Procedure PC-bs-ILP

The following bound selection algorithm PC-bs-ILP is based on the idea that bound choices of two prime constituents need to share an element from {l, u} if and only if they are conjunctively associated. To achieve this the algorithm constructs and solves the system (3.1) for any given PC decomposition P of ϕ. As a result, the algorithm fixes the bound choice B of each prime constituent in the PC decomposition P of ϕ.

Steps 1–2 of the algorithm solve the 0-1 ILP system (3.1) for a PC decom- position P of ϕ. For this we use ILP-construct, which first constructs the system (3.1) without explicitly constructing the conjunctive associativity graph

G(ϕ, P ).

Steps 4–6 ensure that the bound choice B of each Gauss prime constituent is nonempty, and that B is not {l, u}. We ensure that the latter condition holds only to prevent including both (−∞, (), ∅) and (∞, (), ∅) into E when P contains exclusively Gauss prime constituents, because in such a case only one of ±∞ is needed.

Algorithm PC-bs-ILP(P, ϕ, x).

Input: a PC decomposition P of ϕ w.r.t. x.

Output: a PC decomposition P0 of ϕ w.r.t. x containing only prime constituents with fixed bound choices B.

1. (f, C) := ILP-construct(P, ϕ, x) 2. Compute a solution s of the system C.

3. For each µ ∈ P at position π with type Υ ∈ {COGAUSS, AT} do 3.1. B := ∅

3.2. If s(xl

π) = 1, then B := B ∪ {l}.

3.3. If s(xu

π) = 1, then B := B ∪ {u}.

3.4. Fix the bound choice of µ to B. 4. Bg:= {l}

5. If there exists a variable xu

π such that s(xuπ) = 1, then Bg:= {u}.

6. For each µ ∈ P with type Υ = GAUSS do 6.1. Fix the bound choice of µ to Bg.

7. Return P .

Algorithm ILP-construct(P, ϕ, x).

Input: a PC decomposition P of ϕ w.r.t. x.

Output: a system C of 0-1 ILP constraints for ϕ and P as in (3.1). 1. (C, P0) := ILP-subroutine P, ϕ, x, ()

2. For each µ = (π, Υ, c, F , B) ∈ P0 do 2.1. C := C ∪ {xlπ+ xuπ ≥ 1}

3. Return C.

Algorithm ILP-subroutine(P, ϕ, x, π).

Input: a PC decomposition P of ϕ w.r.t. x, a position π ∈ Pos(ϕ).

Output: a pair (C, P0), where C is a system of 0-1 ILP constraints, and P0⊆ P is the set of co-Gauss and atomic prime constituents occurring in π(ϕ).

1. If π is a position of a co-Gauss or atomic prime constituent in P , then 1.1. Return (∅, {π}).

2. If π is a position of a Gauss prime constituent in P or a leaf position, then 2.1. Return (∅, ∅).

3. For each child position π|1, . . . , π|n of π do 3.1. (Ci, Pi0) := ILP-subroutine(P, ϕ, x, π|i) 4. C :=Sn i=1Ci 5. P0:=Sn i=1P 0 i

6.1. Return (C, P0).

7. If the top-level operator of π(ϕ) is “∧,” then

7.1. For each pair of PCs µi∈ Pi0 and µj ∈ Pj0 such that i < j, do

7.1.1. Let πi and πj be the positions of µi and µj, respectively.

7.1.2. C := C ∪ {xl πi+ x u πj ≥ 1, x u πi+ x l πj ≥ 1} 7.2. Return (C, P0).

It is easy to see that ILP-construct meets its specification and constructs system (3.1) for a given PC decomposition P of ϕ. Its worst-case running time is quadratic in the size of the structural tree for ϕ because of the loop in step 7 of ILP-subroutine. Consequently, algorithmPC-bs-ILP is correct as well. Reasonable Solutions of 0-1 ILP System C

Algorithm PC-bs-ILP picks in step 2 any solution s of the system C. Therefore, it is correct to assign each variable the value 1 to obtain a solution of C. However, it is of huge practical importance to obtain as small elimination set E as possible. Here we propose a few strategies towards achieving this goal. These strategies are based on the observation that we have to be careful and minimize the number of variables that are assigned the value 1 in a solution s.

