# Notes on Abstract Interpretation

This is my notes on abstract interpretation based on Bruno Blanchet’s Introduction to Abstract Interpretation, Matt Might’s blog post Order theory for computer scientists, Alexandru Salcianu’s Notes on Abstract Interpretation, and Stephen Chong’s CS252r course material.

### Motivation

Most interesting properties of programs are undecidable; they can be reduced to halting problem.
Proof. By contradiction. Assume that there exists a analyzer Halt(P,E) returning true when P terminates on input E, and false otherwise. Consider a program P'(P) = if Halt(P, P) then loop else halt. Now apply P'(P')

• If Halt(P',P') is true, then P'(P') loops. Contradiction.
• If Halt(P',P') is false, then P'(P') halts. Contradiction. $\square$

So we introduce approximations. The approximations must be sound (correct): if the system gives a definite answer, then this answer is true. Sometimes the system answers “maybe”, then the system is not complete.

Abstract interpretation is a theory of approximation. We first formalize the semantics of programs, then its approximations.

### A Simple Language

#### Syntax

Expressions:

E ::= variables
| numbers
| E + E
| E - E
| E * E
| E / E
| E >= E


Commands:

C ::= end
| x := E
| if E goto n
| input x
| print E


A program is a function Prog from integers to commands, indicating which command is at each address in the program.

#### Trace Semantics

The state of programs consists of 1) the program counter which indicates the next command to execute, 2) an environment which contains a partial map from variables to values. We ignore overflows.

Formally:
$$\rho \in Env = Var \rightarrow \mathbb{Z} \\ pc \in PC \\ s \in State = PC \times Env$$

• Semantics of expressions: $\llbracket E \rrbracket \rho \in \mathbb{Z}$. In case of errors (e.g., division by 0), $\llbracket E \rrbracket \rho$ is undefined. We may use $\bot$ to represent that.

$$\llbracket x \rrbracket \rho = \rho(x) \\ \llbracket E_1 + E_2 \rrbracket \rho = \llbracket E_1 \rrbracket \rho + \llbracket E_2 \rrbracket \rho \\ \cdots$$

• Semantics of commands is modeled as a transition system: $\langle pc, \rho \rangle \rightarrow \langle pc’, \rho’ \rangle$.
• If Prog(pc) = x := E, then $\langle pc, \rho \rangle \rightarrow \langle pc+1, \rho[ x \mapsto \llbracket E \rrbracket \rho ] \rangle$, if $\llbracket E \rrbracket \rho$ is defined.
• If Prog(pc) = input x, then $\langle pc, \rho \rangle \rightarrow \langle pc+1, \rho[ x \mapsto v ] \rangle$ for some $v \in \mathbb{Z}$.
• If Prog(pc) = print E, then $\langle pc, \rho \rangle \rightarrow \langle pc+1, \rho \rangle$ if $\llbracket E \rrbracket \rho$ is defined.
• If Prog(pc) = if E goto n, then $\langle pc, \rho \rangle \rightarrow \langle n, \rho \rangle$ if $\llbracket E \rrbracket \rho \neq 0$; $\langle pc, \rho \rangle \rightarrow \langle pc+1, \rho \rangle$ otherwise.

A sequence of state is a trace. A program may have several traces, depending of the user input; so the semantics of a program is a set of traces.

### Approximation

We want to prove properties like “all traces of the program satisfy a given condition”. To do this, we over-approximate the set of traces, and obtain a superset of the set of concrete traces. An analysis is more precise when it yields a smaller superset of the set of concrete trances. All correct approximations are the ones that are greater than the smallest correct one, namely, the set of concrete traces.

#### Order, Lattices, Complete lattices

Definition (Partially ordered set): A partially ordered set is a set $S$ equipped with a binary relation $\leq$ such that 1) $\leq$ is reflexive, 2) $\leq$ is antisymmetric, and 3) $\leq$ is transitive. A binray relation $R$ is antisymmetric if $a R b$ and $b R a$ implies $a = b$.

• $c \in S$ is an upper bound of $X \subseteq S$ iff $\forall c’ \in X, c’ \leq c$.
• $c \in S$ is a lower bound of $X \subseteq S$ iff $\forall c’ \in X, c \leq c’$.
• $c \in S$ is the least upper bound of $X \subseteq S$ iff $\forall c’ \in X, c’ \leq c$ and $\forall c’’ \in S$ s.t. $\forall c’ \in X, c’ \leq c’’$, we have $c \leq c’’$.
• $c \in S$ is the greatest lower bound of $X \subseteq S$ iff $\forall c’ \in X, c \leq c’$ and $\forall c’’ \in S$ s.t. $\forall c’ \in X, c’’ \leq c’$, we have $c’’ \leq c$.

