marked systems and circular splicingmath.unipa.it/fici/pdf/nizza2007marked.pdfcomputing standard...

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 ???

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.

Splicing Systems

S = (A, I, R)

Splicing Systems

S = (A, I, R)

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.

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

Example

CIRCULAR SPLICING

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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)

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

∼hu1u2,∼ku3u4

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

∼hu1u2,∼ku3u4

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

∼hu1u2,∼ku3u4

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).

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).

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)

Computational power

(Fagnot – 04)

Computational power

(Fagnot – 04)

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)

Our problem

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)

∼ha, ∼kb `(a,b)∼hakb (h, k ∈ A∗)

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)

∼ha, ∼kb `(a,b)∼hakb (h, k ∈ A∗)

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.

Marked Systems

Example

I = {a, b, c}, R = {(a, b), (b, c), (c, c)}

Example

I = {a, b, c}, R = {(a, b), (c, c)}

Marked Systems

Example

I = {a, b, c}, R = {(a, b), (b, c), (c, c)}

Example

I = {a, b, c}, R = {(a, b), (c, c)}

Marked Systems

Example

I = {a, b, c}, R = {(a, b), (b, c), (c, c)}

Example

I = {a, b, c}, R = {(a, b), (c, c)}

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∼.

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

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

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∗)

R ∩ (J × J) 6= {(a1, a2), (a2, a3), (a3, a4)}

L(S) = I ∪⋃

J⊆I, J transitive

R ∩ (J × J) 6= {(a1, a2), (a2, a3), (a3, a4)}

L(S) = I ∪⋃

J⊆I, J transitive

R ∩ (J × J) 6= {(a1, a2), (a2, a3), (a3, a4)}

L(S) = {w ∈ I+ : alph(w) ⊆ I is transitive}

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

Regularity

then L(S) = ∪iL(Si) is regular

Else L(S) is not regular

Regularity

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).

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).

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+

Self Splicing

∼hakb `(a,b)∼ha, ∼kb

L(S) = ∼I+

Self Splicing

L(S) = ∼I+

Self Splicing

TheoremLet S = (I, R) be a Marked System with self-splicing.Then

L(S) =⋃

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 regularWe can give an algebraic characterization of L(S)

The language L(S) generated by S is always regular

We can give an algebraic characterization of L(S)

The language L(S) generated by S is always regularWe can give an algebraic characterization of L(S)

Thank you for your attention

marked systems and circular splicingmath.unipa.it/fici/pdf/nizza2007marked.pdfcomputing standard...

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