Upper bounds for the size and the depth of formulae for MOD-functions
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Igor S. Sergeev
Abstract
We obtain new upper bounds for the size and the depth of formulae for some MOD-functions (that is, functions counting n bits modulo m). In particular, the depth of counting n bits modulo 3 is bounded by 2.8 log2n + O(1) in the standard basis {Λ, V,—}; the size of counting modulo 5 is bounded by O(n3.22) in the same basis; the depth of counting modulo 7 is bounded by 2.93 log2n+O(1) in the basis of all binary Boolean functions.
Originally published in Diskretnaya Matematika (2016) 28, №2,108-116 (in Russian).
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 14-01-00671a
Funding statement: Research was supported by RBRF, project № 14-01-00671a.
Acknowledgment
The author thanks the reviewer for valuable remarks, in particular for reminder of the paper [1].
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Articles in the same Issue
- Frontmatter
- An approach to the transformation of periodic sequences
- On the number of functions of k-valued logic which are polynomials modulo composite k
- Upper bounds for the size and the depth of formulae for MOD-functions
- Mean and variance of the number of subfunctions of random Boolean function which are close to the affine functions set
- The second coordinate sequence of the MP-LRS over nontrivial Galois ring of an odd characteristic
- On the asymptotics of the number of repetition-free Boolean functions in the basis {&, ∨, ⊕, ¬}
- On the gate complexity of reversible circuits consisting of NOT, CNOT and 2-CNOT gates
Articles in the same Issue
- Frontmatter
- An approach to the transformation of periodic sequences
- On the number of functions of k-valued logic which are polynomials modulo composite k
- Upper bounds for the size and the depth of formulae for MOD-functions
- Mean and variance of the number of subfunctions of random Boolean function which are close to the affine functions set
- The second coordinate sequence of the MP-LRS over nontrivial Galois ring of an odd characteristic
- On the asymptotics of the number of repetition-free Boolean functions in the basis {&, ∨, ⊕, ¬}
- On the gate complexity of reversible circuits consisting of NOT, CNOT and 2-CNOT gates
