Simple Circuit Extensions for XOR in PTIME

TL;DR

This paper demonstrates that the $XOR$-Simple Extension Problem can be solved in polynomial time by developing a fixed-parameter tractable algorithm, extending previous results on similar Boolean functions and circuit complexity.

Contribution

The work extends the understanding of circuit extension problems by showing $XOR_n$ admits a polynomial-time solution under certain conditions, broadening the class of functions with efficiently solvable extension problems.

Findings

$XOR_n$-Simple Extension Problem is in polynomial time.

Optimal $XOR_n$ circuits are characterized as binary trees of $( eg)XOR_2$ gates.

The algorithm is efficient when circuits are linear, polynomially enumerable, and have bounded fan-out.

Abstract

The Minimum Circuit Size Problem for Partial Functions ($MCSP^*$) is hard assuming the Exponential Time Hypothesis (ETH) (Ilango, 2020). This breakthrough hardness result leveraged a characterization of the optimal $\{\land, \lor, \neg\}$ circuits for $n$-bit $OR$ ($OR_n$) and a reduction from the partial $f$-Simple Extension Problem where $f = OR_n$. It remains open to extend that reduction to show ETH-hardness of total $MCSP$. However, Ilango observed that the total $f$-Simple Extension Problem is easy whenever $f$ is computed by read-once formulas (like $OR_n$). Therefore, extending Ilango's proof to total $MCSP$ would require one to replace $OR_n$ with a slightly more complex but similarly well-understood Boolean function. This work shows that the $f$-Simple Extension problem remains easy when $f$ is the next natural candidate: $XOR_n$. We first develop a fixed-parameter tractable algorithm for the $f$-Simple Extension Problem that is efficient whenever the optimal circuits for $f$ are (1) linear in size, (2) polynomially "few" and efficiently enumerable in the truth-table size (up to isomorphism and permutation of inputs), and (3) all have constant bounded fan-out. $XOR_n$ satisfies all three of these conditions. When $\neg$ gates count towards circuit size, optimal $XOR_n$ circuits are binary trees of $n-1$ subcircuits computing $(\neg)XOR_2$ (Kombarov, 2011). We extend this characterization when $\neg$ gates do not contribute the circuit size. Thus, the $XOR$-Simple Extension Problem is in polynomial time under both measures of circuit complexity.