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1-way multihead quantum finite state automata (1QFA(k)) can be thought of modified version of 1-way quantum finite state automata (1QFA) and k-letter quantum finite state automata (k-letter QFA) respectively. It has been shown by Moore and Crutchfield as well as Konadacs and Watrous that 1QFA can’t accept all regular language. In this paper, we show different language recognizing capabilities of our model 1-way multihead QFAs. New results presented in this paper are the following ones: 1) We show that newly introduced 1-way 2-head quantum finite state automaton (1QFA(2)) structure can accept all unary regular languages. 2) A language which can’t be accepted by 1-way deterministic 2-head finite state automaton (1DFA((2)) can be accepted by 1QFA(2) with bounded error. 3) 1QFA(2) is more powerful than 1-way reversible 2-head finite state automaton (1RMFA(2)) with respect to recognition of language.

Classical finite state automaton is the very basic model of classical finite machine. Likewise a quantum finite state automaton may be seen as basic model of finite state quantum machine. A variety of models of quantum finite state automaton are used. 1-way quantum finite automaton (1QFA) can be seen as the simplest model of quantum automaton .The two most popular models of quantum finite state automaton are quantum finite state automaton introduced by Moore and Crutchfield [

Languages accepted by multitape or multihead finite automaton were introduced in [

A konadacs and J Watrous [

Belovs et al. [

We know that the language

1RMFA(2) ( [

In this section we give different definitions and corresponding results for 1QFA.

One-way quantum finite state automaton can been seen as the simplest model of quantum computation.Quantum finite automata could be of large importance is the fact that quantum memory seems to be very expensive and it is therefore of very much importance to know what can be achieved with limited amounts of quantum resources.

One-way quantum finite state automaton seem to model very well the way very simple quantum processors work (Ambainis and freivalds, 1998), and also the way simple classical/quantum processors are expected to work: the classical part reads an input, picks up the corresponding quantum operator (a transition mapping) and performs it on a quantum memory of fixed size, independent of the size of input. 1QFA are very simple but less powerful than classical 1-way finite automaton.

Measure many quantum finite state automata (1QFA): We consider 1-way quantum finite automata (QFA) as defined in [

Definition 1. Namely, a 1-way QFA is a tuple

1) Q is a finite set of states,

2)

3)

4)

5)

The states in

A superposition of M is any element of

The computation of a QFA starts in the superposition

Theorem 1. Let L be any language recognized by 1QFA with bounded error. Then L is regular.

Proof. The proof is in [

Proposition 1. Given a language

Proof. The proof is in [

Theorem 2. The language

Proof. This is shown in [

The model of 2-way quantum finite state automaton (2QFA) is first introduced by Watrous [

Definition 2. Formally, a 2-way QFA is specified by 6-tuplet

1) Q is a finite set of states.

2)

3)

4)

5)

6)

The states in

The 2QFA satisfies the following conditions (of well-formedness) for any

1) Local probability and orthogonality condition

2) Separability condition I

3) Separability condition II

4) Separability condition III

In order to process an input word _{x} = #x$ and such a tape of length |x| + 2 is circular, i.e., the symbol to the right of $ is #.

For an integer n let C_{x} be the set (of size (n + 2)|Q|) of all possible configuration of M, for inputs of length x.

linearity. Consider the Hilbert space l_{2}(Q), where Q is the set of internal states of a 2QFA M. Suppose that we havea linear operator

and

M is well-formed when

Theorem 3. Every regular language is accepted by a 2QFA.

Proof. The proof has been shown in [

Multi-letter quantum finite state automata has been introduced in [

Definition 3. Formally, a k-letter QFA M is specified by a 5-tuple

1) Q is a finite set of states,

2)

3) _{Q}.

4)

5)

and ^{n} denotes Euclidean space consisting of all n-dimensional complex vectors.

A k-letter QFA M works in the same way as an measure-once 1-way quantum finite state automaton [

To calculate the probability P_{M}(x) that a k-letter QFA accepts an input string

lows that for any

of all

and

which specifies the computing process of M for an input string x. They identify the states in Q with an orthonormal basis of the complex Euclidean space _{a} denote the projection operator on the subspace spanned by Q_{a} where

where

A k-head quantum finite automaton is a quantum finite automaton having a single read only input tape whose inscription is the input word in between two endmarkers. We define 1-way k-head QFA where k heads of the automaton can move to the right or stay on the current tape square but not beyond the endmarkers.

