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APPROCHE THEMATIQUE DU VIEILLISSEMENT

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78 pages
DEUXIEME PARTIE APPROCHE THEMATIQUE DU VIEILLISSEMENT Rapport final – Vieillissement démographique et territoires en Nord-Pas de Calais à l'horizon 2025 1

  • cadre de vie hors travail

  • métropole lille

  • références dans la doctrine sociale de l'eglise

  • société industrielle

  • régional

  • population ouvrière urbaine

  • cultures urbaines

  • pratique sociale

  • action politique


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Numerical Abstract Domains MPRI 2–6: Abstract Interpretation, application to verification and static analysis
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AntoineMine´
3 February 2012
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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
7/8
3
February
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Shortcomings
2012
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Numerical Abstract
ins
non-relational
Domains
domains
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To prove that, e.g.Y≥ −128, we must be able to: representthe propertiesR=XSandR≤ −D, combinethem to deduceSXD, and thenY=SDX.
Iterations in the interval domain (without widening): X]0X]1X]2. . .X]n Y= 0|Y| ≤144|Y| ≤ . .160 .|Y| ≤128 + 16n In fact,Y[128,128]always holds.
X: input signal Y: output signal S output: last R: deltaY-S D for allowed: max.|R|
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Y:=0; whiletrue do X:=[-128,128]; D:=[0,16]; S:=Y; Y:=X; R:=X-S; if R<=-D then Y:=S-D fi; if R>=D then Y:=S+D fi done
Rate limiter
Non-relation domains cannot represent variablerelationships.
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To prove some invariant after theend of a loop, we often need to find aloopinvariant of amore complex form.
p.5/
To find thisioatlnanonler-invariant, we must find arelationalloop invariant at:(I<X<I)(X+I1 [2])(I[1,5000]), and apply the loop exit condition C]JI>=5000K.
relational loop invariant
X:=0; I:=1; whileI<5000 do if ? then X:=X+1 else X:=X-1 fi; I:=I+1 done
A non-relational analysis finds atthatI= 5000 andXZ.
The best invariant is: (I= 5000)(X[4999,4999])(X0 [2]).
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3
February
2012
Numerical
Reminders
Reminders
Abstract
Domains
Antoine
Mine ´
p.
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78
assignment intoVV test,./∈ {=, <, >, <=, >=, <>}
arithmetic expressions:
Fixed finite set of variablesV, with value inI,I∈ {Z,Q,R,M,F}
commands:
variableVV negation binary operation: ∈ {+,,×, } constant range,c,c0I∪ {±∞} cis a shorthand for [c,c]
exp::=V |exp |expexp |[c,c0]
com::=V := exp |exp./0
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