CoToan: Executable Rules and a Representation Contract for Reproducible Game AI Evaluation

• Implementation choices can alter action probabilities or merge states with different outcomes, silently changing the task presented to an agent.Differences in action semantics, time information, or victory configuration also prevent clean attribution of agent behavior differences to algorithms alone.Source: §I · p. 1
• CoToan combines versioned executable rules, a declarative GDL description, an OpenSpiel environment, and reproducible conformance diagnostics.Its design separates task mechanics and settings, arithmetic witnesses, and the agent interface.Source: §I · p. 1
• The executable task definition specifies arithmetic capture, path blocking, capture-value scoring, and ordered terminal conditions, while separating invariant mechanics from victory settings.Source: §I · p. 1
• The cross-representation contract preserves legal actions, transitions, time, and configuration identity between GDL and OpenSpiel, treating arithmetic witnesses as metadata rather than agent actions.Source: §I · p. 1
• Conformance comparisons and controlled diagnostics examine legal actions, successor states, outcomes, witness-level sampling distortion, and state aliasing from omitted time or victory settings.Source: §I · p. 1
• The three research questions test representation agreement, witness-level action sampling, and whether board occupancy and player-to-move suffice under finite horizons and configurable victory rules.Together, these requirements concern consistent transitions, unambiguous action identity, and sufficient adjudication state—not training outcomes or playing strength.Source: §I · pp. 1–2

A. Board and objective

• Each side starts with one piece of each value from 0 through 9; Red moves first by default, and each side’s zero is fixed on the board.Capturing the opponent’s zero wins; other captures add the target’s value to the score and may trigger configured victory.Source: A. Board and objective · p. 2, Figure 1 · p. 2

B. Ordinary movement

• A nonzero piece moves 1 to its value’s number of squares in any straight or diagonal direction, but every traversed square must be empty.It cannot turn, jump, or capture an enemy through ordinary sliding movement.Source: B. Ordinary movement · p. 2
• In the local example, c4-e6 is legal, c4-f7 exceeds the red 2’s range, and c4-e4 is blocked by the piece on d4.The examples distinguish a range violation from a path obstruction.Source: Figure 2 · p. 2, B. Ordinary movement · p. 2

C. Arithmetic capture

• A capture requires an attacker, adjacent friendly support, and an enemy target on the same line; an enabled arithmetic result sets the positive distance from support to target.The intervening squares must be empty, the attacker replaces the target, and the support remains stationary.Source: C. Arithmetic capture · p. 2
• In the example, red 2 on c4 uses red 1 on d4 to capture on f4 because 2 × 1 = 2; the attacker moves to f4 while the support stays.Source: C. Arithmetic capture · p. 2
• Both floor(8/3)=2 and 8 mod 3=2 justify c4xf4, producing the same successor state.A piece on e4 blocks the capture of f4 but can itself be captured using 2-1=1.Source: C. Arithmetic capture · p. 2

D Two witnesses; one action c4xf4

• Two witnesses can justify the single action c4xf4; the witnesses are not separate moves.The figure’s examples are constructed local positions, not claims about reachability.Source: Figure 3 · p. 3, D Two witnesses; one action c4xf4 · p. 3

D. Scoring and victory settings

• Captured values accumulate as scores; a positive threshold enables immediate victory, while threshold zero disables it.At the default 300-ply horizon, the higher score wins and equal scores draw; threshold victory and horizon adjudication are separate settings.Source: D. Scoring and victory settings · p. 3
• Terminal conditions are checked in order: zero capture, threshold victory, horizon adjudication, then no-move loss.Unconditional horizon draw is optional rather than the default; clocks and unbounded duration are not modeled, and each configuration has its own task identifier.Source: D. Scoring and victory settings · p. 3

A. Game description and interoperability

• GDL and OpenSpiel serve complementary roles: CoToan uses GDL’s explicit relational semantics and OpenSpiel’s integration with search and learning code.The paper distinguishes these purposes from PGN/FEN interchange and other board-game description systems.Source: A. Game description and interoperability · p. 3

B. Search, learning and reproducible evaluation

• Search and self-play systems motivate an algorithm-independent environment with well-defined legal actions, observations, and rewards.The cited examples span UCT, MCTS, AlphaGo, AlphaZero, and MuZero.Source: B. Search, learning and reproducible… · p. 3
• CoToan addresses agreement on the task itself through a versioned specification and property-based conformance checks, rather than proposing a new game language or search algorithm.Its horizon diagnostic instantiates an established finite-horizon issue in a reachable position; it is not claimed as a new general result.Source: B. Search, learning and reproducible… · p. 3

A. State and legal actions

• A state records board occupancy, the controller, elapsed plies, and immutable settings for horizon, threshold, horizon rule, operators, and starting player.Well-formed states constrain piece identities, surviving zeros, and elapsed plies; snapshots after both players have crossed a positive threshold are rejected.Source: A. State and legal actions · pp. 3–4
• Captures require friendly support, a positive arithmetic result, an enemy target at the resulting distance, and clear intervening squares.Ordinary moves travel up to the attacker’s value along an empty path; division is floored, and only enabled operators provide capture witnesses.Source: A. State and legal actions · p. 4
• Captured-value scores are derived from surviving opposing material, since captures remove pieces and no rule creates them.The initial sum is 45, and settings determine when accumulated points decide the winner.Source: A. State and legal actions · p. 4

