Knowledge transfer between simple discrete worlds

Silvère Gangloff; Jan Hula

cs.AIOct 10, 2026

Knowledge transfer between simple discrete worlds

Authors: Silvère Gangloff, Jan Hula

Organizations: University of Ostrava

Abstract

Adapting to a new world and reasoning correctly about which outcomes are possible under rules one has not met before is a basic capacity underlying general intelligence. We study it in small discrete worlds: sets of grid configurations consistent with a shared set of rules, concretely Sudoku-like puzzles built from a common base and a handful of extra rules. We train a single depth-recurrent model on many combinations of rules and test it at four levels of novelty: a new puzzle, a new combination of known rules, a pairing of rules never seen together, and a rule instance not seen at all during training. The model receives the active rules either as a symbolic description or as two solved examples, and a model that gets no rule information is used as a baseline. At recombining rules it was trained on, the description matches or exceeds examples ( $99\%$ against $95\%$ on new combinations). On rule instances whose components never co-occurred in training, the description falls below the no-information baseline while examples stay above it. The framework allows us to test many such generalization regimes in more systematic way than benchmarks such as ARC-AGI and provides new ground for evaluating depth-recurrent reasoning models, which are typically tested on saturated Sudoku benchmarks.

Figures & tables

Panel of Figure 1 — Figure 1: A completion problem (Definition 2.3 ): given cells $D$ (left) determine a unique admissible configuration $x$ (right) in some world $W_{N}(\mathcal{A},\mathcal{F})$ . This particular puzzle comes from the concrete family introduced in Section 3 .

	Type	#	Description of the rules of this type in the family
	Unique-occurrence regions - a region where each symbol appears at most once	11	A fixed support (a $2\times 3$ box, its scattered counterpart, a diagonal, a window, the four center or corner cells) contains each symbol at most once.
	Adjacency - cells in a fixed spatial relation must differ	2	Two cells a knight’s move apart ( AntiKnight ) or diagonally adjacent ( AntiKing ) never share a value.
odd	Value-restricted regions - a region confined to a fixed subset of values	4	Every cell of a diagonal takes values in a fixed set $V\subseteq\{1,\ldots,6\}$ (odd or even).
$+k$	Value-shift relations - two regions whose values are related by a fixed offset	32	A pair of rows or columns $a,b$ (one specific pair, or all three pairs of the top and bottom halves) satisfies $b=a+k\bmod 6$ cell by cell, $k\in\{1,\ldots,5\}$ .

Table 1: Rule types in the

49

-instance pool

\mathcal{V}

. Every rule is a set of forbidden patterns on the

6\times 6

grid; the glyph shades the cells one instance of the type constrains. Full per-instance definitions in Appendix A .

Setup	Prefix	Len.	Symbols used	What the model must do
no-information	none	36	7	Infer the rules from the given cells alone; lower bound on how much rule information the puzzle itself carries.
description-based	$4$ rules $\times$ $4$ tokens	52	36	Read the rule identity as symbols and apply it.
example-based	$2$ solved grids ( $72$ tokens)	108	7	Infer the rules from the examples, then apply them; mirrors the ARC-AGI format.

Table 2: The three conditioning setups. Only the prefix placed before the

36

puzzle cells differs.

Panel of Figure 2 — Figure 2: Overview of the setup. (a) Example world: list of rules and schematic representation of these rules. (b) How the rules are communicated: information is appended before the puzzle under diverse forms. (c) Illustration of the forms of novelty. Letters designate rules.

	mixed split		no-mix, draw 1	L3 dial split
Setup	L0	L1	L2	L3 unsupported	L3 supported
no-information	$60.5\pm 0.8$	$65.4\pm 1.6$	$63.1\pm 1.8$	$52.1\pm 2.3$	$56.7$
description-based	$\mathbf{99.2\pm 0.1}$	$\mathbf{98.5\pm 0.2}$	$67.6\pm 5.5$	$34.6\pm 6.4$	$79.0\pm 5.4$
example-based	$95.2\pm 1.2$	$95.4\pm 1.1$	$\mathbf{82.3\pm 1.6}$	$\mathbf{65.0\pm 0.8}$	$\mathbf{90.8\pm 2.3}$

Table 3: Exact-match accuracy in percent, mean

\pm

standard deviation over

4

training seeds, for the three setups at the four novelty levels. L1 is measured on the mixed split and L0 on the held-out puzzles of the validation worlds. L2 is measured on draw

1

of the no-mix split, on worlds containing cross-group pairs. Both L3 columns use the L3 dial split with the same held-out instances and test worlds: unsupported at level

0

(no supporting instance in training), supported at the natural exposure level (

228

supporting worlds). Best value per column in bold.

