$ Chapter 3 — A General Static Analysis Framework Based on a Compositional Semantics
Chapter 3 — A General Static Analysis Framework Based on a Compositional Semantics
09 May 2021

3.2 Abstractions

Definition: Abstract Domain and Abstraction Relation

We call abstract domain a pair made of a set 𝔸 and an ordering relation ⊑ over that set. Given a concrete domain (ℂ, ⊆), an abstraction is defined by an abstract domain (𝔸, ⊑) and an abstraction relation (⊧) ⊆, ℂ × 𝔸, such that,

  • for all \(c \in ℂ, a_0, a_1 \in 𝔸\), if \(c \models a_0\) and \(a_0 \sqsubseteq a_1\), then \(c \models a_1\)

  • for all \(c_0, c_1 ∈ ℂ, a ∈ 𝔸\), if \(c_0 \subseteq c_1\) and \(c_1 \models a\), then \(c_0 \models a\)

The first one interpretation: if \(c\) satisfies the property described by the abstract element \(a_0\), and if the abstract element \(a_1\) expresses a weaker property than that expressed by \(a_0\), then \(c\) should also satisfy the abstract property \(a_1\).

A relation \(\models\) is sometimes replaced by "concretization function" and "abstraction function".

Definition: Concretization Function
A concretization function (or, for short, concretization) is a function\(\gamma: 𝔸 → ℂ\) such that, for any abstract element \(a\), \(\gamma(a)\) satisfies \(a\) (i.e., \(\gamma(a) \models a\)) and \(\gamma(a)\) is the maximum element of \(ℂ\) that satisfies \(a\).

Definition: Abstract Function

Let \(c\) be a concrete element. We say that \(c\) has a best abstraction if and only if there exists an abstract element \(a\) such that

  1. \(a\) is an abstraction of \(c\), and

  2. any other abstraction of \(c\) is greater than \(a\)

If it exists, this element is unique and called the best abstration of \(c\).

An abstraction function (or, for short, abstraction) is a function \(\alpha: ℂ \rightarrow 𝔸\) that maps each concrete element to its best abstraction.


\[ \forall c \in ℂ, a \in 𝔸, c \models a \Leftrightarrow c \subseteq \gamma(a) \]
Definition: Galois Connection

A Galois connection is a pair made of a concretization function \(\gamma\) and an abstraction function \(\alpha\) such that

\[ \forall c \in ℂ, \forall a \in 𝔸, \alpha(c) \sqsubseteq a \Leftrightarrow c \subseteq \gamma(a) \]

Galois connection's interesting properties:

3.2.2 Non-Relational Abstraction

Intuitively, this abstraction

3.2.3 Relational Abstraction

In general, the choice of an efficient computer representation for abstract domains that describe relational constraints is more difficult than in the case of non-releational abstract domains.


3.3 Computable Abstract Semantics

sound analysis

sound analysis

Abstraction of a sequence of commands: compose each abstraction

\[ [C_0;C_1]^{\#}_𝒫(M^{\#}) = [C_1]^{\#}_𝒫([C_0]^{\#}_𝒫(M^{\#})) \]

This can be generalized into the following theorem:

Theorem: Approximation of Compositions

Let \(F_0, F_1: 𝒫(𝕄) \rightarrow 𝒫(𝕄)\) be two monotone functions, and let \(F_0^{\#}, F_1^{\#}: 𝔸 \rightarrow 𝔸\) be two functions that over-approximate them, that is such that \(F_0 \circ \gamma \subseteq \gamma \circ F_0^{\#}\) and \(F_1 \circ \gamma \subseteq \gamma \circ F_1^{\#}\).
Then \(F_0 \circ F_1\) can be over-approximated by \(F_0^{\#} \circ F_1^{\#}\).

  • \(F_1 \circ \gamma \subseteq \gamma \circ F_1^{\#}\) (by the soundness assumption on \(F_1\))

  • \(F_0 \circ F_1 \circ \gamma \subseteq F_0 \circ \gamma \circ F_1^{\#}\) (\(F_0\) is monotone)

  • \(F_0 \circ F_1 \circ \gamma \subseteq \gamma \circ F_0^{\#} \circ F_1^{\#}\) (by the soundness assumption on \(F_0\))

Intuitive understanding: we can decompose the over-approximation of a composition of operations into the composition of over-approximations of each operation.

Abstract Interpretation of Conditional Branching

Semantics of conditional branching:

\[ [\texttt{if}(B)\{C_0\} \texttt{else}\{C_1\}]_𝒫(M) = [C_0]_𝒫(ℱ_B(M)) \cup [C_1]_𝒫(ℱ_{\neg B}(M)) \]

\(ℱ_{B}\): returns the memory states in \(M\) such that the condition \(B\) evaluates to true.

Analysis of Flow Joins

The abstract join operator \(\sqcup^{\#}\) should satisfy the following soundness property

\[ \gamma(M_0^{\#}) \cup \gamma(M_1^{\#}) \subseteq \gamma(M_0^{\#} \sqcup^{\#} M_1^{\#}) \]

\(\sqcup^{\#}\) can be defined as a pointwise application of a join operator \(\sqcup^{\#}_𝒱\), which works in the value abstract domain:

\[ \text{Fro all variables } \texttt{x}, (M_0^{\#} \sqcup^{\#} M_1^{\#})(x) = M_0^{\#}(x) \sqcup^{\#}_𝒱 M_1^{\#}(x) \]

And the definition of \(\sqcup^{\#}_𝒱\) will really depend on the abstract domain.

Analysis of a Conditional Command

Semantics of conditional branching

\[ [\texttt{if}(B)\{C_0\} \texttt{else}\{C_1\}]^{\#}_𝒫(M^{\#}) = [C_0]^{\#}_𝒫(ℱ_B^{\#}(M^{\#})) \sqcup^{\#} [C_1]^{\#}_𝒫(ℱ_{\neg B}^{\#}(M^{\#})) \]

Abstract Interpretation of Loops

Semantics of loops:

\[ [\texttt{while}(B)\{C\}]_𝒫(M) = ℱ_{\neg B} (\cup_{i \ge 0}([C]_𝒫 \circ ℱ_B)^i(M)) \]

Let \(F = [C]_𝒫 \circ ℱ_B\)

Convergence in Finite Height Lattices


The Design of Abstract Interpreter