1 Prelude: Background and Challenges
1.1 A Historical Account of Topological Automata
In the theory of computation,
finitestate automata (finite automata, or even automata) are one of the simplest and most intuitive mathematical models to describe “mechanical procedures,” each of which depicts a finite number of “operations” in order to determine the membership of any given input word to a fixed language. Such procedures have resemblance of physical systems, which make discrete time evolution, contrary to continuous time evolution. Over decades of their study, these machines have found numerous applications in the fields of engineering, physics, biology, and even economy (see, e.g., [13]). In particular, a oneway^{3}^{3}3Here, we use the term “1way” to exclude the use of moves, which are particular transitions of the machine with its tape head staying still, where refers to the empty string. On the contrary, finite automata that make moves are sometimes called 1.5way finite automata. (or realtime) finite automaton reads input symbols one by one and then processes them by changing a status of the automaton’s interior system step by step. This machinery models the computation of streamlined process, in which it receives streamlined input data and processes it piece by piece by applying operations predetermined for each of the input symbols. For such a machine, a computation is a description of a series of “evolutions” of the interior system.To cope with numerous computational problems, various types of finite automata have been proposed as their appropriate computational models in the past literature. In the 1970s, many features of the existing 1way finite automata were generalized into socalled “topological automata” (see [8] for early expositions and references therein). A topology is a mathematical concept of dealing mostly with open sets and continuous maps that preserve the openness of point sets. More general automata were also defined in terms of category in, e.g., [10]. Topological automata embody characteristic features of various types of finite automata and this fact has helped us take a unified approach toward the study of formal languages and automata theory. The analysis of topological features of the topological automata thus guides us to the better understandings of the theory itself.
Back in the 1970s, Brauer (see references in [8]) and Ehrig and Kühnel [8] discussed topological automata as a topological generalization of Mealy machines, which behave as “transducers,” which simply produce outputs from inputs. In contrast, following a discussion of Bozapalidis [5] on a generalization of stochastic functions and quantum functions (see also [25]), Jeandel [15] studied another type of topological automata that behave as “acceptors” of inputs. Jeandel’s model naturally generalizes not only probabilistic finite automata [21] but also measureonce quantum finite automata [18]. The main motivation of Jeandel’s work, nonetheless, was to study a nondeterministic variant of quantum finite automata and he thus used his topological automata to obtain an upperbound of the language recognition power of nondeterministic quantum finite automata. Concerning the types of “inputs” fed into topological automata, in contrast, Ehrig and Kühnel [8] applied a quite general framework to inputs, which are taken from arbitrary compactly generated Hausdorff spaces, whereas Jeandel [15] used the standard framework with finite alphabets and languages generated over them. Jeandel further took “measures” (which assign real numbers to final configurations) to determine the acceptance or rejection of inputs. Since we are more concerned with the computational power of topological automata in comparison with the existing finite automata, we wish to make our model as simple and intuitive as possible by introducing, unlike the use of measures, sets of accepting and rejecting configurations, into which the machine’s interior system finally fall.
Given an input string over a fixed alphabet , the evolution of an interior status of our topological automaton is described in the form of a series of configurations, which constitutes a computation of the machine. A list of transition operators thus serves as a “program”, which completely dictates the behaviors of the machine on each input. Since arbitrary topological spaces can be used as configuration spaces, topological automata are no longer “finitestate” machines; however, they evolve sequentially as they read input symbols one by one until they completely read the entire inputs and final configurations are observed once (referred to as an “observe once” feature). Moreover, our topological automata enjoy a “deterministic” nature in the sense that which transition operators are applied to the current configurations is completely determined by input symbols alone. This gives rise to “1way deterministic topological automata” (or 1dta’s, in short). Although their tape heads move in one direction from the left to the right, 1dta’s turn out to be quite powerful in recognizing formal languages. By extending transition maps to “multivalued” maps, it is possible to consider nondeterministic moves of topological automata [15].
1.2 A New Model of Topological Automata
All the aforementioned models of topological automata are based only on a relatively small rage of appropriately defined topologies, such as compactly generated Hausdorff spaces. We instead wish to study all possible topologies with no initial restrictions other than discrete applications of transition operators.
This paper thus aims at shedding new light on the basic structures of topological automata and the acting roles of their transition operators that force configurations to evolve consecutively. For this purpose, we start our study with a suitable abstraction of 1way finite automata using arbitrary topological spaces for configurations and arbitrary continuous maps for transitions. Such an abstraction serves as a skeleton to construct our topological automata. We call this skeleton an automata base. Since the essential behaviors of topological automata are strongly influenced by the choice of their automata bases, we are mostly concerned with the properties of these automata bases.
In general, the choice of topologies significantly affects the computational power of topological automata. As shown later, the trivial topology induces the language family composed only of and (for each fixed alphabet ) whereas the discrete topology allows topological automata to recognize arbitrary complex languages. All topologies on a fixed space
form a complete lattice; thus, it is possible to classify the topologies according to the endowed power of associated topological automata.
Initially, a study on topological automata should be focused on achieving the following four key goals.

