A topological approach to non-uniform complexity

Journal Pre-proof A Topological Approach to Non-Uniform Complexity

Silke Czarnetzki, Andreas Krebs

PII:

S0890-5401(19)30060-4

DOI:

https://doi.org/10.1016/j.ic.2019.104443

Reference:

YINCO 104443

To appear in:

Information and Computation

Received date:

1 July 2016

Revised date:

17 May 2019

Accepted date:

4 June 2019

Please cite this article as: S. Czarnetzki, A. Krebs, A Topological Approach to Non-Uniform Complexity, Inf. Comput. (2019), 104443, doi: https://doi.org/10.1016/j.ic.2019.104443.

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A Topological Approach to Non-Uniform Complexity Silke Czarnetzki, Andreas Krebs Wilhelm-Schickard-Institut, Universit¨ at T¨ ubingen Sand 13, 72076 T¨ ubingen, Germany [email protected] [email protected]

Abstract We investigate the feasibility of a topological method for proving separations of non-uniform circuit classes. Thereto, we chose a rather simple class of circuits: non-uniform constant size circuit classes with gate types underlying certain restrictions. In particular, we consider gate types admitting for a description through regular and commutative varieties of languages. For instance, some of the most common gate types such as and-, or- and modulogates have such a representation. Given a variety of regular languages V describing the gate types, we proceed by giving an alternative characterisation of the class of languages recognised by constant size circuits with the respective gates. This alternative description mainly relies on the block product principle (or substitution principle), which we extend to work with non-regular languages. The extended version of the block product principle is then used as the main tool to derive ultrafilter equations for the languages recognised by non-uniform constant size circuits with gates described by V, depending on the profinite equations that define the variety of regular languages V. Moreover, we prove that this set of ultrafilter equations is both sound and complete for the respective class of languages. Finally, to show that the derived set of ultrafilter equations is indeed suitable for proving separations and to demonstrate the method, we use them to reprove that constant size CC0 and constant size AC0 are strictly contained in constant size ACC0 . Keywords: Duality, Topology, Algebra, Language theory, Circuit classes

Preprint submitted to Elsevier

Wednesday 21st August, 2019

1. Introduction In Boolean circuit complexity, deriving lower bounds on circuit size and depth has up to now shown to generally be difficult. While there have been results proving lower bounds for specific instances, we still lack methods that are applicable in general. This paper may be seen as an attempt to merge rather recently developed tools for non-regular languages to be used for the description of non-uniform circuit classes and to hopefully pave the way for more involved techniques that are capable of proving separations for larger classes. The main ingredients that we use come from algebra and topology. So far, specifically the understanding of the regular languages within circuit complexity classes was amplified through the availability of the algebraic approach [6, 4, 3, 16, 29]. For a broader overview we refer to the book by Straubing [27]. In particular, the strong connection between circuit classes and classes of logical formulae [13] opened up further methods to investigate circuit classes algebraically: Logical formulae can be decomposed into quantifier and subformula. This decomposition is mirrored on the side of algebra through the block product or substitution principle, for a survey of which we refer to [28]. Through these connections, also circuit classes and algebraic decomposition methods are intrinsically linked. The second ingredient, which is topology, was also previously successfully employed on the regular languages using profinite methods [18]. Here, in particular the availability of profinite equations for varieties of regular languages through Reiterman’s theorem [22] proved useful. For instance, the equation xω = xω+1 , which reads that the idempotent xω generated by some element x of a finite monoid is equal to xω · x characterises all finite aperiodic monoids. Equivalently, it characterises by Eilenberg’s theorem [9] and [24] the star-free languages. As a result of [5, 26], the regular languages in AC0 are defined by the profinite equation (xω−1 y)ω+1 = (xω−1 y)ω where x and y are words of the same length. For an overview see [19]. While the existence of equations is always guaranteed for varieties of regular languages, there is no general method to obtain them. A large step towards such a constructive method was achieved by Almeida and Weil [2]. They were able to derive profinite equations for the wreath- and block product of two varieties of finite monoids in dependence of the equations defining them. This allows to inductively build equations for larger classes from smaller ones, by determining the equations for the separate parts and merging them through the block product. Concretely, the idea of obtaining 2

equations through the block product was also used in [15]. The previously described discoveries are a short summary of how the algebraic and topological approaches fit together on the regular languages. Rather recently, extensions of both approaches have emerged to be employed on non-regular languages classes: For instance, in the case of the algebraic approach, the circuit complexity class TC0 was characterised through an extended block product of (infinite) monoids with additional structure in [14] and further connections of said extended block product and circuit classes were uncovered in [7]. Also the topological approach has been extended to be applicable to non-regular languages. It was a long-known result that the free profinite monoid is the Stone space of the regular languages [21, 1], but Gehrke, Griegorieff and Pin [10] managed to put this fact into a larger context and combine it with equations: Each Boolean algebra of languages (over a fixed alphabet) is defined by a set of so-called ultrafilter equations over the Stone space of all languages (over the same alphabet). Yet a constructive approach such as the one provided by Almeida and Weil to derive equations for non-regular language classes is missing. Up to now, there have even been only very few classes for which concrete ultrafilter equations could be obtained [11, 8]. Hence, this paper pursues two goals: Its subordinate goal is to calculate equations for constant size circuit classes and show that these equations are capable of proving separations of the classes. Here, one may rightfully argue that these separations could have been achieved by much simpler means. Its main goal, however, is to gain better insights into the principles involved in systematically constructing ultrafilter equations with the help of decomposition methods and proving soundness and completeness. We hope that these insights will be helpful to eventually obtain a general result for non-regular language classes similar to [2]. Organisation of the paper Since the paper makes extensive use of topological methods that may not be considered common knowledge in the community, we have tried to cover a large amount of the necessary theory in the preliminaries, without being too extensive. The reader familiar with these subjects might thus want to skip some, if not most of the subsections. References to literature with more detailed information are included. The rest of the paper is structured as follows: In section 5, we recall the definition of a circuit family and define what it means for the gate types to 3

be defined by a variety of (regular and commutative) languages V. This is the key ingredient to be able to translate gate types into (profinite) equations through Reiterman’s theorem later. After examining the structure of constant size circuits with gates defined by V, we turn to the principles of decomposition. Here, the circuits intuitively speaking are decomposed into gate types and wiring of the gates. To imitate the wiring, we define the notion of Ntransduction. Concluding the section, we prove that each language recognised by a constant size circuit family with gate types defined by V is the preimage of a language in V under some N-transduction. In reference to the connections to the block product, we denote this class by V  Parb . The following chapter is concerned with equations and separations: Since V is a variety of regular and commutative languages, it is defined by a set of profinite equations. Theorem 4 states the ultrafilter equations for V  Parb in dependence of the profinite equations defining V. For some concrete gate types, we show that they can be employed to separate constant size AC0 and CC0 from constant size ACC0 . The proofs of soundness and completeness of the equations are moved to separate sections 7 and 8 due to their length. Discussion, conclusion and hints at further research can be found in Section 9. 2. Preliminaries In the following, let A be a finite alphabet and A∗ be the free monoid over A. For a word w ∈ A∗ , we denote by wi the ith letter of w and by |w| the length of w. If X is a set, then P(X) denotes its powerset. 2.1. Varieties The term variety initially stems from the denotation of solutions to algebraic equations in the field of algebraic geometry. We use the notions of varieties of regular languages and varieties of finite monoids. There is indeed a connection between equational theories and varieties of finite monoids, as stated by Reiterman [22]. Varieties of languages are capable of abstracting the concept of Boolean algebras closed under quotients, which form a base for the description by equations formed by Gehrke, Grigorieff and Pin [10]. Varieties of monoids focus on algebraic properties. Both notions are in one to one correspondence [9]. For more detailed information, we refer to the book of Pin [17]. 4

A Boolean algebra of languages is a set B of languages over A∗ that is closed under finite intersections, finite unions and complement. It is closed under quotients, if for each L ∈ B and a ∈ A, the languages a−1 L := {w ∈ A∗ | aw ∈ L} and La−1 := {w ∈ A∗ | wa ∈ L} belong to B. A variety of regular languages is a class of regular languages V such that 1. for each alphabet A, VA is a Boolean algebra closed under quotients over A∗ . 2. for each monoid morphism ϕ : A∗ → B ∗ , the condition L ∈ VB implies ϕ−1 (L) ∈ VA . A variety of finite monoids is a class of finite monoids V closed under taking submonoids, quotients and finite direct products. That is 1. if M ∈ V and N is a submonoid of M , then N ∈ V 2. if M ∈ V and N is a quotient of M , then N ∈ V  3. if (Mi )i∈I is a finite family of monoids in V, then i∈I Mi ∈ V Eilenberg’s variety theorem [9] proves a one to one correspondence between varieties of finite monoids and varieties of regular languages. That is to say that there are correspondences V → V and V → V that define mutually inverse bijective correspondences between varieties of finite monoids and varieties of regular languages. Roughly speaking, the correspondence V → V is given by the variety of all languages recognisable by monoids of V and V → V is given by the class generated by all syntactic monoids of languages of V. 2.2. Topology and the free profinite monoid Topological spaces play a key role in the description of classes of languages through equations. All basic topological notions needed will be covered in this subsection, as well as one topological space of particular importance, namely the free profinite monoid together with its classical definition from computer science. We will point out that this is just a specific instance of a Stone space, as already uncovered by [1] and [21]. These spaces are covered in Subsection 3 and form the basis for the description of non-regular language classes through equations. Let Ω be a set. A topology T on Ω is a set of subsets of Ω with the following properties 1. ∅, Ω ∈ T 5

 2. if Ai ∈ T for all i in some index set I, then i∈I Ai ∈ T (T is closed under arbitrary unions)  3. if Ai ∈ T for i = 1, . . . , n and n ∈ N, then ni=1 Ai ∈ T (T is closed under finite intersections) A set Ω together with a topology T is called a topological space and denoted by (Ω, T ). The elements of T are called open sets, whereas the complements of open sets are called closed sets. Sets that are both open and closed are called clopen. Furthermore, the closure of a set A, denoted by A is the smallest closed set that contains A. A topological space in which every open set is a union of clopen sets is called zero-dimensional. This also implies that its topology is uniquely determined by its clopens. One property that often helps greatly when examining topological spaces is that of compactness. It allows to reduce to finitely many objects, when in need to prove a property for the whole space and is thus often referred to as “the next best thing to finiteness”. We say a topological space (Ω, T ) is compact, if every open cover of Ω has a finite subcover. Formally that means that every arbitrary family of open subsets (Ui )i∈I for some index set I with  Ui Ω= i∈I

has a subfamily (Uj )j∈J , where J ⊆ I is finite, that satisfies  Uj . Ω= j∈J

One property that is also useful when dealing with topological spaces is the ability to separate points. We say a topological space (Ω, T ) is Hausdorff, if for each two points x, y ∈ Ω there exist two disjoint open sets Ox , Oy ∈ T such that x ∈ Ox and y ∈ Oy . Note that a topological space is not necessarily Hausdorff. The trivial topology for instance, which consists only of Ω and ∅, is not. One instance of topological Hausdorff spaces are metric spaces. A metric on a space Ω is a function d : Ω × Ω → [0, ∞) satisfying for all x, y, z ∈ Ω 1. d(x, y) = 0 ⇔ x = y

(d is positive definite) 6

2. d(x, y) = d(y, x) 3. d(x, z) ≤ d(x, y) + d(y, z)

(d is symmetric) (d satisfies the triangle inequality)

