Repetitive Finite Automata With Translucent Letters

While a finite automaton reads its input strictly from left to right, letter by letter, by now many types of automata have been considered in the literature that process their inputs in a different, more involved way. Under this aspect, the most extreme is the jumping finite automaton of Meduna and Zemek [8] (see also [6]), which, after reading a letter, jumps to an arbitrary position of the remaining input. It is known that the jumping finite automaton accepts languages that are not even context-free, like the language $\{\,w\in\{a,b,c\}^{*}\mid|w|_{\_}a=|w|_{\_}b=|w|_{\_}c\,\}$, but at the same time, it does not even accept the finite language $\{ab\}$.

Another example is the restarting automaton as introduced by Jančar, Mráz, Plátek, and Vogel in [7], which processes a given input in cycles. In each cycle, a restarting automaton scans its tape contents from left to right, using a window of a fixed finite size, until it executes a delete/restart operation. Such an operation deletes one or more letters from the current contents of the window, returns the window to the left end of the tape, and resets the automaton to its initial state. If a window of size larger than one is used, these so-called R-automata accept a proper superclass of the regular languages that is incomparable to the context-free and the growing context-sensitive languages with respect to inclusion (see, e.g., [18]). However, with a window of size one, the R-automata accept exactly the regular languages [9].

Finally, there is the (deterministic and nondeterministic) finite automaton with translucent letters (or DFAwtl and NFAwtl) of Nagy and Otto [14], which is equivalent to a cooperating distributed system of stateless deterministic R-automata with windows of size one. For each state $q$ of an NFAwtl, there is a set $\tau(q)$ of translucent letters, which is a subset of the input alphabet that contains those letters that the automaton cannot see when it is in state $q$. Accordingly, in each step, the NFAwtl just reads (and deletes) the first letter from the left which it can see, that is, which is not translucent for the current state. Here, it is important to notice that, in each step, an NFAwtl reads the current tape contents from the very left, that is, after deleting a letter, it returns its head to the first letter of the remaining tape contents. It has been shown that the NFAwtl accepts a class of semi-linear languages that properly contains all rational trace languages, while its deterministic variant, the DFAwtl, is properly less expressive. In fact, the DFAwtl just accepts a class of languages that is incomparable to the rational trace languages with respect to inclusion [13, 15, 16, 17]. Although the NFAwtl is quite expressive, it cannot even accept the deterministic linear language $L_{\_}{2}=\{\,a^{n}b^{n}\mid n\geq 0\,\}$, as such an automaton cannot possibly compare the number of occurrences of the letter $a$ with the number of occurrences of the letter $b$ and ensure, at the same time, that all $a$’s precede the first $b$.

To make up for this shortcoming, a variant of the finite automaton with translucent letters has been proposed in [10], which, after reading and deleting a letter, does not return its head to the first letter of the remaining tape contents, but that rather continues from the position of the letter just deleted. This means that, in general, only a scattered subword of the input has been read and deleted before the head reaches the end of the input. If the computation is now required to halt, either accepting or rejecting, then it can easily be shown that this type of automaton just accepts the class of regular languages. For the right one-way jumping finite automaton of [2, 4], this problem is overcome by cyclically shifting all the translucent letters that are encountered during a computation to the end of the current tape contents. In this way, these letters may be read and deleted at a later stage of the computation. For the type of automaton proposed in [10], a different approach was taken. When the head of the automaton reaches the end of the input, which is marked by a special end-of-tape marker, then the automaton can decide whether to accept, reject, or continue, which means that it changes its state and again reads the remaining tape contents from the beginning.

It has been established that this type of automaton, called a non-returning finite automaton with translucent letters or an nr-NFAwtl, is strictly more expressive than the NFAwtl. This result also holds for the deterministic case, although the deterministic variant, the nr-DFAwtl, is still not sufficiently expressive to accept all rational trace languages. In [11], the nr-DFAwtl and the nr-NFAwtl are compared to the jumping finite automaton, the right one-way jumping finite automaton of [2, 4], and the right-revolving finite automaton of [3], deriving the complete taxonomy of the resulting classes of languages. As it turns out, the nr-DFAwtl can be seen as an extension of the right one-way jumping finite automaton that can detect the end of its input.

