On homomorphic images of the Szilard languages of matrix insertion–deletion systems with matrices of size 2

Abstract

A Szilard language is a well-known tool in formal language theory to express the derivation process in a grammar system or grammar. Matrix InsDel (insertion–deletion) system is a well-known variant of InsDel system, where the idea of matrix control is combined with InsDel systems. The size of a matrix InsDel system is represented by a septuple of integers $(k; p, m, m^{'};$ $q, n, n^{'})$, where k represents maximum number of rules in a matrix, i.e., size of a matrix. The parameters p, m and $m^{'}$ represent the maximal length of the strings inserted, the maximal length of the left context and the maximal length of the right context of the insertion rules, respectively. The parameters $q, n, n^{'}$ represent the same for deletion rules. In this paper, we investigate the Szilard languages of matrix InsDel systems with matrices of size 2. We give examples of regular, context-free and context-sensitive languages which cannot be the Szilard language of any matrix InsDel system. We show that any regular language can be represented as a homomorphic image of Szilard language of matrix InsDel system of size (2; 2, 0, 0; 1, 0, 0). Any linear language, meta-linear language and rational (or regular) closure of linear language can be obtained as the homomorphic image of Szilard language of matrix InsDel systems of size (2; 1, 1, 0; 1, 1, 0), and (2; 1, 0, 1; 1, 0, 1). Moreover, any recursively enumerable language can be obtained as the homomorphic image of the Szilard language of matrix InsDel systems of size (2; 1, 1, 0; 1, 1, 1), (2; 1, 0, 1; 1, 1, 1), (2; 1, 1, 1; 1, 1, 0), (2; 1, 1, 1; 1, 0, 1), (2; 1, 1, 0; 2, 0, 0), (2; 1, 0, 1; 2, 0, 0), (2; 2, 0, 0; 1, 1, 0),  and (2; 2, 0, 0; 1, 0, 1).

Appendix

Theorem 23

$hSZMat_2INS_1^{0,1}DEL_1^{1, 1} = RE.$

Proof

From Theorems 6 and 22. $\square$

Theorem 24

$hSZMat_2INS_1^{1,1}DEL_1^{1,0} = RE.$

Proof

$A = \{\theta S_1 \# \$ \}$.

(II) For $r_i: X \rightarrow bY$:

if $b \in N^{''}$	if $b \in T_1$
$r_{i1}: [(\lambda , \lambda / r_{i_1}, X), (\#, \lambda / r_i^{1}, \$)]$	$r_{i1}^{b}: [(\theta , \lambda / r_{i_1}, X), (\#, \lambda / r_i^{1}, \$)]$
$r_{i2}: [(r_{i_1}, X / \lambda , \lambda ), (\#, \lambda / r_i^{2}, r_i^{1})]$	$r_{i2}^{b}: [(r_{i_1}, X / \lambda , \lambda ), (\#, \lambda / r_i^{2}, r_i^{1})]$
$r_{i3}: [(\lambda , \lambda / b, r_{i_1}), (\#, \lambda / r_i^{3}, r_i^{2})]$	$r_{i3}^{b}: [(\theta , \lambda / Y, r_{i_1}), (\#, \lambda / r_i^{3}, r_i^{2})]$
$r_{i4}: [(b, \lambda / Y, r_{i_1}), (r_i^{3}, r_i^{2} / \lambda , \lambda )]$	$r_{i4}^{b}: [(Y, r_{i_1} / \lambda , \lambda ), (r_i^{3}, r_i^{2} / \lambda , \lambda )]$
$r_{i5}: [ (r_i^{3}, r_i^{1} / \lambda , \lambda )]$	$r_{i5}^{b}: [(r_i^{3}, r_i^{1} / \lambda , \lambda ), (\#, r_i^{3} / \lambda , \lambda )]$
$r_{i6}: [(Y, r_{i_1} / \lambda , \lambda ), (\lambda , r_i^{3} / \lambda , \$)]$

$\square$

Theorem 25

$hSZMat_2INS_1^{1,1}DEL_1^{0,1} = RE.$

Proof

From Theorems 6 and 24. $\square$

Theorem 26

$hSZMat_2INS_1^{1,0}DEL_2^{0,0} = RE.$

Proof

$A = \{\theta H S_1 E\}$.

