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On Szilard languages of labelled insertion grammars *
Paul Prithwineel
To cite this version:
Paul Prithwineel. On Szilard languages of labelled insertion grammars *. 2018. �hal-01909273�
On Szilard languages of labelled insertion grammars
∗
Prithwineel Paul
Electronics and Communication Sciences Unit
Indian Statistical Institute
Kolkata - 700108, India
prithwineelpaul@gmail.com
October 26, 2018
Abstract
In this work we initiate the study of Szilard languages of labelled insertion grammars. It is well-
known that there exist context-free languages which cannot be generated by any insertion grammar.
We show that there exist some regular languages which cannot be Szilard language of any labelled
insertion grammar. But any regular language can be given as a homomorphic image of Szilard
language obtained by a labelled insertion grammar of weight 1. Also, any context-free language can
be obtained as a homomorphic image of Szilard language of a labelled insertion grammar of weight 2.
We show that even though insertion grammars of weight 1 can generate only context-free languages,
there exist some context-sensitive language which can be obtained as Szilard language of a labelled
insertion grammar of weight 1. At the end we show that any recursively enumerable language can
be characterized by the homomorphic image of Szilard language obtained by a labelled insertion
grammar of weight 5.
Insertion grammar, Szilard languages, Labelled insertion grammar, Chomsky hierarchy
1 Introduction
Insertion and deletion operations are well-known in formal language theory. In insertion operation,
a string is inserted in the specified contexts when the insertion rule is applied, i.e., the string uv is
transformed into uxv after application of the insertion rule (u, λ/x, v). Similarly, the deletion operation
removes strings from the specified contexts and the string uxv is transformed into uv after application
of the deletion rule (u, x/λ, v) where u and v are contexts. Ins-Del (i.e., insertion-deletion) systems
work as a language generating device. These systems are powerful and with only finite set of rules
and axioms can characterize recursively enumerable languages. Ins-Del systems and their variants have
∗
prithwineelpaul@gmail.com
1
been investigated in [30, 34, 29, 7, 32, 9, 18, 33, 23]. The study of insertion grammars (semicontextual
grammars) was initiated in [28]. Computational power, closure properties etc. of the insertion systems
have been discussed in [2, 32, 3, 27, 6, 25, 26]. In [5] it was proved that the linear language {a
n
ba
n
|n ≥
1} cannot be generated by any insertion grammar. But any recursively enumerable language can be
generated by insertion grammars of weight 3 when a homomorphism and weak coding is applied [2, 19].
Moreover, in the study of matrix insertion grammars initiated in [12], it has been shown that matrix
insertion grammars can even characterize recursively enumerable languages. The computational power of
the insertion-deletion systems and insertion grammars combined with the parallel distributed computing
models such as P systems, also have been discussed in [20, 31].
The main contribution of the paper is the association of the well-known concept of Szilard languages
with insertion grammars and compare the Szilard languages obtained by these grammars with the family
of languages in Chomsky hierarchy. The idea of Szilard languages is well investigated in formal language
theory and their closure properties, decidability aspects, complexity aspects for Chomsky grammars, ma-
trix grammars, parallel communicating grammar systems, communicating distributive grammar systems
have been investigated in [22, 16, 14, 17, 15]. Also, the idea of derivation languages (as Szilard and
Control languages) has been introduced for DNA and membrane computing models in [13, 21]. In [13],
derivation languages have been associated with splicing systems and in [21] the same were introduced for
splicing P systems.
In this work, we show that there exist some regular languages which cannot be obtained as Szilard
language by any insertion grammar. But some labelled insertion grammars of weight 1 can obtain
context-sensitive languages as a Szilard language. We also show that labelled insertion grammars with
rules of weight 5 can characterize recursively enumerable languages when a morphism is applied and any
regular language can be represented as a homomorphic image of a Szilard language obtained by labelled
insertion grammar of weight 1. In [24], it has been shown that there exist some context-free languages
which cannot be represented as a homomorphic image of any context-free language. But in this paper,
we show that any context-free language can be obtained as a homomorphic image of Szilard language of
a labelled insertion grammar of weight 2.
The paper is organized as follows. In section 2 we recall the basic definitions required for this paper
along with some well-known results of insertion grammars. In section 3, we define labelled insertion
grammar and the main results have been discussed in section 4. The section 5 is conclusive in nature.
2 Preliminaries
For the basic definitions and notions of formal language theory we refer to [1].
Chomsky normal form [1]: For every context-free grammar G, a grammar G
0
= (N, T, S, P ) can be
effectively constructed where the rules in P are of the form A → BC and A → a such that L(G) \ {λ} =
L(G
0
) \ {λ}.
Type-0-grammar: A type-0-grammar is a construct G = (N, T, S, P ) where N is the non-terminal
2
alphabet and T is the terminal alphabet such that N ∩ T = ∅. The starting symbol S ∈ N and the rules
in P are ordered pairs (u, v) where u ∈ (N ∪ T )
∗
N(N ∪ T )
∗
and v ∈ (N ∪ T )
∗
.
Kuroda normal form: Every type-0 grammar G = (N, T, S, P ) is in Kuroda normal form if the rules
of the grammar G has one of the following forms:
A → BC, AB → CD, A → a, A → λ for A, B, C, D ∈ N and a ∈ T.
Homomorphism: A homomorphism is a mapping h from Σ
∗
to ∆
∗
where Σ, ∆ are alphabets, preserving
concatenation, i.e., h(v.w) = h(v).h(w), v, w ∈ Σ
∗
.
