# Questions tagged [boolean-formulas]

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21 questions
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### Reduce $m$-clause 3SAT to PLANAR-3SAT in $O(m^{2-\varepsilon})$

The classic reduction from 3SAT to PLANAR-3SAT requires a removal of $O(m^2)$ crossings from a rectilinear representation of 3SAT with $m$ clauses. However, the crossing number inequality suggests ...
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### Minimum Unsatisfiable Core

Recently in a course I'm taking, we have been learning about the DPLL(T) algorithm for SMT solving. The basic outline goes like this: ...
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### Computing $\operatorname{MAJ}_n$ by $\operatorname{MAJ}_m$ in depth 2

Can the majority of $n$ bits be computed by a depth 2 formula all of whose gates compute the majority of $m$ bits where $m=O(n^c)$ for a constant $c<1$? Such a formula contains $m+1$ gates and $m^2$...
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### What is the complexity of the set of closed boolean tautologies assuming well formed formulae?

Let's define two CFGs: S ::= 0 | 1 | (S+S) | (S*S) S' ::= 0 | 1 | S+S | S*S | (S) And two languages: M = { w | S generates w, and w evaluates to 1 } M'...
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### Conversion between k-SAT and XOR-SAT

According to XOR Satisfiability Solver Module for DPLL Integration by Tero Laitinen, we need $2^{n-1}$ CNF clauses to convert an $n$ literal XOR-SAT clause if we do not want to increase the number of ...
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### May Boolean circuits be exponentially more concise than Boolean formulae?

Consider a family $(f_n)_{1 \leq n}$ of Boolean functions, where $f_n$ is a function on $n$ variables. Consider for every $n$ the smallest Boolean formula $F_n$ describing $f_n$, and the smallest ...
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### Remove unneeded atoms in CNF minimalization (SAT preprocessing)

This might be a very basic question. I am interested in all atoms of a propositional formula that can be removed from a particular formula, while the derived formula has the same satisfiability ...
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### Number of SAT checks that are needed to find all combinations of subset of boolean variables of a propositional formula

Please mind that I sometimes lack formal mathematical knowledge and English is not my first language, so I might miss the right words. Please change the tile if needed. Also, I have choosen this site ...
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### k-CNF ←→ k-DNF conversion to minimize errors

the following problem/question seems fundamental/hard. it appears in some circuit theory proofs, graph theory, and maybe elsewhere. looking for any nontrivial insight. will add various known/nearby ...
I've been having hard time with proving the following claim: Let $f:\{T,F\}^n\rightarrow \{T,F\}$ be a boolean function. Let $size_{DT}(f)$ denote the number of leaves in the smallest (w.r.t the ...
### Boolean formula balancing in $\mathsf{AC^0}$
I am looking for references about the complexity of Boolean formula balancing problem. In particular, Was it known that Boolean formulas can be balanced in $\mathsf{AC^0}$? Is there a simple proof of ...