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Definitions: Let $n \in \mathbb{N}$. If $\alpha$ and $\beta$ are propositional formulas, then we'll call $\alpha$ and $\beta$ independent if neither implies the other, or more formally, if $\lnot (\alpha \rightarrow \beta)$ and $\lnot(\beta \rightarrow \alpha)$ are each satisfiable, perhaps under different variable assignments. The notation $|\phi|$ refers to the length of $\phi$ in characters.

We saw in the answer to a previous question that if a formula $\phi$ has $2^n$ independently varying propositional variables, then any formula logically equivalent to $\phi$ must be of length at least $2^n$. I'll call this answer $A$.

We now attempt to generalize that result from independent variables to independent formulas. We are constrained to consider only those wffs $\phi$ that contain $2^n$ pairwise independent formulas over $n$ variables. Syntactically, these formulas $\alpha_i$ are constrained to occur at the leaf level of $\phi$, so that if a variable were substituted for the formula, it would occur at the syntactic leaf level of $\phi$. No other formula may occur at the leaf level of $\phi$. Neither $\phi$ nor any of the $\alpha_i$ is allowed to be identically true or false.

Question: Can we then conclude that any formula $\varphi$ logically equivalent to $\phi$ must be of exponential length, that is, if $\varphi \leftrightarrow \phi$, then $|\varphi| = \Omega(2^n)$?

Argument: We create $2^n$ new propositional variables

$$p_1,\ldots, p_{2^n}$$

and then define $2^n$ equivalences $p_i \leftrightarrow \alpha_i$ for each of the independent $n$-place formulas $\alpha_i$. At this point, we can just substitute the variable for the corresponding formula and reduce the problem to the one solved in answer $A$.

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  • $\begingroup$ @D.W.: Thanks for writing. No, I did not intend that. $\endgroup$
    – ShyPerson
    Oct 23, 2023 at 17:05
  • $\begingroup$ @Someone: Thanks for your interest in my question. Were there specific terms you had in mind, or were you looking for a more general treatment? $\endgroup$
    – ShyPerson
    Oct 24, 2023 at 0:12
  • $\begingroup$ @Someone: I've flagged your comment complete since I've addressed it here. $\endgroup$
    – ShyPerson
    Oct 24, 2023 at 0:28

1 Answer 1

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No. Consider the formula

$$\phi = \alpha_1 \lor \alpha_2 \lor \dots \lor \alpha_{2^n} \lor \text{True},$$

where the $\alpha_i$ are pairwise independent according to your definition.

$\phi$ meets the requirements of your question ($\phi$ contains $2^n$ pairwise independent formulas $\alpha_i$, all $\alpha_i$'s occur at the leaf level of $\phi$).

However $\phi$ is equivalent to $\psi = \text{True}$, which does not have exponential length.

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  • $\begingroup$ Thanks for working on my question. I’ve edited the question to exclude situations I hadn't intended. Also, $\lnot(\alpha \rightarrow \text{True}) \equiv \text{False}$. Please let me know if I have made any mistake in marking the comments complete. Thanks $\endgroup$
    – ShyPerson
    Oct 24, 2023 at 0:25
  • $\begingroup$ @ShyPerson, changing the question after you receive an answer, in a way that invalidates that existing answer, is generally frowned upon. Instead, it's better to be very careful to state the question accurately the first time; and if you fail, to ask a new question. I believe if you put a little thought into it, you should be able to find a way to modify my example with your updated requirements. Hint: try a logical-or of two formulas, over variables. Can you make the "or" be equivalent to $\text{True}$? $\endgroup$
    – D.W.
    Oct 24, 2023 at 5:15

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