1. Combining Abstract Interpreters
Mooly Sagiv

2. Combining Data Flow Analyzes
Develop new algorithms from old
If I know how to conservatively represent
Pointers
Integers
Do I know how to handle C programs with integers and pointers?

3. Combining Data Flow Analyzes
Develop new algorithms from old
If I know how to conservatively represent
Pointers
Integers
Do I know how to handle C programs with integers and pointers?
Improve the precision of an analysis
Obtain a more efficient analysis

4. Combining Data Flow Analyzers
Lattice constructors
L1  L2
S  L1 …
Galois connection constructors
Constructing the abstract effect of elementary statements
Model the “relevant” parts of the program
Abstract “irrelevant” parts of the program

5. Cartesian Products
A complete lattice (L1, 1) = (L1, , 1, 1, 1, 1)
A complete lattice (L2, 2) = (, , 2, 2, 2, 2)
Define a Poset L = (L1  L2 , ) where
(x1, x2)  (y1, y2) if
x1  y1 and
x2  y2
L is a complete lattice
But what does an element in L represent?

6. Cartesian Products (cont)
A complete lattice (L1, 1) = (L1, , 1, 1, 1, 1)
A complete lattice (L2, 2) = (, , 2, 2, 2, 2)
Complete lattice L = (L1  L2 , )
A concrete lattice C (usually a powerset)
A Galois connection (C, 1 , 1, L1)
A Galois connection (C, 2 , 2, L2)
Define :C L1  L2 and : L1  L2  C ?
Example: Parity  Sign

7. Cartesian Products (cont)
A Galois connection (C, 1 , 1, L1)
A Galois connection (C, 2 , 2, L2)
A Galois connection (C,  , , L1  L2 )
(c) = <1(c), 2(c)>
(<a1, a2>) = 1(a1)  2(a2)
Define
L1st#: L1 L1
L2st#: L2 L2
How to define L1  L2 st#: L1  L2  L1  L2
Preserve soundness
Preserve relative optimality (induced)
Reasonable
Example: Parity  Sign

8. Semantic Reduction Consider a Galois connection (C,  , , A)
An operation op: A  A is a semantic reduction if
For all a  A: op(a)  a and (op(a)) = (a)

9. Component-wise combinations
Combine several analyses into a single analysis
Cartesian products (Direct product)
Independent attribute method
Relational attribute method
Total function space
Monotone function space
Direct tensor product

10. Independent Attribute Method
A Galois connection (C1, 1 , 1, L1)
A Galois connection (C2, 2 , 2, L2)
A Galois connection (C1C2,  , , L1  L2 )
(<c1, c2>) = <1(c1), 2(c2)>
(<a1, a2>) = <1(a1) , 2(a2)>
Define
L1st#: L1 L1
L2st#: L2 L2
How to define L1  L2 st#: L1  L2  L1  L2
Preserve soundness
Preserve relative optimality (induced)

11. Relational Attribute Method
A Galois connection (P(C1), 1 , 1, P(L1)) where 1: C1L1
1 (X) = {1(c) | c  X}
A Galois connection (P(C2), 2 , 2, P(L2)) where 2: C2L2
2 (X) = {2(c) | c  X}
A Galois connection (P(C1C2),  , , P(L1  L2))
(<X1, X2>) = {<1(c1), 2(c2)> | c1  X1, c2  X2}
(<Y1,Y2>) = {<c1 , c2> | 1(c1)  Y1 2(c2)  Y2 }
But how about transformers?

12. Component-wise combinations
Combine several analyses into a single analysis
Cartesian products (Direct product)
Independent attribute method
Relational attribute method
Total function space
Monotone function space
Direct tensor product