Time Varying Splicing Recognizers

Abstract

In this work, we construct splicing recognizers of radius 2 capable of accepting recursively enumerable languages. Cavaliere, Jonoska and Leupold introduced the concept of splicing recognizers in 2006 and constructed splicing recognizers of radius 4 which can accept recursively enumerable languages. It was conjectured that the radius of splicing recognizers, accepting recursively enumerable languages can be decreased. We prove in this paper that the conjecture is true and in fact, radius 2 is sufficient. We also introduce the concept of time varying splicing recognizers and show that these recognizers of radius 1 are capable of accepting recursively enumerable languages.

Appendix

Extended proof of Theorem 2

Simulation of the transition function $r_R: (q_1, a) \rightarrow (q_2, c, +)$:

When the transition function $r_R : (q_{1}, a) \rightarrow (q_{2}, c, +)$ is simulated completely, the marked string, $o_{l}w_{1}(a, q_{1})a_{i}o_{r}$ is transformed into the string $o_{l}w_{1}c$ $(a_{i}, q_{2})o_{r}$ where $w_{1} \in \varGamma ^{+}, a, a_{i}, c \in \varGamma , q_{1}, q_{2} \in Q.$ Similarly, after simulation of the transition function $r_L: (q_1, a) \rightarrow (q_2, c, -)$, the marked string $o_la_i (a, q_1) a_k o_r$ is transformed into $o_l (a_i, q_2) c a_k o_r$.

The above simulation starts with the application of the rule $r_{10}$ to the marked string $o_l w (a, q_1) a_i o_r$. Rule $r_{10}$ adds $Y_{(a, q_{1}) (a_{i}, q_{2}) }^{[r_R]}$ to the left of the right marker $o_r^{'}$ replacing $a_{i}$.

Rule $r_{11}$ adds $Y_{(a, q_{1}) ( a_{i}, q_{2}) }^{[r_R]}$ to the right of the left marker.

Rule $r_{12}$ removes $Y_{(a, q_{1}) ( a_{i}, q_{2}) }^{[r_R]}$ from the left of the right marker and then $r_{13}$ adds $Y_{(a, q_{1})}^{[r_R]}$ in the same place replacing $(a, q_1)$.

Rule $r_{14}$ and $r_{15}$ add $Y_{(a, q_{1})}^{[r_R]}$ to the right of $o_{l}$ and remove $Y_{(a, q_{1})}^{[r_R]}$ from the left of $o_{r}$ respectively.

Similarly, rules $r_{16}, r_{17}$ and $r_{18}$ add $Y_{c}^{[r+1_R]}$ to the left of $o_{r}$, to the right of $o_{l}$ and again replace $Y_{c}^{[r + 1_R]}$ from the left of $o_{r}$ by c respectively.

Again, $r_{19}, r_{20}, r_{21}, r_{22}$ add $Y_{(a_{i}, q_{2})}^{[r+1_R]}$ to the left of $o_{r}$, remove $Y_{c}^{[r+1_R]}$ from the right of $o_{l}$, remove $Y_{(a, q_{1})}^{[r]}$ from the right of $o_{l}$ and remove $Y_{(a, q_{1}) (a_{i} q_{2})}$ from the right of $o_{l}$ respectively.

After application of the rule $r_{23}$, $Y_{(a_{i}, q_{2})}^{[r + 1_R]}$ is replaced by $(a_{i}, q_{2})$.

Application of the rules from $r_{10}$ to $r_{23}$ in order to the marked string $o_{l}w_{1}(a, q_{1})a_{i}o_{r}$ where $w_{1} \in \varGamma ^{+}, a, a_{i} \in \varGamma , q_{1}, q_{2} \in Q$ generates the following marked strings:

• $o_{l}w_{1}(a, q_{1}) Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}o_{r},$

• $o_{l}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1}$ $(a, q_{1})Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}o_{r},$

• $o_{l}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1}(a, q_{1})o_{r},$

• $o_{l}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1}Y_{(a, q_{1})}^{[r_R]}o_{r}$,

• $o_{l}Y_{(a, q_{1})}^{[r_R]}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1} Y_{(a, q_{1})}^{[r_R]} o_{r},$

• $o_{l}Y_{(a, q_{1})}^{[r_R]}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1} o_{r},$

• $o_{l}Y_{(a, q_{1})}^{[r_R]}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1} (a, q_{1}) Y_{c}^{[r+1_R]}o_{r},$

• $o_{l}Y_{c}^{[r+1_R]}Y_{(a, q_{1})}^{[r_R]}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1}Y_{c}^{[r+1_R]}o_{r},$

• $o_{l}Y_{c}^{[r+1_R]}Y_{(a, q_{1})}^{[r_R]}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1}co_{r},$

