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3
votes
0answers
84 views

Shrinkage Exponent of Formulas over the Full Binary Basis

Håstad has shown that the shrinkage exponent of boolean formulas over the De Morgan basis is 2. In other words, if one keeps each variable of the formula alive with probability $p$ and restricts it ...
2
votes
1answer
73 views

Complexity of generating a pseudo-Boolean function

A pseudo-Boolean function is a mapping from $\mathcal{B}^n = \{0, 1\}^n$ to $\mathbb{R}$. Following is a pseudo-Boolean function. $$s_1 s_4 - s_2 s_3 - s_3 s_5 - s_2 s_5 + s_1 + s_4 - s_1 s_3 - ...
3
votes
1answer
83 views

Symbolic Execution of the Quine-McCluskey Algorithim

If I understand correctly, the Quine–McCluskey algorithm will find the minimum boolean formula size for given boolean function. Has there been any attempts to (for lack of a better term) symbolically ...
11
votes
2answers
240 views

What is the complexity of the equivalence problem for read-once decision trees?

A read-once decision tree is defined as follows: $True$ and $False$ are read-once decision trees. If $A$ and $B$ are read-once decision trees and $x$ is a variable not occurring in $A$ and $B$, ...
2
votes
3answers
169 views

How hard is it to find a “well-distributed” subset of models of a propositional formula?

We consider the propositional language $\mathcal{L}_{\mathit{PS}}$ defined over a finite alphabet $\mathit{PS}$ and the usual logical connectives. An interpretation is an assignment $\mathit{PS} ...
12
votes
0answers
232 views

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 ...
1
vote
1answer
75 views

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 } ...
7
votes
1answer
412 views

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 ...
6
votes
1answer
228 views

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 ...
4
votes
3answers
134 views

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 ...
2
votes
2answers
144 views

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 ...
4
votes
0answers
637 views

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 ...
2
votes
2answers
275 views

Size of decision tree for f is polynomial in the DNF size of f and CNF size of f

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 ...
9
votes
1answer
272 views

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 ...