The ordering can be partial, i.e., exists incomparable elements.

Definition (Lattice): A lattice is a poset $(L, \leq)$ s.t. for all $a, b \in L$, $a$ and $b$ have a least upper bound $a \sqcup b$ and a greatest lower bound $a \sqcap b$. We denote the lattice by $(L, \leq, \sqcup, \sqcap)$.

All finite sets in a lattice have least upper bounds and greatest lower bounds, but not necessarily infinite sets.

Definition (Complete lattice): A complete lattice is a poset $(L, \leq)$ s.t. every subset $X$ of $L$ has a least upper bound $\sqcup X$ and a greatest lower bound $\sqcap X$. In particular, $L$ has a least element $\bot = \sqcup \varnothing$ and a greatest element $\top = \sqcap L$. We denote the lattice by $(L, \leq, \bot, \top, \sqcup, \sqcap)$.

Proposition: If $S$ is a set, $(\mathcal{P}(S), \subseteq, \varnothing, S, \cup, \cap)$ is a complete lattice.

Intuitively, $a \leq b$ means $a$ is more precise than $b$.

#### Collecting semantics

Collecting semantics computes the set of possible traces, formally:

$$T_r = \{ s_1 \rightarrow \cdots \rightarrow s_n | s_1 ~\text{ is the initial state,}~ s_i \rightarrow s_{i+1} ~\text{is an allowed transition} \}$$

The set of reachable states is defined by:

$$S_r = \{ s | s_1 \rightarrow \cdots \rightarrow s \in T_r \}$$

The set of all traces, of all states, are ordered by inclusion. They are complete lattice. (In this case, they are concrete lattice.)

#### Galois connections

Definition (Galois connection): Let $(L_1, \leq_1)$ and $(L_2, \leq_2)$ be posets. $(\alpha, \gamma)$ is a Galois connection (or pair of adjoined functions) between $L_1$ and $L_2$ iff $\alpha : L_1 \to L_2$ and $\gamma : L_2 \to L_1$, and $$\forall x \in L_1, \forall y \in L_2, \alpha(x) \leq_2 y \iff x \leq_1 \gamma(y)$$

This is denoted by $(L_1, \leq_1) \underset{\gamma}{\overset{\alpha}{\rightleftharpoons}} (L_2, \leq_2)$, where $(L_1, \leq_1)$ is the concrete lattice, $(L_2, \leq_2)$ is the abstract lattice, $\alpha$ is the abstraction and $\gamma$ is the concretization. We say that $\alpha(x)$ is the most precise approximation of $x \in L_1$ in $L_2$; $\gamma(y)$ is the least precise element of $L_1$ which can be correctly approximated by $y \in L_2$.

###### Examples of Galois connections
• The concrete lattice is $(\mathcal{P}(S), \subseteq)$ for some set $S$. The abstract lattice contains $\bot$, $c$, $\top$, for each $c \in S$, ordered by $\bot \leq c \leq \top$.
The abstraction is defined by $\alpha(\varnothing) = \bot$, $\alpha(\{c\}) = c$, $\alpha(\_) = \top$.
The concretization is defined by $\gamma(\bot) = \varnothing$, $\gamma(c) = \{c\}$, $\gamma(\top) = S$.

• Sign. The concrete lattice is $\mathcal{P}(\mathbb{Z})$. The abstract lattice is $\mathcal{P}(\{-, 0, +\})$, ordered by inclusion.
The abstraction is defined by:
$$\alpha(S) = \begin{cases} \{+\}, & \text{if }S \cap [0, +\infty] \neq \varnothing \\ \{0\}, & \text{if }0 \in S \\ \{-\}, & \text{if }S \cap [-\infty, 0] \neq \varnothing \end{cases}$$ The concretization is defined by:
$$\gamma(\hat{S}) = \begin{cases} [-\infty, 0], & \text{if } - \in \hat{S} \\ \{0\} , & \text{if } 0 \in \hat{S} \\ [0, +\infty], & \text{if } + \in \hat{S} \end{cases}$$

• Interval. The concrete lattice is $\mathcal{P}(\mathbb{Z})$. The abstract lattice is $$\{ \varnothing \} \cup \{ [a, b] | a \leq b, a \in \mathbb{Z} \cup \{- \infty\}, b \in \mathbb{Z} \cup \{+\infty\} \}$$ The ordering is $[a,b] \leq [a’,b’]$ iff $a’ \leq a$ and $b \leq b’$, additionally, $\varnothing \leq [a, b]$.
The abstraction is defined by:
$$\alpha(S) = \begin{cases} \varnothing, & \text{if } S = \varnothing \\ [min S, max S], & \text{otherwise} \end{cases}$$ The concretization is defined by:
$$\gamma(\hat{S}) = \begin{cases} \varnothing, & \text{if } \hat{S} = \varnothing \\ \{a, \dots, b\}, & \text{if } \hat{S} = [a,b] \end{cases}$$

###### Properties

Proposition: $(\alpha, \gamma)$ is a Glaois connection iff $\alpha$ and $\gamma$ are monotone, $(\alpha \circ \gamma)(y) \leq_2 y$ and $x \leq_1 (\gamma \circ \alpha)(x)$.
Proof.

