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Abstract Interpretation

Explore an abstract semantics for a language with integers and multiplication, focusing on the computation of signs. This abstraction is sound and correctly predicts the sign of expressions. The language is then extended to include unary "-" and addition, with the challenge of finding a domain closed under all abstract operations.

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Abstract Interpretation

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  1. Abstract Interpretation

  2. A Tiny Language • Consider a language with only integers and multiplication.

  3. An Abstraction • Define an abstract semantics that computes only the sign of the result.

  4. Soundness • We can show that this abstraction is correct in the sense that it correctly predicts the sign of an expression. • Proof is by structural induction on e.

  5. Abstract Interpretation • This is an abstract interpretation. • Computation in an abstract domain • In this case {+,0,-}. • The abstract semantics is sound • approximates the standard semantics. • The concretization function establishes the connection between the two domains.

  6. Adding - • Extend our language with unary -

  7. Adding + • Adding addition is not so easy. • The abstract values are not closed under addition.

  8. Solution • We need another abstract value to represent a result that can be any integer. • Finding a domain closed under all the abstract operations is often a key design problem.

  9. Extending Other Operations • We also need to extend the other abstract operations to work with T.

  10. Examples Abstract computation loses information No loss of information

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