When we look at the above description of the generalization of the NFAwtl to the nr-NFAwtl, then we realize that this generalization actually consists of two steps:

• •

  The head of the automaton does not return to the left end of the tape after a letter has been read and deleted. This is the non-returning property (in contrast to the returning property of the NFAwtl).

• •

  Once the end-of-tape marker is reached, an nr-NFAwtl may execute a step that changes its state and returns its head to the left end of the tape. We call this the property of being repetitive (in contrast to the non-repetitiveness of the NFAwtl, which must immediately halt as soon as its head reaches the end-of-tape marker).

In this paper, we study the influence that these two properties have on the expressive capacity of the finite automaton with translucent letters in detail. As we consider both, the deterministic and non-deterministic variants of the resulting types of automata, we obtain eight different classes of automata with translucent letters. In addition, we also include the (deterministic and the non-deterministic) right one-way jumping finite automaton in our study. We shall derive the complete taxonomy of the resulting language classes, which will nicely illustrate the effects that the non-returning property and the repetitiveness have.

Actually, we shall only encounter one new language class that has not been considered before: the class $\mathcal{L}(\mbox{\sf RDFAwtl})$ of languages that are accepted by repetitive DFAwtls. After presenting the necessary notation and definitions in Section 2, we shall present our results on repetitive DFAwtls and repetitive NFAwtls in Section 3. Here, it turns out that the repetitive DFAwtl (the RDFAwtl) is strictly more expressive than the DFAwtl, while its nondeterministic variant, the repetitive NFAwtl (the RNFAwtl), is equivalent to the NFAwtl. Then, in Section 4, we consider closure and non-closure properties for the language class $\mathcal{L}({\sf RDFAwtl})$. Finally, in the concluding section, we address the membership problem and some other decision problems for the RDFAwtl in short, and we state some open problems.

First, we restate the definition of the nondeterministic finite automaton with translucent letters and its deterministic variant, the DFAwtl, from [14].

A configuration of $A$ is a word from the set $Q\cdot\Sigma^{*}\cdot\lhd\,\cup\,\{{\sf Accept},{\sf Reject}\}$. A configuration of the form $qw\cdot\lhd$, where $q\in Q$ and $w\in\Sigma^{*}$, expresses the situation that $A$ is in state $q$, its tape contains the word $w$ followed by the sentinel $\lhd$, and the head of $A$ is on the first letter of $w\cdot\lhd$. For an input word $w\in\Sigma^{*}$, a corresponding initial configuration is of the form $q_{\_}0w\cdot\lhd$, where $q_{\_}0\in I$. The NFAwtl $A$ induces the following single-step computation relation on its set of configurations:

Thus, in each step, $A$ reads and deletes the first letter from the left that is not translucent for the current state. In addition, if all letters on the tape are translucent for the current state, then $A$ halts, and it accepts if the current state is final. A word $w\in\Sigma^{*}$ is accepted by $A$ if there exists an initial state $q_{\_}0\in I$ and a computation $q_{\_}0w\cdot\lhd\vdash_{\_}A^{*}{\sf Accept}$, where $\vdash_{\_}A^{*}$ denotes the reflexive transitive closure of the single-step computation relation $\vdash_{\_}A$. Now, $L(A)=\{\,w\in\Sigma^{*}\mid w\mbox{ is accepted by }A\,\}$ is the language accepted by $A$, and $\mathcal{L}({\sf NFAwtl})$ denotes the class of all languages that are accepted by NFAwtls. Analogously, $\mathcal{L}({\sf DFAwtl})$ denotes the class of all languages that are accepted by DFAwtls.

Next, we define the first of the two possible extensions of the automaton with translucent letters that we mentioned above.

Obviously, each NFAwtl can easily be turned into an RNFAwtl for the same language. Of course, the same applies to a DFAwtl, giving an RDFAwtl for the same language. Thus, we have the following inclusion relations.