Understand how various choices of topological spaces and continuous maps affect the computational power of underlying machines by clarifying the strengths and weaknesses of the language recognition power of the machines.

Determine what kinds of topological features of topological automata nicely characterize the existing finite automata of various types by examining the descriptive power of such features.

Explore different types of topological automata to capture fundamental properties (such as closure properties) of formal languages and finite automata.

Find useful applications of topological automata to other fields of science.
Organization of the Paper.
In Section 2, we will formulate our basic model of 1dta’s. These automata are naturally induced from automata bases and they can express numerous types of the existing 1way finite automata. Through Section 4, we will discuss basic properties of the 1dta’s, including closure properties and the elimination of two endmarkers. Following an exploration of such basic properties, we will compare different topologies in Section 5 by measuring how much computational power is endowed to underlying topological automata. In Section 6, we will show that unique features of wellknown topological concepts, such as compactness and equicontinuity, help us characterize 1way deterministic finite automata (or 1dfa’s). We will lay out a necessary and sufficient condition on a topological space for which underlying machines are no more powerful than 1dfa’s. In Section 7, we will consider a nondeterministic variant of our topological automata (called 1nta’s) by introducing multivalued transition operators. It is known that, for weak machine models, nondeterministic machines can be easily simulated by deterministic ones. By formalizing this situation, we will argue what kind of topology makes 1nta’s simulated by 1dta’s.
We strongly hope that this work initiates a systematic study on the significant roles of topologies played by topological automata and also it leads to better understandings of ordinary finite automata in the end.
2 Basics of Topologies and Automata Bases
Oneway deterministic topological automata can represent the existing oneway finite automata of numerous types. We begin our study of such powerful automata by describing their “basic” framework, which we intend to call an automata base. In what follows, we will provide the fundamental notion of automata bases, which are founded solely on topologies.
2.1 Numbers, Sets, and Languages
Let , , and respectively indicate the sets of all integers, of all real numbers, and of all complex numbers. Given a real number , let . We denote by the set of all natural numbers (i.e., nonnegative integers) and set to be . For any two integers and with , an integer interval expresses the set in contrast with a real interval for . We further abbreviate as for any number .
An alphabet refers to a nonempty finite set of “symbols” or “letters.” A string over an alphabet is a finite sequence of symbols in and the length of a string is the total number of symbols used to form . In particular, the empty string is a string of length and is denoted by . Given two strings and over the same alphabet, is an initial segment of if holds for a certain string . For each number , denotes the set of all strings of length exactly ; moreover, we set and a language over is a subset of . A language is called unary (or tally) if it is defined over a singleletter alphabet. Given a language over , we use the same symbol to denote its characteristic function; that is, for any , if , and otherwise. For two languages and over , the notation denotes . In particular, when is a singleton , we write in place of . Similarly, we write for . The reversal of a string with for any is and is denoted by .
