FOL with Update Expressions

Syntactic extension of FOL with if-then-else, function update, sets.

If-Then-Else

ite(p,x,y) = if p then x else y

Function Update

f (a:=b)(x)

f ( (a,b) :=c)(x,y)

Sets

Set variables (not quantified).

Set operations $\cup$ , $\cap$ , $\setminus$ , $\{x,y\}$ .

Syntax Summary

$\begin{equation*}\begin{array}{l} F ::= A \mid F \land F \mid F \lor F \mid \lnot F \mid \exists x.F \mid \forall x.F \\ A ::= R(T,\ldots,T) \mid T=T \mid S \subseteq S \mid S=S \\ T ::= V_o \mid U(T,\ldots,T) \mid ite(F,T,T) \\ U ::= f \mid U((T,\ldots,T):=T) \\ S ::= V_s \mid S \cup S \mid S \cap S \mid S \setminus S \mid \{T,\ldots,T\} \end{array}\end{equation*}$

Elimination of Extensions

f(a:=b)(x) = ite(x=a,b,f(x))

f ( (a,b):=c )(x,y) = ite(x=a & y=b, c,f(x,y))

z=ite(p,x,y) $\leftrightarrow$ (p & z=x) | (~p & z=y)

F(ite(p,x,y)) $\leftrightarrow$ F(z) & z=ite(p,x,y)

S1 = S0 Un {a} $\leftrightarrow$ ALL x. S1(x) ↔ S0(x) | x=a

f = g ALL x. f(x)=g(x)

Substituting Function Symbols

Consider symbolic execution of assignment to local variable

x = e

Evaluate e in current symbolic state, then assign symbolic value of x to result.

Consider

x.f = e;

i.e.

f = f(x:=e)

We need to replace f with f(x:=e).

do we always get syntactically valid formula in FOL extension above?