Vieillissements démographiques et populations âgées dans l'espace régional du Nord Pas de Calais

Antoine Mine

Description

• cours - matière potentielle : la dernière période


• cours - matière potentielle : des dernières décennies

Partie 1 Vieillissements démographiques et populations âgées dans l'espace régional du Nord-Pas-de-Calais Rapport final – Vieillissement démographique et territoires en Nord-Pas de Calais à l'horizon 2025 1

• vieillissement biologique des individus


• proportions de personnes âgées


• allongement de la durée de la vie


• population âgée


• vieillissement démographique


• période


• âge


• personnes âgées vers les littoraux attractifs…

Sujets

• Midi-Pyrénées
• Union européenne
• Europe

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 16
• Langue: Français

Extrait

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Numerical Abstract Domains MPRI 2–6: Abstract Interpretation, application to verification and static analysis

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Bibliography ([exemple])

Floating-pointabstractions

Generalities, notations (reminders)

Someapplicationsof numerical domains

Handlingnon-linear expressions

Presentation of a fewnumerical abstract domains non-relational domains:intervals,congruences linear equalitydomains polyhedradomain(double description) weakly relational domains:zones,octagons

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Selected

applications

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numerical

domains

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while X>=0 do loop invariant? X:=X-1;

Example X:=[0,10]; Y:=100;

done value ofXandY?

Y:=Y+10

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Goal:findintermittentnumericalinvariants (i.e. at each program point, properties of numerical variables true for all executions)

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Variable bounds

Example X:=[0,10]; Y:=100; X∈[0,10],Y= 100 while X>=0 do X∈[0,10],Y∈[100,200] X:=X-1; X∈[−1,9],Y∈[100,200] Y:=Y+10 X∈[−1,9],Y∈[110,210] done X=−1,Y∈[110,210]

Goal:findintermittentnumericalinvariants (i.e. at each program point, properties of numerical variables true for all executions)

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Variable bounds, linear relations and congruences

X:=[0,10]; Y:=100; X∈[0,10],Y= 100 while X>=0 do X∈[0,10],10X+Y∈[100,200]∩10Z X:=X-1; X∈[−1,9],10X+Y∈[90,190]∩10Z Y:=Y+10 X∈[−1,9],10X+Y∈[100,200]∩10Z done X=−1,Y∈[110,210]∩10Z

Hope:findthe strongestintermittentnumericalinvariants (i.e. at each program point,the strongestproperties of numerical variables true for all executions)

Example

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int delay[10], i; for (i=10; i>0; i=i-1) delay[i-1]= 0; while (1){ int y = delay[i]; delay[i]= input(); i = i+1; if (i>=10) i = 0; /* use y */

}

delay line, in C

Some operations areundefinedor dangerous: arithmetic operations can overflow arrays can be accessed out of bounds

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To prove the absence of run-time error: insertverification conditionsh∙iensuring error-freedom

delay line, in C int delay[10], i; for (i=10; i>0;hi−1∈[−2 31 ,2 31 −1]ii=i-1) hi−1∈[0,9]idelay[i-1]= 0; while (1){ int y =hi∈[0,9]idelay[i]; hi∈[0,9]idelay[i]= input(); hi+ 1∈[−2 31 ,2 31 −1]ii = i+1; if (i>=10) i = 0; /* use y */ }

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