Given a set , the notation denotes the power set of , i.e., the set of all subsets of , and expresses . A monoid is a semigroup with an identity in and an associative binary operator on .
2.2 Topologies and Related Notions
Let us review basic terminology in the theory of general topology (or pointset topology). Given a set of points, a topology on is a collection of subsets of , called open sets, such that satisfies the following three axioms: (1) , (2) any (finite or infinite) union of sets in is also in , and (3) any finite intersection of sets in belongs to . Hence, is a subset of . With respect to , the complement of each open set of is called a closed set. Moreover, a clopen set is a set that is both open and closed. Clearly, and are clopen with respect to . A neighborhood of a point in is a set in that contains . Often, we write to indicate a neighborhood of . A topological space is a pair . When is clear from the context, we omit and simply call a topological space. For a practical reason, we implicitly assume that throughout this paper. For two topological spaces and , we say that is finer than (also is coarser than ) if both and . In such a case, we write , or simply if and are clear from the context. For a topological space , a basis of its topology is a set of subsets of such that every open set in is expressed as a union of sets of . In this case, we say that the basis induces the topology . Given two topological spaces and , the product topology (or Tychonoff topology) on the Cartesian product is the topology induced by the basis . We further write for .
Take a point set and consider all possible topologies on . Let denote the collection of all topologies on . This set forms a complete lattice in which the meet and the join of a collection of topologies on correspond to the intersection of all elements in and the meet of the collection of all topologies on that contain every element of .
There are two typical topologies on : the trivial topology and the discrete topology . Notice that any topology on is located between and in the lattice .
A map from a topological space to another topological space is said to be continuous if, for any and any neighborhood of , there exists a neighborhood of satisfying , where . Given a set of continuous maps, the notation denotes the set of all continuous maps in on (i.e., from to itself) together with a certain given topology, expressed as . When is the set of all continuous maps on , we often omit subscript from and .
2.3 Automata Bases
In 1970s, topological automata were sought to take inputs from arbitrary topological spaces (e.g., [8]). Although such a general treatment of topological automata provides a birdeye view of a topological landscape of a standard setting of formal languages and automata theory, as noted in Section 1, we wish to limit our interest within fixed discrete alphabets because our intention is to compare the language recognition power of topological automata with the existing finite automata that deal only with languages over small discrete alphabets.
To discuss structures of topological automata, we first introduce a fundamental notion of “automata base,” which is a skeleton of various topological automata introduced in Section 3. We are now ready to introduce a fundamental concept of automata base used as a foundation to our model of topological automata. A left act over with a continuous map on satisfies that and for any and any .
Automata Bases. A triplet is called an automata base if , , and satisfy the following three conditions.

is a set of certain topological spaces (which are called configuration spaces).

is a set of continuous maps (called transition operators) from any space in to itself for which

is a monoid with a multiplication operator ,

is a continuous map on ,

is a left act over with , and

must be a continuous map on .