We show that 1QFA(2) is more powerful than 1RMFA(2).

Definition 4. A 1-way multihead quantum finite state automaton is a automaton

1) Q is a finite set of states,

2)

3)

4)

5)

^{n} denotes Euclidean space consisting of all n-dimensional complex vectors. So

a mapping of the form

A superposition of M is any element in the Hilbert space l_{2}(Q). For_{2}(Q) can be expressed as a linear combination of vectors.

The transition function ^{st} head, ^{nd} head and so on and moving the heads according to _{2}(Q) defined by

We require all

Consider the Hilbert space l_{2}(Q), where Q is the set of internal states of a 1QFA(k) M. Suppose that we have a linear operator

and

Here

and

for each

The input word w begin with # and ends with $. The input is accepted if and only if the computation halts in an accepting states. It halts when the transition function is not defined for the current situation. In all other cases the input is rejected.

In these section we write transition matrices of different automaton and discuss different properties of these automaton in terms of their transition matrices.

1) Deterministic finite state automaton

A deterministic finite automaton [

1) Q is a finite set of states,

2)

3)

4)

5)

We design a deterministic finite state automaton

The transition matrix of the deterministic finite state automaton is shown in

Here each row of each transition matrix contain exactly one non-zero entry i.e. 1 for deterministic finite state automaton.

2) Non-deterministic finite state automaton

An non-deterministic finite automaton [

1) Q is a finite set of states,

2)

3)

4)

5)

The only difference between an non-deterministic finite state automaton and deterministic finite state auto- maton is the value of

The transition matrix of the deterministic finite state automaton is shown in

There is atleast one row in a transition matrix for non-deterministic automaton which contain more than one non-zero entry.

3) Reversible finite state automaton

An automaton

1) Q is a finite set of states,

2)

3)

4)

5)

A reversible automaton is a finite automaton in which each letter induces a partial one-to-one map from the set of states into itself. A reversible automaton may have several initial or final states. As a consequence, the minimal automaton of a reversible language may not reversible.

We define reversible automaton

The transition matrix of the above automaton is shown in

In case of reversible automaton dot product of any two row is zero and there are no cycles within the transi- tion/output matrix that can’t accessed from one of the input states.

4) Probabilistic finite state automaton

A probabilistic finite state automaton [

1) Q is a finite set of states,

2)

3)

4)

In case of probabilistic finite state automaton we allow the fractional values in transition matrix with the provision that sum of each row give 1 [see

5) Quantum finite state automaton

We consider 1-way quantum finite state automata (QFA) as defined in [

1) Q is a finite set of states,

2)

3)

4)

5)

The states in

The transition matrix of the quantum finite state automaton looks like [

The transition matrix is unitary since the sum of the squares of the norms in each row adds up to 1 and the dot product of any two row is 0.If all matrices only have 0 or 1 entries and the matrices are unitary,then the automaton is deterministic and reversible.

6) 1-way multihead deterministic finite state automaton

A 1-way k-head deterministic finite state automaton is a deterministic finite state automaton with k- independent read heads on a single input tape with the end markers. On each move the machine can si- multaneously read the k input cells scanned by k-heads,move each head one square to the right or keep stationary.

A 1-way multihead deterministic finite state automaton (1DFA(k)) [

1) Q is a finite set of states,

2)

3)

4)

5)

6)

7)

We define a 1DFA(2)

The transition matrix of the above automaton is [see

Each row of transition matrix contain only one 1 which has the same property as deterministic finite state automaton.

7) 1-way Reversible multihead finite state automaton

A 1-way reversible multihead finite state automaton (1REV-DFA(k)) [

1) Q is a finite set of states,

2)

3)

4)

square to the right and 0 means to keep the head on the current square,

5)

6)

7)

Let M be a 1DFA(k) and D be the set of all reachable configuration that occur in any computation of M beginning with an initial configuration and

1) For any two transitions:

and

it holds if

2) There is at most one transition of the form

The non-context free language

The transition matrix of the above automaton is shown in

Dot product of any two row is zero for multihead reversible finite state automaton.