• First assume that $(\alpha, \gamma)$ is a Galois connection.
• By definition $\alpha(x) \leq_2 y$, and by reflexivity $\gamma(y) \leq_1 \gamma(y)$, thus $(\alpha \circ \gamma)(y) \leq_2 y$.
• By definition $x \leq_1 \gamma(y)$, and by reflexivity $\alpha(x) \leq_2 \alpha(x)$, thus $x \leq_1 (\gamma \circ \alpha)(y)$.
• Let $x \leq_1 x’$. We know $x’ \leq_1 (\gamma \circ \alpha)(x’)$, then by transitivity, $x \leq_1 (\gamma \circ \alpha)(x’)$. Thus by definition, $\alpha(x) \leq_2 \alpha(x’)$ and $\alpha$ is monotone.
• Let $y \leq_2 y’$. We know $(\alpha \circ \gamma)(y) \leq_2 y$, then by transitivity, $(\alpha \circ \gamma)(y) \leq_2 y’$. Thus by definition, $\gamma(y) \leq_2 \gamma(y’)$, and $\gamma$ is monotone.
• Then assume $\alpha$ and $\gamma$ are monotone, $(\alpha \circ \gamma)(y) \leq_2 y$ and $x \leq_1 (\gamma \circ \alpha)(x)$.
• Assume that $\alpha(x) \leq_2 y$, since $\gamma$ is monotone, then $\gamma(\alpha(x)) \leq_1 \gamma(y)$. By assumption, $x \leq_1 \gamma(\alpha(x))$, by transitivity, $x \leq_1 \gamma(y)$.
• Assume that $x \leq_1 \gamma(y)$. since $\alpha$ is monotone, then $\alpha(x) \leq_2 \alpha(\gamma(y))$. By assumption, $\alpha(\gamma(y)) \leq_2 y$, by transitivity, $\alpha(x) \leq_2 y$. $\square$

Proposition: Let $(L_1, \leq_1, \bot_1, \top_1, \sqcup_1, \sqcap_1)$ and $(L_2, \leq_2, \bot_2, \top_2, \sqcup_2, \sqcap_2)$ be complete lattices, and $(\alpha, \gamma)$ be a Galois connection between $L_1$ and $L_2$.
(1) Each function in the pair $(\alpha, \gamma)$ uniquely determines the other: $$\alpha(x) = \sqcap_2 \{ y \in L_2 | x \leq_1 \gamma(y) \} \\ \gamma(y) = \sqcup_1 \{ x \in L_1 | \alpha(x) \leq_2 y\}$$ (2) $\alpha$ is a complete join-morphism (i.e., is additive: $\alpha(\sqcup S) = \sqcup \{ \alpha(x) | x \in S \}$), $\alpha(\bot_1) = \bot_2$. $\gamma$ is a complete meet-morphism (i.e., $\gamma(\sqcap S) = \sqcap \{ \gamma(x) | x \in S \}$), $\gamma(\top_2) = \top_1$.
Proof. TODO $\square$

Proposition: Let $(L_1, \leq_1, \bot_1, \top_1, \sqcup_1, \sqcap_1)$ and $(L_2, \leq_2, \bot_2, \top_2, \sqcup_2, \sqcap_2)$ be complete lattices.
(1) If $\alpha : L_1 \to L_2$ is a complete join-morphism and $\gamma(y) = \sqcup_1 \{ x \in L_1 | \alpha(x) \leq_2 y \}$, then $(\alpha, \gamma)$ is a Galois connection.
(2) If $\gamma : L_2 \to L_1$ is a complete meet-morphism and $\alpha(x) = \sqcup_2 \{ y \in L_2 | x \leq_1 \gamma(y) \}$, then $(\alpha, \gamma)$ is a Galois connection.
Proof. TODO $\square$

Proposition: $\alpha$ is surjective iff $\gamma$ is bijective iff $\alpha \circ \gamma = \lambda y.y$.
Proof. TODO $\square$

Remark: Let $\sigma(y) = \sqcap \{ y’ | \gamma(y’) = \gamma(y)\}$, then $\alpha(L_1) = \sigma(L_2)$.
Proof. TODO $\square$

#### Computing abstract semantics

The approach: start from the concrete semantics, lift it to sets to obtain the collecting semantics, and abstract the collecting semantics.

