Marked Systems and Circular Splicing

Transcript

Clelia De Felice
Gabriele Fici
Rosalba Zizza
Dipartimento di Informatica ed Applicazioni
Università di Salerno
FCT 2007 – August 27-30, 2007
Clelia De Felice, Gabriele Fici, Rosalba Zizza
COMPUTING
STANDARD
NATURAL
ALPHABET {0,1}
ALPHABET {A,C,G,T}
CONCATENATION
DNA SPLICING
TURING MACHINES
SPLICING SYSTEMS
CHOMSKY HIERARCHY
???
Splicing Systems
Splicing Systems (Head 87, Paun 96, Pixton 96):
Generate strings on an alphabet starting from an initial set
through rules:
S = (A, I, R)
Splicing Systems
through rules:
S = (A, I, R)
Strings in the initial set can be linear, circular or both.
Splicing Systems
through 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 Paun
splicing systems.
Circular words and languages
Conjugacy equivalence on A∗ :
w ∼ w0
⇔
w = xy , w 0 = 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 words
in C.
C is circular regular ⇔ Lin(C) is regular.
Circular splicing system
Paun Circular Splicing System:
SC = (A, I, R)
A is the alphabet
I ⊆ ∼A∗ is the initial set
R is the set of rules
SC = (A, I, R)
A is the alphabet
A rule in R is of the form r = u1 #u2 $u3 #u4 :
∼u hu , ∼u ku
2
1
4
3
generate
∼u hu u ku
2
1 4
3
(ui , h, k ∈ A∗ )
SC = (A, I, R)
A is the alphabet
∼u hu , ∼u ku
2
1
4
3
generate
∼u hu u ku
2
1 4
3
(ui , h, k ∈ A∗ )
The words u1 u2 and u3 u4 are called the SITES of the rule r
SC = (A, I, R)
A is the alphabet
∼u hu , ∼u ku
2
1
4
3
generate
∼u hu u ku
2
1 4
3
(ui , h, k ∈ A∗ )
The words u1 u2 and u3 u4 are called the SITES of the rule r
Example
r = a#1$cb#b
∼ba, ∼bacb
`r
∼babacb
CIRCULAR SPLICING
Additional hypotheses
R is reflexive:
u1 #u2 $u3 #u4 ∈ R ⇒ u1 #u2 $u1 #u2 , u3 #u4 $u3 #u4 ∈ R
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
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:
∼
hu1 u2 ku3 u4
` u1 #u2 $u3 #u4
∼
hu1 u2 , ∼ku3 u4
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:
∼
hu1 u2 ku3 u4
` u1 #u2 $u3 #u4
∼
hu1 u2 , ∼ku3 u4
Remark
We can assume that R is symmetric (see the definition of
splicing)
The language generated by a Splicing System
Definition
The language generated by a circular splicing system
S = (A, I, R) is the smallest circular language on A containing I
and closed under application of the rules in R.
The language generated by a Splicing System
Definition
The language generated by a circular splicing system
S = (A, I, R) is the smallest circular language on A containing I
and closed under application of the rules in R.
The class of languages generated by finite circular Paun
splicing systems is denoted by C(Fin, Fin).
Computational power
Theorem (Head, Paun, Pixton – 96)
I ∈ Reg ∼, R finite reflexive, self-splicing ⇒ L(I, R) ∈ Reg ∼
(Thus: using additional hypotheses C(Fin, Fin) ⊆ Reg ∼)
Computational power
Without additional hypotheses:
∼an b n
∈ C(Fin, Fin)
(Siromoney, Subramanian, Dare – 92)
∼((aa)∗ b)
∈
/ C(Fin, Fin)
(Bonizzoni, De Felice, Mauri, Zizza – 03)
Computational power
Without additional hypotheses:
∼an b n
∈ C(Fin, Fin)
(Siromoney, Subramanian, Dare – 92)
∼((aa)∗ b)
∈
/ C(Fin, Fin)
(Bonizzoni, De Felice, Mauri, Zizza – 03)
C(Fin, Fin) ⊆ CS ∼
(Fagnot – 04)
Our problem
Problem
Characterize Reg ∼ ∩ C(Fin, Fin)
Our problem
Problem
Solved if |A| = 1. Moreover Reg ∼ ∩ C(Fin, Fin) = C(Fin, Fin)
(Bonizzoni, De Felice, Mauri, Zizza – 04,05)
Our problem
Problem
Solved if |A| = 1. Moreover Reg ∼ ∩ C(Fin, Fin) = C(Fin, Fin)
(Bonizzoni, De Felice, Mauri, Zizza – 04,05)
Partial results if |A| > 1:
Theorem (Bonizzoni, De Felice, Mauri, Zizza – 04)
