Flattening the Transition P Systems with Dissolution Oana - PowerPoint PPT Presentation

Flattening the Transition P Systems with Dissolution Oana Agrigoroaiei Gabriel Ciobanu “A.I.Cuza” University of Ia¸ si, Department of Interdisciplinary Research, Romania Romanian Academy, Institute of Computer Science, Ia¸ si, Romania oanaag@iit.tuiasi.ro, gabriel@info.uaic.ro 11th International Conference on Membrane Computing 24-27 August 2010 Jena, Germany O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 1 / 9

The general idea Π f with 1 membrane Π with m membranes object a in membrane i object ( a , i ) δ appears in membrane i object ( δ, i ) a rule r f rule r without “out” set r f of rules (possible parents) rule r with “out” membrane i dissoluble set D i of rules ( x , i ) → ( x , cPar i ) mpr step in Π f using r f mpr step in Π (including communication) mpr step in Π f using D i diss step in Π – special rule ∇ → 0 to ensure sepa- ration between applying rules from r f and rules from D i Membrane i is dissoluble: exists r ∈ R i , δ ∈ rhs ( r ). Note: i not dissoluble ⇔ i will not dissolve in any evolution step. O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 2 / 9

General Notions: a rule u → v | x 1 ,..., x n , ¬ y 1 ,..., y m has a set of promoters x i and a set of inhibitors y j ; intermediate configuration of a P system of degree m is a vector W = ( w 1 , . . . , w m ) with w i multiset over O or w i = ∗ ; w i = ∗ specifies that membrane i has been dissolved; W = ( w 1 , . . . , w m ) configuration if all w i ( δ ) = 0; for W , V configurations, W = ⇒ T V whenever W → mpr V or W → mpr → δ V O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 3 / 9

Example 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a r 3 : a → δ 3 b r 4 : b → a + δ 4 b r 5 : b → ( c , out ) ( b , a , b , b ) → mpr ( a , δ, a + c + δ, 0) → δ (2 a + c , ∗ , ∗ , 0) → mpr → mpr ( c , ∗ , ∗ , 2 b ) → mpr (3 c , ∗ , ∗ , 0) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 4 / 9

Example 1 a r 1 : a → ( b , in 4 ) r 2 : b → a 2 δ r 3 : a → δ 3 a + c + δ r 4 : b → a + δ 4 r 5 : b → ( c , out ) ( b , a , b , b ) → mpr ( a , δ, a + c + δ, 0) → δ (2 a + c , ∗ , ∗ , 0) → mpr → mpr ( c , ∗ , ∗ , 2 b ) → mpr (3 c , ∗ , ∗ , 0) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 4 / 9

Example 1 2 a + c r 1 : a → ( b , in 4 ) 4 r 5 : b → ( c , out ) ( b , a , b , b ) → mpr ( a , δ, a + c + δ, 0) → δ (2 a + c , ∗ , ∗ , 0) → mpr → mpr ( c , ∗ , ∗ , 2 b ) → mpr (3 c , ∗ , ∗ , 0) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 4 / 9

Example 1 c r 1 : a → ( b , in 4 ) 4 2 b r 5 : b → ( c , out ) ( b , a , b , b ) → mpr ( a , δ, a + c + δ, 0) → δ (2 a + c , ∗ , ∗ , 0) → mpr → mpr ( c , ∗ , ∗ , 2 b ) → mpr (3 c , ∗ , ∗ , 0) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 4 / 9

Example 1 3 c r 1 : a → ( b , in 4 ) 4 r 5 : b → ( c , out ) ( b , a , b , b ) → mpr ( a , δ, a + c + δ, 0) → δ (2 a + c , ∗ , ∗ , 0) → mpr → mpr ( c , ∗ , ∗ , 2 b ) → mpr (3 c , ∗ , ∗ , 0) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 4 / 9

