Diseño y análisis funcional de nuevos análogos de la kp-10 de rata La participación del sistema kiss1/kiss1r a distintos niveles y en funciones de

FGE-DNF assumes that the input,ϕ, is a formula in disjunctive normal form (DNF). A

formula is in DNF iff it is a disjunction of terms where a term is a conjunction of literals. FGE-DNF is a variant of FGEassuming a DNF formula as input. FGE-DNF is applicable to

Scenario2 in Chapter3where a FM is to be synthesized from a list of existing variants of a product.

We describe FGE-DNF according to the pseudocode in Figure 5.2. When describing

runtime, we assume that the DNF formula only contains satisfiable terms. A term is satisfiable if it does not contain a literal and its negation. Unsatisfiable terms can be removed in linear time.|ϕ| is the size of the input and refers to the number of terms in the DNF formula.

Lines1–2, Dead Features: A variable f is dead iff it is negated in every term ofϕ.

This can be checked in linear time in|ϕ| making this step run in O(|ϕ||F|) time.

Lines 3–4, Implications: Implications are detected by checking if ϕ ∧ fi → fj is

consistent. Sinceϕ is in DNF, these implications are found by checking if every term in

ϕ contains either ¬fi or fj. Detecting all implications takes O(|ϕ||F|

2_{) time.}

Line8, Mutual Exclusions: Mutual exclusions are found by checking ifϕ∧ f_i → ¬ fj, or

equivalently,ϕ ∧ fi∧ fj is consistent. Similar to detecting implications, the satisfiability

can be computed in linear time. This process is repeated for every feature pair, so detection of exclusions also takes O(|ϕ||F_|2) time.

Line11–12, Or Groups: IdentifyingOR-groups involves identifying prime implicates

ofϕ. Detecting prime implicates in ϕ is equivalent to detecting prime implicants of ¬ϕ. An implicant C of a propositional formula ϕ is a term such that C is consistent and

C → ϕ is a tautology. C is a prime implicant if it is minimal. If π is a prime implicates of

ϕ, then ¬π is a prime implicant of ¬ϕ. Using this property,OR-groups can be computed

by using a SAT solver to find prime implicants of¬ϕ.

We use a procedure based on Binary Integer Programming (BIP) [Sil97,MFSO97], a

special case of Integer Linear Programming (ILP) that assumes binary domain for variables (i.e., a variable can be 0 or 1), to compute OR-groups. First, given a feature p,

we construct a new formula¬ϕ0

p with the desired properties to identify OR-groups with

parent p:

1. First, negateϕ to create ¬ϕ. Since ϕ is in DNF, ¬ϕ becomes a CNF formula. 2. Detecting OR-groups for parent p involves detecting prime implicates ofϕ ∧ p.

Since we are working with¬ϕ, we detect prime implicants of ¬ϕ ∨ ¬p. We retain only the clauses in¬ϕ that contain ¬p.

3. The members of a valid OR-group must share a common parent, namely p. Using

p. In other words, retain{ f | f ∈ v0_{∧ (v}0_{, p}0_{) ∈ E}0 _{where v}0_{∈ V}0_and∃p0_{∈ V.p ∈}

p0}.

4. A validOR-group has the constraint p→ f₁∨· · ·∨ f_kwhere the variables{ f₁, . . . , f_k}

are positive. Since we are working with¬ϕ, this constraints translates to prime implicants containing only negative literals. For each clause C, remove all variables v∈ C that are positive.

Now we can use the formula¬ϕ0p to build the BIP problem used to identifyOR-groups

for a parent p:

1. Let L be the set of literals in ¬ϕ0

p. For each literal l ∈ L, introduce a Boolean

variable xl to the BIP problem. Note that¬ϕ0 only contains negative literals.

2. For each clause l1∨ · · · ∨ lm in¬ϕ0p add the linear inequality xl1+ · · · + xlm ≥ 1 to the set of constraints.

3. Since a literal l and its negation¬l cannot both be true in a satisfying assignment, a constraint of the form xl + x¬l ≤ 1 is added for each literal in L. These

constraints are not needed if the process for constructing¬ϕ0

p outlined above is

used. ¬ϕ0_p has removed all negative variables, and only positive variables remain in the formula.

4. The objective function is to minimize the number of positive literals: P_l∈Lx_l. The optimal solutions to this BIP correspond to the prime implicants of¬ϕ ∨ ¬p. We demonstrate the BIP translation using the input DNF formula ϕ in Figure 5.3a. Computing the feature groups for featurea, we first construct the CNF formula¬ϕ0

a.

The formula ¬ϕ0

a only includes negated variables in ¬ϕ that implya based on the

implication graph in Figure5.3b. The resulting formula is shown in Figure5.3c. Finally, the resulting BIP translation is in Figure5.3d. The optimal solution to this BIP problem has l_¬b and l_¬c set to 1 and l_¬d set to 0. As a result,{b,c} are a feature group with parenta.