Exclusive and essential sets of implicates of a Boolean function

1. Exclusive and essential sets of implicates of a Boolean function Ondrej Cepek Charles University in Prague, Czech Republic jointly with Endre Boros, Alex Kogan, Petr Kucera, Petr Savicky DIMACS-RUTCOR Seminar on Boolean and Pseudo-Boolean Functions, January 20, 2009

2. Outline • Notation and basic definitions • Exclusive sets • Definition and example • Exclusive sets and CNF minimization • Essential sets • Definition and examples • Duality between CNF representations and essential sets • Essential sets and CNF minimization • Coverable functions

3. Boolean basics • Boolean function on n variables is a mapping {0,1}n → {0,1} • Literals = variables and their negations • Clause = disjunction of literals • Clause C is an implicate of function f if f ≤ C • C is a prime implicate of f if dropping any literal means that C is no longer an implicate of f • CNF, prime CNF, irredundant CNF • Notation: Ip(f) = set of all prime implicates of f

4. Boolean basics • two clauses are resolvable if they have exactly one conflicting literal producing a resolvent • if C1 = A  x , C2 = B  x then R(C1, C2) = A  B • R(S) is a resolution closure of set S of clauses • resolution is complete: Ip(f)  R(S) for any CNF representation S of a function f • Notation : I(f) = R(Ip(f)) • Of course, I(f) is closed under resolution and we will not be interested in implicates of f outside of I(f)

5. Horn Basics • a clause is negative if it contains no positive literals and it is pure Horn if it contains one positive literal • a clause is Horn if it is negative or pure Horn • a CNF is Horn if it consists of Horn clauses • a Boolean function is Horn if it can be represented by a Horn CNF • Fact: f is Horn Ip(f) contains only Horn clauses • Corollary: I(f) also contains only Horn clauses (not true for the set of all implicates)

6. CNF minimization (of # of clauses) • Optimization version: Given a CNF F find a CNF G representing the same function as F and such that G consists of a minimum possible number of clauses. • Decision version: Given a CNF F and a number k does there exists a CNF G representing the same function as F such that G consists of ≤ k clauses? • NPH for general CNFs (SAT is a special case), for Horn CNFs [Ausiello, D’Atri, Sacca 1986], and for cubic Horn CNFs[Boros, Cepek 1994] • Polynomial for acyclic and quasi-acyclic Horn CNFs [Hammer, Kogan 1995]

7. Exclusive sets of implicates • Let f be a Boolean function. Then X  I(f)is an exclusive set of f if for every two resolvable clauses C1, C2 I(f) the following implication holds: R(C1, C2)  X  C1  X andC2  X • Example: f Horn, X = {C  I(f) | C is pure Horn} • Theorem: Let F  I(f) and G  I(f) be two distinct CNFs representing function f and let X  I(f) be an exclusive set of f. Then F  X and G  X represent the same function (called the X-component of f).

8. Exclusive sets and minimization • Corollary: Let F  I(f) and G  I(f) be two distinct CNFs representing function f and let X  I(f) be an exclusive set of f. Then the CNF (F \ X)  (G  X) represents f. • Lemma: Let  = X0  X1  ...  Xt be a chain of exclusive sets of a function f in which R(Xt) = I(f), and let Si  Xi \ Xi-1 be minimal subsets such that R(Xi-1  Si) = R(Xi) for i = 1,...,t. Then S1  …  St is a minimal representation of f.

9. Essential sets of implicates • Let f be a Boolean function. Then X  I(f)is an essential set of f if for every two resolvable clauses C1, C2 I(f) the following implication holds: R(C1, C2)  X  C1  X orC2  X • Example 1: f Horn, X = {C  I(f) | C is negative} • Example 2: t  {0,1}n, X(t) = {C  I(f) | C(t) = 0} • Example 3: S  I(f) such that S = R(S), X = I(f) \ S • Theorem: Let S  I(f) be arbitrary. Then S (as a CNF) represents f if and only if S  X   for every nonempty essential set X  I(f).

10. Essential sets of implicates • Corollary: Let X  I(f) be arbitrary. Then X is a nonempty essential set of f only if X  S   for every CNF representation S  I(f) of the function f. • Theorem: Let X  I(f) be any minimal set such that X  S   for every CNF representation S  I(f) of the function f. Then X is an essential set of f. • Theorem: Let D  I(f) be any maximal set not representing f. Then D = R(D), I(f) \ D is an essential set of f, and moreover I(f) \ D = X(t) for some t. • Corollary: Let X  I(f) be a minimal nonempty essential set of f. Then X = X(t) for some t.

11. Essential sets and minimization • Definition: For a function f let cnf(f) denote the minimum number of clauses in a CNF representation of f and ess(f) the maximum number of pairwise disjoint nonempty essential sets of f. • Corollary: For every function f: ess(f)≤cnf(f). • Conjecture: For every function f: ess(f)=cnf(f). • Definition: For a function f let ess*(f) denote the maximum number of vectors t such that X(t)’s are pairwise disjoint nonempty essential sets of f. • Corollary: For every function f: ess*(f)= ess(f).

12. Essential sets and minimization • Let H be the set of Horn functions. Then the CNF minimization (decision version) for H is in NP. • Assume ess(f)=cnf(f) for every Horn function f. Is then the CNF minimization for H also in co-NP? • Definition: Let s  t be two falsepoints of f. Then we define a clause C(s,t)=(iI(s,t)xi)(iO(s,t)xi) where I(s,t)={i | s[i]=t[i]=1} and O(s,t)={i | s[i]=t[i]=0}. • Lemma: Let s  t be two falsepoints of function f. Then X(s)  X(t)   if and only if C(s,t) is an implicate of f.

13. Essential sets and minimization • Summary: Minimization for H is in NP and it is also NPH so it is NPC. If the conjecture holds for H then minimization for H is in co-NP. Thus NP = co-NP. • Remark: The same is true even for the set H3 of cubic Horn CNFs. • Corollary: Unless NP = co-NP there exists a cubic Horn function f for which ess(f)<cnf(f). • Fact: There is a cubic Horn function on 4 variables for which ess(f) = 4 and cnf(f) = 5. • Definition: A function f is coverable if ess(f)=cnf(f).

14. Open problems • Let Cov = {f | f is coverable}, Horn-Cov = H  Cov. • Recognition of Horn-Cov? If polynomial then CNF minimization for Horn-Cov is in NP  co-NP. • Recognition of Cov? • Minimization for Horn-Cov? Most likely possible if Horn-Cov recognizable. • Minimization for Cov? Hopeless unless SAT is polynomial for Cov.

15. Concluding remarks • All statements made about the set of Horn functions H can be repeated for any tractable class fulfilling: • poly-time recognition • poly-time SAT • closed under partial assignment • contains all prime representations Thank You.