(II) For $r_i: X \rightarrow bY$:

if $b \in N^{''}$	if $b \in T_1$
$r_{i1}: [(\lambda , X / \lambda , \lambda ), (E, \lambda / r_i^{1}, \lambda )]$	$r_{i1}^{b}: [(\lambda , X / \lambda , \lambda ), (E, \lambda / r_i^{1}, \lambda )]$
$r_{i2}: [(H, \lambda / Y_1^{r_i}, \lambda ), (E, \lambda / \#_{r_i}, \lambda )]$	$r_{i2}^{b}: [(H, \lambda / Y_1^{r_i}, \lambda ), (E, \lambda / \#_{r_i}, \lambda )]$
$r_{i3}: [(E, \lambda / \#_{r_i}^{'}, \lambda ), (E, \lambda / r_i^{2}, \lambda )]$	$r_{i3}^{b}: [(E, \lambda / \#_{r_i}^{'}, \lambda ), (E, \lambda / r_i^{2}, \lambda )]$
$r_{i4}: [(E, \lambda / r_i^{3}, \lambda ), (\lambda , \#_{r_i}^{'} \#_{r_i} / \lambda , \lambda )]$	$r_{i4}^{b}: [(E, \lambda / r_i^{3}, \lambda ), (\lambda , \#_{r_i} \#_{r_i}^{'} / \lambda , \lambda )]$
$r_{i5}: [(H, \lambda / b, \lambda ), (\lambda , r_i^{2} / \lambda , \lambda )]$	$r_{i5}^{b}: [(\theta , \lambda / r_i^{'}, \lambda ), (\lambda , r_i^{2} / \lambda , \lambda )]$
$r_{i6}: [(\lambda , H / \lambda , \lambda ), (Y_1^{r_i}, \lambda / Y_2^{r_i}, \lambda )]$	$r_{i6}^{b}: [(\lambda , r_i^{'} H / \lambda , \lambda ), (Y_1^{r_i}, \lambda / Y_2^{r_i}, \lambda )]$
$r_{i7}: [(Y_2^{r_i}, \lambda / H, \lambda ), (\lambda , r_i^{3} r_i^{1} / \lambda , \lambda )]$	$r_{i7}^{b}: [(Y_2^{r_i}, \lambda / H, \lambda ), (\lambda , r_i^{3} r_i^{1} / \lambda , \lambda )]$
$r_{i8}: [(H, \lambda / Y, \lambda ), (\lambda , Y_1^{r_i}Y_2^{r_i} / \lambda , \lambda )]$	$r_{i8}^{b}: [(H, \lambda / Y, \lambda ), (\lambda , Y_1^{r_i}Y_2^{r_i} / \lambda , \lambda )]$

$\square$

Theorem 27

$hSZMat_2INS_1^{0,1}DEL_2^{0,0}= RE.$

Proof

From Theorems 6 and 26. $\square$

Theorem 28

$hSZMat_2INS_2^{0,0}DEL_1^{1,0}= RE.$

Proof

(II) For $r_i: X \rightarrow bY$:

if $b \in N^{''}$	if $b \in T_1$
$r_{i1}: [(\lambda , \lambda / bY, \lambda ), (Y, X / \lambda , \lambda )]$	$r_{i1}^{b}: [(\lambda , \lambda / r_{i_1}Y, \lambda ), (Y, X / \lambda , \lambda )]$
	$r_{i2}^{b}: [(\$, r_{i_1} / \lambda , \lambda )]$

$\square$

Theorem 29

$hSZMat_2INS_2^{0,0}DEL_1^{0,1} = RE.$

Proof

From Theorems 6 and 28. $\square$