## 1 Introduction

It is a well-known fact that all Turing machines (TM) form a countable set as each TM has a finite description. Moreover, since each TM defines (recognizes) a single language as a recognizer, the class of languages recognized by TMs, called recursive enumerable languages, also forms a countable set. On the other hand, all (unary or binary) languages form an uncountable set. Thus, we can easily conclude that there are some languages that cannot be recognized by (associated with) any TM

[13].As a very restricted form of TM, finite state automaton (FSA) [7] reads the input once from left to right and then gives its decision. Their computational power is significantly less, and the class of languages recognized by them is called regular languages, a proper sub-class of recursive enumerable languages. On the other hand, a FSA can be enhanced by making probabilistic choices, called probabilistic finite automaton (PFA) [6]

. On contrary to FSAs or TMs, all PFAs form an uncountable set if they are allowed to use real-valued transition probabilities. Then, one may ask whether the languages recognized by PFAs also form an uncountable set or not.

A FSA can either accept or reject a given input string, and so it is easy to classify the set of all strings into two sets, i.e., the language recognized by the automaton and its complement. However, a PFA defines a probability distribution on the input strings and so, in order to split the set of all strings into two sets, we additionally use a threshold, called cutpoint. That is, a PFA defines (recognizes) a language with a cutpoint, which contains all strings accepted with probability greater than the cutpoint. Such language is called stochastic

[6].A single PFA may define different stochastic languages with different cutpoints. Rabin, in his seminal paper on PFAs [6], presented a two-state PFA using only rational-valued transitions that recognizes uncountably many languages with cutpoints. That is, even a single PFA can define an uncountable set of languages. Since the PFA is single, the uncountability result is based on the fact that there are uncountably many cutpoints.

Similar result was given for a quantum counterpart of PFA, quantum finite automaton (QFA), [9] that a two-state rational-valued QFA over a single letter alphabet (unary) can define an uncountable set of languages. This result is stronger in a way that Rabin’s result was given for binary languages but the quantum result was given for unary languages. Besides, a similar uncountability result for unary PFAs can be obtained only for three states [10].

The minimal binary and unary PFAs and unary QFAs defining uncountably many languages with cutpoints have two, three, and two states, respectively. All these three results were given by using the fact that the cardinality of cutpoints is uncountable. We find it interesting and natural to obtain the same results by using the fact that the cardinality of automata (i.e., algorithms) is uncountable. In other words, we obtain the same result for fixed cutpoints.

In this note, we show that two-state binary PFAs, two-state unary QFAs, and three-state unary PFAs can recognize uncountably many languages with fixed cutpoints.

We present the notations and definitions used throughout this note in the next section. Then, we explain the original proof given by Rabin [6] in Section 3, and we present our modification on Rabin’s proof in Section 4. In Section 5, we explain the quantum version of Rabin’s proof. After this, we present our quantum result in Section 6. Lastly, we explain the known uncountability results for unary PFAs in Section 7, and we present our construction for unary PFAs in Section 8.

## 2 Background

The input alphabet is represented by . The empty string is represented by . The set of all strings defined on is denoted by . Moreover, . For any given string , represents its length, represents its reverse string, and, if , denotes its -th symbol, where . For any binary string , represents its binary decimal representation, i.e., . The accepting probability of an automaton on a given input string is given by . The cutpoints are defined in the interval .

The (probabilistic or quantum) state of an -state automaton is represented by

-dimensional (stochastic or unity-length) column vector. Any given input is read from left to right and symbol by symbol. The computation starts in a single state. The computation is traced by a column vector. For each symbol, an operator is associated. Whenever this symbol is read, the current vector is multiplied from the left by the corresponding linear operator (represented as

-dimensional matrix). Based on the final state vector, the accepting probability of input by the automaton is calculated.For pre- and post-processing, probabilistic and quantum finite automata (PFAs and QFAs, respectively) can read one specific symbol before reading the input and one specific symbol after reading the input, respectively. However, in this paper, the models do not do any pre- or post-processing.

Formally, an -state () PFA is a 5-tuple

where is the set of states, is an

-dimensional (left) stochastic matrix associated to symbol

, is the initial state, and is the set of accepting state(s).The initial probabilistic state is a zero-one stochastic vector having 1 in its -th entry. For empty string, the final probabilistic state is . For a given input , the final probabilistic state is

Then, the accepting probability of by is

Any language that is recognized by a PFA with cutpoint is called stochastic, and the class of stochastic languages forms an uncountable set [6].

We assume the reader is familiar with the basic concepts (i.e., unitary evolution and projective measurements) and notations (i.e., ket-notation) used in quantum computation (see [4] and [8] for a complete and quick reference, respectively).

In the literature, there are many different definitions of QFAs [1]. Here we use the known most restricted version, also so-called Moore-Crutchfield or Measure-Once QFA [3].

