Achievement Set: Theory & Applications

• Achievement Set is the collection of attainable states defined in mathematics as subsums of convergent series and extended to game-based and learning systems.
• In classical analysis, achievement sets range from finite sets to Cantor sets, multi-intervals, and Cantorvals, classified via Kakeya conditions and index comparisons.
• Applied interpretations reformulate achievement sets as in-game trophy catalogs and goal-tracking events, enabling predictive modeling in reinforcement learning and digital analytics.

Searching arXiv for recent and foundational uses of “achievement set” across mathematics, games, and applied ML/HCI. Across the cited literature, the term achievement set denotes several non-equivalent objects. Its classical meaning is the set of all subsums of a convergent series, usually for an absolutely convergent real sequence; in later work it is extended to conditionally convergent series, finite-dimensional vector series, planar subsum sets, hyperspace operators, and several families of achievement-style games. In applied settings, the term is also used for finite sets of in-environment achievements, trophy lists, or realized-goal subsets, as in reinforcement learning, digital games, and self-tracking systems (Głąb et al., 19 Dec 2025, Kula et al., 2023, Zhou et al., 2023, Kwak, 2022, Gordon et al., 2019).

1. Classical subsum-set meaning

For an absolutely convergent series $\sum x_n$, the classical achievement set is the set of all subseries sums,

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$

equivalently

$$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$

This is the basic definition used in the modern survey literature and in recent work on positive convergent series (Głąb et al., 19 Dec 2025, Nowakowski, 19 Dec 2025). For signed real series, the one-dimensional theory reduces to the nonnegative case via

$E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$

so many topological questions can be studied under the assumption $x_n\ge 0$ (Głąb et al., 19 Dec 2025).

Kakeya’s classical results remain foundational in this setting. For an absolutely summable nontrivial real sequence, $E(x_n)$ is compact and perfect; if $|x_n|>\sum_{k=n+1}^\infty |x_k|$ for every $n$, then $E(x_n)$ is homeomorphic to the Cantor set; if $|x_n|\le \sum_{k=n+1}^\infty |x_k|$ for all but finitely many $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$0, then $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$1 is a finite union of closed intervals (Marchwicki, 2017, Głąb et al., 19 Dec 2025). A standard tail notation is

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$2

and much of the theory is governed by the comparison between $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$3 and $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$4 (Moroz, 2024, Głąb et al., 19 Dec 2025).

The survey literature fixes the modern one-dimensional classification: for an absolutely convergent series, the achievement set is exactly one of four types—finite set, multi-interval set, Cantor set, or Cantorval (Głąb et al., 19 Dec 2025). In the positive nonincreasing case, the trichotomy interval / Cantor / Cantorval is often expressed in terms of the indices where $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$5 and $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$6 (Nowakowski, 19 Dec 2025).

2. Topological types, Kakeya conditions, and multigeometric regimes

A central invariant is the set of Kakeya indices

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$7

Classically, $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$8 is a multi-interval set if and only if $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$9 is finite, while $$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$0 is a Cantor set if $$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$1 has finite complement (Głąb et al., 19 Dec 2025). What happens when both $$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$2 and its complement are infinite is subtler. The Guthrie–Nymann example

$$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$3

has achievement set a Cantorval and

$$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$4

whereas

$$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$5

has a Cantor-set achievement set with the same Kakeya pattern (Głąb et al., 19 Dec 2025). Recent work strengthens this non-classifiability: for every infinite $$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$6 with infinite complement, one can realize $$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$7 both with $$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$8 a Cantor set and with $$E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.$$9 a Cantorval (Głąb et al., 19 Dec 2025, Nowakowski, 19 Dec 2025).

Within structured families, sharper parameter criteria are available. For generalized multigeometric series

$E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$0

with $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$1 locally increasing and power bounded at $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$2, the paper “The achievement set of generalized multigeometric sequences” gives explicit threshold constants $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$3 separating interval, non-interval, interval-containing, and Cantor regimes (Karvatskyi et al., 2023). In particular, for sufficiently large $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$4, $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$5 is a compact interval; for sufficiently small $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$6, it is homeomorphic to the Cantor set; and under an arithmetic richness condition on block-subsums, there is an intermediate parameter region where $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$7 is a Cantorval (Karvatskyi et al., 2023).

A distinct recent development is the generalized Ferens framework. If a convergent GF series satisfies the explicit block inequalities labeled (GF1) and (GF2), then its achievement set is a Cantorval; this yields new achievable Cantorvals outside the multigeometric class (Marchwicki et al., 2023). The same paper proves two strong addition theorems: there exists an achievable Cantorval whose $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$8-fold algebraic sum remains a Cantorval for every finite $E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,$9, and for any $x_n\ge 0$0 with $x_n\ge 0$1, there exists an achievable Cantor set $x_n\ge 0$2 such that $x_n\ge 0$3 is a Cantor set for $x_n\ge 0$4, a Cantorval for $x_n\ge 0$5, and an interval for $x_n\ge 0$6 (Marchwicki et al., 2023).

