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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)aD, aa; 0 00 0 2. (antisymmetry)a, aD,[aaaa] =a=a; 0 00 00 3. (transitivity)and, aa, aD,[aaaa] =aa. Definition 2(closure).Given a partial order(D,)and a mappingρ:DD. 1. Wesay thatρis a upper closure operator, if and only if: (a) (idempotence)dD, ρ(ρ(d)) =ρ(d); (b) (extensivity)dD, dρ(d); 0 00 (c) (monotonicity)d, dD, dd=ρ(d)ρ(d).
2. Wesay thatρis a lower closure operator, if and only if: (a) (idempotence)dD, ρ(ρ(d)) =ρ(d); (b) (antiextensivity)dD, ρ(d)d; 0 00 (c) (monotonicity)d, dD, dd=ρ(d)ρ(d). Definition 3(least upper bound).Given a partial order(D,)and a subsetXA, we say thatmD is a least upper bound forX, if and only if: 1. (bound)aX, am; 0 0 2. (leastone) andaD,[aX, aa] =ma. 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 subsetXA, we say that mDis a greatest lower bound forX, if and only if: 1. (bound)aX, ma; 0 0 2. (leastone) andaD,[aX, aa] =am. 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{dX| ∀xX, dx}. 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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