If X ∗ is a cycle closed star language (ex. X regular group code
or X finite with X ∗ closed under conjugacy) then
∼X ∗ ∈ Reg ∼ ∩ C(Fin, Fin)
CSSH
Definition (Ceterchi, Martin-Vide, Subramanian – 04)
A (1, 3)-Circular Semi-simple Splicing System is a finite Paun
circular splicing system in which the rules have the form
(a#1$b#1)
a, b ∈ A
To shorten notation we write the rule above
(a, b)
CSSH
Definition (Ceterchi, Martin-Vide, Subramanian – 04)
A (1, 3)-Circular Semi-simple Splicing System is a finite Paun
circular 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∗ )
Marked Systems
A Marked System is a (1, 3)-CSSH system with
I = SITES(R) = A.
Marked Systems
I = SITES(R) = A.
Example
I = {a, b, c}, R = {(a, b), (b, c), (c, c)}
Marked Systems
I = 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).
Marked Systems
I = 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).
Proposition
Every marked system admits a canonical decomposition in
transitive marked subsystems.
Distance and Diameter
The distance between two letters ai , aj is 1+ the length of the
shortest path in R linking ai and aj .
The diameter of a Marked System is the maximum value of the
distance 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
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
Theorem
If d(S) < 3 then L(S) ∈ Reg ∼. If d(S) > 3 then L(S) ∈
/ Reg ∼.
Regularity when d(S) = 3
Regularity Condition
Let S = (I, R) be a marked system. S satisfies the Regularity
Condition if ∀ J = {a1 , a2 , a3 , a4 } ⊆ I one has
R ∩ (J × J) 6= {(a1 , a2 ), (a2 , a3 ), (a3 , a4 )}
R ∩ (J × J) 6= {(a1 , a2 ), (a2 , a3 ), (a3 , a4 )}
Theorem
L(S) is regular
⇔
S satisfies the Regularity Condition.
R ∩ (J × J) 6= {(a1 , a2 ), (a2 , a3 ), (a3 , a4 )}
Theorem
L(S) is regular
⇔
Moreover, if L(S) is regular we can characterize it:
[
L(S) = I ∪
J⊆I, J
∼
(∩ai ∈J J ∗ ai J ∗ )
transitive
R ∩ (J × J) 6= {(a1 , a2 ), (a2 , a3 ), (a3 , a4 )}
Theorem
L(S) is regular
⇔
Moreover, if L(S) is regular we can characterize it:
L(S) = {w ∈ I + : alph(w) ⊆ I is transitive}
Regularity
Let S = (I, R) be a marked system.
Regularity
We compute the canonical decomposition: S = ∪i Si
Regularity
For each Si we compute its diameter d(Si )
Regularity
If for each i one has:
d(Si ) < 3
or
d(Si ) = 3 + Reg.Cond.
then L(S) = ∪i L(Si ) is regular
Regularity
d(Si ) < 3
or
Else L(S) is not regular
Regularity
d(Si ) < 3
or
Else L(S) is not regular
If L(S) is regular we can give an algebraic characterization of
its structure
Remark
Remark
Let A be an alphabet. There exists a finite number of possible
Marked Systems over A (each one coming with its canonical
decomposition).
Remark
Remark
Let A be an alphabet. There exists a finite number of possible
Marked Systems over A (each one coming with its canonical
decomposition).
So, given a regular circular language C over an alphabet A we
can test if a Marked System S exists such that C = L(S).
Self Splicing
The self-splicing operation:
∼hu
1 u2 ku3 u4
`u1 #u2 $u3 #u4
∼hu
1 u2 ,
∼ku
3 u4
Self Splicing
∼hakb
`(a,b)
∼ha,
∼kb
Self Splicing
∼hakb
`(a,b)
∼ha,
∼kb
Theorem
Let S = (I, R) be a transitive Marked System with self-splicing.
Then
L(S) = ∼I +
Self Splicing
∼hakb
∼ha,
`(a,b)
Theorem
Let S = (I, R) be a
Then
∼kb
Marked System with self-splicing.
[
L(S) =
∼ +
J
J⊆I, J transitive
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
The language L(S) generated by S is always regular
We can give an algebraic characterization of L(S)
Thank you for your attention

Marked Systems and Circular Splicing

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