Exclusion ATS (EATS): Approximate Exclusion in Teams

• Exclusion ATS (EATS) is a framework in team semantics where classical exclusion atoms are extended to approximate exclusion atoms via large subteams.
• The framework establishes a completeness theorem and a polynomial-time decision algorithm, demonstrating practical tractability in noisy and incomplete data.
• Its axiomatization, comprising eight key principles, ensures precise control over coordinate manipulations and approximation monotonicity in error-tolerant reasoning.

Exclusion ATS (EATS) denotes a framework in team semantics in which classical exclusion atoms are extended to approximate exclusion atoms satisfied when the corresponding exact exclusion holds on a large enough subteam. In this setting, a team is a set of assignments and can be seen as a mathematical model of a uni-relational database. The framework consists of a definition of approximate exclusion via large-subteam semantics, a finite sound and complete axiomatization, a completeness theorem for semantic consequences with approximation degree $\epsilon<\tfrac12$, and a polynomial-time algorithm for the finite implication problem. The same results apply to exclusion dependencies in database theory (Häggblom, 2024).

1. Semantic setting and exact exclusion

The underlying semantic objects are teams. Let $\mathcal V$ be a finite set of variables and $M$ a nonempty domain. An assignment is a function $s:\mathcal V\to M$, and a team $T$ over $\mathcal V$ is a finite set of assignments $\{s_1,\ldots,s_k\}$. For a tuple of variables $x=(x_1,\ldots,x_n)$, one writes

$s(x)=(s(x_1),\ldots,s(x_n)).$

The classical exclusion atom $t_1 \# t_2$, where $\mathcal V$0 are same-length tuples of variables, is interpreted by

$\mathcal V$1

Equivalently, no two assignments in $\mathcal V$2, possibly distinct, agree on $\mathcal V$3 and $\mathcal V$4 (Häggblom, 2024).

This exact atom has two structural properties that are central to the approximate theory. First, exclusion atoms are downward closed: if $\mathcal V$5, then any subteam of $\mathcal V$6 also satisfies it. Second, they have the empty-team property. These closure features are what make the large-subteam definition of approximation behave well.

2. Approximate exclusion atoms

Fix $\mathcal V$7. The approximate exclusion atom $\mathcal V$8 is satisfied when a sufficiently large subteam satisfies the exact atom:

$\mathcal V$9

In words, one may discard at most $M$0 assignments so that the remaining team satisfies exact exclusion (Häggblom, 2024).

The parameter $M$1 therefore measures tolerated error at the team level. The case $M$2 recovers the usual exclusion atom exactly. At the opposite extreme, the system includes a derivable total case at error $M$3, reflecting that full discardability trivializes satisfaction. This suggests a controlled interpolation between strict incompatibility and bounded-error reasoning over noisy teams.

A common misconception is to treat approximate exclusion as a probabilistic relaxation in the sense of assigning probabilities directly to collisions. The formal definition is different: the relaxation is combinatorial and subteam-based. Satisfaction is witnessed by the existence of a large exact subteam, not by averaging violations across assignments.

3. Axiomatization and inference system

Derivability from a set $M$4 of premises is written $M$5. The axiomatization is schematic for tuples of appropriate lengths, and the notation $M$6 abbreviates $M$7. The system comprises eight principles (Häggblom, 2024):

• (A1) Irreflexivity of approximate exclusion:

$M$8

• (A2) Symmetry:

$M$9

• (A3) Weakening (adding irrelevant coordinates):

$s:\mathcal V\to M$0

• (A4) Contraction:

$s:\mathcal V\to M$1

• (A5) Permutation of coordinates:

$s:\mathcal V\to M$2

provided $s:\mathcal V\to M$3 and $s:\mathcal V\to M$4.

• (A6) Exclusion-contraction scheme:

$s:\mathcal V\to M$5

• (A7) Team-monotonicity in the error parameter:

$s:\mathcal V\to M$6

• (A8) Trivial totality:

$s:\mathcal V\to M$7

Soundness is immediate by the downward-closed definition and simple counting arguments. The system isolates two distinct kinds of monotonicity. Coordinate manipulations are handled by the structural rules (A3)–(A6), while approximation monotonicity is handled separately by (A7). The axiom (A8) makes explicit that approximation degree $s:\mathcal V\to M$8 collapses the content of exclusion.

4. Completeness and proof strategy

The central completeness statement is formulated for consequences whose approximation degree is not too large. Precisely, if $s:\mathcal V\to M$9 and $T$0 is a possibly infinite set of approximate exclusion atoms such that whenever $T$1 contains atoms of error $T$2, there is a least such error, then

$T$3

This identifies a range of approximation in which semantic consequence and syntactic derivability coincide (Häggblom, 2024).

The proof proceeds by contraposition. Assuming $T$4, the construction builds a counterexample team $T$5 of size $T$6 chosen so that

$T$7

where $T$8 is the next larger error in $T$9; if no such $\mathcal V$0 exists, one takes $\mathcal V$1. In $\mathcal V$2, exactly $\mathcal V$3 assignments witness a forbidden collision between $\mathcal V$4 and $\mathcal V$5, forcing removal of at least $\mathcal V$6 assignments to satisfy $\mathcal V$7, so $\mathcal V$8. The proof then checks that no premise in $\mathcal V$9 fails on $\{s_1,\ldots,s_k\}$0 with its allowed error, using set-representation lemmas mirroring the rules (A3)–(A6) and (A7). Hence $\{s_1,\ldots,s_k\}$1, which yields the desired non-implication (Häggblom, 2024).