Setting	description-based	+ dropout	floor	example-based
L3 dial, level 0 (unsupported)	$34.6\pm 6.4$	$58.8\pm 1.6$	$52.1\pm 2.3$	$65.0\pm 0.8$
no-mix split, L2	$67.6\pm 5.5$	$71.8\pm 2.2$	$63.1\pm 1.8$	$82.3\pm 1.6$
no-mix split, L3	$58.7\pm 3.2$	$66.6\pm 2.1$	$61.7\pm 1.6$	$77.0\pm 2.2$
L3 dial, level 0, L1 (in support)	$98.3\pm 0.9$	$96.5\pm 0.2$	$75.4\pm 0.9$	$98.0\pm 0.1$

Table 4: Description dropout. Exact match in percent, mean

\pm

standard deviation over

4

seeds, evaluated with the description present; floor is the no-information model on the same split. The last row shows the cost on in-support rules. Full results, including further splits and evaluation with the description masked, in Appendix C.3 .

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 3 — Figure 3: Left: the BoxesRect2x3 rule requires the six cells physically inside one $2\times 3$ box (shaded, one possible filling shown) to together contain every symbol exactly once. Right: its scattered counterpart ScatteredRect2x3 instead groups the six cells that share the same position across the grid’s six boxes (here, each box’s top-left corner, shaded) into one such group.

Panel of Figure 4 — Figure 4: Left: MainDiagonal requires the six cells on the $\searrow$ diagonal (shaded, one possible filling shown) to together contain every symbol exactly once. Right: AntiDiagonal imposes the same requirement on the six cells of the $\swarrow$ diagonal instead.

Panel of Figure 5 — Figure 5: Left: AntiKnight forbids the highlighted cell (dark grey) from sharing a value with any cell a chess knight’s-move away (red). Right: AntiKing forbids the same cell from sharing a value with any diagonally-adjacent cell (red).

Panel of Figure 6 — Figure 6: Left: ParityDiagOdd restricts every cell on the main diagonal (shaded) to the odd values $\{1,3,5\}$ ; unlike the earlier diagonal rule, repeats are allowed here, since membership in this fixed set is all that is required. Right: ParityADiagEven applies the analogous restriction to the anti-diagonal, with the even set $\{2,4,6\}$ . The remaining two combinations, ParityDiagEven and ParityADiagOdd , swap which parity set applies to which diagonal.

Panel of Figure 7 — Figure 7: The three window placements (shaded, one possible filling shown), with the $2\times 3$ box tiling from Figure 3 overlaid as a dashed reference grid. Windows6 ’s pair straddles the box boundaries, in the spirit of a real Windoku puzzle. WindowsWrapH and WindowsWrapV instead each wrap a window pair around the grid’s border (matching shade intensity marks the two pieces of the same window, split across the wrap); neither ever coincides with a BoxesRect2x3 box.

Panel of Figure 8 — Figure 8: Both rules cut the grid into two fixed areas and then shift the alphabet value , position by position, by $k$ between them - the areas themselves never move or depend on $k$ ( $x^{\prime}$ denotes $x$ shifted by $k$ in the alphabet, mod $6$ ). Left: RowShiftK{k} cuts the grid into a top and a bottom area (one column shown: top values $a,b,c$ ; the corresponding bottom values must be $a^{\prime},b^{\prime},c^{\prime}$ ); every other column is cut and shifted the same way. Right: ColShiftK{k} is the column-wise analogue, cutting a left and a right area instead.

Panel of Figure 9 — Figure 9: Left: RowLink{0,5,k} links only row $1$ and row $6$ (shaded) - every column’s value in row $6$ must equal its value in row $1$ shifted by $k$ in the alphabet ( $x^{\prime}$ denotes $x{+}k\bmod 6$ ); every other row is unconstrained by this rule. $\mathcal{V}$ includes two more such row pairs, $(2,5)$ and $(3,4)$ (1-indexed), and the column analogues, each at any $k\in\{1,\ldots,5\}$ as a separate instance. Right: ColLink{0,5,k} is the column-wise version, linking only column $1$ and column $6$ .

Panel of Figure 10 — Figure 10: Left: Center4 requires the four center cells (shaded) to all differ from each other - only $4$ of the $6$ symbols are involved, so unlike the region rules above, no symbol is required to appear. Right: Corners4 imposes the same all-different requirement on the four corner cells instead.

Panel of Figure 11 — Figure 11: Two admissible configurations of the world $\{\texttt{BoxesRect2x3},\ \texttt{RowLink(1,6,}k{=}3)\}$ on top of the Latin-square base (thick lines mark boxes; shaded rows are linked, row $6$ equals row $1$ shifted by $3$ ). These are the two solved grids the example-based setup (Section 3.2 ) shows the model before a puzzle from this same world (Figure 1 shows one such puzzle).