The 2025 paper “On a new condition implying that an achievement set is a Cantorval and its applications” introduces the Star Procedure, a recursive sufficient criterion formulated in terms of overlap lengths $x_n\ge 0$7 and auxiliary quantities $x_n\ge 0$8. If the inequalities $x_n\ge 0$9, $E(x_n)$0, and $E(x_n)$1 can be continued indefinitely, then $E(x_n)$2 is a Cantorval (Nowakowski, 19 Dec 2025). This criterion recovers classical examples such as $E(x_n)$3, yields new multigeometric Cantorvals, and proves that Kakeya conditions alone cannot classify the mixed regime more finely than the original Kakeya theorems (Nowakowski, 19 Dec 2025).

3. Conditionally convergent and finite-dimensional extensions

For conditionally convergent series, the real-line picture is simpler than the absolute theory: for a conditionally convergent real series,

$E(x_n)$4

and more generally if a real series is potentially conditionally convergent, then its achievement set is $E(x_n)$5 (Marchwicki, 2017, Bartoszewicz et al., 2016). In finite-dimensional spaces, however, the structure becomes much richer.

The paper “Subsums of conditionally convergent series in finite dimensional spaces” studies

$E(x_n)$6

and the absolute subsum set $E(x_n)$7. Its main theorem states that for pairwise distinct $E(x_n)$8 and nonzero $E(x_n)$9,

$|x_n|>\sum_{k=n+1}^\infty |x_k|$0

if and only if the exponents are pairwise distinct; consequently the full achievement set is also $|x_n|>\sum_{k=n+1}^\infty |x_k|$1 (Marchwicki et al., 2018). A highlighted special case is

$|x_n|>\sum_{k=n+1}^\infty |x_k|$2

which had previously been open (Marchwicki et al., 2018).

The geometry of Levy vectors gives a complementary description in $|x_n|>\sum_{k=n+1}^\infty |x_k|$3. If a conditionally convergent planar series has more than two Levy vectors, then

$|x_n|>\sum_{k=n+1}^\infty |x_k|$4

If it has exactly two Levy vectors, several behaviors are possible: $|x_n|>\sum_{k=n+1}^\infty |x_k|$5 can still equal $|x_n|>\sum_{k=n+1}^\infty |x_k|$6; one can have $|x_n|>\sum_{k=n+1}^\infty |x_k|$7; and there are examples with $|x_n|>\sum_{k=n+1}^\infty |x_k|$8 but achievement set extremely small, even with each vertical section containing at most one point (Glab et al., 2017). This phenomenon is one reason the survey literature treats the two-Levy-vector case as especially delicate (Głąb et al., 19 Dec 2025).

More generally, for a conditionally convergent series $|x_n|>\sum_{k=n+1}^\infty |x_k|$9 with full sum range $n$0 and an absolutely convergent series $n$1, one has the product formula

$n$2

This yields examples such as $n$3, graph-like sets, nonclosed sets, and proper open strips in higher-dimensional conditional theory (Bartoszewicz et al., 2016).

4. Planar achievement sets, hyperspaces, and the spectre

For an absolutely convergent planar series,

$n$4

the basic structural facts are that

$n$5

and $n$6 is compact and centrally symmetric (Kula et al., 2023). Unlike the one-dimensional case, these sets need not be products, and the paper “Achievement sets of series in $n$7” shows that planar sections can realize far more complicated one-dimensional objects.

The main cut theorem states that for every finite $n$8 with $n$9 and every absolutely convergent real sequence $E(x_n)$0 with $E(x_n)$1, there exists a planar series $E(x_n)$2 such that

$E(x_n)$3

where $E(x_n)$4 is the corresponding set of $E(x_n)$5-sums (Kula et al., 2023). A consequence is that an $E(x_n)$6-Cantorval can occur as a horizontal section of a planar achievement set, so the planar theory is not exhausted by product types such as $E(x_n)$7, $E(x_n)$8, $E(x_n)$9, or $|x_n|\le \sum_{k=n+1}^\infty |x_k|$0 (Kula et al., 2023).

A second major innovation is the spectre. For an Abelian group $|x_n|\le \sum_{k=n+1}^\infty |x_k|$1 and $|x_n|\le \sum_{k=n+1}^\infty |x_k|$2,

$|x_n|\le \sum_{k=n+1}^\infty |x_k|$3

On $|x_n|\le \sum_{k=n+1}^\infty |x_k|$4, $|x_n|\le \sum_{k=n+1}^\infty |x_k|$5 coincides with the center of distances, but the spectre is vector-valued and well adapted to $|x_n|\le \sum_{k=n+1}^\infty |x_k|$6 (Kula et al., 2023). For every absolutely convergent series, each term belongs to the spectre of its achievement set: $|x_n|\le \sum_{k=n+1}^\infty |x_k|$7 In the planar case, if $|x_n|\le \sum_{k=n+1}^\infty |x_k|$8, then

$|x_n|\le \sum_{k=n+1}^\infty |x_k|$9

while for the Sierpiński carpet one has

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$00

which implies that the Sierpiński carpet is not the achievement set of any series (Kula et al., 2023).