The classical case is recovered by setting $\{s_1,\ldots,s_k\}$2. In that subcase, the result supplies a completeness theorem for usual exclusion atoms that had been missing from the literature. More specifically, completeness holds for the six rules (A1)–(A6) restricted to $\{s_1,\ldots,s_k\}$3. A plausible implication is that the approximate framework is not merely an extension of the exact theory but also fills a foundational gap in the exact theory itself.

5. Implication problem, tractability, and database interpretation

For finite input $\{s_1,\ldots,s_k\}$4 with $\{s_1,\ldots,s_k\}$5, the implication problem asks whether $\{s_1,\ldots,s_k\}$6. The decision procedure is polynomial time (Häggblom, 2024). Its sketch is:

1. If $\{s_1,\ldots,s_k\}$7 or $\{s_1,\ldots,s_k\}$8 with $\{s_1,\ldots,s_k\}$9, return TRUE.
2. If there exists $x=(x_1,\ldots,x_n)$0 with $x=(x_1,\ldots,x_n)$1, return TRUE.
3. If $x=(x_1,\ldots,x_n)$2, return FALSE.
4. Compute the set representation

$x=(x_1,\ldots,x_n)$3

and for each coordinate $x=(x_1,\ldots,x_n)$4 define

$x=(x_1,\ldots,x_n)$5

1. For each $x=(x_1,\ldots,x_n)$6 with $x=(x_1,\ldots,x_n)$7:
   • let $x=(x_1,\ldots,x_n)$8;
   • if $x=(x_1,\ldots,x_n)$9 or $s(x)=(s(x_1),\ldots,s(x_n)).$0, return TRUE;
   • if

   $s(x)=(s(x_1),\ldots,s(x_n)).$1

   for some $s(x)=(s(x_1),\ldots,s(x_n)).$2, return TRUE.

2. Return FALSE.

Correctness follows from the set-representation lemmas corresponding to (A3)–(A6) and (A7). Each check is polynomial in the size of $s(x)=(s(x_1),\ldots,s(x_n)).$3 and $s(x)=(s(x_1),\ldots,s(x_n)).$4, hence the implication problem is in $s(x)=(s(x_1),\ldots,s(x_n)).$5 (Häggblom, 2024).

The database-theoretic interpretation is immediate. When $s(x)=(s(x_1),\ldots,s(x_n)).$6, exact exclusion atoms are the exclusion dependencies of database theory. The rules (A1)–(A6) at $s(x)=(s(x_1),\ldots,s(x_n)).$7 coincide with the classical Armstrong-style rules—irreflexivity, symmetry, augmentation, transposition, permutation—together with the new contraction rule. Approximate dependencies generalize exact ones by allowing a controlled fraction of errors, thereby modeling noisy or incomplete data. The completeness and polynomial-time decision results therefore lift the classical theory of finite axiomatizability and tractability to the approximate setting (Häggblom, 2024).

6. Other uses of the acronym “EATS”

The acronym “EATS” is overloaded across several research areas. In the present logical setting it refers to Exclusion ATS as introduced through approximate exclusion in team semantics, but the same acronym appears with different meanings in high-energy physics, causal inference, variable selection, and statistical mechanics.

Usage	Domain	Core object
Exclusion ATS	Team semantics, database theory	Approximate exclusion atoms on teams (Häggblom, 2024)
Exclusion Asymptotic Test Statistic	ATLAS statistics	Profile-likelihood-ratio statistic $s(x)=(s(x_1),\ldots,s(x_n)).$8, Asimov dataset, $s(x)=(s(x_1),\ldots,s(x_n)).$9 limits (Casadei, 2011)
Exclusion And Shape Tests	Potential outcomes	Graph-based support restrictions, MIS inequalities, test statistic $t_1 \# t_2$0 (Kaido et al., 24 Dec 2025)
Exclusion Automatic Threshold Selection	Stability selection	Noise-only exclusion threshold $t_1 \# t_2$1 and profile-likelihood elbow for $t_1 \# t_2$2 (Huang et al., 28 May 2025)
Exposition of “Exclusion ATS” from multiple exclusion statistics	Statistical mechanics	Correlated-state exclusion, $t_1 \# t_2$3, generalized Wu-like distribution (Riccardo et al., 2019)

In the ATLAS setting, the method centers on the one-sided profile-likelihood statistic

$t_1 \# t_2$4

together with the Asimov dataset and the $t_1 \# t_2$5 construction for exclusion limits on the signal strength $t_1 \# t_2$6 (Casadei, 2011). In potential-outcomes models, EATS denotes a graph-based framework in which maximal independent set inequalities of the form

$t_1 \# t_2$7

encode sharp testable implications of exclusion and shape restrictions (Kaido et al., 24 Dec 2025). In stability selection, EATS is a data-adaptive thresholding method that first computes a 95th-percentile exclusion cutoff $t_1 \# t_2$8 from a shuffled noise dataset and then applies a profile-likelihood elbow criterion to filtered selection probabilities (Huang et al., 28 May 2025). In the multiple-exclusion-statistics literature, the term is associated with a generalized exclusion framework based on correlated-state counting and an exclusion spectrum function $t_1 \# t_2$9 (Riccardo et al., 2019).

This acronymic multiplicity can generate confusion. In arXiv-facing discourse, the intended meaning is usually determined by the ambient field: team semantics and dependencies point to approximate exclusion atoms; collider statistics point to profile-likelihood exclusion; causal inference points to support-restriction tests; stability selection points to threshold calibration; and many-body statistical mechanics points to correlated exclusion counting.