The flattened P system Π f = ( O f , µ f , R f ) O f = ( O ∪ { δ } ) × { 1 , . . . , m } ∪ {∇} ; µ f is formed of only one membrane; R f = � r ∈ R r f ∪ {∇ → 0 } ∪ � i dissolvable D i ; O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 5 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a The configuration of Π f : r 3 : a → δ 3 b ( b , 1) + ( a , 2) + ( b , 3) + ( b , 4) r 4 : b → a + δ 4 b r 5 : b → ( c , out ) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a The set of rules r f 1 contains only r 3 : a → δ one rule: 3 b ( a , 1) → ( b , 4) | ( δ, 2) , ( δ, 3) , ¬∇ r 4 : b → a + δ 4 b r 5 : b → ( c , out ) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a The set of rules r f 2 also contains r 3 : a → δ only one rule: 3 b ( b , 1) → ( a , 1) | ¬∇ r 4 : b → a + δ 4 b r 5 : b → ( c , out ) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a This also takes place for r f 3 : r 3 : a → δ 3 ( a , 2) → ( δ, 2) + ∇| ¬∇ b and for r f r 4 : b → a + δ 4 : 4 b ( b , 3) → ( a + δ, 3) + ∇| ¬∇ r 5 : b → ( c , out ) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a This also takes place for r f 3 : r 3 : a → δ 3 ( a , 2) → ( δ, 2) + ∇| ¬∇ b and for r f r 4 : b → a + δ 4 : 4 b ( b , 3) → ( a + δ, 3) + ∇| ¬∇ r 5 : b → ( c , out ) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) The set of rules r f 5 contains three r 2 : b → a rules, one for each possible desti- 2 nation of the c from ( c , out ): a r 3 : a → δ ( b , 4) → ( c , 3) | ¬∇ , ( δ, 3) 3 b r 4 : b → a + δ ( b , 4) → ( c , 2) | ( δ, 3) , ¬∇ , ( δ, 2) 4 b r 5 : b → ( c , out ) ( b , 4) → ( c , 1) | ( δ, 3) , ( δ, 2) , ¬∇ O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a r 3 : a → δ We add two sets of rules, D 2 and 3 D 3 , to deal with moving objects b if membrane 2 or membrane 3 is r 4 : b → a + δ dissolved. 4 b r 5 : b → ( c , out ) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a r 3 : a → δ The rules in D 2 are given for all 3 x ∈ O : b r 4 : b → a + δ ( x , 2) → ( x , 1) | ( δ, 2) 4 b r 5 : b → ( c , out ) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b r 1 : a → ( b , in 4 ) r 2 : b → a 2 a The rules in D 3 are given for all r 3 : a → δ x ∈ O : 3 b ( x , 3) → ( x , 2) | ( δ, 3) , ¬ ( δ, 2) r 4 : b → a + δ 4 ( x , 3) → ( x , 1) | ( δ, 3) , ( δ, 2) b r 5 : b → ( c , out ) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π 1 b The special symbol ∇ is always r 1 : a → ( b , in 4 ) produced together with one of r 2 : b → a the ( δ, i ) symbols. By appearing, 2 a it stops the application of any rule from the sets r f . r 3 : a → δ 3 The special rule b r 4 : b → a + δ ∇ → 0 4 b is applied with the rules from sets r 5 : b → ( c , out ) D j and by consuming ∇ it allows for rules from r f to be applied in the next step. O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 6 / 9

Example (continued) Configuration of Π f Configuration of Π 1 1 ( b , 1) + ( a , 2) + ( b , 3) + ( b , 4) b r 1 : a → ( b , in 4 ) r f 1 : ( a , 1) → ( b , 4) | ( δ, 2) , ( δ, 3) , ¬∇ r 2 : b → a r f 2 : ( b , 1) → ( a , 1) | ¬∇ 2 r f a 3 : ( a , 2) → ( δ, 2) + ∇| ¬∇ r 3 : a → δ r f 4 : ( b , 3) → ( a + δ, 3) + ∇| ¬∇ 3 r f 5 : ( b , 4) → ( c , 3) | ¬∇ , ( δ, 3) b ( b , 4) → ( c , 2) | ( δ, 3) , ¬∇ , ( δ, 2) r 4 : b → a + δ ( b , 4) → ( c , 1) | ( δ, 3) , ( δ, 2) , ¬∇ 4 b ∇ → 0 r 5 : b → ( c , out ) D 2 : ( x , 2) → ( x , 1) | ( δ, 2) D 3 : ( x , 3) → ( x , 2) | ( δ, 3) , ¬ ( δ, 2) ( x , 3) → ( x , 1) | ( δ, 3) , ( δ, 2) W 0 = ( b , a , b , b ) flat ( W 0 ) = ( b , 1)+( a , 2)+( b , 3)+( b , 4) O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 7 / 9

Example (continued) Configuration of Π f Configuration of Π 1 1 ( b , 1) + ( a , 2) + ( b , 3) + ( b , 4) b r 1 : a → ( b , in 4 ) r f 1 : ( a , 1) → ( b , 4) | ( δ, 2) , ( δ, 3) , ¬∇ r 2 : b → a r f 2 : ( b , 1) → ( a , 1) | ¬∇ 2 r f a 3 : ( a , 2) → ( δ, 2) + ∇| ¬∇ r 3 : a → δ r f 4 : ( b , 3) → ( a + δ, 3) + ∇| ¬∇ 3 r f 5 : ( b , 4) → ( c , 3) | ¬∇ , ( δ, 3) b ( b , 4) → ( c , 2) | ( δ, 3) , ¬∇ , ( δ, 2) r 4 : b → a + δ ( b , 4) → ( c , 1) | ( δ, 3) , ( δ, 2) , ¬∇ 4 b ∇ → 0 r 5 : b → ( c , out ) D 2 : ( x , 2) → ( x , 1) | ( δ, 2) D 3 : ( x , 3) → ( x , 2) | ( δ, 3) , ¬ ( δ, 2) ( x , 3) → ( x , 1) | ( δ, 3) , ( δ, 2) ( b , a , b , b ) → mpr ( a , δ, a + c + δ, 0) ( b , 1) + ( a , 2) + ( b , 3) + ( b , 4) → mpr ( a , 1) + ( δ, 2) + ( a + c + δ, 3) + 2 ∇ O. Agrigoroaiei, G. Ciobanu Flattening P Systems with Dissolution CMC 2010 7 / 9