TelAviv University Programming Languages Seminar

1
Measuring the Precision of Abstract
Interpretations

• Alessandra Di Pierro and Herbert Wiklicky
• Moderator Roman Manevich

2
Motivation
Neil D.Jones and Flemming Nielson.Abstract
interpretation A semantic-based tool for program
analysis
One shortcoming of the development ... is that
a correct analysis may be so imprecise as to be
practically useless. ... The notion of
correctness is topological in nature but we
would ideally like something that was a bit more
metric in nature so that we could express how
imprecise a correct analysis is. Unfortunately
no one has been able to develop an adequate
metric for these purposes.
3
Outline

• Probabilistic abstract interpretation
• Construction of probabilistic abstract
  interpretations
• Measuring probabilistic abstract interpretations
• Examples
• Related Work

4
Probabilistic Semantics

• Probabilistic programs
• coin flipping determines flow of information
  (probabilistic execution)
• Distribution of results (r,prob(r))
• Generalize deterministic programs

5
Classical Abstract Interpretation

• Introduced by Patrick and Radhia Cousot 1977
• Two semantics
• Reference standard (concrete) semantics
• Approximate (abstract) semantics
• Over two domains lattices
• The relation between them given byGalois
  Connection

6
Complete Lattices Reminder

• A poset (L, ? ) is a complete lattice if every
  subset has least and upper bounds
• L (L, ?) (L, ?, ?, ?, ?, ?)
• ? ? ? ? L
• ? ? L ? ?
• Lemma For every poset (L, ? ) the following
  conditions are equivalent
• L is a complete lattice
• Every subset of L has a least upper bound
• Every subset of L has a greatest lower bound

7
Galois Connections Revisited

• Definition 1
• Let C(C,?) and D(D, ?) be two posets
• If ?C?D and ? D?C s.t for all c?C and
  d?Dc??(d) iff ?(c)?d
• (C,?,?,D) forms a Galois Connection

8
Galois Connections Revisited
D
C
d
?
?(d)
?(c)
?
c
9
Galois Connections Revisited

• Definition 2
• Let C(C,?) and D(D, ?) be two posets
• If ?C?D and ? D?C s.t ? and ? are monotone ?
  ? ? is reductive? ? ? is extensive
• (C,?,?,D) forms a Galois Connection

10
Galois Connections Revisited
D
C
d
?
?(?(d))
?(?(c))
?
c
11
Galois Connections Revisited
C
C
???
12
Galois Connections Revisited
D
D
???
13
Probabilistic Abstract Interpretation

• Probabilistic semantics
• Two semantics
• Reference standard (concrete) semantics
• Approximate (abstract) semantics
• Over two (probabilistic) domains vector spaces
• The relation between them given by Moore-Penrose
  pseudo-inverse

14
Projections

• A linear operator TV?V is a projection if T?TT
• If T(V)W then for every v?Vvv?v?
• v? will be abstracted T(v?)0v? will be
  preserved T(v?)v?

15
Projection Example
y
?((x,y))x?(x)(x,0)
x
16
Adjoint Orthogonal Operators

• If T is a linear operator
• exists unique T such that ?T(u),v??u,T(v)?
• TTTTI ? TT-1
• ?T(u),T(v)??u,v?
• ?T(u)??u?

17
Orthogonal Projections

• T is an orthogonal projection iff TT
• For a linear map ?V?VThe map of ? ?(V)?W (a
  subspace of V)
• There is a unique orthogonal projection??(v)W?W
  s.t. ??(v)(W)?(V)

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Classic ? Probabilistic

• ?,? Monotone ? Linear
• ???(c)?c ? ?????
• ???(d)?d ? ?????

19
Moore-Penrose Pseudo Inverse

• C, D finite dimensional vector spaces
• ?C?D
• ??D?C is the (unique) Moore-Penrose Pseudo
  Inverse iff
• ?????
• ?????

20
Alternative Characterization

• ?????? (also holds for GC)
• ?????? (also holds for GC)
• (???)???
• (???)???

21
Probabilistic Abstract Interpretation

• C,D two probabilistic domains
• ?C?D and ? D?C linear maps
• ?,? moore-penrose pseudo-inverse

22
Projection as Approximation

• Let ?WV?V be an orthogonal projection.
• For every x?V ?W(x) is the unique vector in
  Ws.t. ?x- ?W(x)? is minimal

23
Construction of ProbabilisticAbstract
Interpretations

• Probabilistic induced analysis
• Vector space lifting
• General construction
• Infinite dimensional abstractions

24
Vector Space Lifting

• Recasting cpo based semantics in vector space
  setting
• Similar to power space
• If C is a cpo V(C)?xcc xc??, c?C

25
General (Classic) Construction
?
A
A
?
f
f
?
B
B
?
26
General (Classic) Construction

• f A ? B is correct approximation of f
  iff? ? f ?B f ? ?
• Best iff f ? ? f ? ?
• Complete iff? ? f f ? ?

27
Measuring ProbabilisticAbstract Interpretations

• Pseudo-quotient?(? ? f ) ? (f ? ?)? B ?
  B
• (A,B, f ) is correct iff ????1complete iff
  ???1

28
Examples

• Rule of sign
• Multiplication
• Addition
• Cast out of nine

29
Rule of Signs
30
Rule of Signs Multiplication
?
V(Z2)
V(Sign2)
?
?
?
V(Z)
V(Sign)
31
Lifting Multiplication

• V(Sign) R3- ? (1,0,0) 0 ? (0,1,0) ?
  (0,0,1)
• V(Sign2) R6- ? - ? (1,0,0,0,0,0) - ? 0
  ? (0,1,0,0,0,0) 0 ? ? (0,0,1,0,0,0) 0 ? 0
  ? (0,0,0,1,0,0) 0 ? ? (0,0,0,0,1,0) ?
  ? (0,0,0,0,0,1)

(? ? ?) ? (? ? ?2) id ???1
32
Rule of Signs Addition

• Qn(? ? ) ? ( ? ?2) for truncated
  computations in -n,n converges to
• (1.0 0.0 0.0 0.0)(0.0 1.0 0.0 0.0)(0.0 0.0
  1.0 0.0)(0.5 0.0 0.5 0.0)
• ???1.5

33
Cast by d

• eval(abc) eval(a,b,c) - true if abc-
  false otherwise
• ?N3?D3
• eval D3?B ?false, ?true

34
Cast by d
?
V(D3)
V(N3)
eval
eval
id
V(B)
V(B)
35
Cast by d

• SdV(B)?V(B)true ? (1,0)false ? (0,1)
• Numerical experiments gave?S2?11.93 vs.
  ?S9?11.72

36
Related Work

• Concurrent Constraint Programming Towards
  Probabilistic Abstract Interpretation /
  Alessandra Di Pierro and Herbert Wiklicky
• Making Abstract Interpretations Complete
  /Roberto Giacobazzi, Francesco Ranzato
• Data Flow Frequency Analysis / Ramalingam

37
The End