###### Values and Environments

The correctness of the abstraction $\hat{x}$ of a value $x$ is $\alpha(x) \leq \hat{x}$. For an operator, such as $+$, is a correct abstraction iff $\alpha(a) \leq \hat{a}$ and $\alpha(b) \leq \hat{b}$, then $\alpha(a+b) \leq \hat{a} \hat{+} \hat{b}$.

The abstraction of environments can be defined in two steps:

• Abstract sets of environments to mappings from variables to sets of integers: $$\mathcal{P}(Var \to \mathbb{Z}) \underset{\gamma_{R1}}{\overset{\alpha_{R1}}{\rightleftharpoons}} Var \to \mathcal{P}(\mathbb{Z})$$ The abstraction is defined by $\alpha_{R1} (R_1) = \lambda x . \{ \rho(x) | \rho \in R_1 \}$.
The concretization is defined by $\gamma_{R1}(\hat{R_1}) = \{ \lambda x . y | y \in \hat{R_1}(x) \}$.

• Then abstract each result into $L$ by $(\alpha, \gamma)$: $$Var \to \mathcal{P}(\mathbb{Z}) \underset{\gamma_{R2}}{\overset{\alpha_{R2}}{\rightleftharpoons}} Var \to L$$ The abstraction is defined by $\alpha_{R2}(R_2) = \lambda x . \alpha(R_2(x))$.
The concretization is defined by $\gamma_{R2}(\hat{R_2}) = \lambda x . \gamma(\hat{R_2}(x))$.

• Finally, the Galois connection of environments $$\mathcal{P}(Var \to \mathbb{Z}) \underset{\gamma_R}{\overset{\alpha_R}{\rightleftharpoons}} Var \to L$$ is obtained by composing the two Galois connections: $\alpha_R = \alpha_{R_2} \circ \alpha_{R_1}$ and $\gamma_R = \gamma_{R_1} \circ \gamma_{R_2}$.

To summarize:

Semantics Collecting Semantics Abstraction Abstract Semantics
Values $\mathbb{Z}$ $(\mathcal{P}(\mathbb{Z}), \subseteq)$ $\underset{\gamma}{\overset{\alpha}{\rightleftharpoons}}$ $(L, \leq)$
Operators $+ : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ $+ : \mathcal{P}(\mathbb{Z}) \times \mathcal{P}(\mathbb{Z}) \to \mathcal{P}(\mathbb{Z})$ $\hat{a} ~\hat{+}~ \hat{b} = \alpha(\gamma(\hat{a}) + \gamma(\hat{b}))$
Environments $\rho : Var \to \mathbb{Z}$ $(R : \mathcal{P}(Var \to \mathbb{Z}), \subseteq)$ $\underset{\gamma_R}{\overset{\alpha_R}{\rightleftharpoons}}$ $\hat{R} : Var \to (L, \leq)$
###### Expressions
Semantics Collecting Semantics Abstract Semantics
$\llbracket E \rrbracket \rho \in \mathbb{Z}$ $\llbracket E \rrbracket R \in \mathcal{P}(\mathbb{Z})$ $\widehat{\llbracket E \rrbracket} \hat{R} \in L$
$\llbracket x \rrbracket \rho = \rho(x)$ $\widehat{\llbracket x \rrbracket} \hat{R} = \hat{R}(x)$
$\llbracket E_1 + E_2 \rrbracket \rho = \llbracket E_1 \rrbracket \rho + \llbracket E_2 \rrbracket \rho$ $\widehat{\llbracket E_1 + E_2 \rrbracket} \hat{R} = \widehat{\llbracket E_1 \rrbracket}\hat{R} ~\hat{+}~ \widehat{\llbracket E_2 \rrbracket}\hat{R}$

The correctness of the abstract semantics of expressions: if $\alpha_R(R) \leq \hat{R}$ then $\alpha(\llbracket E \rrbracket R) \leq \widehat{\llbracket E \rrbracket} \hat{R}$.
Proof: By induction on expression. TODO $\square$

###### Traces

The abstract trace semantics is defined as follows:

• If Prog(pc) = x := E, then $\langle pc, \hat{R} \rangle \to \langle pc+1, \hat{R}[x \mapsto \widehat{\llbracket E \rrbracket} \hat{R} ]\rangle$.
• If Prog(pc) = input x, then $\langle pc, \hat{R} \rangle \to \langle pc+1, \hat{R}[x \mapsto \top] \rangle$
• If Prog(pc) = print x, then $\langle pc, \hat{R} \rangle \to \langle pc+1, \hat{R} \rangle$
• If Prog(pc) = if E goto n, then
• $\langle pc, \hat{R} \rangle \to \langle n, \hat{R} \rangle$
• $\langle pc, \hat{R} \rangle \to \langle pc+1, \hat{R} \rangle$

The transition for conditional is non-deterministic. Sometimes we can have a better analysis by refining the values from E.

