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Revision #1 to TR18-192 | 13th December 2018 23:17

Circuit Depth Reductions

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Revision #1
Authors: Alexander Golovnev, Alexander Kulikov
Accepted on: 13th December 2018 23:17
Downloads: 1
Keywords: 


Abstract:

The best known circuit lower bounds against unrestricted circuits remained around $3n$ for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than $5n$. In this work, we suggest a first non-gate-elimination approach for obtaining circuit lower bounds. Namely, we prove that every (unbounded-depth) circuit of size $s$ can be expressed as an OR of $2^{s/3.9}$ $16$-CNFs. While this structural result does not immediately lead to new lower bounds, it suggests a new avenue of attack on them.

Our result complements the classical depth reduction result of Valiant which asserts that logarithmic-depth circuits of linear size can be computed by an OR of $2^{\varepsilon n}$ $n^{\delta}$-CNFs. It is known that no such graph-theoretic reduction can work for circuits of super-logarithmic depth. We overcome this limitation (for small circuits) by taking into account both the graph-theoretic and functional properties of circuits.

We show that qualitative improvements of the following pseudorandom constructions imply qualitative (from $3n$ to $\omega(n)$) improvement of size lower bounds for log-depth circuits via Valiant's reduction: dispersers for varieties, correlation with constant degree polynomials, matrix rigidity, and hardness for depth-$3$ circuits with constant bottom fan-in. On the other hand, now even modest quantitative improvements of the known constructions give elementary proofs of quantitatively stronger circuit lower bounds ($3.9n$ instead of $3n$).


Paper:

TR18-192 | 12th November 2018 18:02

Circuit Depth Reductions





TR18-192
Authors: Alexander Golovnev, Alexander Kulikov
Publication: 12th November 2018 18:34
Downloads: 202
Keywords: 


Abstract:

The best known circuit lower bounds against unrestricted circuits remained around $3n$ for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than $5n$. In this work, we suggest a first non-gate-elimination approach for obtaining circuit lower bounds. Namely, we prove that every (unbounded-depth) circuit of size $s$ can be expressed as an OR of $2^{s/3.9}$ $16$-CNFs. While this structural result does not immediately lead to new lower bounds, it suggests a new avenue of attack on them.

Our result complements the classical depth reduction result of Valiant which asserts that logarithmic-depth circuits of linear size can be computed by an OR of $2^{\varepsilon n}$ $n^{\delta}$-CNFs. It is known that no such graph-theoretic reduction can work for circuits of super-logarithmic depth. We overcome this limitation (for small circuits) by taking into account both the graph-theoretic and functional properties of circuits.

We show that qualitative improvements of the following pseudorandom constructions imply qualitative (from $3n$ to $\omega(n)$) improvement of size lower bounds for log-depth circuits via Valiant's reduction: dispersers for varieties, correlation with constant degree polynomials, matrix rigidity, and hardness for depth-$3$ circuits with constant bottom fan-in. On the other hand, now even modest quantitative improvements of the known constructions give elementary proofs of quantitatively stronger circuit lower bounds ($3.9n$ instead of $3n$).



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