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. 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. 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: I p (f) = set of all prime implicates of f

4. 4 Boolean basics  two clauses are resolvable if they have exactly one conflicting literal producing a resolvent  if C 1 = A  x, C 2 = B   x then R(C 1, C 2 ) = A  B  R(S) is a resolution closure of set S of clauses  resolution is complete: I p (f)  R(S) for any CNF representation S of a function f  Notation : I(f) = R(I p (f))  Of course, I(f) is closed under resolution and we will not be interested in implicates of f outside of I(f)

5. 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  I p (f) contains only Horn clauses  Corollary: I(f) also contains only Horn clauses (not true for the set of all implicates)

6. 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. 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 C 1, C 2  I(f) the following implication holds: R(C 1, C 2 )  X  C 1  X and C 2  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. 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  = X 0  X 1 ...  X t be a chain of exclusive sets of a function f in which R(X t ) = I(f), and let S i  X i \ X i-1 be minimal subsets such that R(X i-1  S i ) = R(X i ) for i = 1,...,t. Then S 1  …  S t is a minimal representation of f.

9. 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 C 1, C 2  I(f) the following implication holds: R(C 1, C 2 )  X  C 1  X or C 2  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. 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. 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. 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)  x i )  (  i  O(s,t) x i ) 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. 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 H 3 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. 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. 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.