The hyperspace paper “Spectre operator, achievement sets and sets of $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$01-sums in a hyperspace of compact sets” places these objects inside $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$02 with the Hausdorff metric. It proves that the family $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$03 of achievement sets in $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$04 is closed and hence nowhere dense, whereas the family $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$05 of sets of $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$06-sums is not closed (Nowakowski et al., 2 Nov 2025). The same paper establishes monotonicity properties of the spectre along finite partial-sum sets $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$07 and tail achievement sets $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$08, and proves planar analogues of the first two gap lemmas while showing that the one-dimensional third-gap phenomenon fails in $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$09 (Nowakowski et al., 2 Nov 2025).

5. Achievement sets in games and combinatorial structures

In several game-theoretic literatures, “achievement” refers not to subsums but to the family of positions satisfying a winning target. The resulting “achievement set” is therefore positional rather than analytic.

In convex-geometry game theory, the game $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$10 is played on a finite convex geometry $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$11 with closure operator

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$12

The relevant achievement family is

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$13

that is, the generating positions whose convex closure already contains the winning set $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$14 (McCoy et al., 2020). The paper “Impartial Achievement Games on Convex Geometries” develops a structure theory based on maximally non-generating sets, intersection subsets, and parity-sensitive structure classes, and derives explicit nim-value formulas for extreme-point targets, vertex geometries of trees, and affine geometries in one dimension (McCoy et al., 2020).

In positional game theory, the paper “A unified convention for achievement positional games” formalizes an achievement positional game as

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$15

where $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$16 and $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$17 are the blue and red winning hyperedges of the two players (Galliot et al., 23 Mar 2025). Here an “achievement set” is naturally a player-specific winning hyperedge, or more broadly the player’s winning family $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$18 or $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$19. The paper proves that many Maker–Maker principles extend to this asymmetric setting and classifies the complexity of deciding whether Left wins as first player: the problem is in $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$20 for $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$21, NP-hard for $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$22, coNP-complete for $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$23, and PSPACE-complete for $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$24 (Galliot et al., 23 Mar 2025).

Other combinatorial achievement games make the same shift from subsums to target families. In the weak polyomino set $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$25-achievement game, the maker tries to occupy cells congruent to one of a target set of polyominoes, and the paper proves that every winning team of cardinality $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$26 is simpler than a specific unbounded “super winner” $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$27 (Fisher et al., 2010). In the general position achievement game on graphs, the evolving chosen-vertex set must remain a general position set, and the last legal move wins (Klavžar et al., 2021). These uses are terminologically consistent with the idea of “achieving” membership in a target family, but they are structurally distinct from the subsum-set tradition.

6. Applied reinterpretations in tracking, games, and reinforcement learning

Applied literatures use the phrase in still different ways. In self-tracking, the MyFitnessPal study defines the “achievement set” as the subset of weight-loss goals that are eventually realized: for a goal weight $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$28, a goal is achieved if at some later time the user logs a weight less than or equal to $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$29 (Gordon et al., 2019). On that operational definition, the study of 1,413,431 users reports that only 18.2% of weight-loss goals are achieved, and that a Random Forest model using the first 7 days of behavior predicts eventual achievement with 79% ROC AUC (Gordon et al., 2019). This usage is not a set of subsums but a realized-goal subset of observed user-goal episodes.

In reinforcement learning, an achievement-based environment is formalized as a Markov Decision Process with Achievements

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$30

where $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$31 is the finite internal set of achievements and $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$32 is the achievement completion function (Zhou et al., 2023). SEA first learns an embedding of reward-triggering transitions with the determinant loss

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$33

then clusters the known achievements and recovers a dependency DAG using

$$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$34

after which a graph-based controller uses the recovered structure for exploration (Zhou et al., 2023). In this setting, the achievement set is a latent but finite set of reusable events rather than a topological subset of $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$35.

In digital-game analytics, the PlayStation study treats a game’s trophy list as its achievement set. At platform scale, the dataset contains 13,792 games and 377,938 trophies, with trophy-point structure Bronze $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$36, Silver $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$37, Gold $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$38, Platinum $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$39 (Kwak, 2022). The paper identifies strong platform conventions: trophy-score totals cluster around 300 and 1,200, “L” games are defined by total score $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$40, “H” games by total score $$E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},$$41, and larger games typically have exactly one Platinum trophy (Kwak, 2022). Here the achievement set is a designed finite catalog rather than a generated mathematical set.

Taken together, these applied uses suggest a broader semantic pattern: “achievement set” often denotes the finite family of achievements available in a system, or the subset of them realized by an agent. A plausible implication is that the mathematical and applied literatures share the vocabulary of attainable states while differing sharply in ontology: subsums in analysis, winning positions in games, and labeled goals or events in HCI and RL.