The following simple example illustrates the way in which a repetitive finite automaton with translucent letters works.

Finally, we come to the second extension of the automaton with translucent letters that we mentioned in the introduction.

The following result states that the non-repetitive non-returning finite automata with translucent letters of Def. 5 are very weak in that they just accept the regular languages.

It was actually this observation that led us to define the nr-NFAwtl and the nr-DFAwtl in [10].

Thus, an nr-NFAwtl is both at the same time, non-returning in the sense of Def. 5 and repetitive in the sense of Def. 2. In particular, the nr-NFAwtl (nr-DFAwtl) should not be confused with the nr-nr-NFAwtl (nr-nr-DFAwtl) of Def. 5. The latter has been introduced here only to complete the picture. Theorem 6 above shows that they are not really interesting types of automata with translucent letters. Finally, we should also mention another related type of automaton, the right one-way jumping finite automaton.

Actually, as defined in [2, 4], the (N)ROWJFA does not have an end-of-tape marker, but it is obvious that our definition is equivalent to the original one. We just introduced this end-of-tape marker to ensure consistency with our other types of automata. We see that the (N)ROWJFA overcomes the problem of processing letters that are skipped over by cyclically shifting these letters to the right so that they can be read later. The following results are known concerning the various types of automata introduced above.

In addition, $\mathcal{L}(\mbox{\sf ROWJFA})$ is incomparable under inclusion to $\mathcal{L}(\mbox{\sf DFAwtl})$ and $\mathcal{L}(\mbox{\sf NFAwtl})$, and $\mathcal{L}(\mbox{\sf NROWJFA})$ is incomparable to $\mathcal{L}(\mbox{\sf DFAwtl})$, $\mathcal{L}(\mbox{\sf NFAwtl})$, and $\mathcal{L}(\mbox{\sf nr-DFAwtl})$. Thus, it remains to compare the deterministic and nondeterministic repetitive finite automata with translucent letters to these other types of automata.

We claim that the language $L_{\_}{\vee,c}=L(A_{\_}{\vee,c})$ of Example 4 is not accepted by any DFAwtl. As each DFAwtl can be simulated by an RDFAwtl, this shows that $\mathcal{L}({\sf DFAwtl})\subsetneq\mathcal{L}({\sf RDFAwtl})$.

Let $n>2m$, and let $w=a^{n}b^{n}c\in L_{\_}{\vee,c}$. Then the computation of $A$ on input $w$ is accepting, that is, it is of the form

where $w_{\_}r\in(\tau(q_{\_}{i_{\_}r}))^{*}$ and $q_{\_}{i_{\_}r}\in F$. If $|w_{\_}r|_{\_}a>0$, then $A$ would also accept on input $a^{n+1}b^{n}c\not\in L_{\_}{\vee,c}$, if $|w_{\_}r|_{\_}b>0$, then $A$ would also accept on input $a^{n}b^{n+1}c\not\in L_{\_}{\vee,c}$, and if $|w_{\_}r|_{\_}c>0$, then $A$ would also accept on input $a^{n}b^{n}\not\in L_{\_}{\vee,c}$. Hence, it follows that $w_{\_}r=\lambda$, that is, the accepting computation above consists of $2n+1$ transition steps, each of which deletes a letter, and the final accepting step. In particular, the only occurrence of the letter $c$ is read and deleted during the above computation, that is, there exist an index $j$ and integers $r,s\geq 0$ such that $r+s=j$ and

We now distinguish several cases.

1. (1)

   Assume that $a,b\in\tau(q_{\_}{i_{\_}j})$. Then $A$ executes the following computation on input $a^{n}b^{2n}$:

   $$q_{\_}0a^{n}b^{2n}\cdot\lhd\vdash_{\_}A^{j}q_{\_}{i_{\_}j}a^{n-r}b^{2n-s}\cdot\lhd\vdash_{\_}A\left\{\begin{array}[]{ll}{\sf Accept},&\mbox{if }q_{\_}{i_{\_}j}\in F,\\ {\sf Reject},&\mbox{if }q_{\_}{i_{\_}j}\not\in F.\end{array}\right.$$