The tuple (Ω, d) then is a metric space. If the metric is clear from the definition, we will omit to mention d and say that Ω is a metric space. A metric can be seen as a function that assigns a distance to two points and induces a topology on a space in the following way. The topology induced by a set E ⊆ P(Ω) is the smallest topology that contains E. Let d be a metric, x ∈ Ω and  > 0. The -ball centred at x is defined by B (x) = {y ∈ Ω | d(x, y) < } and the set {B (x) | x ∈ Ω,  > 0} induces a topology on Ω. One can verify that this definition makes any metric space Hausdorff. It is often useful to define a topological space just by its base, rather than giving the whole topology. This is due to that fact that many properties need only be proven for a base in order to hold for the whole topology. Another crucial notion is that of convergence, which for metric spaces intuitively translates to getting arbitrarily close to some point of the space. We say a sequence (xn )n∈N of points in Ω converges to a point x ∈ Ω with respect to some topology, if for each open set O that contains x, there exists an N ∈ N such that xn ∈ O for all n ≥ N . On a set that is equipped with the trivial topology, every sequence converges to every point. If the space is compact, then every sequence has a converging subsequence. If the space is Hausdorff, then each sequence converges to at most one point. Note that even though the elements of a sequence might get arbitrarily close with respect to some metric, there need not exist a convergence point. One of the standard examples being Q with the usual euclidean metric. The sequence (1 − n1 )n does not converge in Q, but it converges in R to e. The following two definitions formalise the properties described in the example. A sequence (xn )n∈N is called a Cauchy sequence with respect to some metric d, if for each  > 0 there exists an N ∈ N such that for all n, m ≥ N we have d(xn , xm ) < . Every sequence, that converges to some point in a metric space is a Cauchy sequence, but not every Cauchy sequence needs necessarily converge. We say a metric space is complete, if every Cauchy sequence converges. Each metric space has an associated complete space, which can be constructed in the following way: Define the distance of two Cauchy sequences  y) = limn→∞ d(xn , yn ). Let [Ω] x = (xn )n∈N and y = (yn )n∈N by setting d(x, 7

denote the space of all Cauchy sequences in Ω. Then d is not a metric on [Ω], since d(x, y) = 0 does not imply x = y, but it defines a metric on the  y) = 0. space [Ω]/∼ where x ∼ y if and only if d(x,  is a complete metric space, also called the compleThe space ([Ω]/∼ , d)  tion of Ω, often denoted by Ω. For instance R can be constructed as the completion of Q with the euclidean distance as metric.  are equivalence classes of Cauchy sequences. FurtherThe elements of Ω  by mapping an element x to the class of more, Ω can be embedded into Ω Cauchy sequences, that are eventually constant and equal to x. By abuse of notation, but for the sake of readability, we will often identify and write  instead of explicitly mentioning the embedding. The same holds for x ∈ Ω, subsets of Ω. The completion has Ω as a dense subspace, meaning that every  is the limit of a sequence in Ω. In a more apprehensive way, element of Ω we can think of the completion as the set together with all its convergence points. Together with these definitions, we can define the free profinite monoid over an alphabet A. We say that a finite monoid M separates two words x and y, if there is a morphism ϕ : A∗ → M , with ϕ(x) = ϕ(y). The function d : A∗ × A∗ → [0, ∞) defined by d(x, y) = max{2−|M | | M separates x and y} then is a metric on A∗ and thus induces a topology. The topology itself is not of great interest, since it is discrete, i.e. every singleton is open. The ∗ is the completion of A∗ with respect to d. free profinite monoid A ∗ by The free monoid A∗ , as stated before, can be embedded into A mapping a word w to the class of Cauchy sequences, that are eventually constant and equal to w. ∗ , that is not an element of A∗ is the limit An instance of an element of A n! of the sequence a , denoted by aω . The free profinite monoid entails useful properties. It is a compact and Hausdorff space and it holds information about the regular languages, as a ∗ is clopen [1]. language L ⊆ A∗ is regular if and only if its closure L ⊆ A The notion of continuity formalises the property of preserving limits under functions. We say that a function g : X → Y between two topological spaces is continuous if for every open set U of Y , g −1 (U ) is open in X. This translates to the sentence that a function is continuous if preimages of open 8

sets are again open. Since the preimage is closed under complement an equivalent definition is that preimages of closed sets are closed. If X and Y are countable, this is equivalent to saying that for each sequence (xn )n∈N with limn→∞ xn = x ∈ X, the property limn→∞ g(xn ) = g(x) holds. The free profinite monoid can also be defined in more categorical terms by its unique property. For each morphism ϕ : A∗ → M into a finite monoid M equipped with the discrete topology there exists a unique continuous ∗ → M , with ϕ(a) extension ϕˆ : A ˆ = ϕ(a) for each a ∈ A. 3. Stone duality Stone duality forms the foundation for the description of our circuit classes. While this chapter does not depend on the previous ones, it is necessary to introduce the notions from Stone duality and equations, to be able to draw the connection back to circuit classes in the following chapters. In general, a Stone space is a space which is compact, Hausdorff and zero-dimensional, which means that the topology has a basis of clopen sets. Both Pippenger [21] and Almeida [1] observed that the free profinite monoid is an instance of a Stone space. These spaces are in direct correspondence to Boolean algebras [25], which makes them particularly interesting for the investigation of circuit complexity classes. For the free profinite monoid, the corresponding Boolean algebra is that of all regular languages. We give a constructive reproof of the observation of Almeida and Pippenger in this subsection. This is done, since the constructions used in there will play a key role in later chapters. Concretely, a Stone space can be characterised through the following objects. Let B be a Boolean algebra over a set X. An ultrafilter of B is a nonempty subset γ of B that satisfies 1. 2. 3. 4.

∅∈ / γ, if L ∈ γ and K ⊇ L, then K ∈ γ, (γ is closed under extension) if L, K ∈ γ then K ∩ L ∈ γ, (γ is closed under finite intersections) for each L ∈ B, either L ∈ γ or Lc ∈ γ (ultrafilter condition)

Throughout the document, we will often also be dealing with (nonempty) subsets of B that are not ultrafilters, but satisfy conditions 1 to 3. Such a set is called proper filter. Since ultrafilters are maximal filters with respect to inclusion, by Zorn’s Lemma, any proper filter can be extended to an (non-unique) ultrafilter. A property, which will later be of use. 9

In topology, ultrafilters are often used as a generalisation of sequences to uncountable spaces. Terms, such as convergence, can be defined likewise, such that they agree with the definitions on sequences for countable spaces. This correspondence is used and stated more concretely in the proof of theorem 1. The ultrafilter condition is often referred to as the property that ultrafilters “have to choose”. For any partition of elements from the Boolean algebra they have to contain exactly one partition element. Intuitively speaking, this makes them similar to decision trees and describes atomic properties of the Boolean algebra. Lemma 1. Let B be a Boolean algebra and μ an ultrafilter of B. Further, let L = {L0 , . . . , Ln−1 } be a finite partition of A∗ of languages from B. Then there exists an i ∈ {0, . . . , n − 1} such that Li ∈ μ. Proof. Since L is a partition of A∗ , we have that n−1 

Li ∈ μ.

i=0

/ μ holds. Since μ Suppose that for each i ∈ {0, . . . , n − 1} the condition Li ∈ is an ultrafilter, this implies that for all i ∈ {0, . . . , n − 1}, Lci ∈ μ. Observe that n−1  Lj = Lci i=0 i=j

and thus Lj ∈ μ, which is a contradiction to the assumption. Each Boolean algebra has an associated zero-dimensional compact Hausdorff space S(B), called its Stone space. The elements of S(B) are the ul = {γ ∈ S(B) | trafilters of B, where the topology is induced by the sets L  L ∈ γ} for L ∈ B. In particular, the sets L are the clopen sets of S(B). This gives rise to the duality between Boolean algebras and Stone spaces in the sense that each Boolean algebra has an associated Stone space and each Stone space in turn defines a Boolean algebra via its set of clopens. Therefore, we also refer to the Stone space of S(B) as the dual space of B. Moreover, the dual space contains the underlying set X as a dense subset. Its embedding is given by x → {L ∈ B | x ∈ L}. 10

That this is well-defined follows from the fact that the set on the right satisfies conditions 1. − 4. and thus is an ultrafilter of B. For each L ∈ B  is the closure of the language L with respect to this inclusion. the set L Let Reg be the variety of all regular languages. As mentioned, that the Stone space of the regular languages S(RegA ) is no other than the free ∗ is no new result. This correspondence justifies that profinite monoid A any profinite word may be seen as an ultrafilter of Reg. Since we need to transform concrete profinite words into ultrafilters in later sections, we give the precise construction in the following reproof and will use it in later sections, to obtain the desired separations of circuit classes. For instance, using the construction in the proof, the profinite word aω will correspond to the following set aω = {L ∈ Reg | ∃n0 ∈ N ∀n ≥ n0 : an! ∈ L}. We say two spaces are homeomorphic if there exists a bijection f such that both f and the inverse function of f are continuous with respect to the topology the spaces are equipped with. Informally speaking, this says the spaces are equipped with the same topology. ∗ is homeomorphic to S(RegA ). Theorem 1 ([21, 1]). The space A Proof. For a Cauchy sequence u with respect to the introduced metric on ∗ , define the set A Fu = {L ∈ RegA | ∃n0 ∈ N ∀n ≥ n0 : un ∈ L}. To see that Fu defines an ultrafilter of RegA observe that Fu is a non-empty set closed under intersections and upsets. What is left to show is that for any regular language L, the set Fu contains either L or its complement. For that let L be a regular language and h : A∗ → M a morphism into a finite monoid recognising L, such that L = h−1 (K) for some K ⊆ M . Since u is a Cauchy sequence, there exists an N ∈ N such that for all n, m ≥ N we have d(un , um ) < 2−|M | and thus h(un ) = h(um ). Since either h(uN ) ∈ K or h(uN ) ∈ K c and Lc = h−1 (K c ), this implies that either un ∈ L or un ∈ Lc for all n ≥ N . Thus, Fu is an ultrafilter of RegA . Furthermore, let u and v be two equivalent Cauchy sequences. Then, by definition u ∼ v if and only if for each morphism h : A∗ → M into a finite monoid, there exists an N ∈ N such that for all n, m ≥ N we have d(un , vm ) < 2−|M | . By a similar argument to the one above, this is implies Fu = F v . 11

Denote by [u] the equivalence class of u. Then the mapping ∗ → S(RegA ) φ:A [u] → Fu is well-defined. We claim that this mapping is a bijection. To show, that it is surjective, let γ ∈ S(RegA ). The set {L | L ∈ γ} ∗ and, since γ is an ultrafilter, any two is a collection of closed subsets of A ∗ is a sets in the collection have non-empty intersection. Furthermore, A compact space, and thus  F = L = ∅. L∈γ

Recall that this is due to the fact that in a compact space,  any collection of closedsubsets F satisfying for each finite G ⊆ F that G∈G G = ∅ also satisfies G∈F G = ∅. Hence there exists a sequence u such that [u] ∈ F . Let L ∈ γ. By definition of F , we obtain that [u] ∈ L and thus there exists an n0 ∈ N such that for all n ≥ n0 we have un ∈ L. But this is exactly saying that L ∈ Fu . Hence γ ⊆ Fu and since ultrafilters are maximal γ = Fu . Thus the mapping is surjective. Furthermore, it is injective. Suppose that there exist two Cauchy sequences u and v with [u] = [v]. Then there exists a L ∈ RegA such that [u] ∈ L but [v] ∈ / L. Since γ is an ultrafilter, either L or Lc is an element of γ which implies that either [u] ∈ F or [v] ∈ F . Hence F is a singleton. Thus, φ is a bijective mapping and there is a well-defined map ∗ ψ : S(RegA ) → A  γ → L L∈γ

By construction, these maps are mutually inverse and  = ψ({γ | L ∈ γ}) = L ψ(L) and similarly

 φ(L) = L.