Panel of Figure 12 — Figure 12: Exposure dials (Section 3.4 ). (a) L2 accuracy on draw $1$ of the no-mix split as training worlds mixing the two rule groups are added; the test worlds are the same at every level and the training sets are nested. (b) L3 accuracy on the held-out column links as training worlds containing a supporting column link are added; held-out instances and test worlds are the same at every level. Points are means over $4$ seeds, bars are one standard deviation. The diamond is the description-based model trained with description dropout; hollow squares are the example-based setup on a second draw of the training worlds.

mixing	$\rho$	$c(\rho)$	L2			L3
worlds			description-based	example-based	floor	description-based	example-based	floor
0	0	0.00	$67.6\pm 5.5$	$82.3\pm 1.6$	62.1	$58.7\pm 3.2$	$77.0\pm 2.2$	59.7
5	0.01	0.05	$70.0\pm 3.2$	$88.6\pm 0.8$	63.8	$63.6\pm 4.8$	$79.3\pm 0.9$	61.9
25	0.05	0.34	$89.2\pm 0.6$	$92.6\pm 0.5$	64.0	$70.5\pm 2.5$	$78.8\pm 1.4$	62.4
100	0.20	0.64	$96.6\pm 1.4$	$95.8\pm 0.3$	67.7	$73.4\pm 0.8$	$81.8\pm 0.3$	61.2
373	0.746	0.86	$98.5\pm 1.0$	$97.0\pm 0.3$	70.4	$77.8\pm 3.3$	$82.0\pm 1.4$	60.9

Appendix

Table 5: L2 exposure dial, draw

1

of the no-mix split. Exact match in percent, mean

\pm

standard deviation over

4

seeds. Test sets are identical at every level; training sets are nested. L0 accuracy is

97.8

99.5\%

for the description-based and example-based setups at every level.

support	supporting	L3			L1
worlds	instances (of 10)	description-based	example-based	floor	description-based	example-based	floor
0	0	$34.6\pm 6.4$	$65.0\pm 0.8$	52.1	$98.3\pm 0.9$	$98.0\pm 0.1$	75.4
5	5	$48.5\pm 2.9$	$70.0\pm 0.6$	52.5	$98.5\pm 0.6$	$98.1\pm 0.3$	75.5
25	10	$48.0\pm 7.4$	$77.0\pm 1.1$	53.2	$98.4\pm 0.1$	$97.9\pm 0.2$	74.1
100	10	$79.0\pm 7.1$	$86.6\pm 1.6$	57.7	$98.2\pm 0.4$	$97.8\pm 0.3$	74.0
228	10	$79.0\pm 5.4$	$90.8\pm 2.3$	56.7	$97.8\pm 0.9$	$97.6\pm 0.2$	68.8

Appendix

Table 6: L3 exposure dial, draw

1

. Five column links are held out at every level; the level is the number of training worlds containing one of the ten remaining column links (supporting instances). Exact match in percent, mean

\pm

standard deviation over

4

seeds. Test sets are identical at every level. Second training draw at levels

0

25

and

228

(

2

seeds): example-based

68.2\pm 0.3

81.8\pm 2.1

91.9\pm 1.4

; description-based

39.8\pm 0.2

69.2\pm 1.8

84.4\pm 3.3

Split	Draw	description-based	example-based	floor
axis	1	$36.4\pm 3.7$ / 94.0	$59.4\pm 1.8$ / 97.0	45.4 / 66.3
	2	$37.3\pm 3.2$ / 98.6	$60.3\pm 1.4$ / 97.8	50.4 / 69.5
	3	$35.8\pm 2.1$ / 98.0	$64.0\pm 0.2$ / 97.8	50.8 / 76.8
row-pair	1	$44.9\pm 3.5$ / 97.3	$72.2\pm 0.5$ / 97.0	57.7 / 67.4
	2	$41.4\pm 4.8$ / 97.8	$69.6\pm 1.3$ / 96.4	55.9 / 68.6
	3	$44.9\pm 0.4$ / 98.8	$69.4\pm 1.6$ / 97.5	61.6 / 66.8

Appendix

Table 7: Further held-out sets for L3, three draws each. Each cell gives L3 / L1 exact match in percent (for the no-mix split, L3 / L2). Mean

\pm

standard deviation over

4

seeds. The mixed split has one draw: description-based

73.4\pm 2.1

, example-based

81.2\pm 0.9

, floor

55.9\pm 0.3

on L3.