• $o_{l}Y_{c}^{[r+1_R]}Y_{(a, q_{1})}^{[r_R]}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1}c Y_{(a_{i}, q_{2})}^{[r+1_R]}o_{r},$

• $o_{l}Y_{(a, q_{1})}^{[r_R]}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1}c Y_{(a_{i}, q_{2})}^{[r+1_R]}o_{r},$

• $o_{l}Y_{(a, q_{1}) (a_{i}, q_{2})}^{[r_R]}w_{1}c Y_{(a_{i}, q_{2})}^{[r+1_R]}o_{r},$

• $o_{l}w_{1}c Y_{(a_{i}, q_{2})}^{[r+1_R]}o_{r},$

• $o_{l}w_{1}c (a_{i}, q_{2})o_{r},$ where $w_{1} \in \varGamma ^{+}, a_{i}, c \in \varGamma , q_{2} \in$ Q.

Simulation of the transition function $r_L: (q_1, a) \rightarrow (q_2, c, -)$: The transition function $r_L : (q_{1}, a) \rightarrow (q_{2}, c, -)$ is also simulated in the similar manner and the marked string $o_{l}w_{1}a_{i}(a, q_{1})a_{j}o_{r}$ is transformed into $o_{l}w_{1}(a_{i}, q_{2})ca_{j}o_{r}$ where $w_{1} \in \varGamma ^{*}, a_{i}, a_{j}, a, c \in \varGamma , q_{1}, q_{2} \in Q$ after simulation is complete.

The rules $r_{24}, r_{25}$ add $Z_{(a_{i}, q_{2}) (a, q_{1}) a_{j}}^{[r_L]}$ to the left of $o_{r}$ and to the right of $o_{l}$ respectively.

Rules $r_{26}, r_{29}, r_{32}$ remove $Z_{(a_{i}, q_{2}) (a, q_{1}) a_{j}}^{[r_L]}$, $Z_{(a_{i}, q_{2}) (a, q_{1}) }^{[r_L]}$, $Z_{a_{i}}^{[r_L]}$ from the left of $o_{r}$ respectively.

Similarly, rules $r_{34}, r_{35}, r_{42}, r_{43}, r_{44}$ remove $Z_{a_{i}}^{[r_L]},$ $Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}, Z_{c}^{[r+1]}, Z_{(a_{i}, q_{2})}^{[r+1_L]},$ $Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}$ from the right of $o_{l}$ respectively.

Rules $r_{27}, r_{30}, r_{37}, r_{40}$ replaces $(a, q_{1}), a_{i}, Z_{(a_{i}, q_{2})}^{[r_L]},$ $Z_{c}^{[r+1_L]}$ by $Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}, Z_{a_{i}}^{[r_L]},$ $(a_{i}, q_{2}), c, a_{j}$ respectively in the left of $o_{r}.$

Also, rules $r_{28}, r_{31}, r_{36}$ and $r_{39}$ add $Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}, Z_{a_{i}}^{[r_L]},$ $Z_{(a_{i}, q_{2})}^{[r+1_L]}$ and $Z_{c}^{[r+1_L]}$ to the right of $o_{l}$ respectively.

The rules $r_{33}, r_{38}$ and $r_{41}$ add $Z_{(a_{i}, q_{2})}^{[r_L]}, Z_{c}^{r+1_L]}$ and $Z_{a_{j}}^{[r+1_L]}$ to the left of $o_{r}$ respectively.

Rule $r_{45}$ replaces $Z_{a_j}^{[r+1_L]}$ by $a_j$ in the left of $o_r$.

Hence if the rules from $r_{24}$ to $r_{45}$ are applied in order the transition function $r_L : (q_{1}, a) \rightarrow (q_{2}, c, -)$ is simulated.

The marked strings obtained after application of the rules from $r_{24}$ to $r_{45}$ in order to the string $o_{l}w_{1}a_{i}$ $(a, q_{1})a_{j}o_{r}$ where $w_{1} \in \varGamma ^{*}, a_{i}, a_{j}, a, c \in \varGamma , q_{1} \in Q$ are as follows:

• $o_{l}w_{1}a_{i}(a, q_{1})Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}a_{i}(a, q_{1})Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}a_{i}(a, q_{1})o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}a_{i}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}a_{i}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}a_{i}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}Z_{a_{i}}^{[r_L]}o_{r}$

• $o_{l}Z_{a_{i}}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}Z_{a_{i}}^{[r_L]}o_{r}$