∗ and S(RegA ) are known Thus, clopen sets get mapped to clopens. Both A to be compact, Hausdorff and zero-dimensional. These conditions are sufficient for the topology being uniquely determined by the clopen sets. Thus, the two spaces are homeomorphic. 12

Another Stone space that is of special interest, is the dual of the powerset ˇ of a set X. It is called the Stone-Cech compactification of X and is denoted by βX. An important property of βX is that any map f : X → K into a compact Hausdorff space K, can be extended uniquely to a continuous map βf : βX → K. Furthermore any map f : X → Y has a unique continuous extension, also denoted by βf : βX → βY . It is defined by the equivalence L ∈ βf (γ) ⇔ f −1 (L) ∈ γ for all L ∈ P(Y ) and γ ∈ βX. Another useful property that originates from Stone duality is that if B is a Boolean algebra and C is a subalgebra of B, then the associated Stone space S(C) is a quotient of S(B). If there is an embedding of C in B then there is a surjective map between the Stone spaces. embedding

C −−−−−−→ B quotient

S(C) ←−−−− S(B) Since any Boolean algebra B is a subalgebra of the powerset P(X), we know that its Stone space S(B) will be a quotient of β(X). Note that an ultrafilter of a subalgebra C of B, when seen as a collection of sets, is not an ultrafilter of B, since the fourth property is violated. However, it still satisfies conditions one to three and is hence a proper filter of B. As a special case of this, any profinite word may be seen as a filter of βA∗ . Again, for example, aω may be seen as an ultrafilter of S(RegA ) and thus is a filter of βA∗ . Hence, any point of S(C) defines a filter on B. An even weaker property holds. It will often be necessary to show that there exist two ultrafilters having certain properties. For these properties it will be sufficient to know that the filter contains a specific collection of subsets. A collection of subsets is said to have the finite intersection property, if any two sets have non-empty intersection. We call such a set a filter base. By the same argument, any filter base can be extended to an ultrafilter. We will use this method to construct ultrafilters from filter bases to obtain ultrafilters with certain properties. The set of all filters of B, denoted by F(B) is isomorphic to the space of all closed subsets of S(B), the Vietoris space of S(B), which is denoted by V(S(B)). The Vietoris space itself is equipped with a topology such that it is compact and Hausdorff. 13

Since its points are closed sets, the Vietoris space provides information when an ultrafilter is not precisely determined by some set, but can be extended to various ultrafilters. 4. Equations One of the main motivations for the description of circuit classes through equations is that equations provide a general approach to show separations. The intuition on how equations enable us to separate and why topology plays a key role can be described by a few examples concerning monoids and regular languages. 4.1. A reason for equations For a fixed alphabet A, let LA∗ (Com) be the class of (commutative) languages recognised by commutative finite monoids, which means every language of this class is a preimage of some morphism from A∗ into a commutative, finite monoid. How do we show that a regular language does not belong to that class? The idea behind that is quite simple. For example let a, b ∈ A. Since any morphism into a commutative monoid will map ab to the same element as ba, for all L ∈ LA∗ (Com), the equivalence ab ∈ L ⇔ ba ∈ L holds. Thus, any language that does contain ab but not ba is not an element of L ∈ LA∗ (Com). Knowing that this equivalence holds for all languages recognised by commutative monoids, say we identify the two words and treat them as one and the same, denoting this by [ab ↔ ba]. Now, a language L that contains ab but not ba will, after identifying the two words, have a word in common with its complement, as illustrated in Figure 1. Furthermore, any language that has non-empty intersection with its complement after identifying ab and ba, does not belong to LA∗ (Com). We can generalise this by saying that we now not only identify ab and ba, but any two words u, v ∈ A∗ that satisfy for all L ∈ LA∗ (Com) the equivalence u ∈ L ⇔ v ∈ L. Verifying that this is an equivalence relation between words, we obtain a quotient of A∗ after identifying the words. Then, a language L does not belong to LA∗ (Com) if L and its complement have non-empty intersection after identifying the words as described. 14

L ba

ab

L

Lc Lc

[ab ↔ ba]

Figure 1: When identifying ab and ba then L and its complement have non-empty intersection.

So the equations, as for instance [ab ↔ ba], hold information about which points to check if we want to prove non-membership to LA∗ (Com). However, for more complex classes this simple attempt of identifying explicit words will fail. Already when considering languages recognised by aperiodic monoids - monoids such that for each element x there exists an n ∈ N with xn = xn+1 - we cannot identify two explicit words. If we look at the unary alphabet {a}, then for each language L recognised by some aperiodic monoid, there will be an n ∈ N such that an ∈ L ⇔ an+1 ∈ L. However, this n might get arbitrarily large, depending on the language. For example, all languages LN = {an | n ≥ N } for N ∈ N are languages recognised by aperiodic monoids. Thus we will not find two words u and v, such that u ∈ L ⇔ v ∈ L for all languages L recognised by aperiodic monoids. However, we know there always exists such an n. So in a sense, we could try to say that for some arbitrarily large ω, the equivalence aω ∈ L ⇔ aω+1 ∈ L holds for all languages recognised by aperiodic monoids. This still requires a proper definition, as the two objects are no longer words and thus cannot be elements of L. However, if we interpret aω as a limit of words in a space where limit points exist, we can define L to be the language L together with all its limit points and write aω ∈ L ⇔ aω+1 ∈ L. Then we can identify aω and aω+1 and the separation method will be applicable again in this new space, as illustrated in Figure 2. Again, this provides a quotient of the space containing the limit points and the equations tell us, which points are candidates for being in the intersection. The free profinite monoid in the regular setting and general Stone spaces 15

L

L aω+1

Lc

aω

L

Lc Lc

[aω ↔ aω+1 ]

Figure 2: After adding limit points and identifying them we can apply the non-empty intersection method again, where L denotes the language L together with the new points and Lc respectively.

in the non-regular setting allow us to formalise the procedure of adding limit points. 4.2. Equations on Stone spaces Equations for Stone spaces are the main focus of our description of circuit classes. The intuition for equations on Stone spaces stems from the relation between Boolean subalgebras and quotients of Stone spaces. If C is a subalgebra of B, then S(C) is a quotient of S(B), hence one space is a continuous projection of the other. Basically, giving equations is describing the kernel of said projection. For a Boolean algebra B ⊆ P(A∗ ), we denote the canonical projection from β(A∗ ) onto S(B) by πB . Definition 1 (Equation). An ultrafilter equation is a tuple (μ, ν) ∈ βA∗ × βA∗ . Let B be a Boolean algebra. We say that B satisfies the equation (μ, ν) if πB (μ) = πB (ν). With respect to some Boolean algebra B we say that [μ ↔ ν] holds. Lemma 2. Let B be a subalgebra of P(A∗ ). For μ, ν ∈ βA∗ we have πB (μ) = πB (ν) if and only if for all L ∈ B the equivalence L ∈ μ ⇔ L ∈ ν holds. Proof. The projection πB is given by πB (μ) = {L ∈ μ | L ∈ B} and thus the equivalence holds. The following Theorem is the result of Gehrke, Grigorieff and Pin, which stated the connection between Boolean algebras of languages and equations on βA∗ . 16

Theorem 2 ([10]). Every Boolean algebra of languages is defined by a set of ultrafilter equations of the form [μ ↔ ν], where μ and ν are ultrafilters on the set of words. We say a set of equations is sound for a Boolean algebra B, if all L in B satisfy all equations and complete, if a language in A∗ satisfying all equations is in B. In the setting of regular languages, such a connection has been known since Reiterman [22] formulated and proved his theorem that each variety of regular languages is defined by a set of profinite equations. Combining the results of Reiterman and Gehrke, Grigorieff and Pin states that each Boolean algebra of regular languages over A∗ is defined by a set of ultrafilter equations, where the ultrafilters are elements of the Stone space of the regular languages. Profinite equations, or ultrafilter equations on S(RegA ) admit for a useful equivalent definition, since the free profinite monoid is, as the name indicates, a monoid. This allows for a characterisation of equations through morphisms. Let V be a variety of finite monoids and VA the corresponding Boolean algebra. Then VA satisfies the profinite equation u = v if and only if for each morphism ϕ : A∗ → M into a monoid M of V, the equality ϕ(u)  = ϕ(v)  holds. 5. Constant Size Circuits Families and the Block Product 5.1. Circuits over words Circuits are used to measure the computational complexity of Boolean functions. We recall briefly the basic definitions and then turn to the description of languages recognised by specific constant size circuit families. Definition 2 (Boolean function). A i-ary Boolean function f is a map from {0, 1}i to {0, 1}. We say a collection of Boolean functions (fi )i∈N is a family of Boolean functions, if for each i ∈ N, fi is an i-ary Boolean function. We say a Boolean function f is symmetric, if for each permutation σ of {0, . . . , i − 1}, the equality f (x0 , . . . , xi−1 ) = f (xσ(0) , . . . , xσ(i−1) ) holds. Respectively, a family of Boolean functions is called symmetric, if each fi is. In the following definition, it is necessary to allow both Boolean functions and families of Boolean functions, in order to obtain a suitable notion for the finiteness of a base. 17

Definition 3 (Base). A base is a set containing both Boolean functions and families of Boolean functions. We say that a base is symmetric, if all of its members are. For the definition of circuit, directed graphs are a central component. Instead of speaking of the in- and out-degree of a node, in circuit complexity, these notions are often referred to as fan-in and fan-out. In the following, we only consider circuits which are defined over symmetric bases. This is not a limiting restriction, since most interesting circuit classes consider only symmetric bases. Definition 4 (Circuit). Let A be a finite alphabet, and B be a symmetric base. A circuit over the base B for words in An is an acyclic directed graph with a unique node with fan-out 0. Nodes with fan-in 0 are called inputs and are labelled by 1,0 or xi = a, where i ∈ {0, . . . , n − 1} and a ∈ A. All other nodes are called gates. Among these, the unique node with fan-out 0 is called the output gate. Each gate with fan-in i is labelled by an i-ary Boolean function or by a family of Boolean functions of the base B. We say that the size of a circuit is the number of its gates. Recall that inputs were not counted as gates and thus do not affect the size of the circuit. This will later become crucial for circuit families of constant size, in order to be able to accept words of arbitrary length. The depth of a circuit is the maximal length of a path from an input to its output gate. The evaluation of a word w ∈ An for a circuit is defined inductively, as usual. The value of the input 0 is always 0, and the value of 1 is 1. The value of the input labelled by xi = a is 1 if wi = a, and 0 otherwise. The value of a gate with i predecessors is its label function f or respectively fi , if the label f is a family of Boolean functions, applied to the values of the predecessors. This is well defined as we consider only bases containing symmetric functions or families, for which the order of the input is insignificant. We say a circuit accepts a word w ∈ A∗ , if its output gate evaluates to 1 for the word w. For a small example of a circuit that accepts all words of length three containing more 1s than 0s, see Figure 3. Please note that, as opposed to other common definitions, our definition of circuit family over some base does not require the base to be finite, but we limit each circuit family to use only a finite number of elements from the base. This allows to define circuit classes like ACC0 directly over an infinite base instead of defining ACC0 as a union of ACC0 [p], each defined over a finite base. 18

