marked systems and circular splicingmath.unipa.it/fici/pdf/nizza2007marked.pdfcomputing standard...
TRANSCRIPT
Marked Systems and Circular Splicing
Clelia De Felice Gabriele Fici Rosalba Zizza
Dipartimento di Informatica ed ApplicazioniUniversità di Salerno
Laboratoire I3S - Université de Nice-Sophia Antipolis – November 28, 2007
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
COMPUTING
STANDARD NATURAL
ALPHABET {0,1} ALPHABET {A,C,G,T}
CONCATENATION DNA SPLICING
TURING MACHINES SPLICING SYSTEMS
CHOMSKY HIERARCHY ???
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Splicing Systems
Splicing Systems (Head 87, Paun 96, Pixton 96):
Generate strings on an alphabet starting from an initial setthrough rules:
S = (A, I, R)
Strings in the initial set can be linear, circular or both.
We deal with finite (i.e. I and R both finite) circular Paunsplicing systems.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Splicing Systems
Splicing Systems (Head 87, Paun 96, Pixton 96):
Generate strings on an alphabet starting from an initial setthrough rules:
S = (A, I, R)
Strings in the initial set can be linear, circular or both.
We deal with finite (i.e. I and R both finite) circular Paunsplicing systems.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Splicing Systems
Splicing Systems (Head 87, Paun 96, Pixton 96):
Generate strings on an alphabet starting from an initial setthrough rules:
S = (A, I, R)
Strings in the initial set can be linear, circular or both.
We deal with finite (i.e. I and R both finite) circular Paunsplicing systems.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Circular words and languages
Conjugacy equivalence on A∗:
w ∼ w ′ ⇔ w = xy , w ′ = yx (x , y ∈ A∗)
Example: abbc ∼ bcab
A circular word ∼w ∈ ∼A∗ is a conjugacy class.
A circular language is C ⊆ ∼A∗.
Lin(C) ⊆ A∗ is the set of all linearizations of circular wordsin C.
C is circular regular ⇔ Lin(C) is regular.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Circular splicing system
Paun Circular Splicing System: SC = (A, I, R)
A is the alphabetI ⊆ ∼A∗ is the initial setR is the set of rules
A rule in R is of the form r = u1#u2$u3#u4:
∼u2hu1,∼u4ku3 generate ∼u2hu1u4ku3 (ui , h, k ∈ A∗)
The words u1u2 and u3u4 are called the SITES of the rule r
Example
r = a#1$cb#b ∼ba,∼bacb ` r∼babacb
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Circular splicing system
Paun Circular Splicing System: SC = (A, I, R)
A is the alphabetI ⊆ ∼A∗ is the initial setR is the set of rules
A rule in R is of the form r = u1#u2$u3#u4:
∼u2hu1,∼u4ku3 generate ∼u2hu1u4ku3 (ui , h, k ∈ A∗)
The words u1u2 and u3u4 are called the SITES of the rule r
Example
r = a#1$cb#b ∼ba,∼bacb ` r∼babacb
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Circular splicing system
Paun Circular Splicing System: SC = (A, I, R)
A is the alphabetI ⊆ ∼A∗ is the initial setR is the set of rules
A rule in R is of the form r = u1#u2$u3#u4:
∼u2hu1,∼u4ku3 generate ∼u2hu1u4ku3 (ui , h, k ∈ A∗)
The words u1u2 and u3u4 are called the SITES of the rule r
Example
r = a#1$cb#b ∼ba,∼bacb ` r∼babacb
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Circular splicing system
Paun Circular Splicing System: SC = (A, I, R)
A is the alphabetI ⊆ ∼A∗ is the initial setR is the set of rules
A rule in R is of the form r = u1#u2$u3#u4:
∼u2hu1,∼u4ku3 generate ∼u2hu1u4ku3 (ui , h, k ∈ A∗)
The words u1u2 and u3u4 are called the SITES of the rule r
Example
r = a#1$cb#b ∼ba,∼bacb ` r∼babacb
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
