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Consider propositional logic over the connectives $\land$, $\lor$, and $\lnot$. We have two well-formed formulas $\varphi$ and $\sigma$ which are equivalent: $\varphi \leftrightarrow \sigma$. The formula $\sigma$ has no occurrences of the negation symbol ($\lnot$). Now let's push negation inward to reduce $\varphi$ to Negation Normal Form and call that normal form $\varphi'$. We note that $\varphi' \leftrightarrow \sigma$.

EDIT: We further stipulate that neither $\varphi$ nor $\sigma$ contain tautologies or contradictions.

Two questions:

  1. Can we conclude that $\varphi'$ has no occurrences of the negation symbol ($\lnot$)? If we convert both $\varphi'$ and $\sigma$ to disjunctive normal form for instance, neither should have negation symbols since we're given that $\sigma$ doesn't.
  2. Can we conclude that the length of $\varphi'$ is bounded above by some polynomial function of the length of $\varphi$? That is, the length of $\varphi'$ won't be exponentially larger than the length of $\varphi$? If not, please provide a counterexample.
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  1. No. Consider $\sigma = P$, and $\varphi = P \lor (P \land \neg P)$. These are equivalent, and since $\varphi$ is already in NNF, $\varphi'=\varphi$
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  • $\begingroup$ I've edited the question to address your counterexample. $\endgroup$
    – ShyPerson
    Jun 20, 2019 at 2:31
  • $\begingroup$ @ShyPerson Hmm, how exactly is 'contains a tautology or contradiction' defined? For example, does $(P \land Q) \lor (P \land \neg Q)$ (which is equivalent to $P$) 'contain a tautology or contradiction'? $\endgroup$
    – Bram28
    Jun 20, 2019 at 11:50
  • $\begingroup$ Let's try this: it 'contains a tautology or contradiction' if valid propositional rules of inference can reduce it to a shorter logically equivalent formula. So your example $(P \land Q) \lor (P \land \lnot Q)$ satisfies this definition. $\endgroup$
    – ShyPerson
    Jun 21, 2019 at 4:14

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