Definition 5 (Circuit family). We say (Ci )i∈N is a family of circuits over a symmetric base B for words in A∗ if there is a finite base F ⊆ B such that for every i ∈ N the circuit Ci is a circuit over the base F for words in Ai . A circuit family recognises a language L ⊆ A∗ , if for each w ∈ L, the circuit C|w| accepts w. x1 x2 x3 The size (resp. depth) of a circuit family (Ci )i∈N is a map that maps i to the size (resp. depth) of ∧ ∨ Ci . ¬ A circuit class is a collection of families of circuits over a common base together with additional ∧ restrictions on the size and depth of the circuit families. ∨ For example, the circuit class known as NC0 consists of polynomial size, constant depth circuit y1 families over the base {∧2 , ∨2 , ¬1 }. This base conFigure 3: Example of sists only of Boolean functions and hence the fan-in a circuit with inputs of gates over such a base is bounded. This imx1 , x2 , x3 , and,or and plies that NC0 is equal to the class of constant negation gates and size circuits over that base. As we are interested output y1 . in constant size circuits that should access all inputs we consider bases that contain families of Boolean functions. The circuit class AC0 consists of polynomial size, constant depth circuit families over the base {∧ = (∧i )i∈N , ∨ = (∨i )i∈N , ¬1 }, where the base contains both Boolean functions and families of Boolean functions. The class CC0 consists of polynomial size, constant depth circuit families over the base {modp = (modp )i∈N | p ∈ N}, where a function in the family modp evaluates to 1 if and only if the number of 1s in the input is divisible by p. And lastly, the circuit class ACC0 consists of polynomial size, constant depth circuit families over the union of the bases of AC0 and CC0 . 5.2. Circuits over Bases Defined by Varieties Our primary goal is to describe the classes of languages accepted by constant size circuit families (that meet our restriction of symmetric gates) through equations. Since equations are a concept that works entirely on Boolean algebras of languages and not directly on circuits, we are now going to give an alternative characterisation of those classes of languages that is no longer dependent on circuits. Instead, it relies entirely on varieties of regular 19

languages. This characterisation will later pave the way to determine the equations for said classes. We begin by drawing a connection between languages and Boolean functions. Definition 6 (Family of Boolean functions defined by a language). A language L ⊆ {0, 1}∗ in a natural way defines a family of Boolean functions, denoted by f L = (fiL )i∈N where fiL (x0 , . . . , xi−1 ) = 1 iff x0 . . . xi−1 ∈ L. This definition allows us to describe Boolean functions through languages and subsequently allows us to describe the base of a circuit class, which consists of possibly infinitely many Boolean functions, through a variety of regular languages in the following way. Definition 7 (Bases defined by a variety of regular languages). Given a variety V of commutative regular languages, V{0,1} is a collection of languages in {0, 1}∗ and each of these languages defines a family of symmetric Boolean functions. We call the set B(V) = {f L | L ∈ V{0,1} } the base defined by V. Recall that a base was permitted to consist of an infinite number of (families of) Boolean functions, but a circuit family over an (infinite) base is only allowed to use a finite subset of the elements in the base. For our purpose, we consider bases generated by varieties of regular languages, where the languages are regular and commutative. The commutativity ensures that we are dealing with symmetric bases only, as required earlier. This is not much of a limitation as many gate types correspond to commutative regular languages, which is illustrated in the table on the left of Figure 4, whereas the right table displays that also typical circuit classes use bases consisting of such gate types. Gate Type Language ∧ ∨ modp

1∗ {0, 1}∗ 1{0, 1}∗ 0∗ ((10∗ )p )∗

Circuit class

Base

AC0 CC0 ACC0

{∧, ∨} {modp | p ∈ N} {∧, ∨, modp | p ∈ N}

Figure 4: On the left: Typical gate types and the languages they are defined by. On the right: Typical circuit classes

We now show that each constant size circuit over a base defined by a variety of regular commutative languages V admits for a circuit accepting 20

the same languages. This circuit can be written as one layer of gates from V accessing the inputs with an NC0 circuit below, as depicted in Figure 5.2. Note that negation gates can be neglected in the NC0 part, since they can be moved to the top of the circuit by de Morgan’s laws and a variety of regular languages contains always both a language and its complement. x1

x2

f L1

f L2

x3

...

...

xn

f Lk

NC0

Figure 5: Schematic drawing of a simplified circuit with gates defined by languages L1 , . . . , Lk from a variety of regular languages V.

Lemma 3. Let V be a variety of regular commutative languages. The languages recognised by constant size circuits over the base B(V) are recognised by circuits of the following form: The top layer of gates, accessing the inputs, consists of gates labelled by B(V). All gates from other layers are labelled either by ∨2 or ∧2 . Proof. Let (Ci )i∈N be a constant size family of circuits with gates from B(V). Then there exists a k ∈ N such that for each i ∈ N the size of the circuit Ci is at most k. Moreover, by definition there is a l ∈ N and finitely many (families of) Boolean functions f1 , . . . , fl such that for each i ∈ N, the gates of Ci are labelled by f1 , . . . , fl ∈ B(V). As this base is defined by V, which is a variety of regular and commutative languages, for each j = 1, . . . , l there exists a regular language Lj ∈ V{0,1} such that fj = f Lj . Since each of those languages is regular, the set of left- and right quotients {s−1 Lj t−1 | s, t ∈ {0, 1}∗ } is finite for each j = 1, . . . , l. Hence also the set F = {s−1 Lj t−1 | j = 1, . . . , n and s, t ∈ {0, 1}∗ } is finite and we let m ∈ N be the size of it. 21

We are now going to prove that for each i ∈ N there exists a circuit Di of the desired form and with labels defined by F accepting precisely the same languages as Ci and whose size is only dependent on k and m. Recall that since both k and m are independent of i, this will ultimately make the family (Di )i∈N constant size. For the sake of simplicity, we omit the i, writing C for Ci and D for Di (where i is fixed but arbitrary). We show the above claim by induction over the depth of C. If the depth of C is one, then C consists of a single gate labelled by f Lj for some j ∈ {1, . . . , l}, which is directly connected to the inputs and no other gates. Hence C is already of the desired form and equal to D. Assume now that the depth of C is is larger than one. Then the predecessors of the output gate g of C are either inputs or gates in C, which may themselves be considered the outputs of a circuit (which is a subgraph of C) with depth strictly smaller than C. We construct D from C as follows, where the individual steps are also displayed graphically in Figure 6: 1. By induction we can replace the circuits in C whose output gates are the predecessors of g with circuits of the desired form (i.e. the top layer of gates has labels defined by F and all other gates are ∨2 or ∧2 ). 2. Say g is labelled by L = Lj for some j ∈ {1, . . . , l}. Then the set of all left and right quotients {s−1 Lt−1 | s, t ∈ {0, 1}∗ } is contained in F. Let I be the set of inputs, which are directly connected to g. We proceed by removing g from the already modified circuit and adding for each quotient of L, a gate labeled by said quotient. Each of these new gates is connected to all the inputs in I. Hence now only the topmost layer of gates has labels from B(V). Observe the set of leftand right quotients corresponds to the set of elements of the syntactic monoid of L. Hence intuitively the new gates determine the element of the syntactic monoid that the parts of the input belonging to I evaluate to. 3. Since L is symmetric, we can now compute the result of g, which depends only on whether the substring of inputs in I followed by the outputs of the inductively replaced circuits, is in L. Since the syntactic monoid ML of L is finite and we have already determined the element of ML to which the inputs at I evaluate, we can do the rest of the computation in ML through basic case distinctions an NC0 circuit is capable of. Observe that the size of this circuit is only dependant on k and m but independent of the input size. As usual, the negations can 22

be moved to the top by De Morgan’s law, where they are combined with the gates from B(V) in the sense that for each L ∈ V, also Lc ∈ V. ...

xi 1

...

x i2

B(V)

xi3

xi 1

...

x i2

B(V) + NC0

B(V)

...

xi3

B(V) + NC0

g

g

Original circuit, g is labelled by f L for some L ∈ F

1. Subcircuits replaced by induction

xi1

...

x i2

B(V) + NC0

Q1

...

xi1

xi3

xi2

B(V) + NC0

B(V) + NC0

Qi

...

Q1

Qr

...

xi3

B(V) + NC0

Qi

Qr

NC0 2. Connect input to quotients of L: Q1 , . . . , Qr

3. Add NC0 circuit, to determine value of g

Figure 6: Illustration of the process of moving all gates from B(V) to the topmost layer. Newly added or modified parts are gray.

Hence, the newly constructed circuit D has the desired form and is of constant size, which completes the proof. Note that a constant size circuit over the base V can recognise more than a Boolean combination of languages from V. The wiring of the input to the gates makes the significant difference, since not necessarily all inputs are wired to each gate. 23

Since we consider non-uniform circuit classes, it is possible to only wire, for instance, inputs in prime positions to a gate. Hence we are able to query with a circuit, if only the inputs in specific positions belong to V. It is due to that fact that the recognised classes of languages are non-regular. 5.3. The Block Product We previously introduced bases for circuits that were defined by a variety of regular languages. This section deals with an operation that reflects the non-uniform wiring of the inputs of the circuit on the side of the recognised languages. Thereto, we define a unary operation ·  Parb on varieties of regular languages mapping a variety of commutative regular languages V to the class of languages V  Parb recognised by constant size circuits over the base V. The intuition and notation for the operation comes from the algebraic background where similar ideas have been used on the algebraic side in [7]. The principles intrinsically rely on the block product for finite monoids, a binary operation which plays an important role in the decomposition of finite monoids. One of the well-known contributions concerning the block product being the paper by Rhodes and Tilson [23]. Since the block product exposes strong links to classes of logic and most circuit classes admit for a characterisation through logic, this implicitly ties the block product to circuit classes. Since we are dealing with non-regular language classes, it was necessary to relate to a modified version of the block product for infinite monoids. In this paper however, we omit the algebraic definition of the block product of infinite monoids but rather define a mechanism similar to the block product principle, that provides the same languages as recognised by the block product, and that is purely defined on the language side. For more details on the algebraic and logic side for varieties of regular languages we refer to a survey about the block product principle [28] or for the non-regular case together with connections to circuit classes to [14]. Here we will restrict from the block product principle to the unary operation ·  Parb , that provides the correct languages for our circuit classes. Denote by |·| : A∗ → N the natural morphism that maps each word to its length. We say a mapping f : A∗ → B ∗ is length preserving, if |f (u)| = |u| for all u ∈ A∗ . Definition 8 (N-transduction). Let D be a finite partition of N2 . By [(i, j)]D denote the equivalence class that (i, j) belongs to. Then an N24

transduction is a length preserving map τD : A∗ → (A × D)∗ , where (τD (w))i = (wi , [(|wi |)]D ) = (wi , [(i − 1, |w| − i)]D ). These N-transductions will be essential to the description of constant size circuit classes. In dependence of a (regular, commutative) variety V, which may be seen as describing the gate types, we define the following class of languages that is eventually going to be related to constant size circuit classes over the base V. Definition 9 (V  Parb ). Let V be a variety of regular languages, then we define V  Parb as the class of languages, where (V  Parb )A is the Boolean algebra generated by the languages τD−1 (L) for all finite partitions D of N2 , and L ∈ VA×D . Observe that this class of languages also contains non-regular languages, as the next example illustrates: Example 1. Consider the language {an bn | n ∈ N} and the partition D of N2 consisting of the sets L = {(i, j) | i < j}, R = {(i, j) | i > j} and M = {(i, j) | i = j}. If the letter wi is in the left half of the word, it gets mapped to (wi , L) by τD , if it is exactly in the middle to (wi , M ) and otherwise to (wi , R). Hence {an bn | n ∈ N} = τD−1 (((a, L)|(b, R))∗ ). The following theorem is essential for the further investigation of equations for constant size circuit classes. Intuitively, it can be explained as follows: The variety of regular languages V determines the gate types of the circuits. According to Lemma 3, we may visualise a constant size circuit with gates from V as an NC0 circuit with exactly one layer of gates from V wired to the inputs on top. Now an N-transduction is responsible for describing precisely the wiring of the inputs to the gates of V. Theorem 3. Let V be a variety of commutative regular languages. The languages accepted by constant size circuit families over a base defined by the variety V, are exactly the languages in V  Parb .