is a set of observable pairs , both of which are clopen^{4}^{4}4In this paper, we demand the clopenness of and . It is, however, possible to require only the openness. sets in a certain space in (where and are respectively called by an accepting space and a rejecting space).
We often say that an automata base is reasonable if , , and are all nonempty. In the rest of this paper, we will deal only with reasonable automata bases.
It is often convenient to deal with without . A pair is conveniently called a subautomata base. For operators in and a point of , we simply write or even for and we also abbreviate as . Note that for every . Given an operator , we say that is closed under if holds for any pair . Given a “property”^{5}^{5}5This informal term “property” is used in a general sense throughout this paper, not limited to “topological properties,” which usually means the “invariance under homeomorphisms.” associated with topological spaces, we say that satisfies if all topological spaces in satisfy .
3 OneWay Deterministic Topological Automata
We formally describe our model of oneway deterministic topological automata (or 1dta’s, for short), which are based on the choice of automata bases. To demonstrate the expressiveness of 1dta’s, we show how the existing finite automata of various types can be redefined in terms of our topological automata.
3.1 Basic Models of 1dta’s
Formally, let us introduce our model of topological automata, each of which reads input symbols taken from a fixed alphabet, modifies configurations in a deterministic manner, and finally observes the final configurations to determine the acceptance or rejection of the given inputs. Concerning the last step of observation, a similar situation appears in quantum computation as “measurement.” Our model is “observe once” because we observe only the final configuration after a computation terminates instead of observing configurations at every move of the automaton.
Hereafter, let denote an arbitrary reasonable automata base. Here, we use two endmarkers (left endmarker) and (right endmarker) that surround each input string as .
Framework of 1dta’s. Assuming an arbitrary input alphabet with , let us define a basic model of our topological automata. An 1way (observeonce^{6}^{6}6It is possible to consider an observemany model of 1dta in which, at each step, the 1dta checks if, at each step, the current configuration falls into . For a further discussion, refer to Section 8.) deterministic topological automaton with the endmarkers (succinctly called a 1dta) is a tuple , where is an extended alphabet, is a configuration space in with a certain topology on , is the initial configuration in , each is a transition operator in acting on , and is an observable pair in for satisfying the exclusion principle: and are disjoint (i.e., ). For convenience, let .
Notice that the use of the endmarkers helps us avoid an introduction of a special transition operator associated with (the empty string). Without endmarker, by contrast, 1dta’s must process an input string with no knowledge of the end of the string.
Our definition of 1dta’s is different from the existing ones stated in Section 1 in the following ways. Ehrig and Kühnel [8] took compactly generated Hausdorff spaces for our and . Jeandel [15] took a metric space for and also used a measure, which maps to instead of our observable pair . As another possible formulation of transition operators, we may be able to use a single map instead of a series . Such a map was used by, e.g., Ehrig and Kühnel [8]; however, as pointed out in [8], is no longer continuous, and thus we need additional requirements.
Configurations and Computation. Let be an input string of length in . We set to be an endmarked input string, which includes (left endmarker) and (right endmarker). This can be considered as a string over .
The machine works as follows. A configuration of on is a point of . A configuration in (resp., ) is called an accepting configuration (resp., a rejecting configuration). Both accepting and rejecting configurations are collectively called halting configurations. We begin with the initial configuration , which is the th configuration of on . At the st step, we apply and obtain . For an index , we assume that is the th configuration of on . At Step (), the st configuration is obtained from by applying the operator corresponding to ; that is, . For any series , we abbreviate as the multiplication . Notice that, since is a monoid with the multiplication, also belongs to . The final configuration is obtained from by and it coincides with . The obtained series of configurations, , is called a computation of on the input . When a 1dta has no endmarker, by contrast, a computation is simply generated by for each , and the final configuration coincides with .
Acceptance and Rejection. Finally, we determine whether a 1dta accepts and rejects each input string by checking whether the final configuration falls into and , respectively. To be precise, we say that accepts (resp., rejects) if (resp., ). We say that recognizes if, for every string , the following two conditions are met: (1) if , then accepts and (2) if , then rejects . We use the notation to denote the language that is recognized by .