Weak coding: A weak coding is a morphism which maps each letter onto a letter or empty string.
Szilard languages [1]: Let G = (N, T, S, P ) be Chomsky grammar and F be an alphabet such that
the cardinality of the sets F and P is same. Let f be a mapping from P to F such that for each p ∈ P
a unique label f(p) is associated with p and is called the label of the rule p. A derivation in G is called
successful if a string over T is generated staring from S. With each successful derivation of G, a string
over F can be associated if labels of the any successful derivation are concatenated sequentially. The
language generated in this manner is called Szilard language of the grammar G and is denoted by SZ(G).
Example 1. Let G = ({S}, {a, b}, S, {S → aSb, S → ab}) be a context-free grammar. The rules are
labelled in the following manner: f
1
: S → aSb, f
2
: S → ab. Hence, the Szilard language obtained by the
grammar is SZ(G) = {f
n
1
f
2
| n ≥ 0}.
The family of finite, linear, regular, context-free, context-sensitive and recursively enumerable lan-
guages is denoted by F IN, LIN, REG, CF, CS, RE respectively.
Insertion grammar [3]: An insertion grammar is a construct G = (V, A, P ) where V is the set of
alphabets, A is the set of initial strings and P is the set of insertion rules.
Let G be an insertion grammar, then the relation ⇒ is defined in the following manner:
w ⇒ z if and only if w = w
1
uvw
2
, z = w
1
uxvw
2
for (u, λ/x, v) ∈ P, w
1
, w
2
∈ V
∗
.
The language generated by the insertion grammar G is:
L(G) = {z ∈ V
∗
|w ⇒
∗
z, w ∈ S}.
Moreover, an insertion grammar G is called of weight n [2] if and only if
n = max{|u| | (u, λ/x, v) ∈ P or (v, λ/x, u) ∈ P, x ∈ V
∗
}.
The family of languages generated by the insertion grammars of weight n is denoted as INS
n
and
union of all these families is denoted as IN S
∞
.
The followings are well known results in insertion grammars [28, 27, 6, 26]:
(1)F IN ⊂ INS
1
⊂ IN S
2
⊂ IN S
3
. . . ⊂ INS
∞
⊂ CS.
(2) REG is incomparable with all families INS
n
, n ≥ 1 and REG ⊂ INS
∞
.
(3) INS
1
⊂ CF but CF is incomparable with all INS
n
, n ≥ 2 and INS
∞
. Also INS
2
contains
non-semilinear languages.
(4) LIN is incomparable with all INS
n
, n ≥ 0 and INS
∞
.
(5) Each regular language is the homomorphic image of a language in INS
1
.
3
The following characterization of recursively enumerable language was proved by Kari and Sosik in [2] and
Onodera in [19] independently where S
3
denotes the family of languages generated by insertion grammars
of weight at most 3:
Theorem 2.1. For each recursively enumerable language L there exists a morphism h, a weak coding g
and a language L
1
∈ S
3
such that L = g(h
−1
(L
1
)).
3 Labelled insertion grammar
A labelled insertion grammar is a construct Γ = (V
1
, A
1
, P
1
, Lab) where V
1
∩ Lab = ∅ and the rules of P
1
are labelled in one-to-one manner with the elements from the set Lab. A derivation of insertion grammar
is called a terminal derivation if it is as follows:
x
0
⇒
a
1
x
1
⇒
a
2
x
2
⇒
a
3
. . . ⇒
a
n
x
n
where x
0
∈ A
1
and no rule of Γ is applicable to x
n
.
If the labels of the applied insertion rules in the above terminal derivation are concatenated in the
order of application, a string over Lab is obtained and the set of all such strings forms a language which
is different from the language generated by the insertion grammar. It is called Szilard language of the
labelled insertion grammar Γ. From the above derivation, the string a
1
a
2
. . . a
n
∈ SZINS
m
(Γ) where
m = max{|u| | (u, λ/α, v) ∈ P
1
or (v, λ/α, u) ∈ P
1
}.
The notation SZINS
m
(Γ) denotes the Szilard language of the labelled insertion grammar Γ of weight
m. The family of Szilard languages SZINS
m
(Γ) of the labelled insertion grammars with insertion rules
of size m, is denoted as SZINS
m
. When m is not specified, it is replaced by ∗.
In the next section, we discuss the main results of the Szilard languages of the insertion grammars
with respect to the weight. At first, we prove that there exist some regular languages which cannot be
obtained as a Szilard language by any labelled insertion grammar. But any regular language can be
given as homomorphic image of a Szilard language of a labelled insertion grammar of weight 1. Also, any
context-free language can be given as a homomorphic image of Szilard language of a labelled insertion
grammar of weight 2. Furthermore, any recursively enumerable language can be characterized by Szilard
language of the labelled insertion grammar of weight 4 when a homomorphism is applied.
4 The Main Results
It is very well-known that languages such as {aa} are not a Szilard language of any Chomsky grammar.
But we show that it can be Szilard language of a labelled insertion grammar.
Theorem 4.1. {aa} is a Szilard language of a labelled insertion grammar.
Proof. We construct a labelled insertion grammar Γ such that SZINS
m
(Γ) = {aa}. Let Γ = (V
1
, A
1
, R
1
,
Lab) be a labelled insertion grammar where, V
1
= {S, X
a
, Y }, T
1
= {S, Y }, A = {SY SY }, R = {a :
(S, λ/X
a
, Y )}, Lab = {a}.
4