• $o_{l}Z_{a_{i}}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}o_{r}$

• $o_{l}Z_{a_{i}}^{[r]}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}Z_{(a_{i}, q_{2})}^{[r+1_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})}^{[r_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}Z_{(a_{i}, q_{2})}^{[r+1_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}Z_{(a_{i}, q_{2})}^{[r+1_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})}^{[r+1_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}Z_{(a_{i}, q_{2})}^{[r+1_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})}^{[r+1_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}(a_{i}, q_{2})o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})}^{[r+1_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}(a_{i}, q_{2})Z_{c}^{[r+1_L]}o_{r}$

• $o_{l}Z_{c}^{[r+1_L]}Z_{(a_{i}, q_{2})}^{[r+1_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}(a_{i}, q_{2})Z_{c}^{[r+1_L]}o_{r}$

• $o_{l}Z_{c}^{[r+1_L]}Z_{(a_{i}, q_{2})}^{[r+1_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}(a_{i}, q_{2})co_{r}$

• $o_{l}Z_{c}^{[r+1_L]}Z_{(a_{i}, q_{2})}^{[r+1_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}(a_{i}, q_{2})cZ_{a_{j}}^{[r+1_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})}^{[r+1_L]}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}(a_{i}, q_{2})cZ_{a_{j}}^{[r+1_L]}o_{r}$

• $o_{l}Z_{(a_{i}, q_{2})(a, q_{1})a_{j}}^{[r_L]}w_{1}(a_{i}, q_{2})cZ_{a_{j}}^{[r+1_L]}o_{r}$

• $o_{l}w_{1}(a_{i}, q_{2})cZ_{a_{j}}^{[r+1_L]}o_{r}$

• $o_{l}w_{1}(a_{i}, q_{2})ca_{j}o_{r}$, where $w_{1} \in \varGamma ^{*}, a_{i}, a_{j}, a \in \varGamma , q_{1}, q_{2}$ $\in Q.$

Extended proof of Theorem 3

Simulation of the transition function $r_R: (q_1, a) \rightarrow (q_2, c, +)$: $r_R : (q_{1}, a) \rightarrow (q_{2}, c, +)$ is simulated by application of the rules from $r_{15}$ to $r_{26}$ in order to the marked string $o_{l}w_{1}(a, q_{1})a_{j}o_{r}, w_{1} \in \varGamma ^{+}, a, a_{j} \in \varGamma , q_{1} \in Q$ and the following marked strings are generated:

• $o_{l}Y_{c(a_{j}, q_{2})}^{[r_R]}(a, q_{1})a_{j}o_{r}$ (The string $o_{l}w_{1}X_{1}^{[r_R]}$ is also produced with it. In the next step they are spliced together.)

• $o_{l}w_{1}Y_{c(a_{j}, q_{2})}^{[r_R]}(a, q_{1})a_{j}o_{r}$

• $o_{l}w_{1}Y_{c(a_{j}, q_{2})}^{[r_R]}o_{r}$

• $o_{l}Y_{c(a_{j}, q_{2})}^{[r_R]}w_{1}Y_{c(a_{j}, q_{2})}^{[r_R]}o_{r}$

• $o_{l}Y_{c}^{[r_R]} Y_{c(a_{j}, q_{2})}^{[r_R]}w_{1}Y_{c(a_{j}, q_{2})}^{[r_R]}o_{r}$

• $o_{l}Y_{c}^{[r_R]} Y_{c(a_{j}, q_{2})}^{[r_R]}w_{1}o_{r}$

• $o_{l}Y_{c}^{[r_R]} Y_{c(a_{j}, q_{2})}^{[r_R]}w_{1}Y_{c}^{[r_R]}o_{r}$

• $o_{l} Y_{c(a_{j}, q_{2})}^{[r_R]}w_{1}Y_{c}^{[r_R]}o_{r}$

• $o_{l} Y_{c(a_{j}, q_{2})}^{[r_R]}w_{1}co_{r}$

• $o_{l} Y_{c(a_{j}, q_{2})}^{[r_R]}w_{1}cY_{(a_{j}, q_{2})}^{[r_R]}o_{r}$

• $o_{l}w_{1}cY_{(a_{j}, q_{2})}^{[r_R]}o_{r}$

• $o_{l}w_{1}c(a_{j}, q_{2})o_{r}$.