25

Proof. We first prove that every language recognised by some constant size circuit family over the base V is indeed contained in V  Parb . Since by Lemma 3, each language recognised by constant size circuits over the base V is a Boolean combination of languages recognised by depth one circuits over the base V and V  Parb is a Boolean algebra by definition, it suffices to show that each language recognised by depth one circuits over the base V is in V  Parb . Hence let T ⊆ A∗ be a language accepted by the depth one circuit family C = (Ci )i∈N over a base defined by V. Let F = {L ∈ V | f L is a label in C.} be the finite set of languages defining the labels of C. We are now going to define a finite partition D of N2 and a language K ⊆ (A × D)∗ such that T = τD−1 (K). Recall that the inputs of some circuit were labelled either by the constants 0 or 1 or by a ∈ A, where those decisive for the acceptance of a word are the ones labelled by a ∈ A. For L ∈ F, n ∈ N and v ∈ N|A| (where we denote by va ∈ N for a ∈ A the component of v at a) we define ⎫ ⎧

i ∈ {1, . . . , n} and for each a ∈ A pre- ⎪ ⎪ ⎬ ⎨

cisely va inputs are labelled by xi = a DL,n,v = (i − 1, n − i) . ⎪

and connected to a gate labelled by f L ⎪ ⎭ ⎩

in the circuit Cn .  Let DL,v = n∈N DL,n,v . We are now going to use these sets to form a finite partition of N2 which will define an N-transduction. Before, however, we need without loss of generality to make some assumptions: Observe that since L is commutative the function f L is symmetric and we may rearrange the order of the connections to the gate labelled by f L without changing the output of the gate. Hence we may assume that if an input xi is connected j-times to some gate, then those connections are consecutive without any other connections from different inputs in between. The connections from xi are labeled with some a ∈ A and this label determines if the value passed to the gate is going to be 1 or 0. We also assume that the connections are grouped with respect to their labels, which is again possible since L is commutative. Moreover, since L is regular, by a pumping argument, there is some number NL ∈ N such that if a connection with label a ∈ A appears more than NL times, then there is some number j ≤ NL such that the value of the gate remains unaltered, if we assume it appears only j times. With these assumptions being justified, one may now observe that due to the fact that C has depth one and each Ci has exactly one gate, the sets 26

DL,v are pairwise disjoint and hence the set D = {DL,v | L ∈ F, and 0 ≤ va ≤ NL for each a ∈ A.} is a finite partition of N2 . This partition defines an N-transduction τD , which for some word w ∈ A∗ maps the i-th letter wi to (wi , DL,v ) if and only if (i − 1, |w| − i) ∈ DL,v and which is the case if for each a ∈ A there are precisely va connections labelled by a from the input xi to the gate which is labelled by f L in C|w| . We are now going to define the language K ⊆ (A × D)∗ . For that, define the morphism h : (A × D)∗ → {0, 1}∗ 

(a, DL,v ) → 1va 0(

b=a

vb )

 and let K = L∈F h−1 (L). To prove that indeed T = τD−1 (K), let w ∈ A∗ . Now by definition of K, w ∈ τD−1 (K) if and only if there is some L ∈ F such that h(τD (w)) ∈ L. We prove that this is the case if and only if w ∈ T , which is the case if w is accepted by C|w| . Let L|w| ∈ F be the language such that the single gate of C|w| is labelled by f L|w| . Then, by construction (τD (w))i = (a, DL|w| ,vi ) for some vi ∈ {0, . . . , NL|w| }|A| . It follows from the previous observations that h((τD (w))i ) contains precisely as many 1s and 0s as wi contributes to the single gate of C|w| which is labelled by f L|w| . Hence h(τD (w)) ∈ L|w| if and only if C|w| accepts w and we may conclude that T = τD−1 (K). Since K is a union of inverse morphic images of languages from V, we conclude that also K ∈ V which proves one direction of the claim. For the other direction it suffices to show that for each finite partition D of N2 and each language K ∈ VA×D , the language τD−1 (K) is recognised by a constant size circuit family over the base B(V), since the languages recognised by such constant size circuits form a Boolean algebra. Let MK be the syntactic monoid of K and ηK : (A × D)∗ → MK its syntactic morphism. Observe that since V is a variety of regular languages, any language recognised by MK is in V (see for instance [20]). For each

27

m ∈ MK define a morphism hm : (A × D)∗ → {0, 1}∗  1 if ηK (a, P ) = m (a, P ) → 0 otherwise. Observe by defining the morphism ϕm : {0, 1}∗ → MK sending 1 to m and 0 to the neutral element of MK , we obtain that ϕ−1 (m) ∈ V{0,1} . Denote by #ηmK (w) the total amount of letters (a, P ) occurring in w for which ηK (a, P ) = m. Since K is commutative, we have  ηK ηK (w) = m#m (w) m∈MK ηK

n and (ϕm ◦ hm )(w) = m#m (w) . Define Cm,n = ϕ−1 m (m ) for m ∈ MK and n ∈ N. Observe that since MK is finite, there are finitely many distinct languages Cm,n and by the previous observation, K is a Boolean combination of languages of the form h−1 m (Cm,n ). To abbreviate a bit, we call the finitely many distinct languages Cm,n in the Boolean combination C1 , . . . , Cl and address the corresponding morphisms hm and ϕm (for appropriate m) by h1 , . . . , hl and ϕ1 , . . . , ϕl . Since C1 , . . . , Cl are recognised by a morphism into MK , all of them are elements of V{0,1} . Let f1 , . . . , fl be the Boolean functions corresponding to C1 , . . . , Cl . Observe that for a fixed word length n and a word w of that length, the second component of (τD (w))j is always equal to some Pn,j , independently of the rest of w (besides its length). The circuit we construct for word length n thus has f1 , . . . , fl as gates and for each a ∈ A, there is a wire from the input xj to each of the gates fi , for which hi (a, Pj,n ) = 1, querying whether xj = a. The gates f1 , . . . , fl are then connected, mirroring the Boolean combination from before. By construction, this circuit accepts a word w, if and only if τD (w) ∈ K, which completes the proof.

6. Equations for the block product While the theorem of Gehrke, Griegorieff and Pin provides the existence of ultrafilter equations for (V  Parb )A , it does not answer the question on how to obtain them. More precisely, it is unclear how to obtain a set of equations that is not trivial. In this section, we are going to give a set of 28

non-trivial equations that are both sound and complete and we will show that this set is feasible to prove separations for the classes. The set of ultrafilter equations for (V  Parb )A will be dependent on the profinite equations that define the variety of regular languages V in the sense that if u = v is a profinite equations for V, then there are ultrafilters γu and γv such that [γu ↔ γv ] is an equation for (V  Parb )A . As a corollary to the main theorem that defines the equations, we show their applicability by proving separations. To concretely define the equations that hold for (V  Parb )A , we define a function that gets as arguments a word w, another word s and a vector of positions p, such that the positions of p in w are substituted by the letters of s. Naturally, such a substitution only makes sense for some inputs. For instance the positions in p should not exceed the length of the word w. For technical reasons, each element of the vector of positions will be a tuple containing the distance of the position from the beginning and the end of the word. For an element p ∈ (N2 )∗ let pi = (xi , yi ). We define the set of correct substitutions ⎧

⎫

|s| = |p| , ⎨ ⎬

D = (w, s, p) ∈ A∗ × A∗ × (N2 )∗

∀i : |w| − 1 = xi + yi , . ⎩

x0 < . . . < x|p|−1 < |w| ⎭ Note that for any element (w, s, p) ∈ D, the second components of p are determined by w and the first components of p. Thus, the substitution function in the following can be defined by solely relying on the first components. Given a word w = w0 . . . wm−1 ∈ A∗ of length m and k, l with 0 ≤ k ≤ l < m, we define w[k,l] = wk . . . wl . Let n be the length of s, then the function s : A∗ × A∗ × (N2 )∗ → A∗ is defined as  w[0,x0 −1] s0 w[x0 +1,x1 −1] s1 . . . sn−1 w[xn−1 +1,m−1] if (w, s, p) ∈ D, s(w, s, p) = w otherwise. Furthermore, define the function that maps the second component to its length as λ : A∗ × A∗ × (N2 )∗ → A∗ × N × (N2 )∗ (w, s, p) → (w, |s| , p) 29

and let π2 : A∗ × A∗ × (N2 )∗ → A∗ with π2 (w, s, p) = s be the projection on the second component. As the function s substitutes letters in certain positions, we need the following definition in order to define which positions of p are “indistinguishable” by a language in (V  Parb )A . Define the mapping πc : A∗ × A∗ × (N2 )∗ → P(N2 ) that maps the third component onto its content, given by πc (w, s, p) = {p0 , . . . , p|p| }. Note that any finite subset of N2 is a finite subset of β(N2 ) and thus πc can be interpreted as a mapping into the space F(N2 ) of all filters of P(N2 ), by sending it to the intersection of all ultrafilters containing the set. Furthermore, β(N2 ), which contains all ultrafilters of P(N2 ) can be seen as a subspace of F(N2 ), which is homeomorphic to Vietoris of β(N2 ). Then there exists ˇ and extension of πc denoted by βπc , known as the Stone-Cech extension ∗ ∗ 2 ∗ 2 βπc : β(A × A × (N ) ) → F(N ) (as implicitly defined on page 13). Together with these definitions we can formulate the theorem that provides a set of equations for (V  Parb )A . Theorem 4. The variety (V  Parb )A is defined by the equations [βs(γu ) ↔ βs(γv )]. where [u = v] is a profinite equation that holds on V and γu , γv ∈ β(A∗ × A∗ × (N2 )∗ ) satisfying (1) βλ(γu ) = βλ(γv ) (2) u ⊆ βπ2 (γu ) and v ⊆ βπ2 (γv ) (3) βπc (γu ) = βπc (γv ) ∈ β(N2 ) Proof. We prove soundness of these equations in Section 7 and completeness in Section 8 6.1. Proving separations To prove separations, it is necessary to give two ultrafilters that form an equation and a language violating that equation. The following definitions and lemmata all serve the purpose to later be able to construct ultrafilters having certain properties such that they form an equation of (V  Parb )A . The constructing is done via filter bases. As mentioned, any filter base can be extended to an ultrafilter. Since a filter base only needs to satisfy the finite intersection property, it is in general much easier to give a filter base than to straight up define an ultrafilter. Hence the following statements all revolve around proving that an ultrafilter containing some specific filter base has a certain property. 30

Lemma 4. Let X, Y be two sets, γ ∈ βX and g : X → Y a function. Then, for each α ∈ βY the following conditions are equivalent 1. βg(γ) = α 2. {g −1 (L) | L ∈ α} ⊆ γ Proof. (1) implies (2). Let L ∈ α. Then also L ∈ βg(γ) and by definition of βg(γ) this implies g −1 (L) ∈ γ. Thus (2) holds. (2) implies (1). Let L ∈ α. Then g −1 (L) ∈ γ and hence L ∈ βg(γ), which implies that α ⊆ βg(γ). Since α is an ultrafilter and thus maximal, we have α = βg(γ). We call the set {g −1 (L) | L ∈ α} pullback of α by g, denoted by g −1 (α). We will often use the fact that adding certain sets to a pullback of some ultrafilter will still yield a filter base. Lemma 5. Let X, Y be two sets, γ ∈ βX and g : X → Y a function and α ∈ βY . For some set R ⊆ X, the condition g(R) ∈ α implies that g −1 (α) ∪ {R} is a filter base. Proof. We have to show that no element of g −1 (α) has empty intersection with R. Since g(R) is an element of α, we have g(R) ∩ L = ∅ for each L ∈ α. This of course implies that g −1 (L) ∩ R = ∅ for all L ∈ α and thus yields the claim. Lemma 6. Let γ ∈ β(A∗ × A∗ × (N2 )∗ ) with k ≥ 1. Then, for each α ∈ β(N2 ), the following conditions are equivalent: 1. βπc (γ) = α 2. {A∗ × A∗ × P ∗ | P ∈ α} ⊆ γ Furthermore, these conditions hold for γ with respect to some α if and only if  3. For each partition {P1 , . . . , Pn } of N2 , we have ni=1 (A∗ ×A∗ ×Pi∗ ) ∈ γ Proof. Since A∗ × A∗ × P ∗ = πc−1 (P ), (1) and (2) are equivalent by Lemma 4. For the second assertion, suppose there is an α ∈ β(N) such that conditions (1) and (2) hold. Let {P1 , . . . , Pn } be a partition of N2 . Then n 2 i=0 Pi = N and the fact that α is an ultrafilter implies that Pk ∈ α for some k ∈ {1, . . . , n} and thus A∗ × A∗ × Pk∗ ∈ γ by condition (2). Since γ is an upset, (3) holds. 31

