between the s of the pattern. This corresponds to checking where lists these finitely many bad distances. ∎

Lemma 9.

Suppose has decidable word problem and is not abelian. Then many-one reduces to the word problem of .


For , let be any effective list of elements of satisfying (using the fact has decidable word problem), and consider where (using that is not abelian) and the action of is

We have if and only if . Namely, if then acts nontrivially on where is the unique configuration satisfying , while if then for all either or , and in either case a direct computation shows for all . ∎

Lemma 10.

For any torsion groups , we have .


Let . If with , then with . Let , so . If has order for all , then , and thus , and we have shown as claimed.

So suppose that , and consider the action on . The action of shifts around, and based on its contents multiplies from the right by elements of . For any fixed , implies that there exists with , such that for all . Since has order at most , we have , concluding the proof. ∎

Theorem 6.

Let be . For any f.g. torsion group there exists a recursively presented torsion group with , such that the word problem of conjunctively reduces to with exponential rate and many-one reduces to the word problem of .


By the previous lemmas, if for finitely-generated groups and which have decidable word problems, and is nonabelian with bounded torsion, then

  • is recursively presented (Lemma 8),

  • because and by the choice of generators (Lemma 6),

  • (Lemma 10),

  • conjunctively reduces to with exponential rate (Lemma 8),

  • many-one reduces to (Lemma 9).

In the previous theorem, the implicit constants for can be taken to be  (for the lower bound) and  (for the upper bound, by setting ). Of course, if has bounded torsion, so does .

Theorem 7.

Let be a total recursive function. Then there exists a recursively presented torsion group with bounded torsion, whose word problem is -impredictable.


Let be a -impredictable set (Theorem 5) and apply the previous theorem to obtain a recursively presented torsion group with bounded torsion, such that the word problem of conjunctively reduces to with exponential rate and many-one reduces to the word problem of . Then in particular the word problem of wtt-reduces to with exponential rate . By Lemma 4, the word problem of is -impredictable. In particular, it is -impredictable by Lemma 3. ∎

5 Application: Four heads are better than three

We first define group-walking automata and the subshifts they recognize. By we mean the projection to the th coordinate of a finite Cartesian product. For a finite set write for the subshift on a group clear from context containing those satisfying .

Definition 3.

Let be a finite alphabet. A -headed group-walking automaton on the full shift is a tuple , where are state sets not containing the symbol , and are finite clopen subsets of the product subshift , and is a cellular automaton satisfying and

for all , and .

For a -headed automaton as above, we denote by the subshift

We denote by the class of all subshifts for -headed automata . We also write .

This definition may seem cryptic on a first reading. Its details are unraveled in [15] (see [14] for a discussion of possible variants). Our interpretation of is that a -headed group-walking automaton can define , for a particular (in our opinion natural) way of defining subshifts by such automata.

Key points are that interpreting the th track of as giving the position or a head and its current state, is a local rule that tells how the heads move on configurations, and the assumptions imply that all heads are always present, are initialized in (roughly) the same position (described by ), have to join together to reject a configuration (described by ), and cannot communicate over distances (because is a cellular automaton).

The following observation is essentially Proposition 2 in [15], though here we “complement” the separating subshift, because the result of [15] cannot be used with recursively presented groups.

Lemma 11.

Let be a finitely-generated torsion group, let , and suppose that there exists a superexponential function such that the word problem of is -impredictable. Then

We sketch the proof from [15], for our complemented definitions.

Proof sketch.

Let be superexponential and recursive, let be such that the word problem of is -impredictable for , and let be the corresponding function, so

For each , let be the configuration where if and only if . Define , and let be the smallest subshift containing the configurations with , i.e. the forbidden patterns are (encodings of) the complement of the word problem of (together with the recursive set of patterns ensuring ). We clearly have .

The crucial observation in [15] was that for any fixed three-headed automaton , the function eventually bounds how far the -projections of the heads can be from each other during valid runs on a configuration , which means that we can construct a Turing machine that, given access to an oracle for the initial segment of the word problem (we add an extra since we linearize the word problem), we can simulate all runs of on the configurations (observe that there are essentially only different starting positions that need to be considered), halting if and only if one of them halts (i.e. the finite clopen set is entered and the configuration is forbidden).

Suppose for a contradiction that is a three-headed automaton that defines . Letting be the Turing machine described above which simulates , we have for all large enough that

but by the definition of there exist arbitrarily large such that

a contradiction.

On the other hand since it is intrinsically in the sense of [15], by a similar proof as in [15], observing that using an oracle for the word problem of we can easily forbid all with . ∎

Four heads are now seen to be better than three:

Proof of Theorem 3.

By Theorem 7, there exists a group which has bounded torsion and has -impredictable word problem. Such a group is -impredictable for some superexponential function , where , and thus the previous lemma implies . ∎

Remark 2.

For maximal “automaticity”,111Using the Grigorchuk group or another torsion automata group, the statement of our main result is somewhat amusing, in that it states that a group of finite-state automata acting on a subshift on an automata group admits subshifts definable by group-walking automata with four heads but not three. Of course, we do not claim that this is the only way to prove the result, in fact Conjecture 1 suggests the contrary. or to avoid using f.g. bounded torsion groups (whose infiniteness is rather difficult to verify for mortals), one can replace the use of bounded torsion groups by any torsion group with recursive torsion function. For example using the Grigorchuk group as in the construction of (then by [5, Theorem VIII.70]), and using Theorem 6 directly, the previous proof goes through with the same exponential tower. Automata groups in the sense of [4] cannot have bounded torsion by Zelmanov’s theorem [5, 16, 17].


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