To do this one first constructs a linear objective function κ in the variables S

π∈V{x l

π, xuπ}, where V is the set of vertices of the graph G(ϕ, P ) = (V, E),

i.e., the set of positions of all co-Gauss and atomic prime constituents in P . Minimizing κ w.r.t. the system C then prevents unnecessary 1-assignments in s. A simple objective function κ that ensures this is:

κ = X

π∈V

xl_π+ xu_π. (3.2)

Here we point to the fact that it is straightforward to adjust ILP-construct to also return this objective function κ for P and to minimize κ w.r.t. C in step 2 of PC-bs-ILP afterwards.

In practice one indeed wants to use an efficient 0-1 ILP solver [1] or a SAT solver with optimization functionality [40] for finding a solution of C with the smallest possible value of κ.

On the theoretical side observe that the facts that each constraint of (3.1) contains exactly two variables and that the objective function κ from (3.2) is linear ensure that this optimization problem can be reduced to the minimum weighted vertex cover problem. This is a well-studied combinatorial optimiza- tion problem [4, 5, 57, 24]. However, it is possible that the special structure of the graph—that always comes to existence from G(ϕ, P )—allows for more efficient approximation and kernelization algorithms.

Now we return to our Example 65 where we showed three colorings χl,

χu, and χ. The values of the objective function κ from (3.2) for these three

colorings are 5, 5, and 6, respectively. Observe, however, that the cost 5 of χl

does not really correspond with the produced structural test points: The prime constituent at position π4 added one to the objective function value (because

s(xlπ4) = 1 and s(x

l

π4) = 0 in this case) even though it does not produce any test point.

Therefore, it can make sense to multiply the variables in (3.2) by weights

wl_π and wu_π, respectively. Intuitively, it should be then possible to estimate the costs of produced test points more precisely. Consequently, such an adjusted κ should guide a solver towards such solutions of the system C that are expected to yield “nicer” elimination sets after PC-to-TPs.

As an example that it can make a significant difference for a prime con- stituent whether its lower bound candidate solutions or upper bound candidate solutions are used to obtain test points we consider the atomic formula f ≤ 0, where f = x3_{+ ax}2

+ bx + c and a, b, c ∈ Z[u]. For simplicity we do not use clustering. Since the leading coefficient of f is positive, there are only four potential real 3-types of f . What these real types look like was discussed in Subsection 2.5.2. The following parametric root descriptions cover weak lower bounds of Φ(f ≤ 0, a) for any parameter values a ∈ Rm_:

f, (3, 2)
, f, (4, 2)
.

At the same time, parametric root descriptions covering weak upper bounds of Φ(f ≤ 0, a) for any parameter values a ∈ Rmare

f, (1, 1)
, f, (2, 1)
, f, (3, 1)
, f, (3, 2)
, f, (4, 1)
, f, (4, 3)
.

Recall that, for example, the root specification (3, 1) of f represents the first (counting from the left to the right) real root of f hai whenever f hai is of real type (−1, 0, 1, 0, 1). Assuming that the costs, i.e., the time and the length of the obtained formula, of substituting a parametric root description f, (t, r)
are the same regardless of the root specification (t, r) one would set wl

πto 2 and

wu

π to 6. Therefore, xlπ+ xuπ would be replaced with 2xlπ+ 6xuπ in the objective

function (3.2). Here π is the position of the atomic prime constituent f ≤ 0 in the formula ϕ.

For our Example 65 a weighted objective function κ based on the approach we have just described would look as follows:

κ = xl_π 1+ x u π1+ x l π2+ 0 · x u π2+ x l π3+ 0 · x u π3+ 0 · x l π4+ x u π4+ 0 · x l π5+ x u π5. The costs of the three admissible colorings χl, χu, and χ with respect to κ are

then 3, 3, and 2, respectively.

One could go even further and analyze the degree of f , the degrees of the parameters occurring in f , the number of recursive lower-degree virtual substi- tutions and lengths of the virtual substitution formula schemes of Appendix A to adjust the weights wl

πand wπuof prime constituents in P even more precisely.

However, in most cases counting the number of root specifications producing a test point, as we have shown above, should already yield a good approximation of the actual costs of a prime constituent when its lower/upper bound candidate solutions are taken.