The correctness of the abstrace trace semantics is expressed by: if $\langle pc_1, \rho_1 \to pc_2, \rho_2 \rangle$, and $\alpha_R(\{ \rho_1 \}) \le \hat{R_1}$, then there exists $\hat{R_2}$ s.t. $\langle pc_1, \hat{R_1} \rangle \to \langle pc_2, \hat{R_2} \rangle$ and $\alpha_R(\{ \rho_2 \}) \leq \hat{R_2}$.

###### States

The next part is the abstract semantics of set of states.

Semantics Collecting Semantics Abstraction Abstract Semantics
$s:PC \times (Var \to \mathbb{Z})$ $S:\mathcal{P}(PC \times (Var \to \mathbb{Z}))$ $\underset{\gamma_S}{\overset{\alpha_S}{\rightleftharpoons}}$ $\hat{S}:PC \to (\{ \bot \} \cup (Var \to L))$

For abstraction on environments, we defined $$\alpha’_R(R) = \begin{cases} \bot & \text{if } R = \varnothing \\ \alpha_R(S) & \text{otherwise} \end{cases}$$ Then the abstraction is defined by $$\alpha_S(S) = \lambda pc . \alpha’_R(\{ \rho | (pc, \rho) \in S \})$$

For concretization on environments, we define $$\gamma’_R(\hat{R}) = \begin{cases} \varnothing & \text{if } \hat{R} = \bot \\ \gamma_R(\hat{R}) & \text{otherwise} \end{cases}$$ Then the concretization is defined by $$\gamma_S(\hat{S}) = \{ (pc, \rho) | \rho \in \gamma’_S(\hat{S}(pc))\}$$

The function $next$ is defined as given a set of (concrete) states returns the set of next states.

$$next(S) = \{ s’ | s \in S, s \to s’ \}$$

The corresponding abstract function $\widehat{next}$ satisfies $\widehat{next} \ge \alpha_S \circ next \circ \gamma_S$, which is defined as:

$$\alpha_S \circ next \circ \gamma_S(\hat{S}) = \lambda pc_2 . \alpha’_R(\{ \rho_2 | \rho_1 \in \gamma’_R(\hat{S}(pc_1)), \langle pc_1, \rho_1 \rangle \to \langle pc_2, \rho_2 \rangle \})$$

We have $\langle pc_1, rho_1 \rangle \to \langle pc_2, rho_2 \rangle$ and $\alpha_R(\{ \rho_1 \}) \leq \hat{S}(pc_1)$, so by correctness of the abstract transition relation, $\exists \hat{R_2} \text{ s.t. } \langle pc_1, \hat{S}(pc_1) \rangle \to \langle pc_2, \hat{R_2}\rangle$ and $\alpha_R(\{ \rho_2 \}) \leq \hat{R_2}$. So

$$\alpha_S \circ next \circ \gamma_S(\hat{S}) = \lambda pc_2 . \sqcup \{ \hat{R_2} | \exists pc_1 . \langle pc_1, \hat{S}(pc_1) \rangle \to \langle pc_2, \hat{R_2}\rangle\}$$

And then $\widehat{next}$ is defined as:

$$\widehat{next}(\hat{S}) = \lambda pc_2 . \sqcup \{ \hat{R_2} | \exists pc_1 . \langle pc_1, \hat{S}(pc_1) \rangle \to \langle pc_2, \hat{R_2}\rangle\}$$

#### Combination of Abstractions

##### Product

Let $L_i, L_i’$ ($i \in I$) be the complete lattices, where $I$ is a set of labels. If $(\alpha_i, \gamma_i)$ is a Galois connection:
$$(L_i, \leq_i) \underset{\gamma_i}{\overset{\alpha_i}{\rightleftharpoons}} (L_i’, \leq_i’)$$ Then we have a Galois connection:
$$(\times_{i \in I} L_i, \leq) \underset{\gamma}{\overset{\alpha}{\rightleftharpoons}} (\times_{i \in I} L_i’, \leq’)$$ If $a, b \in \times_{i \in I} L_i$, then $a \leq b$ iff $\forall i \in I, a_i \leq_i b_i$ (similarly for $\leq’$), and $\alpha(\langle x_i, … \rangle) = \langle \alpha_i(x_i), … \rangle$, and $\gamma(\langle x_i, … \rangle) = \langle \gamma_i(x_i), … \rangle$.

