# Depth Reduction for Circuits of Unbounded Fan-In

@article{Allender1994DepthRF, title={Depth Reduction for Circuits of Unbounded Fan-In}, author={Eric Allender and Ulrich Hertrampf}, journal={Inf. Comput.}, year={1994}, volume={112}, pages={217-238} }

We prove that constant depth circuits of size nlogO(1)n over the basis AND, OR, PARITY are no more powerful than circuits of this size with depth four. Similar techniques are used to obtain several other depth reduction theorems; in particular, we show every set in AC0 can be recognized by a family of depth-three threshold circuits of size nlogO(1)n. The size bound nlogO(1)n is optimal when considering depth reduction over AND, OR, and PARITY. Most of our results hold for both the uniform and…

## 46 Citations

### Upper and lower bounds for some depth-3 circuit classes

- Computer Science, MathematicsProceedings of Computational Complexity. Twelfth Annual IEEE Conference
- 1997

It is shown that the fan-in of the AND gates can be reduced toO(logn) in the case wherem is unbounded, and to a constant in the cases where m is constant, and an exponential lower bound is proved if OR gates are also permitted on level two andm is a constant prime power.

### Depth-Reduction for Composites

- Computer Science2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

A new depth-reduction construction is obtained, which implies a super-exponential improvement in the depth lower bound separating NEXP from non-uniform ACC, and it is shown that every circuit with AND, OR, NOT, and MODm gates can be reduced to a depth-2, SYM-AND circuit.

### Isolating an Odd Number of Elements and Applications in Complexity Theory

- Computer ScienceTheory of Computing Systems
- 1998

Using this construction, this work improves bounds on several results in complexity theory that originally used a construction due to Valiant and Vazirani, and obtains better bounds on polynomials which approximate boolean functions.

### Depth-d Threshold Circuits vs. Depth-(d + 1) AND-OR Trees

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2022

It is shown that under a modified version of their projection procedure, any depth-d threshold circuit with n1+γ wires simplifies to a near-trivial function, whereas an appropriately parameterized AND-OR tree of depth d + 1 maintains structure.

### Depth reduction for noncommutative arithmetic circuits

- Computer Science, MathematicsSTOC
- 1993

The results show that OptSACi is contained in AC1, and other results relating Boolean and arithmetic circuit complexity are proved.

### Circuits with composite moduli

- Computer Science
- 2016

The main result is that every ACC circuit of polynomial size and depth d can be reduced to a depth-2 circuit SYM◦AND of size 2(logn)O(d) , which improves exponentially the previously best-known construction by Yao-Beigel-Tarui, which has size blowup 2 2O( d) .

### On circuit complexity classes and iterated matrix multiplication

- Computer Science, Mathematics
- 2012

It is shown that extremely modest-sounding lower bounds for certain problems can lead to non-trivial derandomization results, and there are families of polynomials having small depth-two arithmetic circuits that cannot be expressed by algebraic branching programs of width two.

### Depth Reduction for Noncommutative Arithmetic Circuits (extended Abstract)

- Computer Science, Mathematics
- 1993

The results show that OptSAC 1 is contained in AC 1, and other results relating Boolean and arithmetic circuit complexity are proved.

### Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1995

### A Uniform Circuit Lower Bound for the Permanent

- Mathematics, Computer ScienceSIAM J. Comput.
- 1994

The authors show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the…

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