We define to be the family of all languages, each of which is defined over a certain alphabet and is recognized by a certain 1dta.
Two 1dta’s and having the common and are said to be (computationally) equivalent if . This relation satisfies reflexivity, symmetry, and transitivity.
For two topological spaces and , is homeomorphic to by a map if (i) is a bijection, (ii) is continuous, and (iii) the inverse function is continuous. This function is called a homeomorphism. Let be any reasonable automata base. For each index , let denote an arbitrary 1dta. We say that and are homeomorphic if (i) is homeomorphic to via a certain homeomorphism with , (ii) and satisfy that implies , and (iii) for each , is homeomorphic to by (i.e., restricted to ).
As shown below, two homeomorphic 1dta’s must recognize exactly the same language.
Lemma 3.1
Let be any reasonable automata base. Let be any 1dta for each . If is homeomorphic to , then and are computationally equivalent.
Proof.
For two given 1dta’s and , let be a homeomorphism from to . We intend to show that . Let be any input string. By induction, it is possible to prove that, for any and any , iff , provided that and . Thus, if , then and thus follows, where . This implies and . Therefore, we obtain . Since is homeomorphic to via , it follows that . This yields the conclusion that .
By a similar argument, we also conclude that implies . ∎
Hence, we can freely identify all 1dta’s that are homeomorphic to each other.
3.2 Conventional Finite Automata are 1dta’s
Our topologicalautomata framework naturally extends the existing 1way finite automata of various types. To see this fact, let us demonstrate that typical models of 1way finite automata can be nicely fit into our framework. This demonstrates the usefulness of our computational model.
As concrete examples, we here consider only the following types of wellknown finite automata used in the past literature. To relate to the definition of topological automata, all the existing finite automata discussed below are assumed to equip with two endmarkers and .
(i) Deterministic Finite Automata. A 1way deterministic finite automaton (or a 1dfa) with two endmarkers can be viewed as a special case of 1dta when equals with the discrete topology, contains all maps from to for each , and contains all nonempty partitions of for each . Languages recognized by 1dfa’s are called regular and denotes the set of all regular languages.
(ii) Probabilistic Finite Automata [21]. A stochastic matrix is a nonnegativereal matrix in which every column^{7}^{7}7Unlike the standard definition, in accordance with our topological automata, we apply each stochastic matrix to column vectors from the left, not from the right in the early literature. sums up to exactly . A boundederror 1way probabilistic finite automaton (or 1pfa) is a special case of 1dta, where (in which each point of
is expressed as a column vector),
is composed of all stochastic matrices, and is the set of all pairs , each of which is defined as the inverse images of projections onto unit basis vectors in . The notationdenotes the set of all languages recognized by 1pfa’s with boundederror probability. When we consider unboundederror probability, we write
to denote the set of all stochastic languages (i.e., languages that are recognized by 1pfa’s with unboundederror probability). It is known that [21] and since is in .(iii) MeasureOnce Quantum Finite Automata [18]. A boundederror measureonce 1way quantum finite automaton (or mo1qfa, in short) is a quantum extension of boundederror 1pfa, which is allowed to measure the inner state of the mo1qfa only once after reading off all input symbols. exactly a 1dta, where is a set of spaces for , consists of unitary matrices, contains all pairs such that and for two projections and a constant , where denotes the norm. Write to denote the collection of all languages recognized by mo1qfa’s with boundederror probability. Note that coincides with [6].
(iv) MeasureMany Quantum Finite Automata [16]. A boundederror measuremany 1way quantum finite automaton (or mm1qfa) is a variant of mo1qfa, which makes a measurement every time the mm1qfa reads an input symbol. Each mm1qfa can be described as a 1dta, where contains all sets of the form and consists of all maps defined in [26] as
(1) 
where if and if , for a certain unitary matrix and 3 projections , , and onto the spaces spanned by different sets of basis vectors. Concerning boundederror 1qfa’s, we set and for each constant . Let express the set of all such pairs . For basic properties of , see [26, Appendix]. We write to denote the collection of all languages recognized by 1qfa’s with boundederror probability. It is known that [16].
(v) Quantum Finite Automata with Mixed States and Superoperators [1, 9, 23] (see also a survey [4]). A 1way quantum finite automaton with mixed states and superoperators (or simply, a 1qfa) extends mo1qfa’s and mm1qfa’s in computational power. Let denote the set of complex matrices, let , let for a set satisfying
(the identity matrix). Let
