Contents - Student subdomain for University of Bath

3.5. EQUATIONS AND INEQUALITIES 107
S 3 projects onto [−1, 1], which is the union of three sets in (3.58). We have to
decompose S 3 into four sets:
S 3,1 = {(−1, 0)}, S 3,2 = {(1, 0)},
S 3,3 = {(x, y) | x 2 + y 2 − 1 = 0 ∧ y > 0},
S 3,4 = {(x, y) | x 2 + y 2 − 1 = 0 ∧ y < 0}.
S 1 splits into eight sets, one above each of (−∞, −1) and (1, ∞) and two above
each of the other components of (3.58). It is obvious that this is the minimal
refinement of the original decomposition to possess a cylindric decomposition.
Furthermore in this case no linear transformation of the axes can reduce this.
If we wanted a sampled decomposition, we could choose x-coordinates of −2,
−1, 0, 1 and 2, and y-coordinates to match, from {0, ±1, ±2}.
Cylindricity is fundamental to solving problem 3 via the following two propositions.
Proposition 42 Let
∃x n . . . ∃x m+1 P (x 1 , . . . , x n ) (3.59)
be an existentially quantified formula, D be a sampled algebraic decomposition
of R n which is sign-invariant for all the polynomials occurring in P , and D ′
be a sampled algebraic decomposition of R m such that D is cylindrical over D ′ .
Then a quantifier-free form of (3.59) is
∨
Def(C ′ ). (3.60)
Proposition 43 Let
C ′ ∈D ′ ∃C∈Cyl(C ′ )P (Sample(C))
∀x n . . . ∀x m+1 P (x 1 , . . . , x n ) (3.61)
be a universally quantified formula, D be a sampled algebraic decomposition of
R n which is sign-invariant for all the polynomials occurring in P , and D ′ be
a sampled algebraic decomposition of R m such that D is cylindrical over D ′ .
Then a quantifier-free form of (3.61) is
∨
Def(C ′ ). (3.62)
C ′ ∈D ′ ∀C∈Cyl(C ′ )P (Sample(C))
These two propositions lead to a solution of problem 3.
Theorem 25 ([Col75]) Let x 0 , . . . , x k be sets of variables, with x i = (x i,1 , . . . ,
x i,ni ), and let N i = ∑ i
j=0 n j. Let P (x 0 , . . . , x k ) be a semi-algebraic proposition,
D i be an algebraic decomposition of R Ni such that each D i is cylindric over D i−1
and D k is sign-invariant for all the polynomials in P . Then a quantifier-free
form of
Q k x k . . . Q 1 x 1 P (x 0 , . . . , x k ) (3.63)