CIRCULAR SPLICING
� �� � � �� � �� �
� �� � �
� �� � � �� � �� �
� �� � �
�� � � � � �� �
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Additional hypotheses
R is reflexive:
u1#u2$u3#u4 ∈ R ⇒ u1#u2$u1#u2, u3#u4$u3#u4 ∈ R
R is symmetric:
u1#u2$u3#u4 ∈ R ⇒ u3#u4$u1#u2 ∈ R
Self-splicing:∼hu1u2ku3u4 ` u1#u2$u3#u4
∼hu1u2,∼ku3u4
RemarkWe can assume that R is symmetric (see the definition ofsplicing)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Additional hypotheses
R is reflexive:
u1#u2$u3#u4 ∈ R ⇒ u1#u2$u1#u2, u3#u4$u3#u4 ∈ R
R is symmetric:
u1#u2$u3#u4 ∈ R ⇒ u3#u4$u1#u2 ∈ R
Self-splicing:∼hu1u2ku3u4 ` u1#u2$u3#u4
∼hu1u2,∼ku3u4
RemarkWe can assume that R is symmetric (see the definition ofsplicing)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Additional hypotheses
R is reflexive:
u1#u2$u3#u4 ∈ R ⇒ u1#u2$u1#u2, u3#u4$u3#u4 ∈ R
R is symmetric:
u1#u2$u3#u4 ∈ R ⇒ u3#u4$u1#u2 ∈ R
Self-splicing:∼hu1u2ku3u4 ` u1#u2$u3#u4
∼hu1u2,∼ku3u4
RemarkWe can assume that R is symmetric (see the definition ofsplicing)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Additional hypotheses
R is reflexive:
u1#u2$u3#u4 ∈ R ⇒ u1#u2$u1#u2, u3#u4$u3#u4 ∈ R
R is symmetric:
u1#u2$u3#u4 ∈ R ⇒ u3#u4$u1#u2 ∈ R
Self-splicing:∼hu1u2ku3u4 ` u1#u2$u3#u4
∼hu1u2,∼ku3u4
RemarkWe can assume that R is symmetric (see the definition ofsplicing)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
The language generated by a Splicing System
DefinitionThe language generated by a circular splicing systemS = (A, I, R) is the smallest circular language on A containing Iand closed under application of the rules in R.
The class of languages generated by finite circular Paunsplicing systems is denoted by C(Fin, Fin).
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
The language generated by a Splicing System
DefinitionThe language generated by a circular splicing systemS = (A, I, R) is the smallest circular language on A containing Iand closed under application of the rules in R.
The class of languages generated by finite circular Paunsplicing systems is denoted by C(Fin, Fin).
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Computational power
Theorem (Head, Paun, Pixton – 96)
I ∈ Reg∼, R finite reflexive, self-splicing ⇒ L(I, R) ∈ Reg∼
(Thus: using additional hypotheses C(Fin, Fin) ⊆ Reg∼)
Without additional hypotheses:∼anbn ∈ C(Fin, Fin)(Siromoney, Subramanian, Dare – 92)∼((aa)∗b) /∈ C(Fin, Fin)(Bonizzoni, De Felice, Mauri, Zizza – 03)
C(Fin, Fin) ⊆ CS∼
(Fagnot – 04)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Computational power
Theorem (Head, Paun, Pixton – 96)
I ∈ Reg∼, R finite reflexive, self-splicing ⇒ L(I, R) ∈ Reg∼
(Thus: using additional hypotheses C(Fin, Fin) ⊆ Reg∼)
Without additional hypotheses:∼anbn ∈ C(Fin, Fin)(Siromoney, Subramanian, Dare – 92)∼((aa)∗b) /∈ C(Fin, Fin)(Bonizzoni, De Felice, Mauri, Zizza – 03)
C(Fin, Fin) ⊆ CS∼
(Fagnot – 04)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Computational power
Theorem (Head, Paun, Pixton – 96)
I ∈ Reg∼, R finite reflexive, self-splicing ⇒ L(I, R) ∈ Reg∼
(Thus: using additional hypotheses C(Fin, Fin) ⊆ Reg∼)
Without additional hypotheses:∼anbn ∈ C(Fin, Fin)(Siromoney, Subramanian, Dare – 92)∼((aa)∗b) /∈ C(Fin, Fin)(Bonizzoni, De Felice, Mauri, Zizza – 03)
C(Fin, Fin) ⊆ CS∼
(Fagnot – 04)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Our problem
ProblemCharacterize Reg∼ ∩ C(Fin, Fin)
Solved if |A| = 1. Moreover Reg∼ ∩ C(Fin, Fin) = C(Fin, Fin)(Bonizzoni, De Felice, Mauri, Zizza – 04,05)