##### Sets

Let $L$ be a complete lattice, and $S$ be a set. Let $f : S \to L$ be a function. Then we have Galois connection:
$$(\mathcal{P}(S), \subseteq) \underset{\gamma}{\overset{\alpha}{\rightleftharpoons}} (L, \leq)$$ where $\alpha(x) = \sqcup\{ f(x’) | x’ \in x \}$ and $\gamma(y) = \{ x’ | f(x’) \leq y \}$.

##### Functions with fixed input
1. Let $L$ and $L’$ be complete lattices, and $S$ be a set. If $(\alpha, \gamma)$ is a Galois connection:
$$(L, \leq) \underset{\gamma}{\overset{\alpha}{\rightleftharpoons}} (L’, \leq’)$$ Then we have a Glaois connection:
$$(S \to L, \leq) \underset{\gamma’}{\overset{\alpha’}{\rightleftharpoons}} (S \to L’, \leq’)$$ The function lattices are ordered pointwise: $f \leq f’$ iff $\forall x \in S, f(x) \leq f’(x)$, $\alpha’(f) = \lambda x . \alpha(f(x))$, and $\gamma’(f) = \lambda x . \gamma(f(x))$.

2. Let $L$ and $L’$ be complete lattices, and $S$ be a set. If $(\alpha, \gamma)$ is a Galois connection:
$$(L, \leq) \underset{\gamma}{\overset{\alpha}{\rightleftharpoons}} (L’, \leq’)$$ Then we have a Glaois connection:
$$(S \to L, \leq) \underset{\gamma’}{\overset{\alpha’}{\rightleftharpoons}} (L’, \leq’)$$ The function lattices are ordered pointwise: $f \leq f’$ iff $\forall x \in S, f(x) \leq f’(x)$, $\alpha’(f) = \sqcup \{ \alpha(f(x)) | x \in S \}$, and $\gamma’(y) = \{ f | \forall x \in S, f(x) \leq \gamma(y) \}$.

These are two ways to abstract over arrays. The first abstraction uses one abstract value for each element in the array; the second abstract uses one abstract element for the whole array.

##### Functions with abstracted input
1. Let $(L_1, \leq_1, \bot_1, \top_1, \sqcup_1, \sqcap_1)$ and $(L_2, \leq_2, \bot_2, \top_2, \sqcup_2, \sqcap_2)$ be complete lattices. The set of monotone functions $(L_1 \overset{m}{\to} L_2, \leq_f, \bot_f, \top_f, \sqcup_f, \sqcap_f)$ ordered pointwise is a complete lattice, where

$$f_1 \leq_f f_2 \Leftrightarrow \forall x \in X, f_1(x) \leq f_2(x)$$ $$\bot_f = \lambda x . \bot, \top_f = \lambda x. \top$$ $$\sqcup_f F = \lambda x . \sqcup \{ f(x) | f \in F \}, \sqcap_f F = \lambda x . \sqcap \{ f(x) | f \in F \}$$

1. Let $(L_1, \leq_1, \bot_1, \top_1, \sqcup_1, \sqcap_1)$ and $(L_2, \leq_2, \bot_2, \top_2, \sqcup_2, \sqcap_2)$ be complete lattices. A function $f : L_1 \to L_2$ is additive iff $\forall S \subseteq L_1$, $f(\sqcup S) = \sqcup \{ f(x) | x \in S \}$. The set of additive functions $(L_1 \overset{a}{\to} L_2, \leq_f, \bot_f, \top_f, \sqcup_f, \sqcap_f)$ ordered pointwise is a complete lattice, where

$$f_1 \leq_f f_2 \Leftrightarrow \forall x \in X, f_1(x) \leq f_2(x)$$ $$\bot_f = \lambda x . \bot, \top_f = \lambda x . \top$$ $$\sqcup_f F = \lambda x . \sqcup \{ f(x) | f \in F \}, \sqcap_f F = \sqcup_f \{ g | \forall f \in F, g \leq_f f \}$$

TODO

#### Composition of abstractions

Let $L_1$, $L_2$, $L_3$ be complete lattices. If $(\alpha, \gamma)$ is a Galois connection between $L_1$ and $L_2$, and $(\alpha’, \gamma’)$ is a Galois connection between $L_2$ and $L_3$, then we have Galois conection between $L_1$ and $L_3$:

$$(L_1, \leq_1) \underset{\gamma \circ \gamma’}{\overset{\alpha’ \circ \alpha}{\rightleftharpoons}} (L_3, \leq_3)$$ Example: signs as abstraction of intervals.