and be projections onto the spaces spanned by disjoint sets of basis vectors. Each pair is defined as and for any constant , where is the trace of a square matrix .(vi) Deterministic Pushdown Automata. A 1way deterministic pushdown automaton (or a 1dpda) can be seen as a 1dta when satisfies the following properties. Let , where is a distinguished bottom marker not in . Let be composed of all maps of the form for 2 functions and , where and . Intuitively, simulates a series of moves in which reads one symbol and then makes a single nonmove followed by a certain number of moves. Let consist of all pairs with and , where and are partitions of . We write for the class of these languages.
4 Basic Properties of 1dta’s
We have introduced the model of 1dta’s in Section 3 and this computational model has an ability to characterize the existing finite automata of various types, as shown in Section 3.2. Next, we will explore basic properties of those 1dta’s and their language families .
4.1 Elimination of Endmarkers
In many cases, it is possible to eliminate two endmarkers from a 1dta without changing the language recognized by . A simple way to do so is to modify the initial configuration, say, of to a new initial configuration using an operator of . Even if we stick to the same , a slight modification of all operators of provides the same effect as shown in Lemma 4.1.
We say that a set of operators is continuously invertible if every operator in is invertible and its inverse is in and continuous.
Lemma 4.1
Let be any reasonable automata base. Assume that is continuously invertible. For every 1dta , there exists another equivalent 1dta with the same , , , , and but no leftendmarker.
Proof.
We define a new set of operators for each symbol . Define for any and let . The desired reads an input and behaves exactly as does by applying . ∎
We can eliminate as well by slightly changing observable pairs of as stated in Lemma 4.2. The set is closed under inverse image of operations in if, for any operator and for any pair , the pair also belongs to , where .
Lemma 4.2
Let be any reasonable automata base. Assume that is closed under inverse images of operators in . For every 1dta , there exists its equivalent 1dta with the same , , , and but no rightendmarker.
Proof.
Let be any 1dta in the premise of the lemma. We define a new observable pair of by setting and . Clearly, and are disjoint because so are and . Note that and are written as and . Since is closed under inverse images of operators in , it follows that for any . In particular, since , we conclude that . ∎
It turns out that the endmarker elimination is quite costly by placing heavy restrictions on an automata base . It is not clear, however, under what “minimal” condition we can eliminate the endmarkers.
4.2 Closure Properties
Let us discuss closure properties of a family 1DTA. First, we start with the closure property under inverse homomorphism, which turns out to be satisfied by every family . Given two alphabets and , a homomorphism is a function from to with its extension satisfying that and for any and any . We say that a language family is closed under inverse homomorphism if, for any language and any homomorphism , the set also belongs to .
Lemma 4.3
For any reasonable automata base , is closed under inverse homomorphism.
Proof.
Let and denote two alphabets and consider a homomorphism and its extension to . For any given automata base stated in the premise of the lemma, take a 1dta that recognizes a language .
Let us define a new 1dta , which intends to recognize . Initially, we set and . For each symbol , we define . The definition of implies for any string . It then follows that, for each input , . We thus conclude that iff . From this equivalence, follows. ∎
Boolean closures are quite fundamental properties. To state our claims regarding those closure properties, we start with new terminology to use. We say that is symmetric if, for any pair , also belongs to . We consider the “product” of two topological spaces and by setting and by taking the associated product topology . An automata base is said to be closed under product if, for any and any , and are homeomorphic to certain in and in , respectively.
Next, we say that is closed under acceptunion product if, for any and any , and are homeomorphic to certain sets and for which belongs to , respectively. Similarly, we can define the notion of the closure under rejectunion product by swapping the roles of two subscripts “acc” and “rej” in the above definition.
Lemma 4.4
Let be any reasonable automata base.

If is symmetric, then is closed under complementation.

If is closed under product and is closed under acceptunion product, then is closed under union.

If is closed under product and is closed under rejectunion product, then is closed under intersection.
The language family is closed under neither union nor intersection. Similarly, is not closed under union [3]. Since, as shown in Section 3.2, 1dta’s can characterize those language families, the premises of Lemma 4.4 may not be removed.
Proof of Lemma 4.4. (1) The closure property of under complementation can be obtained simply by exchanging between and since is symmetric.
(2) For each , we take a language over recognized by a certain 1dta . Let . For and , we consider defined by and . Moreover, set , , and . For any initial segment of , . It thus follows that (i) iff either or and (ii) iff both