Partial results if |A| > 1:
Theorem (Bonizzoni, De Felice, Mauri, Zizza – 04)If X ∗ is a cycle closed star language (ex. X regular group codeor X finite with X ∗ closed under conjugacy) then∼X ∗ ∈ Reg∼ ∩ C(Fin, Fin)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Our problem
ProblemCharacterize Reg∼ ∩ C(Fin, Fin)
Solved if |A| = 1. Moreover Reg∼ ∩ C(Fin, Fin) = C(Fin, Fin)(Bonizzoni, De Felice, Mauri, Zizza – 04,05)
Partial results if |A| > 1:
Theorem (Bonizzoni, De Felice, Mauri, Zizza – 04)If X ∗ is a cycle closed star language (ex. X regular group codeor X finite with X ∗ closed under conjugacy) then∼X ∗ ∈ Reg∼ ∩ C(Fin, Fin)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Our problem
ProblemCharacterize Reg∼ ∩ C(Fin, Fin)
Solved if |A| = 1. Moreover Reg∼ ∩ C(Fin, Fin) = C(Fin, Fin)(Bonizzoni, De Felice, Mauri, Zizza – 04,05)
Partial results if |A| > 1:
Theorem (Bonizzoni, De Felice, Mauri, Zizza – 04)If X ∗ is a cycle closed star language (ex. X regular group codeor X finite with X ∗ closed under conjugacy) then∼X ∗ ∈ Reg∼ ∩ C(Fin, Fin)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
CSSH
Definition (Ceterchi, Martin-Vide, Subramanian – 04)
A (1, 3)-Circular Semi-simple Splicing System is a finite Pauncircular splicing system in which the rules have the form
(a#1$b#1) a, b ∈ A
To shorten notation we write the rule above (a, b)
So:
∼ha, ∼kb `(a,b)∼hakb (h, k ∈ A∗)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
CSSH
Definition (Ceterchi, Martin-Vide, Subramanian – 04)
A (1, 3)-Circular Semi-simple Splicing System is a finite Pauncircular splicing system in which the rules have the form
(a#1$b#1) a, b ∈ A
To shorten notation we write the rule above (a, b)
So:
∼ha, ∼kb `(a,b)∼hakb (h, k ∈ A∗)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Marked Systems
A Marked System is a (1, 3)-CSSH system withI = SITES(R) = A.
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
Example
I = {a, b, c}, R = {(a, b), (c, c)}
The first one is transitive (all letters are "linked" by rules).
PropositionEvery marked system admits a canonical decomposition intransitive marked subsystems.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Marked Systems
A Marked System is a (1, 3)-CSSH system withI = SITES(R) = A.
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
Example
I = {a, b, c}, R = {(a, b), (c, c)}
The first one is transitive (all letters are "linked" by rules).
PropositionEvery marked system admits a canonical decomposition intransitive marked subsystems.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Marked Systems
A Marked System is a (1, 3)-CSSH system withI = SITES(R) = A.
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
Example
I = {a, b, c}, R = {(a, b), (c, c)}
The first one is transitive (all letters are "linked" by rules).
PropositionEvery marked system admits a canonical decomposition intransitive marked subsystems.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Marked Systems
A Marked System is a (1, 3)-CSSH system withI = SITES(R) = A.
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
Example
I = {a, b, c}, R = {(a, b), (c, c)}
The first one is transitive (all letters are "linked" by rules).
PropositionEvery marked system admits a canonical decomposition intransitive marked subsystems.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Distance and Diameter
The distance between two letters ai , aj is 1+ the length of theshortest path in R linking ai and aj .
