LARA

This is an old revision of the document!

Definition of Set Constraints

Set constraints are a logic used in program analysis. In set constraints, each variable denotes a set of ground terms.

Let $L$ be a language of function symbols and constants. We write $f \in L$ for an element of $L$. Let $GT$ be the set of ground terms in language $L$.

Syntax of Set Constraints

\[ \begin{array}{l} 

S ::= v | S \cap S | S \cup S | S \setminus S | f(S, ..., S) | f^{-i}(S) \\
F ::= S \subseteq S \mid F \land F

\end{array} \] where

  • $v$ - set variable
  • $\cap,\cup,\setminus,\subseteq$ - standard set operations and relations
  • $f \in L$ - function symbol
  • in $f^{-i}$, $i$ is an integer between 1 and arity of $f$

Semantic

  • $v$ - set variable, subset of $GT$
  • $\cap,\cup,\setminus,\subseteq$ - standard set operations and relations
  • $f(S_1,\ldots,S_n) = \{ f(t_1,\ldots,t_n) \mid t_1 \in S_1,\ldots, t_n \in S_n \}$
  • $f^{-i}(S) = \{ t_i \mid f(t_1,\ldots,t_n) \in S \}$

Example1

\[ 

  f(\{a,f(b,b)\},\{b,c\}) = \{ f(a,b),f(a,c),f(f(b,b),b), f(f(b,b),c) \}

\]

Example2

\[ 

  f^{-2}(\{a,f(a,b),f(f(a,a),f(c,c))\}) = \{b,f(c,c)\}

\]

Example3

What is the least solution of constraints \[ \begin{array}{l} 

a \subseteq S \\
f(f(S)) \subseteq S

\end{array} \] where $a$ is constant and $f$ is unary function symbol.

Example4

What is the least solution of constraints \[ \begin{array}{l} 

a \subseteq S \\
black(S) \subseteq S \\
red(black(S)) \subseteq S

\end{array} \] where $a$ is constant, $red,back$ are unary function symbols.

Example5

Let $L = \{a_1,\ldots,a_n, not, and, or, implies\}$ where

  • $a_1,\ldots,a_n$ are constants representing propositional constants
  • $not$ is unary function symbol representing negation in abstract syntax tree
  • $and,or,implies$ are binary function symbols representing $\land,\lor,\rightarrow$ in syntax tree

Then the set of ground terms $GT$ represents propositional formulas.

What does the least solution of these constraints represent (where $S$,$P$ are set variables): \[ \begin{array}{l} 

a_1 \subseteq P \\
\ldots \\
a_n \subseteq P \\
P \subseteq S \\
not(P) \subseteq S \\
and(S,S) \subseteq S \\
or(S,S) \subseteq S

\end{array} \]

Existence of Solutions

We were able to talk about “the least solution” because previous examples can be rewritten into form \[ 

(S_1,\ldots,S_n) = F(S_1,\ldots,S_n)

\] where $F (2^{S})^n \to (2^{S})^n$ is a $\sqcup$-morphism (and therefore monotonic and $\omega$-continuous). The least solution can therefore be computed by fixpoint iteration (but it may contain infinite sets).

Example

Previous example can be rewritten as \[ \begin{array}{l} 

P = P \cup a_1 \cup \ldots a_n \\
S = S \cup P \cup not(P) \cup and(S,S) \cup or(S,S)

\end{array} \]

Is every set expression in set constraints monotonic?

Conditional Constraints

Let us simplify this expression: \[ 

  f^{-2}(f(S,T)) =

\] 

 = f^{-2}(\{ f(s,t) \mid s \in S, t \in T \})

\] 

 = \{ s_1 \in S \mid \exists t_1 \in T \}

\] 

 = \left\{ \begin{array}{rl} 
     S, \mbox{ if } T \neq \emptyset \\
     \emptyset, \mbox{ if } T = \emptyset
            \end{array}
   \right.

\]