Is division by zero included in QF_NRA?
The SMT-LIB standard is confusing in this matter. The paper where the standard is defined simply does not discuss this point, in fact NRA and QF_NRA do not appear anywhere in that document. Some information is provided on the standard website. Reals are defined as including:
- all terms of the form (/ m n) or (/ (- m) n) where
- m is a numeral other than 0,
- n is a numeral other than 0 and 1,
- as integers, m and n have no common factors besides 1.
This explicitly excludes zero from the denominator when it comes to constant values. However, later, division is defined as:
- / as a total function that coincides with the real division function
for all inputs x and y where y is non-zero,
This is followed up by a note:
Since in SMT-LIB logic all function symbols are interpreted as total
functions, terms of the form (/ t 0) *are* meaningful in every
instance of Reals. However, the declaration imposes no constraints
on their value. This means in particular that
- for every instance theory T and
- for every closed terms t1 and t2 of sort Real,
there is a model of T that satisfies (= t1 (/ t2 0)).
This is seemingly contradictory, because the first quote says that (/ m 0) is not a number in QV_NRA, but the latter quote says that / is a function such that (= t1 (/ t2 0)) is satisfiable for any t1 and t2.
The de-facto reality on the ground is that division by zero seems to be included in SMT-LIB, despite the statement that (/ m n) is only a Real number if n is nonzero. This is related to a previous question of mine: y=1/x, x=0 satisfiable in the reals?