Suppose that γ satisfies (3) and let α = {P ⊆ N2 | A∗ × A∗ × P ∗ ∈ γ}. Then α is an upset closed under intersection. For each P ⊆ N2 , the partition {P, P c } forces either A∗ × A∗ × P ∗ ∈ γ or A∗ × A∗ × (P c )∗ ∈ γ. Thus α is an ultrafilter and by the equivalence of (1) and (2) we have βπc (γ) = α. Together with these observations, we are now ready to prove separations using the obtained ultrafilter equations. The topological method has the advantage that there is no need for probabilistic methods to find specific inputs for the circuits to be fixed or swapped. In order to apply it to show separations, we use the following recipe: Let u = v be a profinite equation that holds for the gate types of some circuit class and L a language of which we want to show that it is not contained in that class. We show the existence of two ultrafilters γu and γv satisfying conditions 1. to 3. from Theorem 4, such that L ∈ βs(γu ) and L ∈ / βs(γv ). Since according to Theorem 4, [βs(γu ) ↔ βs(γv )] is an equation for the circuit class, that means L violates the equation and is hence not contained in it. To show the existence of γu and γv , Lemmata 4, 5 and 6 are used to construct two filter bases, which γu (resp. γv ) eventually extend, ensuring that both ultrafilters satisfy the necessary conditions. The profinite equations displayed in Figure 7 will be used for separations and hold for the languages defining the respective gate types: For instance, the syntactic monoid of the language defining a modp gate is the cyclic group Zp and hence has only one idempotent, which justifies the profinite equation xω = y ω . All other equations are equally easy to verify. They will be used in the proof of the next Corollary, following the recipe for separation above. Gates

Profinite Identities

Circuit Class (const. size)

{∧, ∨} {∧, ∨, modp | p ∈ N} {modp | p ∈ N}

xy = yx x2 y = xy 2 xy = yx xy = yx xω = y ω

AC0 ACC0 CC0

Figure 7: Gate types and the profinite equations they satisfy.

Corollary 1. The following relations hold: Constant size CC0 is strictly contained in constant size ACC0 and constant size AC0 is strictly contained in constant size ACC0 . Also constant size CC0 and constant size AC0 are not comparable. 32

Proof. We prove the first claim by showing that LAND = 1∗ is not contained in constant size CC0 and the second claim by showing that LPARITY = (0∗ 10∗ 1)∗ 0∗ , which is the language consisting of all words with an even number of 1s, is not contained in constant size AC0 . Evidently both languages are in constant size ACC0 . In the following, we use the recipe described above. In the following, we let A = {0, 1}. For the first claim, take the profinite equation [0ω = 1ω ], that holds for the variety providing us with the gate types of CC0 . Consider the filter base Fω = {A∗ × {n! | n ≥ N } × (N2 )∗ | N ∈ N} and the filter base  n  FP = A∗ × N × Pi∗ | {P0 , . . . , Pn } is a partition of N2 . i=0

It is easy to see that both sets are filter bases and that also their union, denoted by F, is a filter base. Hence there exists an ultrafilter μ ∈ β(A∗ × N × N∗ ) containing F. Next, consider the subset of A∗ × A∗ × (N2 )∗ : 1∗ × {1n! | n ∈ N} × (N2 )∗ . Adding this set to the set λ−1 (μ) is a filter base according to Lemma 5. We denote this filter base by F1ω . By F0ω we denote the result of adding to the base λ−1 (μ) the set 1∗ × {0n! | n ∈ N} × (N2 )∗ , which yields a filter base again by Lemma 5. Let γ0ω be an ultrafilter containing F0ω and γ1ω be an ultrafilter containing F1ω . We show that γ1ω and γ0ω satisfy the conditions of Theorem 4. Since both ultrafilters contain λ−1 (μ), by Lemma 4 we obtain βλ(γ1ω ) = μ = βλ(γ0ω ), ensuring the first condition. For the second condition, recall that 0ω = {L ∈ Reg | ∃n0 ∈ N ∀n ≥ n0 : 0n! ∈ L}. Since λ−1 (Fω ) is contained in γ0ω and so is 1∗ × {0n! | n ∈ N} × (N2 )∗ , it follows from the closure of ultrafilters under intersection that 0ω ⊆ βπ2 (γ0ω ). The same argumentation holds for 1ω ⊆ βπ2 (γ1ω ). 33

For the third condition, we have that by Lemma 6 and the choice of the filter base FP , we obtain βπc (γ1ω ) = βπc (γ0ω ). Thus both ultrafilters satisfy conditions 1.-3. of Theorem 4, such that [βs(γ0ω ) ↔ βs(γ1ω )] is an equation for CC0 . By definition of s, we obtain s(1∗ × {1n! | n ∈ N} × (N2 )∗ ) ⊆ LAND and since ultrafilters are closed under extension 1∗ × {1n! | n ∈ N} × (N2 )∗ ⊆ s−1 (LAND ) ∈ γ1ω . This again by definition of βs implies that LAND ∈ βs(γ1ω ). However, s(1∗ × {0n! | n ∈ N} × (N2 )∗ ) ⊆ LcAND , which shows by the same argumentation as before that LcAND ∈ βs(γ0ω ) implying that LAND ∈ / βs(γ0ω ). Thus, LAND violates the equation [βs(γ0ω ) ↔ βs(γ1ω )] which holds for constant size CC0 , proving that LAND = 1∗ is not in constant size CC0 . For the second claim, we use the profinite equation [110 = 100] satisfied by the variety corresponding to the gate types of AC0 . We abbreviate the proof a bit, since the argumentation is essentially the same as before. Adding the set A∗ × {3} × N∗ to the filter base FP from before yields a filter base which is is contained in an ultrafilter ν ∈ β(A∗ × N × N∗ ). Consider the sets S110 = 0∗ × {110} × (N2 )∗ and S100 = 0∗ × {100} × (N2 )∗ . We let F110 (resp. F100 ) be the filter base resulting from adding S110 (resp. S110 ) to the set λ−1 (ν). Let γ110 be an ultrafilter containing F110 and γ100 be an ultrafilter containing F100 . By the same arguments as before, both ultrafilters satisfy conditions 1.-3. of Theorem 4 and thus [βs(γ110 ) ↔ βs(γ100 )] is an equation satisfied by constant size AC0 . Since s(S110 ) ⊆ LPARITY and s(S100 ) ⊆ LcPARITY , we obtain LPARITY ∈ βs(γ110 ) but LPARITY ∈ / βs(γ100 ) 0 and thus it is not in constant size AC . 7. Proof of Soundness The following Lemma provides a set of equations that define precisely the kernel of the projection π(V  Parb )A , which is used as an intermediate tool to prove the soundness of the equations in Theorem 4. Lemma 7. Let μ, ν ∈ βA∗ . Then for each partition D of N2 the Boolean algebra V(A×D) satisfies the equation [βτD (μ) ↔ βτD (ν)] if and only if [μ ↔ ν] is an equation of (V  Parb )A . Proof. Let μ, ν ∈ βA∗ such that [βτD (μ) ↔ βτD (ν)] holds for all partitions D of N2 and let L ∈ (V  Parb )A be a generator of the Boolean algebra. 34

Recall that by definition there exists a partition D of N2 and a language S ∈ V(A×D) such that L = τD−1 (S). Then L ∈ μ ⇔ τD−1 (S) ∈ μ ⇔ S ∈ βτD (μ) ⇔ S ∈ βτD (ν) ⇔ τD−1 (S) ∈ μ ⇔ L ∈ ν. This proves both directions of the claim. We are now ready to state the proof of the main theorem: Let u, v be two profinite words such that u = v is a profinite equation for V and let γu , γv ∈ β(A∗ ×A∗ ×(N2 )∗ ) be the associated ultrafilters satisfying the conditions of Theorem 4. We have to show that [βs(γu ) ↔ βs(γv )] is an equation of (V  Parb )A . By Lemma 7 this is equivalent to [βτD (βs(γu )) ↔ βτD (βs(γv ))] being an equation of V for each partition D of N2 . Thus, by Lemma 2 it suffices to show that for each partition D and each language L ∈ V(A×D) , the equivalence s−1 (τD−1 (L)) ∈ γu ⇔ s−1 (τD−1 (L)) ∈ γv holds. For P ∈ D, define the morphism hP : A∗ → (A × D)∗ a → (a, P ). −1 Let E be the Boolean algebra generated by the set {h−1 P (x L) | x ∈ (A × D)∗ }. Since L is a regular language, it has only finitely many left quotients x−1 L and hence E is finite. Moreover, as V is a variety of regular languages and as such closed under inverse morphisms and quotients, VA contains E. Since E is a finite Boolean algebra, its atoms – the minimal elements with respect to inclusion – form a partition of A∗ , which we denote by At(E). By Lemma 1, at least one of the elements of At(E) is contained in u (resp. v). Now, since u = v is a profinite equation which holds for V, this implies L ∈ u ⇔ L ∈ v for all regular languages and in particular for each element of At(E). Hence u and v contain precisely the same elements of At(E). We denote by Eu,v ∈ At(E) some element which is contained in both u and 35

v. Thus, by the second condition in Theorem 4, Eu,v ∈ u ⊆ βπ2 (γu ) and Eu,v ∈ v ⊆ βπ2 (γv ). To show that s−1 (τD−1 (L)) is contained in γu if and only if it is contained in γv , by upward closure of filters it is enough to consider the subset s−1 (τD−1 (L)) ∩ π2−1 (Eu,v ). This subset can be rewritten as the following join decomposition 



(s−1 (τD−1 (L)) ∩ π2−1 (Eu,v ) ∩ Dc )∪  (πc−1 (P ) ∩ s−1 (τD−1 (L)) ∩ π2−1 (Eu,v ) ∩ D) ,

P ∈D

where D, as defined in Section 6, denotes the points where the function s is not just a projection on the first component. Since λ−1 (λ(D)) = D and condition 1. states that βλ(γu ) = βλ(γv ), we have that D ∈ γu ⇔ D ∈ γv . We show for each set of the join separately that it is contained in γu if and only if it is contained in γv . This is sufficient, as both γu and γv are ultrafilters and the sets of the decomposition are disjoint. • The set s−1 (τD−1 (L)) ∩ π2−1 (Eu,v ) ∩ Dc : By definition of s it is the identity on the first component when restricted to Dc . Denote by π1 : A∗ × A∗ × (N2 )∗ → A∗ the projection on the first component. We obtain s−1 (τD−1 (L)) ∩ π2−1 (Eu,v ) ∩ Dc = π1−1 (τD−1 (L)) ∩ π2−1 (Eu,v ) ∩ Dc . Since π1 factors through λ and condition 1. holds, we conclude that the set belongs to γu if and only if it belongs to γv . • The set s−1 (τD−1 (L)) ∩ π2−1 (Eu,v ) ∩ πc−1 (P ) ∩ D for P ∈ D: Observe that the above set is equal to {(w, s, p) ∈ D | τD (s(w, s, p)) ∈ L, πc (w, s, p) ∈ P and s ∈ Eu,v }. For a word w ∈ A∗ and Q ⊆ (N2 ) with Q∩{(0, |w|−1), (1, |w|−2), . . . , (|w|− 1, 0)} = {(x0 , y0 ), . . . , (xk , yk )} where x1 < x2 < . . . < xk define w[Q] = wx1 . . . wxk 36

