Rule based modeling and application to biomolecular networks Abstract interpretation of protein protein interactions networks

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Rule-based modeling and application to biomolecular networks Abstract interpretation of protein-protein interactions networks Questions set Jerome Feret LIENS (INRIA,ENS,CNRS) 1 Abstract Interpretation Definition 1 (partial order). A partial order (D,≤) is given by a set D and a binary relation ≤ ? D?D such that: 1. (reflexivity) ?a ? D, a ≤ a; 2. (antisymmetry) ?a, a? ? D, [a ≤ a? ? a? ≤ a] =? a = a?; 3. (transitivity) and ?a, a?, a” ? D, [a ≤ a? ? a? ≤ a??] =? a ≤ a”. Definition 2 (closure). Given a partial order (D,≤) and a mapping ? : D ? D. 1. We say that ? is a upper closure operator, if and only if: (a) (idempotence) ?d ? D, ?(?(d)) = ?(d); (b) (extensivity) ?d ? D, d ≤ ?(d); (c) (monotonicity) ?d, d? ? D, d ≤ d? =? ?(d) ≤ ?(d?). 2. We say that ? is a lower closure operator, if and only if: (a) (idempotence) ?d ? D, ?(?(d)) = ?(d); (b) (antiextensivity) ?d ? D, ?

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Rule-based modeling and application to biomolecular networks Abstract interpretation of protein-protein interactions networks Questions set

J´eroˆmeFeret

´ LIENS (INRIA,ENS,CNRS)

1 AbstractInterpretation Deﬁnition 1(partial order).A partial order(D,≤)is given by a setDand a binary relation≤ ∈D×D such that: 1. (reﬂexivity)∀a∈D, a≤a; 0 00 0 2. (antisymmetry)∀a, a∈D,[a≤a∧a≤a] =⇒a=a; 0 00 00 3. (transitivity)and∀, aa, a”∈D,[a≤a∧a≤a] =⇒a≤a”. Deﬁnition 2(closure).Given a partial order(D,≤)and a mappingρ:D→D. 1. Wesay thatρis a upper closure operator, if and only if: (a) (idempotence)∀d∈D, ρ(ρ(d)) =ρ(d); (b) (extensivity)∀d∈D, d≤ρ(d); 0 00 (c) (monotonicity)∀d, d∈D, d≤d=⇒ρ(d)≤ρ(d).

2. Wesay thatρis a lower closure operator, if and only if: (a) (idempotence)∀d∈D, ρ(ρ(d)) =ρ(d); (b) (antiextensivity)∀d∈D, ρ(d)≤d; 0 00 (c) (monotonicity)∀d, d∈D, d≤d=⇒ρ(d)≤ρ(d). Deﬁnition 3(least upper bound).Given a partial order(D,≤)and a subsetX⊆A, we say thatm∈D is a least upper bound forX, if and only if: 1. (bound)∀a∈X, a≤m; 0 0 2. (leastone) and∀a∈D,[∀a∈X, a≤a] =⇒m≤a. By antisymmetry, if it exists a least upper bound is unique, thus we call it the least upper bound. Deﬁnition 4(greatest lower bound).Given a partial order(D,≤)and a subsetX⊆A, we say that m∈Dis a greatest lower bound forX, if and only if: 1. (bound)∀a∈X, m≤a; 0 0 2. (leastone) and∀a∈D,[∀a∈X, a≤a] =⇒a≤m. By antisymmetry, if it exists a greatest lower bound is unique, thus we call it the greatest lower bound. Deﬁnition 5(complete lattice).Given a partial order(D,≤), we say thatDis a complete lattice if any subsetXhas a least upper boundtX. In a complete lattice, any subsetXhas a greatest lower bounduX. Moreover, u(X) =t{d∈X| ∀x∈X, d≤x}. The element>=t(D)is the greatest element ofD, and the element⊥=t(∅)is the least element. A complete lattice is usually denoted by(D,≤,⊥,>,t,u).

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