The diameter of a Marked System is the maximum value of thedistance between two different letters (or 2 if |I| = 1).
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
d(a, c) = 3 d(a, b) = 2 d(a, a) = 3 d(c, c) = 2 d(S) = 3
TheoremIf d(S) < 3 then L(S) ∈ Reg∼. If d(S) > 3 then L(S) /∈ Reg∼.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Distance and Diameter
The distance between two letters ai , aj is 1+ the length of theshortest path in R linking ai and aj .
The diameter of a Marked System is the maximum value of thedistance between two different letters (or 2 if |I| = 1).
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
d(a, c) = 3 d(a, b) = 2 d(a, a) = 3 d(c, c) = 2 d(S) = 3
TheoremIf d(S) < 3 then L(S) ∈ Reg∼. If d(S) > 3 then L(S) /∈ Reg∼.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Distance and Diameter
The distance between two letters ai , aj is 1+ the length of theshortest path in R linking ai and aj .
The diameter of a Marked System is the maximum value of thedistance between two different letters (or 2 if |I| = 1).
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
d(a, c) = 3 d(a, b) = 2 d(a, a) = 3 d(c, c) = 2 d(S) = 3
TheoremIf d(S) < 3 then L(S) ∈ Reg∼. If d(S) > 3 then L(S) /∈ Reg∼.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Distance and Diameter
The distance between two letters ai , aj is 1+ the length of theshortest path in R linking ai and aj .
The diameter of a Marked System is the maximum value of thedistance between two different letters (or 2 if |I| = 1).
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
d(a, c) = 3 d(a, b) = 2 d(a, a) = 3 d(c, c) = 2 d(S) = 3
TheoremIf d(S) < 3 then L(S) ∈ Reg∼. If d(S) > 3 then L(S) /∈ Reg∼.
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity when d(S) = 3
Regularity Condition
Let S = (I, R) be a marked system. S satisfies the RegularityCondition if ∀ J = {a1, a2, a3, a4} ⊆ I one has
R ∩ (J × J) 6= {(a1, a2), (a2, a3), (a3, a4)}
TheoremL(S) is regular ⇔ S satisfies the Regularity Condition.
Moreover, if L(S) is regular we can characterize it:
L(S) = I ∪⋃
J⊆I, J transitive
∼(∩ai∈JJ∗aiJ∗)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity when d(S) = 3
Regularity Condition
Let S = (I, R) be a marked system. S satisfies the RegularityCondition if ∀ J = {a1, a2, a3, a4} ⊆ I one has
R ∩ (J × J) 6= {(a1, a2), (a2, a3), (a3, a4)}
TheoremL(S) is regular ⇔ S satisfies the Regularity Condition.
Moreover, if L(S) is regular we can characterize it:
L(S) = I ∪⋃
J⊆I, J transitive
∼(∩ai∈JJ∗aiJ∗)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity when d(S) = 3
Regularity Condition
Let S = (I, R) be a marked system. S satisfies the RegularityCondition if ∀ J = {a1, a2, a3, a4} ⊆ I one has
R ∩ (J × J) 6= {(a1, a2), (a2, a3), (a3, a4)}
TheoremL(S) is regular ⇔ S satisfies the Regularity Condition.
Moreover, if L(S) is regular we can characterize it:
L(S) = I ∪⋃
J⊆I, J transitive
∼(∩ai∈JJ∗aiJ∗)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity when d(S) = 3
Regularity Condition
Let S = (I, R) be a marked system. S satisfies the RegularityCondition if ∀ J = {a1, a2, a3, a4} ⊆ I one has
R ∩ (J × J) 6= {(a1, a2), (a2, a3), (a3, a4)}
TheoremL(S) is regular ⇔ S satisfies the Regularity Condition.
Moreover, if L(S) is regular we can characterize it:
L(S) = {w ∈ I+ : alph(w) ⊆ I is transitive}
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity
Let S = (I, R) be a marked system.