Setting	description-based	+ dropout	floor (36 / 52 tokens)	example-based
L3 dial, level 0 (unsupported)	$34.6\pm 6.4$	$58.8\pm 1.6$	$52.1\pm 2.3$ / $53.2\pm 0.4$	$65.0\pm 0.8$
axis draw 1 (unsupported)	$36.4\pm 3.7$	$53.9\pm 2.2$	$45.4\pm 0.1$ / $46.0\pm 2.0$	$59.4\pm 1.8$
no-mix draw 1, L2	$67.6\pm 5.5$	$71.8\pm 2.2$	$63.1\pm 1.8$ / $64.0\pm 1.3$	$82.3\pm 1.6$
no-mix draw 1, L3	$58.7\pm 3.2$	$66.6\pm 2.1$	$61.7\pm 1.6$ / $62.9\pm 0.2$	$77.0\pm 2.2$
L3 dial, level 0, L1	$98.3\pm 0.9$	$96.5\pm 0.2$	$75.4\pm 0.9$ / $77.7\pm 0.5$	$98.0\pm 0.1$

Appendix

Table 8: Description dropout, evaluated with the description present. Exact match in percent, mean

\pm

standard deviation over

4

seeds. The floor column gives the no-information model at

36

tokens and, after the slash, at

52

tokens (padded to the description-based sequence length).

Setting	Level	description-based, masked	+ dropout, masked	floor at 52 tokens
L3 dial, level 0	L3	$28.0\pm 2.9$	$52.4\pm 2.0$	53.2
	L1	$37.4\pm 4.8$	$78.4\pm 0.6$	77.7
	L0	$33.3\pm 3.8$	$74.7\pm 0.5$	72.8

Appendix

Table 9: Evaluation with the description replaced by null tokens at test time, exact match in percent. Mean

\pm

standard deviation over

4

seeds. On the L3 dial at level

0

, the dropout model matches the

52

-token no-information model to within

2

points; the ordinary description-based model collapses.

Control	axis	row-pair	no-mix	mixed
description-based at 52 tokens (reference)	36.4	44.9	58.7	73.4
description-based padded to 108 tokens	$27.9\pm 1.6$	$55.3\pm 5.0$	$58.5\pm 1.4$	$67.5\pm 3.8^{a}$
no-information at 36 tokens	45.4	57.7	61.7	55.9
no-information at 52 tokens	$46.0\pm 2.0$	$58.8\pm 0.0$	$62.9\pm 0.2$	not run
composed rule vectors	$36.0\pm 5.8$	$57.9\pm 10.2$	$53.7\pm 5.1^{b}$	$70.8^{c}$

Appendix

Table 10: Length and mechanism controls, draw

1

of each split, L3 exact match in percent,

4

seeds. Parameter counts:

1.09

M at

36

tokens,

1.11

M at

52

1.25

M at

108

. Padding the description-based prefix to the example-based length moves the three hard splits by

-8.5

+10.4

and

-0.2

points, with no consistent sign. a Screening recipe,

2

seeds; the unpadded reference at that recipe is

67.9

. b L2 on the same split:

58.1\pm 8.2

. c Screening recipe,

2

seeds.

Mechanism	L0	L1	L3
composed rule vectors	99.1	98.3	70.8
cross-attention from cells to description tokens	98.5	97.2	65.6
description tokens in the sequence (padded)	99.0	96.2	67.5
spatial field (did not converge)	92.4	91.5	60.5
generated per-layer modulation (FiLM; failed)	69.9	73.7	57.7

Appendix

Table 11: Injection mechanisms for the rule description, mixed split, screening recipe (

12

epochs, cosine decay),

2

seeds. Exact match in percent at L0 / L1 / L3. The three working mechanisms are within

2

points of each other on L1.

	old split	new split
old training code	65.8	95.0
new training code	66.5	98.5

Appendix

Table 12: Where the earlier L1 result for the description-based setup came from: training code (old / new) by split (old / new), L1 exact match in percent,

2

seeds per cell except the top-left one (

25

runs).

Exposure	Setup	value shift (links)	uniqueness	parity	aggregate
$0\%$	description-based	75.8	66.3	34.2	58.7
$0\%$	example-based	93.2	74.6	63.1	77.0
$74.6\%$	description-based	98.3	83.8	51.2	77.8
$74.6\%$	example-based	96.8	79.6	69.6	82.0

Appendix

Table 13: L3 exact match in percent by rule family of the held-out instance, L2-dial split,

4

seeds, ten held-out worlds per family, at the lowest and the natural exposure level.

Panel of Figure 13 — Figure 13: The reasoning engine, used unchanged across the three setups; only the input sequence (top box) differs between them. $f$ is the $2$ -layer, $256$ -dimensional MLP-Mixer-style block stack described next, shared between the two update equations.

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Abstract

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