The requirement πc (w, s, p) ∈ P provides the information that all letters of p are in the same equivalence class P of D and by (w, s, p) ∈ D, we obtain |p| = |s|. The condition we need to examine more closely is τD (s(w, s, p)) ∈ L. We split the image of τD ◦ s into the part where the substitution is happening and the remaining part. Fix a tuple (w, s, p) and let S be the set of substituted positions, that is S = {p0 , . . . , p|p|−1 }. Let r be the remaining part in (A × D)∗ defined by r = (τD (s(w, s, p)))[S c ]. Recall that the morphism hP sends a letter a to (a, P ). For s ∈ A∗ , let sP = hP (s) = (s0 , P ) . . . (s|s|−1 , P ). Then sP = (τD (s(w, s, p)))[S]. Hence τD (s(w, s, p)) is a shuffle of r and sP . As L is a commutative language, a shuffle of r and sP belongs to L if and only if rsP ∈ L. This allows us to rewrite τD (s(w, s, p)) ∈ L as rsP ∈ L. By construction of Eu,v we have for every s, t ∈ Eu,v and x ∈ (A × D)∗ that xsP ∈ L iff xtP ∈ L. Hence τD (s(w, s, p)) ∈ L is independent of the choice of s ∈ Eu,v as long as it has the same length as p, so the tuple (w, s, p) remains in D. Finally we can rewrite our set above as an intersection of the sets: {(w, s, p) ∈ A∗ × A∗ × (N2 )∗ | ∃t ∈ Eu,v : |t| = |p| = |s| and rtP ∈ L}∩ πc−1 (P ) ∩ π2−1 (Eu,v ) ∩ D which is equal to   λ−1 {(w, l, p) ∈ A∗ × N × (N2 )∗ | ∃t ∈ Eu,v : |t| = |p| = l and rtP ∈ L} ∩ πc−1 (P ) ∩ π2−1 (Eu,v ) ∩ D. By definition of Eu,v , the set π2−1 (Eu,v ) it is an element of both γu and γv and for all other sets we have by the conditions on the filters that they belong to γu if and only if they belong to γv . 8. Proof of Completeness We will show that the families of equations we obtained in the previous section suffice to characterise (V  Parb )A . Many of the proof techniques 37

used in this section follow the example of [12] and were adjusted to work for the block product and arbitrary (but commutative) varieties of regular languages. For an profinite equation u = v of V denote by E[u=v] the set of all ultrafilter equations [βs(γu ) ↔ βs(γv )] where γu and γv satisfy the conditions of Theorem 4. We will then show that any language L ∈ P(A∗ ) which satisfies all equations E[u=v] for every equation u = v of V belongs to (V  Parb )A . Since we required V to be commutative, the set E[ab=ba] will always be among the sets of equations defining (V  Parb )A , which is an essential part for the proof of completeness. A technique used frequently in the following proofs was already employed in Corollary 1: We build two ultrafilters γu and γv by subsequently adding sets to some filter base, such that finally through the Lemmata 4, 5 and 6 they satisfy the required properties for Theorem 4 to be applicable. Let w ∈ A∗ and i, j < |w|. By w · (ij) denote the word that results when exchanging the letters in positions i and j of w. For L ⊆ A∗ , define a relation RL on N2 by (i, n)RL (j, m) iff i + n = j + m or ∀w ∈ A∗ |w| > i, j

: w ∈ L ⇔ w · (ij) ∈ L

Lemma 8. If a language L of A∗ satisfies all the equations E[ab=ba] for all a, b ∈ A, then RL contains an equivalence relation of finite index. Proof. For (a, b) ∈ A2 , define the sets ⎫

⎧

i < j < |w|, ⎪ ⎪ ⎬

⎨ ∗ ∗ 2 ∗ wi = a, wj = b, Sab = (w, ab, (i, |w| − i)(j, |w| − j)) ∈ A × A × (N ) ⎪

w ∈ L, ⎪ ⎭ ⎩

w · (ij) ∈ /L ⎫

i < j < |w|, ⎪ ⎬

w = a, wj = b, Sba = (w, ba, (i, |w| − i)(j, |w| − j)) ∈ A∗ × A∗ × (N2 )∗ i ⎪

w ∈ L, ⎪ ⎭ ⎩

w · (ij) ∈ /L ⎧ ⎪ ⎨

and

there exists w ∈ A∗ such that 2 2 ((i, n), (j, m)) ∈ (N ) (w, ab, (i, n)(j, m)) ∈ Sab or

(w, ab, (j, m)(i, n)) ∈ S

 Mab =

ab

38

 .

 Then we have RLc = (a,b)∈A2 Mab . We show by contraposition that for all (a, b) ∈ A2 there is a finite partition {P1 , . . . , Pn } of N2 such that the corresponding equivalence relation θab is disjoint from Mab . Taking the refinement of all these equivalence relations will then provide us with the desired equivalence relation contained in RL . Now, suppose that for each finite partition {P1 , . . . , Pn } of N2 , n   2 Pi = ∅. (∗) Mab ∩ i=1

Under this premise, we will construct two ultrafilters γab and γba , satisfying conditions 1. - 3. of Theorem 4 and show that L does not satisfy the equation [βs(γab ) ↔ βs(γba )]. The set  n  A∗ × N × Pi∗ | {P1 , . . . , Pn } is a partition of N2 F= i=1

is a filter base on A∗ × N × (N2 )∗ . By condition (∗), λ(Sab ) does not have empty intersection with any of the elements of F. Thus we can extend the filter base F by λ(Sab ), which is equal to λ(Sba ), and obtain an extended filter base. Let μ ∈ β(A∗ × N × (N2 )∗ ) be an ultrafilter containing the extended filter base. Then Fab = λ−1 (μ) ∪ A∗ × {ab} × (N2 )∗ is again a filter base on A∗ ×A∗ ×(N2 )∗ . To see this, we must consider that μ was required to contain λ(Sab ). Since any L ∈ μ has non-empty intersection with λ(Sab ), their projection on the second component will contain |ab| as an element and thus no preimage of any L ∈ μ will have empty intersection with A∗ × {ab} × (N2 )∗ . Furthermore, any ultrafilter containing Fab will also contain Sab , since λ−1 (λ(Sab )) ∩ A∗ × {ab} × (N2 )∗ = Sab . By the same argument, Fba = λ−1 (μ) ∪ A∗ × {ba} × (N2 )∗ is a filter base and any ultrafilter containing Fba will also contain Sba . Let γab be an ultrafilter containing Fab and respectively γba an ultrafilter containing Fba . Note that Sba ∈ / γab and Sab ∈ / γba , since the two sets have empty intersection. By Lemma 4 and Lemma 6 the two ultrafilters satisfy βπc (γab ) = βπc (γba ) ∈ β(N2 ) and βλ(γab ) = βλ(γba ). 39

Since ultrafilters are upsets and γab contains A∗ × {ab} × (N2 )∗ , we have that {ab} is a subset of βπ2 (γab ) and respectively {ba} ⊆ βπ2 (γba ). By definition s(Sab ) ⊆ L or equivalently Sab ⊆ s−1 (L) and Sba ⊆ s−1 (Lc ). Thus L ∈ βs(γab ) but L ∈ / βs(γba ). By contraposition, if L satisfies E[ab=ba] , then there is an equivalence relation θab of finite index which is disjoint from  Mab . Setting θ = a,b∈A θab , we see that θ is an equivalence relation of finite index contained in RL since   c θ= θab ⊆ Mab = RL a,b∈A

a,b∈A

A direct consequence is the following Corollary, which makes use of the fact that each finite equivalence class can be split into singletons and still yields an equivalence relation. Observe that the idea of splitting the finite equivalence classes into singletons is originally from [11] (Corollary 4.3). Corollary 2. If a language L of A∗ satisfies the equations E[ab=ba] for all a, b ∈ A, then RL contains an equivalence relation of finite index for which each finite equivalence class is a singleton. We use the Eilenberg correspondence between varieties of regular languages and varieties of finite monoids. By V denote the variety of finite monoids associated with V. By Hom(A∗ , V) denote the set of all morphisms from A∗ into a monoid of V. Hom(A∗ , V) is countable, since all monoids are finite and hence there are countably many morphisms into monoids of V. Then there exists a bijection φHom : N → Hom(A∗ , V). As a shorthand define hi := φHom (i), where hi : A∗ → Mi and Mi ∈ V. The space N × Hom(A∗ , V) thus is countable too and hence any family of words of A∗ indexed by N × Hom(A∗ , V) is a sequence. Let φ : N × Hom(A∗ , V) → N be a bijection and (sn )n∈N be a sequence of words. For n ∈ N and h ∈ Hom(A∗ , V), define s(n, h) := sφ(n,h) . Lemma 9. Let (sn )n∈N and (tn )n∈N be two sequences of words of A∗ satisfying the property h(s(n, h)) = h(t(n, h)) for all h ∈ Hom(A∗ , V) and n ∈ N. Then for each N ∈ N there exists a fixed morphism ϕN ∈ Hom(A∗ , V), such that for all i ≤ N : hi (s(N, ϕN )) = hi (t(N, ϕN )) Proof. Let N ∈ N. By Mi denote the monoid that hi maps into. Define ϕN : A∗ → M0 ×. . . ×MN −1 by ϕN (w) = (h0 (w), . . . , hN −1 (w)). This makes 40

ϕN a morphism of Hom(A∗ , V), since V is closed under finite products. The condition ϕN (s(N, ϕN )) = ϕN (t(N, ϕN )) then implies hi (s(N, ϕN )) = hi (t(N, ϕN )). Note that we may choose a bijection φ, that satisfies the property that for n ≤ m, we have φ(n, h) ≤ φ(m, h) for all morphisms h ∈ Hom(A∗ , V). This allows for the following Corollary. Corollary 3. Let (sn )n∈N and (tn )n∈N be two sequences of words of A∗ satisfying the property h(s(n, h)) = h(t(n, h)) for all h ∈ Hom(A∗ , V) and n ∈ N. Then there exist subsequences (smn )n∈N and (tmn )n∈N such that for all i ≤ mn : hi (smn ) = hi (tmn ). Proof. For N ∈ N define ϕN as in Lemma 9. Setting mN := φ(N, ϕN ) proves the claim. For a word w ∈ A∗ and P ⊆ N2 with {(0, |w| − 1), (1, |w| − 2), . . . , (|w| − 1, 0)} ∩ P = {(x1 , y1 ), . . . , (xk , yk )} where x1 < x2 < . . . < xk define w[P ] = wx1 . . . wxk and

P [w] = (x1 , y1 ) . . . (xk , yk ) ∈ (N2 )∗ .