Formally, an -state () QFA is a 5-tuple

where is the set of states, is an

-dimensional unitary matrix associated to symbol

, is the initial state, and is the set of accepting state(s).The initial quantum state is a zero-one unit vector having 1 in its -th entry. For empty string, the final quantum state is . For a given input , the final quantum state is

Then, the accepting probability of by is

which is obtained by making a measurement in computational basis at the end of the computation.

Any language recognized by a QFA (even if it is the most general variant of QFA) with cutpoint was shown to be stochastic [1], and vice versa in most of the cases (any stochastic language can be recognized by almost all variants of QFAs [2, 1]).

A single-letter alphabet and any automaton defined over it is called unary. Similarly, any two-letter alphabet and any automaton defined over it is called binary.

## 3 Rabin’s proof

We start with the proof given by Rabin. We name the PFA presented by Rabin as , which has two states, say and . The computation starts in and the only accepting state is . The PFA operates on binary strings, defined on . Let be a given input. (It is clear that .)

We trace the computation of by a 2-dimensional column stochastic vector (probabilistic state), which is at the beginning. After reading symbol and , applies the following stochastic operators to its probabilistic state

respectively.

The probabilistic state after reading is

Then, is the second entry of . By using induction, we can easily see that . Thus, for any rational number between 0 and 1, say , there exists at least one binary string such that .

Let be two real-valued cutpoints between 0 and 1. Since the rational numbers are dense on real numbers, there exists at least one binary string, say , such that . Thus we can conclude that with cutpoint recognizes the language that is a superset of the language recognized by with cutpoint . More generally, for any given cutpoint, recognizes a different language. Since there are uncountably many cutpoints, recognizes uncountably many stochastic languages (, and so the class of stochastic languages forms an uncountable set).

In this proof, the existence of uncountably many stochastic languages is shown based on a single PFA, and so the result follows from the fact that there are uncountably many cutpoints. We find it interesting and natural to obtain the same result by using the fact that there are uncountably many PFAs. In other words, we fix the cutpoint and show that there is a set of uncountably many PFAs such that each PFA recognizes a different language with this fixed cutpoint.

We note that the result for 3-state PFAs is trivial since it is known that (e.g., see [5]) if language is defined by an -state PFA with cutpoint , then for any given , there always exists an -state PFA, say , recognizing with cutpoint .

## 4 Our modification on Rabin’s proof

Here we show that, for any given nonzero cutpoint, two-state PFAs can define uncountably many languages.^{1}^{1}1The construction here was given by the first author when he was an 11-th grade student in high school. We believe that our proof is more elegant since it is based on the existence of uncountably many PFAs (algorithms).

###### Lemma 1.

For a given , there exists a 2-state PFA accepting any non-empty binary string with probability .

###### Proof.

We use the PFA given by Rabin after modifying the transition matrix for symbol . The PFA has two states: the first one is the initial and the only accepting state is the second one. The transition matrices for symbols 0 and 1 are

respectively. We use induction to prove our Lemma.

Base case: After reading string or , the probabilistic state is

respectively, and so the accepting probability for string 0 or 1 is 0 or , respectively.

Inductive step: Assume that, after reading , the probabilistic state is

Then, after reading , the new probabilistic state is

and hence the accepting probability is . Similarly, after reading , the new probabilistic state is

and hence the accepting probability is , where is the value making the column summation to 1.

We conclude that for any given input string , . ∎

###### Theorem 1.

For any given cutpoint , 2-state PFAs recognize uncountably many stochastic languages with cutpoint .

###### Proof.

Let be two real numbers. Then, there exists a string such that , where is the PFA given by Rabin. Then, we can easily follow these two inequalities

Due to Lemma 1, we can conclude that

Thus, the string is in the language recognized by with cutpoint , and, it is not in the language recognized by with cutpoint . Therefore, for any given two different real numbers between 0 and 1, there exist two PFAs such that they recognize different languages with cutpoint . ∎

###### Corollary 1.

2-state PFAs can recognize uncountably many stochastic languages with cutpoint .

## 5 Quantum version of Rabin’s result

The quantum version of Rabin’s result was given for 2-state unary real-valued QFAs [9], defined as , where is an irrational number and is the counter-clockwise rotation with angle on plane. It is clear that is a real-valued unitary matrix. Moreover, all quantum states on plane form the unit circle.

We fix such that . The automaton starts in , and rotates on plane with angle for each input symbol . Let be the quantum state after reading symbols, i.e., . Then, the accepting probability of is

Since is irrational, , the set of all quantum states that can be in, is dense on the unit circle. Similarly, , the accepting probabilities of all inputs by , is dense on . Therefore, for any given two cutpoints , there is always an input such that

Therefore, the automaton recognizes a different language for each cutpoint in . In other words, the class of languages recognized by with cutpoints forms an uncountable set.

## 6 Our result for unary QFAs

We use the same automata family given in the previous section by restricting irrational parameter .

Let and be two different irrational numbers in . Then, their digit by digit binary representations are as follows:

and

where .

Since and are different, then there exists a minimal such that .

