Combining Abstract Interpreters

1
Combining Abstract Interpreters

• Sumit Gulwani
• Microsoft Research
• Redmond, Group

Ashish Tiwari SRI
RAD
2
Motivation
a1 0 a2 0 b1 1 b2 F(1) c1 2
c2 2
a1 a11 a2 a22 b1 F(b1) b2
F(b2) c1 F(2c1-c2) c2 F(c2)
True
b1lt b2
False

• Abstract interpretation over the abstractions of
  linear arithmetic and uninterpreted functions can
  verify the first and second assertions
  respectively.
• Third assertion can be verified only over the
  combined abstraction.

Assert(a22a1) Assert(b2 F(b1)) Assert(c2c1)
3
Outline

• Logical product combination of lattices
• Abstract interpreter for logical product lattice
• Join operator
• Existential quantification operator
• Correctness and Complexity

4
Logical Product of Lattices

• A lattice L consists of a domain DL and partial
  order ¹L.
• A lattice L is a logical lattice over theory T if
• DL finite conjunctions of atomic facts over T
• E ¹L E iff E )T E
• Let L1 and L2 be logical lattices over T1 and T2
  resp. Then logical product of L1 and L2 is L1L2,
  where
• DL1L2 finite conjunctions of atomic facts over
  T1 T2
• E ¹L1L2 E iff E )T1 T2 E
• and AlienTerms(E) µ
  Terms(E)

5
Outline

• Logical product combination of lattices
• Abstract interpreter for logical product lattice
• Join operator
• Existential quantification operator
• Correctness and Complexity

6
Abstract Interpreter for L1L2
E
E2
E1
E
p

x g
False
True
E
E
E1
E2
Conditional Node
Assignment Node
Join Node
E JoinL1L2(E1,E2) We show how to get
JoinL1L2 from JoinL1 and JoinL2.
E EQL1L2(E, x) E Ex/x Æ
x(gx/x) We show how to get EQL1L2 from EQL1
and EQL2.
E1 MeetL1L2(E, p) E2 E MeetL1L2(E,E) E
Æ E
7
Outline

• Logical product combination of lattices
• Abstract interpreter for logical product lattice
• Join operator
• Existential quantification operator
• Correctness and Complexity

8
Background Combining Decision Procedures
y1 4y3 F(2y2-y1) Æ y1F(y1) Æ y2F(F(y1))
y1 4y3
Purification
a12y2-y1 y1 4y3 a2 y1 y2 y1 a2
a2F(a1) y1F(y1) Æ y2F(F(y1)) y1 a1
Saturation
y1 4y3
This classic algorithm was given by Nelson and
Oppen in 1979.
9
Join Operator

• If E JoinL(E1,E2), then E is the least upper
  bound of E1 and E2 in lattice L
• Examples
• Joinla(z0 Æ y10, z5 Æ y5) zy10 Æ 0z 5
• Joinuf(za Æ yF(a), zb Æ yF(b)) yF(z)
• Joinlauf(za-1 Æ yF(a), zb-1 Æ yF(b)) ?

10
Join Operator

• If E JoinL(E1,E2), then E is the least upper
  bound of E1 and E2 in lattice L
• Examples
• Joinla(z0 Æ y10, z5 Æ y5) zy10 Æ 0z 5
• Joinuf(za Æ yF(a), zb Æ yF(b)) yF(z)
• Joinlauf(za-1 Æ yF(a), zb-1 Æ yF(b))
  yF(1z)
• We next show how to construct JoinL1L2 using
  JoinL1 and JoinL2.

11

Combining Join Operators
za-1 Æ yF(a)
zb-1 Æ yF(b)
Joinufla
za-1 aha,bi
yF(a) aha,bi
zb-1 bha,bi
yF(b) bha,bi
Joinuf
Joinla
ha,bi1z
yF(ha,bi)
EQufla
ha,bi
yF(1z)
12
Outline

• Logical product combination of lattices
• Abstract interpreter for logical product lattice
• Join operator
• Existential quantification operator
• Correctness and Complexity

13
Existential Quantification Operator

• If E EQL(E,V), then E is the least (i.e., most
  precise) element in lattice L such that
• E ¹L E
• Vars(E) Å V
• Examples
• EQla(xa Æ ay, a) x y
• EQuf(xF(a) Æ yF2(a), a) yF(x)
• EQlauf(aby Æ zc1 Æ aF2(b) Æ cF(b),
  a,b,c) ?

14
Existential Quantification Operator

• If E EQL(E,V), then E is the least (i.e., most
  precise) element in lattice L such that
• E ¹L E
• Vars(E) Å V
• Examples
• EQla(xa Æ ay, a) x y
• EQuf(xF(a) Æ yF2(a), a) yF(x)
• EQlauf(aby Æ zc1 Æ aF2(b) Æ cF(b),
  a,b,c) F(z-1)y
• We can construct EQL1L2 using EQL1 and EQL2.

15

Combining Existential Quantification Operators
aby Æ zc1 Æ aF2(b) Æ cF(b)
a, b, c
EQufla
aby Æ zc1
aF2(b) Æ cF(b)
Defla
Defuf
b
EQla
EQuf
c ? z-1 a ?F(z-1)
a y Æ zc1
a F(c)
Substitute
F(z-1) y
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Outline

• Logical product combination of lattices
• Abstract interpreter for logical product lattice
• Join operator
• Existential Quantification operator
• Correctness and Complexity

17
Correctness

• Our algorithms for JoinL1L2 and EQL1L2 are
  sound.
• They are complete when the underlying theories T1
  and T2 are convex, stably infinite, and disjoint.
• Proof of correctness is non-trivial.

18
Computational Complexity

• Complexity of JoinL1L2 and EQL1L2 is worst-case
  quadratic in complexity of JoinL1, JoinL2, EQL1,
  EQL2.
• Steps required for fixed-point computation
• DL(E) max of elements in a chain above E in
  lattice L
• DL1 L2(E) DL1(E1) DL2(E2) AlienTerms(E)
• where E1 and E2 are purified and saturated
  components of E.

19
Conclusion and Future Work

• Defined combination L1L2 of two lattices L1 and
  L2.
• This logical product is more precise than reduced
  product.
• Described abstract interpretation operators for
  L1L2 in terms of corresponding operators for L1
  and L2.
• Lends modularity to design implementation of
  abstract interpreters.
• Future Work
• Handle non-convex theories (eg. arrays) more
  precisely.
• Handle non-atomic facts involving negation
  disjunction.
• Perform experiments.