We compute the canonical decomposition: S = ∪iSi
For each Si we compute its diameter d(Si)
If for each i one has:
d(Si) < 3 or d(Si) = 3 + Reg.Cond.
then L(S) = ∪iL(Si) is regularElse L(S) is not regular
If L(S) is regular we can give an algebraic characterization ofits structure
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity
Let S = (I, R) be a marked system.
We compute the canonical decomposition: S = ∪iSi
For each Si we compute its diameter d(Si)
If for each i one has:
d(Si) < 3 or d(Si) = 3 + Reg.Cond.
then L(S) = ∪iL(Si) is regularElse L(S) is not regular
If L(S) is regular we can give an algebraic characterization ofits structure
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity
Let S = (I, R) be a marked system.
We compute the canonical decomposition: S = ∪iSi
For each Si we compute its diameter d(Si)
If for each i one has:
d(Si) < 3 or d(Si) = 3 + Reg.Cond.
then L(S) = ∪iL(Si) is regularElse L(S) is not regular
If L(S) is regular we can give an algebraic characterization ofits structure
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity
Let S = (I, R) be a marked system.
We compute the canonical decomposition: S = ∪iSi
For each Si we compute its diameter d(Si)
If for each i one has:
d(Si) < 3 or d(Si) = 3 + Reg.Cond.
then L(S) = ∪iL(Si) is regular
Else L(S) is not regular
If L(S) is regular we can give an algebraic characterization ofits structure
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity
Let S = (I, R) be a marked system.
We compute the canonical decomposition: S = ∪iSi
For each Si we compute its diameter d(Si)
If for each i one has:
d(Si) < 3 or d(Si) = 3 + Reg.Cond.
then L(S) = ∪iL(Si) is regularElse L(S) is not regular
If L(S) is regular we can give an algebraic characterization ofits structure
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity
Let S = (I, R) be a marked system.
We compute the canonical decomposition: S = ∪iSi
For each Si we compute its diameter d(Si)
If for each i one has:
d(Si) < 3 or d(Si) = 3 + Reg.Cond.
then L(S) = ∪iL(Si) is regularElse L(S) is not regular
If L(S) is regular we can give an algebraic characterization ofits structure
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Remark
RemarkLet A be an alphabet. There exists a finite number of possibleMarked Systems over A (each one coming with its canonicaldecomposition).
So, given a regular circular language C over an alphabet A wecan test if a Marked System S exists such that C = L(S).
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Remark
RemarkLet A be an alphabet. There exists a finite number of possibleMarked Systems over A (each one coming with its canonicaldecomposition).
So, given a regular circular language C over an alphabet A wecan test if a Marked System S exists such that C = L(S).
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Self Splicing
The self-splicing operation:
∼hu1u2ku3u4 `u1#u2$u3#u4∼hu1u2,
∼ku3u4
TheoremLet S = (I, R) be a transitive Marked System with self-splicing.Then
L(S) = ∼I+
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Self Splicing
The self-splicing operation:
∼hakb `(a,b)∼ha, ∼kb
TheoremLet S = (I, R) be a transitive Marked System with self-splicing.Then
L(S) = ∼I+
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Self Splicing
The self-splicing operation:
∼hakb `(a,b)∼ha, ∼kb
TheoremLet S = (I, R) be a transitive Marked System with self-splicing.Then
L(S) = ∼I+
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Self Splicing
The self-splicing operation:
∼hakb `(a,b)∼ha, ∼kb
TheoremLet S = (I, R) be a Marked System with self-splicing.Then
L(S) =⋃
J⊆I, J transitive
∼J+
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity with self-splicing
Let S = (I, R) be a Marked System with self-splicing.
The language L(S) generated by S is always regularWe can give an algebraic characterization of L(S)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity with self-splicing
Let S = (I, R) be a Marked System with self-splicing.
The language L(S) generated by S is always regular
We can give an algebraic characterization of L(S)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing
Regularity with self-splicing
Let S = (I, R) be a Marked System with self-splicing.
The language L(S) generated by S is always regularWe can give an algebraic characterization of L(S)
Clelia De Felice, Gabriele Fici, Rosalba Zizza Marked Systems and Circular Splicing