Suppose that and . We use the input of length , say . After reading , and rotate by angles

The angles and are congruent to

modulo , respectively, where . We can rewrite and as

The quantum states of and lie in the same quadrant after reading the input . Here the values of and determine the number of a quadrant. But, in any case, we can have either

or

In the 1st and 3rd quadrants, we have and as the quantum state of is closer to -axis and the quantum state of is closer to -axis. In the 2nd and 4th quadrants, we have and as the quantum state of is closer to -axis and the quantum state of is closer to -axis. We listed all cases in the following table.

The case in which and is symmetric. By using the same arguments, we obtain the above table after interchanging the last two headers ( and ). Therefore, we can conclude the following result.

###### Theorem 2.

For any given two irrational numbers and in , the QFAs and recognize different languages with cutpoint .

###### Corollary 2.

The class of languages recognized by 2-state unary real-valued QFAs with cutpoint forms an uncountable set.

## 7 Unary PFAs

Rabin’s proof was given for binary 2-state PFAs. Unary 2-state PFAs can recognize only a few regular languages with cutpoints [5, 10]. Besides, any unary -state PFA can recognize at most nonregular languages with cutpoints [5]. Therefore, there is no direct counterpart of Rabin’s result for unary PFAs. However, we can still show that unary PFAs can define uncountably many stochastic languages. The proof was given first for a family of 4-state unary PFAs [9], and then for a family of 3-state unary PFAs [10].

The former result was already given for a fixed cutpoint () (by combining the quantum result given in Section 5 and Turakainen conversion technique [11, 12].) The latter result was given for the pairs of PFAs and cutpoints, i.e., , and so the proof is still based on the cardinality of cutpoints.

In this section, we give the details of the latter result, and then, in the next section, we present our construction for the fixed cutpoint.

For each , is a 3-state unary PFA over the alphabet . The first state is the starting state and the last state is the only accepting state. The single transition matrix for symbol is

The eigenvalues of

arewhere and are complex conjugates of each other.

For the input , the accepting probability is calculated as

In other words, is equal to .

Due to the Cayley-Hamilton theorem and simplicity of eigenvalues, each entry of is a linear combination of , , and . Thus, we can write as

where the coefficients of and are also conjugates of each other. By using the following initial conditions

the coefficients can be calculated as

The polar forms of and are respectively

where . The polar forms of and are respectively

where . Then, we can rewrite as

where , , and are all positive values.

By picking the cutpoint , we can have

Thus, is greater than the cutpoint if and only if is positive. Remark that since ,

(1) |

###### Fact 1.

[10] For any with , the language recognized by with cutpoint is different than the language recognized by with cutpoint .

###### Proof.

We can conclude the proof by showing the existence of a string, say , such that and have different signs.

By checking the values of on the interval , we can easily see that, if , then . Due to Eq. 1, we know that is always less than . Besides, . Thus, we can say that there exists an integer such that

(2) |

The real line can be partitioned into intervals of length , in which the function does not change its sign — all borderline points are attached to “negative” intervals:

Let and . We know that is greater than and less than . Therefore, and lie in the consecutive intervals, and so they have different signs. Then, is a string that separates both languages. ∎

## 8 Our construction for unary PFAs

We present our construction^{2}^{2}2The construction here was given by the first author when he was a 12-th grade student in high school. for 3-state unary PFAs that can define uncountably many languages with a fixed cutpoint (e.g., ). Similarly to binary PFA case, we introduce another parameter in matrix . Besides, we restrict the interval of with . Let . The new matrix is defined as

Even though is not a stochastic matrix in general, its eigenvalues are the identical to the eigenvalues of :

Similarly to the calculations of , we can write as

for real values , , and . By using the initial conditions

the coefficients can easily be found as

In other words, , or equivalently

(3) |

This time, since we restrict , we have

(4) |

Interestingly, is a stochastic matrix:

Thus, we can define our new unary 3-state PFA, say , by using as the single transition matrix. The initial state is the first state, and the only accepting state is the third state. Therefore, by Eq. 3,

where . By picking , we can get . Let be the PFA where . Then, we can say that is greater than if and only if .

###### Theorem 3.

For any with , the language recognized by with cutpoint is different than the language recognized by with cutpoint .

###### Proof.

Due to Eq. 4, we know that is always less than , and so is less than . Moreover, is less than . Thus, we can say that there exists an integer such that

(5) |

Let and . We know that is greater than and less than . Therefore, as explained in Fact 1, and have different signs. Then, is the string that separates both languages. ∎

###### Corollary 3.

The languages recognized by the family of 3-state unary PFAs with cutpoint form an uncountable set.

## Acknowledgements

Dimitrijevs was partially supported by University of Latvia projects AAP2016/B032 “Innovative information technologies” and ZD2018/20546 “For development of scientific activity of Faculty of Computing”. Yakaryılmaz was partially supported by Akadēmiskā personāla atjaunotne un kompetenču pilnveide Latvijas Universitātē līguma Nr. 8.2.2.0/18/A/010 LU reģistrācijas Nr. ESS2018/289 and ERC Advanced Grant MQC.

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