Simple Circuit Extensions for XOR in PTIME

TL;DR

The paper analyzes the f-Simple Extension problem as a pathway to understanding ETH-hardness of the Minimum Circuit Size Problem (MCSP). It proves that for XOR, the f-Simple Extension problem is solvable in polynomial time under two circuit-measure frameworks by giving a fixed-parameter tractable algorithm and a detailed structural characterization of optimal XOR circuits, notably that optimal XOR_n circuits form binary trees of (¬)XOR_2 blocks with constant fan-out. This XOR-centric result suggests that extending ETH-hardness from MCSP* to total MCSP via simple extensions is unlikely via XOR and motivates seeking other base functions with richer optimal-circuit structure, or alternative reduction frameworks. The work also develops a framework for encoding and decoding simple-extension circuits (Y-tree decompositions) and discusses how Levin reductions tie these constructions to explicit hardness reductions from BPIS, highlighting both the potential and the limits of this approach in pushing ETH-hardness results for total MCSP. Overall, the paper provides a rigorous, parameterized pathway to analyze f-Simple Extension problems, clarifies XOR’s role as a barrier to ETH-hardness in this avenue, and outlines concrete directions for identifying viable base functions to pursue ETH-hardness of total MCSP via simple extensions.

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.

Figures (5)

• Figure 1: An example of the binary tree structure of optimal circuits computing $\XOR_6$. The left sub-figure depicts possible $(\neg)\XOR_2$ blocks in the Red'kin and DeMorgan Bases. Notice each optimal Red'kin circuit is an optimal DeMorgan circuit, but not vice-versa. The right sub-figure depicts that the arrangement of $\XOR_2$ blocks that make up $\XOR_6$ circuits are shared by both bases.
• Figure 2: An example of a Y-Tree Decomposition of size three.
• Figure 3: The structure of our paper. Sections 1 and 2 form an extended abstract. An incoming arrow indicates that the material depends on the previous section.
• Figure 4: An example of applying the passing simplification $1 \land \gamma \to \gamma$. Notice that $\gamma$ inherits the fanout from $\alpha$, and $\alpha$ can then be garbage collected.
• Figure 5: An application of a pruning rule where $\gamma$ itself is a constant. Since $\mathsf{fo}(\alpha) \geq 1$, we have $\phi_{t+1} = \phi_t + \mathsf{fo}'(\alpha) - 2 = \phi_t + \mathsf{fo}(\alpha) - 2 \geq \phi_t + 1 - 2 = \phi_t - 1.$

Theorems & Definitions (21)

• Definition : Simple Extension
• Definition 1: Y-Tree Decomposition
• Definition 2
• Definition 3
• Lemma 4
• Lemma 5
• Lemma 6
• Definition 7: Induced Subcircuit rooted at $\alpha$
• Definition 8
• Definition 9