8) 1-way multihead quantum finite state automaton

1-way multihead quantum finite state automaton is a 1-way k-head quantum finite state automaton where k-heads of the automaton can move to the right or stay on the current tape square but not beyond the end markers.The language

The transition matrix of the above automaton is shown in

The sum of the square of the norms in each row adds up to 1 and dot product of any two row is zero formultihead quantum finite state automaton.

In this section we show that 1QFA(k) has more language recognizing power than 1QFA. 1QFA(2) can recognize regular language

Theorem 4. 1QFA(2) can accept all unary regular languages.

Proof. In [

finitestate automaton where the transition matrices are only 0 and 1 entry, it is essentially a 1-way reversible multihead finite state automaton. So 1-way 2-head quantum finite state automaton accept all unary language.

Example 1. A 1-way 2-head quantum finite state automaton is a automaton

Let,

Define:

The automaton acts as follows: Initially both heads of the automaton M are at #. After reading the input symbols, the automaton M remain at state

For both cases the automaton M move to state

Consider a string w not in L. As w is not in L the heads of the automaton M will arrived in such a way that for that particular position of heads and state, no transition rules are defined. So, for a string w, which M does not accept, there is no sequence of transitions that makes M to its final state after consumption of w. So, M rejects with probability 1. Each pairs of

Example 2. A 1-way 2-head quantum finite state automaton is a automaton

Let,

Define:

The automaton acts as follows: Initially both heads of the automaton M are at #. After reading the input symbols, the automaton M remains at

Consider a string w not in L. As w is not in L the heads of the automaton M will arrived in such a way that for that particular position of heads and state, no transition rules are defined. So, for a string w, which M does not accept,there is no sequence of transitions that makes M to its final state after consumption of w. So, M rejects with probability 1. Condition of unitarity is satisfied for all pairs of

Theorem 5. 1QFA(2) is more powerful than 1QFA with respect to recognition of language.

Proof. In Theorem 2 it was proved that the language

Theorem 6. Given a language

Proof. In [

Example 3. A 1-way 2-head quantum finite state automaton is a automaton

Let,

Define:

The automaton acts as follows: at each reading of the symbol

position of two states

each reading of the symbol the automaton guesses x to be the first character which does not match. Thus if the guess is rightthen the path corresponding to ^{th} letter does not match with theend.

Then at the k^{th} depth

palindrome and does notbelong to L then no matter what depth we traversed

Theorem 7.

Proof. The language

Theorem 8. For every 1-way reversible 2-head finite state automaton M which accepts a language L, thereexists a 1-way 2-head quantum finite state automaton M’ which accepts the same language L.

Proof. We know that the transition matrix of 1-way reversible multihead finite state automaton has the following properties:

1) Dot product of any two row is zero for 1-way reversible multihead finite state automaton.

2) All matrices only have 0 or 1 entries.

Therefore the above two properties of the transition matrix ensures that the transition matrix is also unitary. As a result given a 1-way reversible 2-head finite state automaton M we get a 1-way 2-head quantum finite state automaton M’ which has the same transition matrix,same set of states, same set of accepting states and start state as M. as the transition matrix, start state and accepting states of M and M’ are same,they accept the same language.

Theorem 9. The set of all languages accepted by 1-way reversible 2-head finite state automata (1RMFA(2)) is a proper subset of set of all language accepted by 1-way 2-head quantum finite state automata. (1QFA(2))

Proof. Theorem 8 tells us that for every 1RMFA(2) which accept a language L there exist 1QFA(2) which accept the same language. So, the set of all languages accepted by 1RMFA(2) is a subset of set of all languages accepted by 1QFA(2). From ( [

Corollary 1. 1QFA(2) is computationally more powerful than 1RMFA(2).

In this paper, we studied characteristics of 1QFA(k) with their language accepting capability. There are still many non-regular context free context sensitive languages accepted by 1QFA(k) other than shown in this paper. We show that

Research of Debayan Ganguly is funded by the Council of Scientific Industrial Research (CSIR). This support is greatly appreciated.

Debayan Ganguly,Kingshuk Chatterjee,Kumar Sankar Ray, (2016) 1-Way Multihead Quantum Finite State Automata. Applied Mathematics,07,1005-1022. doi: 10.4236/am.2016.79088