TODO

TODO

#### Loops

##### Fixpoints

Definition (Monotonic function): Given $(L_1, \leq_1)$ and $(L_2, \leq_2)$, a function $f : L_1 \to L_2$ is monotonic (order preserving) if $x \leq_1 x’$ implies $f(x) \leq_2 f(x’)$.

Definition (Continuous function): Given lattices $(L_1, \leq_1)$ and $(L_2, \leq_2)$, a function $f : L_1 \to L_2$ is Scott-continuous if $S \subseteq L_1$ implies $f(\sqcup S) = \sqcup (f(S))$, where $f(S) = \{ f(x) | x \in S \}$. Scott-continuous functions are also monotonic.

Definition (Fixpoint): A fixpoint of a function $f$ is an element x s.t. $f(x) = x$.

Given a complete lattice $(L, \leq)$ and a function $f : L \to L$, we can divide the set $L$ into regions:

• $\text{Fix}(f) = \{ x | x = f(x) \}$ - the fixed point region
• $\text{Asc}(f) = \{ x | x \leq f(x) \}$ - the ascending (extensive) region
• $\text{Des}(f) = \{ x | x \geq f(x) \}$ - the descending (reductive) region
• $\text{Fix}(f) = \text{Asc}(f) \cap \text{Des}(f)$

Theorem (Tarski): The set of fixpoints of a monotone function $f$ on a complete lattice $(L, \leq, \bot, \top, \sqcup, \sqcap)$ is a non-empty complete lattice, ordered by $\leq$. The least fixpoint of $f$ is $\text{lfp}(f) = \sqcap \{ x | x \geq f(x) \}$, and its greatest fixpoint is $\text{gfp}(f) = \sqcup \{ x | x \leq f(x) \}$.
Proof: We show that $a = \sqcap \{ x | f(x) \leq x \}$ is the least fixpoint of $f$.
By definition, $f$ is monotonic and $a \leq x$, so $f(a) \leq f(x)$; $a \in \{ x | f(x) \leq x \}$, so $f(a) \leq a$. Then $f(a) \leq \{ x | f(x) \leq x \}$, so $a \leq f(a)$. Then $a = f(a)$, so $a$ is a fixpoint of $f$.
Suppose that exists another fixpoint $x’$ s.t. $f(x’) = x’$. Since $x’ \in \{ x | f(x) \leq x \}$, so $a \leq x$, then $a$ is the least fixpoint of $f$. $\square$

Equivalently, $\text{lfp}(f) = \sqcap \text{Des}(f) = \sqcap \text{Fix}(f) \in \text{Fix}(f)$, and $\text{gfp}(f) = \sqcup \text{Asc}(f) = \sqcup \text{Fix}(f) \in \text{Fix}(f)$.

The least fixpoint can be obtained by a transfinite iteration of $f$ from $\bot$. $f^{\lambda}(\bot)$ where $\lambda$ is an ordinal, is defined by
$$f^0(\bot) = \bot$$ $$f^{\lambda + 1}(\bot) = f(f^{\lambda}(\bot))$$ $$f^{\lambda} = \sqcup \{ f^\beta(\bot) | \beta \lt \lambda \}$$

The transfinite sequence eventually stabilizes, and its limit is $\text{lfp}(f)$.

##### CPO

Definition (Chain): Let $(S, \leq)$ be a partially ordered set. A chain $C = (x_n)_{n \in \mathbb{N}}$ is a monotone sequence of elements of $S$: $x_0 \leq x_1 \leq \cdots \leq x_n \leq x_{n+1} \leq \cdots$.

Definition (CPO): A complete partial order is a poset $(S, \leq)$ s.t. $S$ has a least element $\bot$ and every chain $C$ has a least upper bound $\sqcup C$. A complete lattice is also a CPO.

Theorem (Kleene): Let $(S, \leq)$ be a CPO. Let $f: S \to S$ be a continuous function. Then $f$ has a least fixpoint $\text{lfp}(f) = \sqcup \{ f^i(\bot) | i \in \mathbb{N} \}$.

##### Fixpoint collecting semantics

The set of reachable states can be computed as fixpoint of function $next$. Let $F_s(S) = S_0 \cup next(S)$, where $S_0$ is the initial state. The set of reachable states is then $S_r = \text{lfp}(F_s) = next^*(S_0) = \cup \{ next^*(S_0) | n \in \mathbb{N} \}$.