Simulation of the transition function $r_L: (q_1, a) \rightarrow (q_2, c, -)$: The transition function $r_L: (q_{1}, a) \rightarrow (q_{2}, c, -)$ is simulated by application of the rules from $r_{27}$ to $r_{49}$ in order. But unlike the previous case, different rules are required to simulate $r_L: (q_{1}, a) \rightarrow (q_{2}, c, -)$ depending on the length of the strings between the markers, i.e., if the string between the markers while simulating the transition function is of length 3, then rules $r_{27}, r_{29}, r_{31}, r_{32}, r_{34}, r_{35},$ $r_{37}, r_{39}, r_{40},$ $r_{42},$ $r_{43}, r_{44}, r_{45}, r_{46},$ $r_{47}, r_{48}, r_{49}$ are applied in order. If the string is of length greater than 3, then the rules $r_{28}, r_{30}, r_{31},$ $r_{33}, r_{34},$ $r_{35}, r_{36}, r_{38}, r_{41},$ $r_{42}, r_{43},$ $r_{44}, r_{45},$ $r_{46}, r_{47}, r_{48}, r_{49}$ are applied in order. In both cases, strings of the form $o_{l}w_{1}a_{i}(a, q_{1})a_{j}o_{r}$ are transformed into $o_{l}w_{1}$ $(a_{i}, q_{2})ca_{j}o_{r}$ where $w_{1} \in \varGamma ^{*}, a, c,$ $a_{i}, a_{j} \in \varGamma , q_{1}, q_{2} \in Q.$

So the marked strings generated during simulation of the transition function $r_L: (q_{1}, a) \rightarrow (q_{2}, c, -)$ are mentioned below:

• $o_l Z_{(a_i, q_2)ca_k}^{[r_L]} a_i (a, q_1) a_k o_r$

• $o_l w_1 Z_{(a_i, q_2) c a_k}^{[r_L]} a_i (a, q_1) a_k o_r$

• $o_l w_1 Z_{(a_i, q_2) c a_k}^{[r_L]} o_r$

• $o_l Z_{(a_j, q_2)}^{[r_L]} w_1 Z_{(a_i, q_2) c a_k}^{[r_L]} o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} w_1 Z_{(a_i, q_2) c a_k}^{[r_L]}o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} w_1 Z_{(a_i, q_2) c a_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} o_r$

• $o_l Z_{(a_i, q_2) c a_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} w_1 Z_{(a_i, q_2)}^{[r_L]} o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} w_1 Z_{(a_i, q_2)}^{[r_L]} o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} w_1 (a_i, q_2) o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} w_1 (a_i, q_2) Z_{ca_k}^{[r_L]} o_r$

• $o_l Z_{(a_i, q_2)}^{[r_L]} w_1 (a_i, q_2) Z_{ca_k}^{[r_L]} o_r$

• $o_l w_1 (a_i, q_2) Z_{ca_k}^{[r_L]} o_r$

• $o_l Z_{a_k}^{[r_L]} w_1 (a_i, q_2) Z_{ca_k}^{[r_L]} o_r$

• $o_l Z_{a_k}^{[r_L]} w_1 (a_i, q_2) c o_r$

• $o_l Z_{a_k}^{[r_L]} w_1 (a_i, q_2) c Z_{a_k}^{[r_L]} o_r$

• $o_l w_1 (a_i, q_2) c Z_{a_k}^{[r_L]} o_r$

• $o_l w_1 (a_i, q_2) c a_k o_r$

If the marked string is $o_l a_i (a, q_1) a_k o_r$, then the following marked strings are generated during simulation.

• $o_l Z_{(a_i, q_2)ca_k}^{[r_L]} a_i (a, q_1) a_k o_r$

• $o_l Z_{(a_i, q_2) c a_k}^{[r_L]} a_i (a, q_1) a_k o_r$

• $o_l Z_{(a_i, q_2) c a_k}^{[r_L]} o_r$

• $o_l Z_{(a_i, q_2)}^{[r_L]} Z_{(a_i, q_2) c a_k}^{[r_L]} o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} Z_{(a_i, q_2) c a_k}^{[r_L]}o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} Z_{(a_i, q_2) c a_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} o_r$

• $o_l Z_{(a_i, q_2) c a_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} (a_i, q_2) o_r$

• $o_l Z_{ca_k}^{[r_L]} Z_{(a_i, q_2)}^{[r_L]} (a_i, q_2) Z_{ca_k}^{[r_L]} o_r$

• $o_l Z_{(a_i, q_2)}^{[r_L]} (a_i, q_2) Z_{ca_k}^{[r_L]} o_r$

• $o_l (a_i, q_2) Z_{ca_k}^{[r_L]} o_r$

• $o_l Z_{a_k}^{[r_L]} (a_i, q_2) Z_{ca_k}^{[r_L]} o_r$

• $o_l Z_{a_k}^{[r_L]} (a_i, q_2) c o_r$

• $o_l Z_{a_k}^{[r_L]} (a_i, q_2) c Z_{a_k}^{[r_L]} o_r$

• $o_l (a_i, q_2) c Z_{a_k}^{[r_L]} o_r$

• $o_l (a_i, q_2) c a_k o_r$