Metatheory - University of Cambridge

3. Normal forms 27
3.4 Cleaning up DNF sentences
I have offered two different proofs of the DNF Theorem. They also described
two different algorithms for putting sentences into DNF. But these algorithms
do not always leave you with the most elegant output. Indeed, just as we can
‘clean up’ the output of the algorithm for calculating interpolants (see §2.2), we
can ‘clean up’ the output of our algorithms for calculating DNF-equivalents.
Here are the clean-up rules; I leave it as an exercise to justify them. First,
we rewrite our DNF sentence using the relaxed notational conventions of fx C.
Now:
Rule 1: remove repetitious conjuncts. If any disjunct contains the same conjunct
more than once, remove all but one instance of that conjunct (respecting
bracketing conventions as appropriate).
Rule 2: remove overly-specific disjuncts. Delete any disjunct which entails any
other disjunct. (If two or more disjuncts entail each other, delete all but the
first of those disjuncts.)
Example: the output of our example in §3.3 was:
(A ∧ B) ∨ (A ∧ ¬C) ∨ (¬D ∧ D) ∨ (¬E ∧ D) ∨ (¬D ∧ E) ∨ (¬E ∧ E)
Noting that a contradiction entails everything, we should remove both contradictory
disjuncts, obtaining (still with relaxed conventions):
(A ∧ B) ∨ (A ∧ ¬C) ∨ (¬E ∧ D) ∨ (¬D ∧ E)
Rule 3: invoke excluded middle. Suppose there is a disjunct whose conjuncts
are just A 1 , . . . , A m and B (in any order), and another disjunct whose conjuncts
are just A 1 , . . . , A m and ¬B (in any order), so that the two disjuncts disagree
only on whether whether B should be prefixed with a negation. In that case,
delete both disjuncts and replace them with the simpler disjunct (A 1 ∧. . .∧A m ).
(However, if we have just B and ¬B as disjuncts, then delete the entire sentence
and leave the tautology (B ∨ ¬B).)
Example: the output of the example in §3.2 was:
(A ∧ B ∧ C) ∨ (A ∧ ¬B ∧ C) ∨ (¬A ∧ ¬B ∧ C) ∨ (¬A ∧ ¬B ∧ ¬C)
which can be simplified to:
(A ∧ C) ∨ (¬A ∧ ¬B)
3.5 Conjunctive Normal Form
So far in this chapter, I have discussed disjunctive normal form. Given the
duality of disjunction and conjunction (see §2.3), it may not come as a surprise
to hear that there is also such a thing as conjunctive normal form (CNF).
The definition of CNF is exactly analogous to the definition of DNF. So, a
sentence is in CNF iff it meets all of the following conditions:
(cnf1) No connectives occur in the sentence other than negations, conjunctions
and disjunctions;