The set of possible traces also can be computed as fixpoint. Let $F_t(T) = S_0 \cup \{ t \to s’ | t \in T, T \text{ ends in } s, s \to s’ \}$. The set of possible traces is $T_r = \text{lfp}(F_t)$.

##### Abstraction of fixpoints

Proposition: Let $F$ be a monotone function on a complete lattice $F: L_1 \to L_1$, and $(\alpha, \gamma)$ be a Galois connection between $L_1$ and $L_2$. Then $\gamma(\text{lfp}(\alpha \circ F \circ \gamma)) \geq \text{lfp}(F)$, i.e., $\text{lfp}(\alpha \circ F \circ \gamma) \geq \alpha(\text{lfp}(F))$.
Proof: TODO $\square$

##### Widening

Definition (Upper bound operator): An upper bound operator is a function $\check{\sqcup} : L \times L \to L$ s.t. $\forall x, y \in L$, $x \leq x \check{\sqcup} y$ and $y \leq x \check{\sqcup} y$.

Let $(l_n)_n$ be a sequence of elements of $L$. Define the sequence $(l_n^\check{\sqcup})_n$ as:
$$l_n^\check{\sqcup} = \begin{cases} l_0 & \text{if } n = 0 \\ l_{n-1}^\check{\sqcup} \check{\sqcup} l_n & \text{if } n > 0 \end{cases}$$

$(l_n^\check{\sqcup})_n$ is an ascending chain, and $l_n^\check{\sqcup} \geq \{ l_0, l_1, \cdots, l_n\}$ for all $n$.

Example: let $int$ be an arbitrary fixed element of interval, then we can construct an upper bound operator:

$$int_1 \check{\sqcup}^{int} int_2 = \begin{cases} int_1 \sqcup int_2 & \text{if } int_1 \leq int \vee int_2 \leq int_1 \\ [-\infty, +\infty] & \text{otherwise} \end{cases}$$
So, $[1,2] \check{\sqcup}^{[0, 2]} [2, 3] = [1, 3]$ (because $[1,2]\leq[0,2]$), and $[2,3] \check{\sqcup}^{[0,2]} [1, 2] = [-\infty, +\infty]$ (because neither $[2,3] \leq [0,2]$ or $[1,2] \leq [0, 2]$).

Definition (Widening Operator): A function $\triangledown : L \times L \to L$ is a widening operator iff

• $\triangledown$ is an upper bound operator, i.e., $\forall l_1, l_2, l_1 \leq l_1 \triangledown l_2 \wedge l2 \leq l_1 \triangledown l_2$
• forall ascending chains $(l_n)_n$ the ascending chain $(l_n^{\triangledown})_n$ eventually stablizes, where

$$l_n^{\triangledown} = \begin{cases} l_n & \text{if } n = 0 \\ l_{n-1}^{\triangledown} \triangledown l_n & \text{otherwise} \end{cases}$$

Proposition: For a monotonic function $f : L \to L$ ($L$ is a complete lattice) and a widening operator $\triangledown$ on $L$, we define the sequence $(f_n^{\triangledown})_n$:

$$f_\triangledown^n = \begin{cases} \bot & \text{if } n = 0 \\ f_{n-1}^\triangledown & \text{if } n > 0 \wedge f(f_{n-1}^\triangledown) \leq f_{n-1}^\triangledown \\ f_{n-1}^\triangledown ~ \triangledown ~ f(f_{n-1}^\triangledown) & \text{otherwise} \end{cases}$$

Then $(f_n^{\triangledown})_n$ is an ascending chain and eventually stabilizes at a value $f_m^{\triangledown}$ such that $f(f_m^{\triangledown}) \leq f_m^{\triangledown}$. Tarski theorem gives us $f_m^\triangledown \geq \text{lfp}(f)$.

##### Narrowing

Definition (Narrowing Operator): A function $\Delta : L \times L \to L$ is a narrowing operator iff

• $\Delta$ is a lower bound operator, i.e., $\forall l_1, l_2, l_2 \leq l_1 \implies l_2 \leq l_1 \Delta l_2 \leq l_1$
• for all descending chains $(l_n)_n$ (e.g., $l_0 \geq l_1 \geq \cdots l_n$), the descending chain $x_0 = l_0, \cdots, x_{ i+1 } = x_i Δ l_{i+1}$ is not strictly descending.

Proposition: For a monoton function $f: X \times X \to X$ and a narrowing operator $\Delta$, if $x_0 \geq \text{lfp}(f)$ and $x_0 \geq f(x_0)$, then the descending chain $x_{i+1} = x_i \Delta f(x_i)$ eventually stabilizes, and its limit is greater or equal to $\text{lfp}(f)$.
Proof: TODO $\square$