   As $a^{n}b^{2n}\in L_{\_}{\vee,c}$, we see that the latter computation must be accepting, that is, $q_{\_}{i_{\_}j}\in F$. Thus, $A$ can also execute the following accepting computation:

   $$q_{\_}0a^{r+s+1}b^{r+s+1}\cdot\lhd\vdash_{\_}A^{j}q_{\_}{i_{\_}j}a^{s+1}b^{r+1}\cdot\lhd\vdash_{\_}A{\sf Accept},$$

   which, however, contradicts the fact that $a^{r+s+1}b^{r+s+1}\not\in L_{\_}{\vee,c}$.

2. (2)

   Assume that $a\not\in\tau(q_{\_}{i_{\_}j})$, but $b\in\tau(q_{\_}{i_{\_}j})$. Then, $r=n$ and $s\leq n$, that is, $a^{n-r}b^{n-s}c=b^{n-s}c$. Now, $A$ executes the following computation on input $a^{n}b^{2n}$:

   $$q_{\_}0a^{n}b^{2n}\cdot\lhd\vdash_{\_}A^{j}q_{\_}{i_{\_}j}a^{n-r}b^{2n-s}\cdot\lhd=q_{\_}{i_{\_}j}b^{2n-s}\cdot\lhd\vdash_{\_}A\left\{\begin{array}[]{ll}{\sf Accept},&\mbox{if }q_{\_}{i_{\_}j}\in F,\\ {\sf Reject},&\mbox{if }q_{\_}{i_{\_}j}\not\in F.\end{array}\right.$$

   As $a^{n}b^{2n}\in L_{\_}{\vee,c}$, we see that the latter computation must be accepting, that is, $q_{\_}{i_{\_}j}\in F$. Thus, $A$ can also execute the following accepting computation:

   $$q_{\_}0a^{n}b^{3n+s}\cdot\lhd\vdash_{\_}A^{j}q_{\_}{i_{\_}j}b^{3n}\cdot\lhd\vdash_{\_}A{\sf Accept},$$

   which, however, contradicts the fact that $a^{n}b^{3n+s}\not\in L_{\_}{\vee,c}$.

3. (3)

   Assume that $b\not\in\tau(q_{\_}{i_{\_}j})$. Then $r\leq n$ and $s=n$, that is, $a^{n-r}b^{n-s}c=a^{n-r}c$. As $n>m$, there exist a state $q$ and integers $k_{\_}0,k_{\_}1,t_{\_}0\geq 0$ and $t_{\_}1\geq 1$ such that the accepting computation above has the form

   $$q_{\_}0a^{n}b^{n}c\cdot\lhd\vdash_{\_}A^{*}qa^{n-k_{\_}0}b^{n-t_{\_}0}c\cdot\lhd\vdash_{\_}A^{+}qa^{n-k_{\_}0-k_{\_}1}b^{n-t_{\_}0-t_{\_}1}c\cdot\lhd\vdash_{\_}A^{*}q_{\_}{i_{\_}j}a^{n-r}c\cdot\lhd\vdash_{\_}A^{*}{\sf Accept}.$$

   Hence, we also obtain the following accepting computation:

   $$q_{\_}0a^{n+k_{\_}1}b^{n+t_{\_}1}c\cdot\lhd\vdash_{\_}A^{*}qa^{n+k_{\_}1-k_{\_}0}b^{n+t_{\_}1-t_{\_}0}c\cdot\lhd\vdash_{\_}A^{+}qa^{n-k_{\_}0}b^{n-t_{\_}0}c\cdot\lhd\vdash_{\_}A^{*}{\sf Accept}.$$

   This implies that $a^{n+k_{\_}1}b^{n+t_{\_}1}c\in L_{\_}{\vee,c}$, which yields $k_{\_}1=t_{\_}1$.

