On the Complexity of the Conditional Independence Implication Problem With Bounded Cardinalities

TL;DR

The paper addresses CI implication under bounded variable cardinalities and proves co-NEXPTIME-hardness, even when all variables are binary. It introduces a two-stage reduction: first from Binary Bounded Tiling to Periodic Bounded Tiling, then from Periodic Bounded Tiling to a bounded CI implication via Li-style affine existential information predicates, including UNIF_k and related constructions. The result situates the bounded CI problem as co-NEXPTIME-hard while known upper bounds place it in EXPSPACE, highlighting a gap and underscoring the surprising difficulty of CI reasoning under bounded domains. The work advances understanding of the complexity of CI reasoning with cardinality constraints and informs the theoretical limits of logic-based statistical inference and related AI systems.

Abstract

We show that the conditional independence (CI) implication problem with bounded cardinalities, which asks whether a given CI implication holds for all discrete random variables with given cardinalities, is co-NEXPTIME-hard. The problem remains co-NEXPTIME-hard if all variables are binary. The reduction goes from a variant of the tiling problem and is based on a prior construction used by Cheuk Ting Li to show the undecidability of a related problem where the cardinality of some variables remains unbounded. The CI implication problem with bounded cardinalities is known to be in EXPSPACE, as its negation can be stated as an existential first-order logic formula over the reals of size exponential with regard to the size of the input.

Figures (5)

• Figure 1: A tiling implementation of a binary counter which shifts its position on every decrement, with the starting and final tile in the top-right and bottom-left corners respectively. This example counts down from 5 (101 in binary). Every tile except for the final $\star$ is of the form $a_b^c$, where $a$ is the bit value (possibly blank), $b$ is the value of the bit directly to the right (or blank if there is none), and $c$ is optionally $*$ if a borrow operation is required. The shaded tiles of the top row function in the same manner, but they are "memorized" within the tiling system such that the placement of the top-right tile forces the top row to write out the binary initial value. The tiles in the lower rows are chosen deterministically based on their right and top neighbor. Finally, the $\star$ tile only occurs when the tile above is blank and the one to the right requires a borrow, which indicates that the counter has just gone below zero. In order to be unable to further count down, we disallow any tiles being below or to the left of tile $\star$. The size of the tiling is $(k' + b + 2) \times (k' + 2)$, where $b$ is the number of bits and $k'$ is the initial value; however, this could be modified such that the final tiling has size $(k' + b + 2) \times (k' + b + 2)$ by padding with $b$ dummy rows at the top. For sufficiently large $k$, we can always efficiently find $b, k'$ such that $k' + b + 2 = k$.
• Figure 2: Modified adjacency relation for the system $\mathcal{D}"$ -- only adjacencies marked by $\bullet$ are permitted, as well as all adjacencies from the original tiling system. The asterisk denotes any tiles from the system $\mathcal{D}'$, while $s, f$ represent any tile from the initial and final subset of tiles, respectively ($S$ and $F$ defined above).
• Figure 3: An example "border" created by the above tiling, with tiles from the original set not shown and $s, f$ representing tiles from $S, F$ respectively. The dashed rectangle represents the actual rectangle being tiled, while the tiles outside are periodic copies added to better illustrate the construction.
• Figure 4: Visualization of the tori which are a product of the cycles created by $(X_1, X_2)$ and $(Y_1, Y_2)$, with each vertex corresponding to a quadruple of the values of $(X_1, X_2, Y_1, Y_2)$. The axes show these cycles --- a quadruple's $(X_1, X_2)$ (resp. $(Y_1, Y_2)$) values are determined by projecting onto the horizontal (resp. vertical) axis. Additionally, the left torus shows highlighted edges which arise when all but one variable (of $X_1$, $X_2$, $Y_1$, $Y_2$, $Z$) are fixed as well as an example face which arises when $Z$ and two other variables are fixed. The right torus has each face labeled with its type and each vertex labeled with its group -- these are used in Section \ref{['section:tiles']} in order to restrict allowed labelings of the vertices. The second "corresponding" torus and the $Z$ axis are omitted for clarity.
• Figure 5: Left: Li's representation of a $4 \times 4$ torus coloring and the $4 \times 4$ tiling that it yields. The conversion from 16 vertices to 32 gives the tiling an additional diagonal periodicity. Combined with the fact that this does not work for non-square $n \times m$ tilings (without arranging them into a $\text{lcm}(n, m) \times \text{lcm}(n, m)$ square), this proves problematic for restricting the size of a periodic tiling. Right: the corresponding $8 \times 8$ torus labeling in our representation. Each tile has 4 labels, one for each corner of the tile (indicated by the arrow in this case). The tiles do not need to be Wang tiles.

Theorems & Definitions (20)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9