Lemma 10. Let L be a language of A satisfying all the equations E[u=v] . Let θ be an equivalence relation of finite index contained in RL and let P be an infinite equivalence class of θ. Then there exists an n ∈ N and a morphism h : A∗ → M into a finite monoid M ∈ V such that for all s, t ∈ A∗ , if 1. n ≤ |s| = |t|, 2. si = ti for all i ∈ /P 3. h(s[P ]) = h(t[P ]) then s ∈ L ⇔ t ∈ L. Proof. By contraposition: Assume that for each n ∈ N and each morphism h : A∗ → M into a finite monoid M ∈ V, there exist s(n, h), t(n, h) ∈ A∗ satisfying 1.-3., but s(n, h) ∈ L and t(n, h) ∈ / L. Roughly, we are going to do the following: The words s(n, h) (resp. ∗ with limits say s t(n, h)) are used to define converging sequences in A (resp. t). These limits are shown to be a profinite equation of V, that is s = t. This profinite equation is then utilised to construct ultrafilters γs and 41

γt in the usual way, such that [βs(γs ) ↔ βs(γt )] is an equation in E[s=t] , but L ∈ βs(γs ) and L ∈ / βs(γt ). As this is a contradiction to the requirement that L satisfies all equations E[u=v] , the claim then follows. First, we turn to constructing the mentioned sequences. Recall that the map φ : N × Hom(A∗ , V) → N is a bijection. Let sn = s(φ−1 (n)) and tn = t(φ−1 (n)). Considering the sequences (sn [P ])n∈N and (tn [P ])n∈N , condition 3. provides h(s(n, h)[P ]) = h(t(n, h)[P ]). By Corollary 3 there exist subsequences (smn [P ])n∈N and (tmn [P ])n∈N such that for all i ≤ mn : hi (smn [P ]) = hi (tmn [P ]). ∗ , both (smn [P ])n∈N and (tmn [P ])n∈N As A∗ can be embedded into A define sequences in the free profinite monoid. Moreover, the free profinite monoid is compact and hence every sequence has a convergent subsequence. Thus, there exists a set J ⊆ {mn | n ∈ N} such that (sj [P ])j∈J converges and a set I ⊆ J such that both (si [P ])i∈I and (ti [P ])i∈I converge. Define s := limi∈I si [P ] and t := limi∈I ti [P ]. We claim that s = t is an ˆ ˆ equation of V, which is h(s) = h(t) for every morphism h : A∗ → M into a ˆ denotes its unique continuous extension to A ∗ ). monoid M ∈ V (where h ∗ Let h ∈ Hom(A , V), then there exists an i0 ∈ N such that h = hi0 . By choice of the sequences, for all i > i0 we have h(si [P ]) = h(ti [P ]), which ˆ ˆ ˆ is continuous. This proves that s = t is indeed implies h(s) = h(t), since h an equation of V. To define the previously mentioned ultrafilters, let Ts = {(sn , sn [P ], P [sn ]) | n ∈ N} and Tt = {(sn , tn [P ], P [sn ]) | n ∈ N} By condition 2. for any n ∈ N we obtain s(sn , sn [P ], P [sn ]) = sn ∈ L and s(sn , tn [P ], P [sn ]) = tn ∈ / L and thus s(Ts ) ⊆ L and s(Tt ) ⊆ Lc . We claim that there exist two ultrafilters γs and γt satisfying i. βλ(γs ) = βλ(γt ) ii. s ⊆ βπ2 (γs ) and t ⊆ βπ2 (γt ) iii. βπc (γs ) = βπc (γt ) ∈ β(N2 ) such that L ∈ βs(γs ) and Lc ∈ βs(γt ). To ensure property iii., let α ∈ β(N2 ) with P ∈ α. Recall that πc mapped a triple from A∗ × A∗ × (N2 )∗ to the content of the third component. We 42

denote by πc,λ the map from A∗ × N × (N2 )∗ sending an element to the content of its third component. Thus the pullback of α by πc,λ provides a filter base on A∗ × N × (N2 )∗ . ∗

2 ∗

−1 πc,λ (α)

A × N × (N ) ←−−−− N2 Furthermore we have that λ(Ts ) = λ(Tt ), which implies that πc,λ (Ts ) = −1 πc,λ (Tt ) = P . Thus, adding the set {λ(Ts )} to πc,λ (α) still yields a filter ∗ 2 ∗ base by Lemma 5. Let μ ∈ β(A × N × (N ) ) be an ultrafilter containing the extended filter base. Observe that the map πc,λ factors through λ by πc,λ ◦ λ = πc and thus −1 πc (α) ⊆ λ−1 (μ). This ensures that any ultrafilter γ containing λ−1 (μ) will satisfy βλ(γ) = μ and βπc (γ) = α by Lemma 4. λ−1 (μ)

−1 πc,λ (α)

A∗ × A∗ × (N2 )∗ ←−−− A∗ × N × (N2 )∗ ←−−−− N2 Since λ(Ts ) ∈ μ, the sets Fs = λ−1 (μ) ∪ π2−1 (s) ∪ {Ts } and Ft = λ−1 (μ) ∪ π2−1 (t) ∪ {Tt } are both filter bases. Any ultrafilter γs containing Fs and γt containing Ft will satisfy conditions i. − iii.. Let γs and γt be two such ultrafilters. Then Ts ∈ γs and since s(Ts ) ⊆ L, we obtain Ts ⊆ s−1 (L) and thus s−1 (L) ∈ γs and by Tt ⊆ Lc , s−1 (Lc ) ∈ γt . Thus L ∈ βs(γs ) and L ∈ / βs(γt ). By contraposition, the claim holds. Lemma 11. Let L be a language of A∗ satisfying all the equations E[u=v] and let θ be an equivalence relation of finite index contained in RL . Then there exists an n ∈ N and a morphism h : A∗ → M into a finite monoid M ∈ V such that for all s, t ∈ A∗ , if n ≤ |s| = |t| and h(s[P ]) = h(t[P ]) for each θ equivalence class P , then s ∈ L ⇔ t ∈ L. Proof. Let L satisfy the equations E[ab=ba] . Then by Corollary 2, RL contains an equivalence relation of finite index θ for which each finite equivalence class is a singleton. Let P1 , . . . , Pr be the equivalence classes of θ. For each i ∈ {1, . . . , r} with Pi infinite, according to Lemma 10 there exist ni ∈ N and a morphism hi such that the conditions stated there hold. Furthermore define n = max{ni | Pi is infinite } and h(w) = (h1 (w), . . . , hr (w)). Again, 43

h is a morphism into a monoid of V, since V is closed under finite products. Now let s, t ∈ A∗ , with n ≤ |s| = |t| and h(s[P ]) = h(t[P ]) for each θ equivalence class P . We define words wi ∈ A∗ for i = 0, . . . , n and j = 0, . . . , |s| by  sj if j ∈ Pk and i < k (wi )j = tj otherwise. By construction we have w0 = s, wn = t and Lemma 10 applies to each pair wi−1 , wi with i ∈ {1, . . . , n} and thus wi−1 ∈ L ⇔ wi ∈ L. It follows that s ∈ L ⇔ t ∈ L. For N ∈ N denote by A≥N the set of all words of length greater or equal to N , that is A≥N = {w ∈ A∗ | |w| ≥ N } Theorem 5. If L ∈ P(A∗ ) satisfies all the equations E[u=v] , then L ∈ (V  Parb )A . Proof. Let h : A∗ → M be a morphism into a monoid of V. For P ⊆ N2 and m ∈ M define the set LP,m = {w ∈ A∗ | h(w[P ]) = m}. Since M ∈ V, the language R = h−1 (m) is an element of VA . Let D = {P, P c } be a partition of N2 . Define the morphism eD : (A × D)∗ → A∗ (a, P ) → a (a, P c ) →  Since VA is closed under inverse morphisms, e−1 D (R) ∈ VA×D . Then −1 −1 LP,m = {w ∈ A∗ | τD (w) ∈ e−1 D (R)} = τD (eD (R))

is an element of (V  Parb )A . In particular, for each N ∈ N the language LP,m ∪ A≥N is in (V  Parb )A , which can be seen easily, when we recall that (V  Parb )A corresponds to constant size circuits and that we can modify the circuit such that it recognises only words from a certain length on. With this consideration and the previous observations, we are ready to prove that L ∈ (V  Parb )A . By Corollary 2, the relation RL contains an equivalence relation θ of finite index for which each finite equivalence class is a singleton. Let P1 , . . . , Pr be the corresponding partition of N2 . By Lemma 11, there exists an N ∈ N and a morphism h : A∗ → M into a monoid M ∈ V such that for m ∈ M and  r   Lm = LPi ,m ∩ A≥N i=1

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either Lm ⊆ L or Lm ⊆ Lc . ≥N =  This implies that there exists some Q ⊆ M such that L ∩ A L . Since (V  P ) contains all finite languages, L is a Boolean arb A m∈Q m combination of languages in (V  Parb )A . 9. Conclusion We have presented a method to describe circuit classes by equations, given an abstract description of the gate types. The tools and techniques used originate from algebra and topology and have previously been used on regular language classes. Due to recent developments in generalising these methods to non-regular classes, they are now powerful enough to describe circuit classes. But the knowledge that they are powerful enough itself is not sufficient, as we require a constructive mechanism behind these descriptions. Since non-uniform circuit classes are by definition not finitely presentable, this seemed to be impossible. Nevertheless, we were able to find a description of small but natural circuit classes via equations. While we are aware that the examined circuit classes are significantly small and separation results could have been shown by less intricate means, the calculations served merely as a sanity check and proof of concept, in order to be aware of where the difficulties of this approach lie, if we were to opt for the description of larger classes. This description seems helpful as it easily allows to prove the nonmembership of a language to some circuit class. Another advantage is the possibility of using Zorn’s Lemma for the extension of filter bases to ultrafilters, which prevents us from having to use probabilistic arguments in many places. Also in Lemma 10 we use purely topological arguments of convergence, for which it is unclear how this could be achieved purely combinatorially. The results we acquired are not so different from the results about equations for varieties of regular languages by Almeida and Weil [2]. Since the used transductions exhibit similarities to a derived category, this gives hope that their results can be used as a roadmap for further research. In [7] it was shown that a certain restricted version of the block product of our constant size circuit classes would actually yield linear size circuit classes (over the same base). Here having equations for all languages captured by this circuit class, not just the regular ones, would pay off greatly. By showing that a padded version of a language is not in a linear circuit 45

class we could already prove that PARITY is not in a polynomial size circuit class. Equations for non-regular language classes could be used to overcome previous bounds. The separation results in the corollary can easily be extended to show that a padded version of those languages is not contained in these circuit classes. A different approach would be to examine the way the block product was used here. The evaluation of a circuit is equivalent to a program over finite monoids. While the program itself has little computational power, it allows non-uniform operations like our N-transducers. The finite monoid itself corresponds loosely speaking to the computational power of the gates of the circuit, which was handled by our variety of regular languages V. For general circuit classes one would need to consider larger varieties of finite monoids containing also non-commutative monoids. While the methods here seem to be extendable to non-commutative varieties of finite monoids, the more complicated problem remaining is to find an extension of the block product that corresponds to polynomial programs over these monoids. Acknowledgements ´ Pin for suggestions and Special thanks go to Mai Gehrke and Jean-Eric comments on the early draft of this paper. Especially the Vietoris construction was pointed out by Mai Gehrke. References [1] Jorge Almeida. Finite Semigroups and Universal Algebra. World Scientific Publishing Co. Inc., Singapore, 1995. [2] Jorge Almeida and Pascal Weil. Profinite categories and semidirect products. Journal of Pure and Applied Algebra, 123:1–50, 1998. [3] David A. Mix Barrington and Howard Straubing. Lower bounds for modular counting by circuits with modular gates. In LATIN ’95: Theoretical Informatics, Second Latin American Symposium, Valpara´ıso, Chile, April 3-7, 1995, Proceedings, pages 60–71, 1995. [4] David A. Mix Barrington and Howard Straubing. Superlinear lower bounds for bounded-width branching programs. Journal of Computer and System Sciences, 50(3):374–381, 1995. [5] David A. Mix Barrington, Howard Straubing, and Denis Th´erien. Non-uniform automata over groups. Inf. Comput., 89(2):109–132, 1990. [6] David A. Mix Barrington and Denis Th´erien. Non-uniform automata over groups. In Automata, Languages and Programming, 14th International Colloquium, ICALP87, Karlsruhe, Germany, July 13-17, 1987, Proceedings, pages 163–173, 1987. [7] Christoph Behle, Andreas Krebs, and Mark Mercer. Linear circuits, two-variable logic and weakly blocked monoids. Theor. Comput. Sci., 501:20–33, 2013.

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