   Now we consider the computations of $A$ on input $a^{n+\nu\cdot t_{\_}1}b^{n+\nu\cdot t_{\_}1}$ for all $\nu\geq 0$:

   $$q_{\_}0a^{n+\nu\cdot t_{\_}1}b^{n+\nu\cdot t_{\_}1}\cdot\lhd\vdash_{\_}A^{*}q_{\_}{i_{\_}j}a^{n-r}\cdot\lhd.$$

   As $a^{n+\nu\cdot t_{\_}1}b^{2n+2\nu\cdot t_{\_}1}\in L_{\_}{\vee,c}$, we see that the computation of $A$ that begins with the configuration $q_{\_}{i_{\_}j}a^{n-r}b^{n+\nu\cdot t_{\_}1}$ leads to acceptance for all $\nu\geq 0$. Hence, we obtain

   $$q_{\_}0a^{n}b^{n}b^{n+t_{\_}1}\cdot\lhd\vdash_{\_}A^{*}q_{\_}{i_{\_}j}a^{n-r}b^{n+t_{\_}1}\cdot\lhd\vdash_{\_}A^{*}{\sf Accept},$$

   but we have $a^{n}b^{n}b^{n+t_{\_}1}\not\in L_{\_}{\vee,c}$, as $t_{\_}1>0$, a contradiction.

As this covers all cases, we see that the language $L_{\_}{\vee,c}$ is indeed not accepted by any DFAwtl. $\Box$

On the other hand, it can be shown quite easily that the RDFAwtl (the RNFAwtl) is a special case of the nr-DFAwtl (the nr-NFAwtl).

Assume that $qw\cdot\lhd$ is a configuration of $A$, that is, $q\in Q$ and $w\in\Sigma^{*}$. From the definition of the computation relation $\vdash_{\_}A$, we see that there are four different cases that we must consider.

First, if $w=uav$ for some words $u\in(\tau(q))^{*}$, $v\in\Sigma^{*}$, and a letter $a\in(\Sigma\smallsetminus\tau(q))$, then $A$ executes a transition from $\delta(q,a)$.

• •

  If $p\in\delta(q,a)$, then $qw\cdot\lhd=quav\cdot\lhd\vdash_{\_}Apuv\cdot\lhd$ is a possible step of $A$. In this case, $B$ can execute the following sequence of steps:

  $$qw\cdot\lhd=quav\cdot\lhd\vdash_{\_}Bup^{\prime}v\cdot\lhd\vdash_{\_}Bpuv\cdot\lhd.$$

• •

  On the other hand, if $\delta(q,a)=\emptyset$, then $A$ halts and rejects. However, in this case, also $\delta_{\_}B(q,a)=\emptyset$, and hence, $B$ halts and rejects as well.

Finally, if $w\in(\tau(q))^{*}$, then $A$ halts.

• •

  If $\delta(q,\lhd)={\sf Accept}$, then $\delta_{\_}B(q,\lhd)={\sf Accept}$, too.

• •

  If $\delta(q,\lhd)=\emptyset$, then $A$ rejects. In this case, $\delta_{\_}B(q,\lhd)=\emptyset$, that is, $B$ halts and rejects as well.

It follows that $L(A)\subseteq L(B)$.

Conversely, if $w\in L(B)$, then it is easily verified that each accepting computation of $B$ on input $w$ is just a simulation of an accepting computation of $A$ on input $w$. Thus, we see that $L(B)=L(A)$.

Finally, the above definition of $B$ shows that $B$ is deterministic, if $A$ is. $\Box$

The language

is a rational trace language, and as such, it is accepted by an NFAwtl. However, as proved in [11], this language is not accepted by any nr-DFAwtl. Hence, $L_{\_}\vee$ is not accepted by any RDFAwtl, either. Thus, we immediately obtain the following non-inclusion result.

As defined above, an RNFAwtl $A=(Q,\Sigma,\lhd,\tau,I,\delta)$ may run into an infinite computation. Just assume that $q$ is a state of $A$, $w\in(\tau(q))^{*}$, and $q\in\delta(q,\lhd)$. Then $qw\cdot\lhd\vdash_{\_}Aqw\cdot\lhd\vdash_{\_}Aqw\cdot\lhd$, and so forth. However, we can avoid this by converting $A$ into an equivalent RNFAwtl $B$ as follows.