[Paper Review] Circuits with arbitrary gates for random operators

Stasys Jukna, Georg Schnitger|arXiv (Cornell University)|Apr 29, 2010

Complexity and Algorithms in Graphs15 references3 citations

TL;DR

This paper establishes strong lower bounds on circuit complexity for computing random and linear Boolean operators using arbitrary gates. It proves that almost all n-operators require Ω(n²) wires in general circuits, and linear operators require Ω(n²/log n) wires in depth-2 circuits when middle-layer or output gates are restricted to linear functions, using Kolmogorov complexity and information-theoretic arguments.

ABSTRACT

We consider boolean circuits computing n-operators f:{0,1}^n --> {0,1}^n. As gates we allow arbitrary boolean functions; neither fanin nor fanout of gates is restricted. An operator is linear if it computes n linear forms, that is, computes a matrix-vector product y=Ax over GF(2). We prove the existence of n-operators requiring about n^2 wires in any circuit, and linear n-operators requiring about n^2/\log n wires in depth-2 circuits, if either all output gates or all gates on the middle layer are linear.

Motivation & Objective

• To resolve the open question of whether random n-operators require quadratic-sized circuits when arbitrary gates are allowed.
• To analyze the complexity of linear operators f_A(x) = Ax over GF(2) in depth-2 circuits with restricted gate types.
• To establish tight lower bounds on wire count using incompressibility and information-theoretic techniques.
• To show that even with arbitrary gates, certain operators inherently require near-quadratic wiring.

Proposed method

• Uses a counting argument to upper bound the number of n-operators computable with L wires, showing that L must be Ω(n²) to cover all 2^{n2^n} operators.
• Applies an incompressibility argument based on Kolmogorov complexity to show that some matrices A require n² bits to describe, implying high circuit complexity.
• Restricts circuit analysis to depth-2 structures and assumes no direct input-to-output wires to simplify encoding.
• Encodes the circuit using O(L log n) bits by storing adjacency matrices B and C, and the behavior of output functions on basis vectors of image subspaces.
• Uses the linearity of output functions on image subspaces to reconstruct the full operator from basis evaluations.
• Leverages the fact that linear functions on subspaces can be fully described by their values on a basis, reducing description length.

Experimental results

Research questions

• RQ1 Do random n-operators require Ω(n²) wires in general circuits with arbitrary gates?
• RQ2 Can linear operators f_A(x) = Ax over GF(2) be computed with o(n²/log n) wires in depth-2 circuits when middle-layer gates are linear?
• RQ3 Is the complexity of linear operators in depth-2 circuits inherently high even when arbitrary gates are allowed on the middle layer?
• RQ4 Can information-theoretic or Kolmogorov complexity arguments establish strong lower bounds in general circuit models?
• RQ5 What is the minimal wire count required to compute a random or linear operator in depth-2 circuits with structural constraints on gate types?

Key findings

• For almost all n-operators, the minimum wire count s(f) is Ω(n²), showing that quadratic complexity is necessary even with arbitrary gates.
• Linear n-operators require Ω(n²/log n) wires in depth-2 circuits when all output gates are linear.
• Linear n-operators require Ω(n²/log n) wires in depth-2 circuits when all middle-layer gates are linear.
• The lower bound holds under the assumption that the circuit uses no direct input-to-output wires, which is without loss of generality for optimal circuits.
• The proof technique relies on encoding the circuit using O(L log n) bits, where L is the number of wires, and comparing this to the Kolmogorov complexity of the matrix A.
• The authors conjecture that the same Ω(n²/log n) lower bound holds even without restrictions on gate types in depth-2 circuits.