Exclusive%20and%20essential%20sets%20of%20implicates%20of%20a%20Boolean%20function

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
• 0,1n ? 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)

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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)

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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, DAtri, Sacca 1986 , and
  for cubic Horn CNFs Boros, Cepek 1994
• Polynomial for acyclic and quasi-acyclic Horn
  CNFs Hammer, Kogan 1995

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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 and C2 ? 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).

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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.

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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 or C2 ? X
• Example 1 f Horn, X C ? I(f) C is
  negative
• Example 2 t ? 0,1n, 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).

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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.

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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).

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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 siti1 and
  O(s,t)i siti0.
• 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.

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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) lt 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).

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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.

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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.