B. Actions are transitions, not arithmetic witnesses

• Actions use source and destination as identity; arithmetic witnesses remain explanatory metadata, not separate agent actions.The GDL term is (move sSRC sDST), while the OpenSpiel ID is 99*SRC+DST with illegal entries masked.Source: B. Actions are transitions, not… · p. 4
• Collapsing captures with identical endpoints preserves their successor relation but not uniform sampling over arithmetic witnesses.An endpoint action’s witness count can therefore change its probability under witness-level sampling.Source: B. Actions are transitions, not… · p. 4
• The endpoint contract prevents arithmetic multiplicity from becoming an extra agent decision and a potential policy confound.Source: B. Actions are transitions, not… · p. 4

C. Declarative encoding and one-step correspondence

• GDL encodes dynamic facts, geometry, arithmetic pairs, enabled operators, scores, and victory settings in finite rules without engine-specific arithmetic extensions.Candidate actions precede terminal status, avoiding a legal/terminal cycle.Source: C. Declarative encoding and one-step… · p. 4
• For fixed settings and a well-formed nonterminal state, legal actions correspond bijectively, and the two representations produce matching successors and terminal outcomes.The proof sketch matches movement predicates, updates, score totals, and ordered adjudication tests.Source: C. Declarative encoding and one-step… · p. 4
• Executable tests check the representation definitions, generated facts, and adapter behavior rather than relying on the correspondence argument alone.Source: C. Declarative encoding and one-step… · p. 4

D. Observation and framework contract

• The OpenSpiel observation includes identity planes, controller, remaining budget, and configuration scalars; snapshots preserve complete settings and reject mismatched imports.Structural validation is not a reachability proof.Source: D. Observation and framework contract · pp. 4–5

E. Reusing search agents

• Search agents can use OpenSpiel’s legal_actions, child or clone/apply_action, is_terminal, and returns methods, while shared cache keys distinguish configuration, board, controller, and remaining budget.A port preserves behavior only when rules, evaluation conventions, and action handling agree.Source: E. Reusing search agents · p. 5

A. Protocol and reproducibility

• The baseline protocol compares complete move sets, rotation/color-swap behavior, and snapshot round trips on 300 reachable samples and 300 structurally valid synthetic positions.The reachable samples come from ten trajectories of up to 30 plies; samples need not be distinct.Source: A. Protocol and reproducibility · p. 5
• Additional checks compare 70 selected GDL fixtures, audit nine configurations across 210 valid states, and exhaust all 32 operator masks and 512 subsets of Blue’s nonzero material.The protocol also checks histories, observations, configuration-mismatch rejection, and usability with one MCTS and one alpha-beta decision.Source: A. Protocol and reproducibility · p. 5

B. RQ1: Cross-representation conformance

• The nine-configuration audit compared 210 valid states and one successor for each of 158 nonterminal cases across Python, GDL, and OpenSpiel; depth-one and depth-two counts were 87 and 7457 regression fingerprints.The audit also included 45 simulation tests; the depth counts are not game-tree size estimates.Source: B. RQ1: Cross-representation conformance · p. 5

C. RQ2: Arithmetic witnesses change action sampling

• Across 66 nonterminal fixtures, 6023 endpoint-action instances became 6031 witness-labeled instances; this diagnostic is not a prevalence estimate for normal play.Six fixtures contained duplicated endpoint actions.Source: C. RQ2: Arithmetic witnesses change… · p. 5
• In Fig. 3D, uniform endpoint sampling selects c4xf4 with probability 1/43, while uniform witness sampling selects it with probability 2/44=1/22 despite the identical board update.The controlled position has 43 endpoint actions and 44 witness-labeled actions.Source: C. RQ2: Arithmetic witnesses change… · p. 5

D. RQ3: Horizon omission aliases reachable states

• At plies 3 and 299, identical occupancy, controller, and legal actions lead to different outcomes after the same move: a nonterminal state at ply 4 and a draw at ply 300.The late draw follows from equal-score adjudication, not an unconditional draw rule; the budget channel separates the states.Source: D. RQ3: Horizon omission aliases… · p. 5
• With the same occupancy, controller, and ply, score adjudication yields a Red win while the optional draw setting yields a draw; threshold 9 ends the game where threshold 0 does not.These diagnostics establish information requirements, not a measured learning benefit.Source: D. RQ3: Horizon omission aliases… · p. 6

E. Limitations and validity

• The specification covers supported configurations rather than every historical or tournament variant, and its correspondence argument is not mechanized.Tests do not exhaust reachable states; GDL checks use a subset translator and clingo rather than an independent GGP competition runtime.Source: E. Limitations and validity · p. 6
• Synthetic fixtures and short-horizon simulations target correctness, while the diagnostics do not quantify effects on learning strength, game balance, or educational outcomes.The authors propose a subsequent agent study with matched computational budgets, exchanged colors, disclosed seeds, and reported uncertainty,.Source: E. Limitations and validity · p. 6

VI. CONCLUSION

• CoToan found no mismatches in evaluated cases, and its diagnostics show that reproducible evaluation requires consistent action identity, sufficient state information, and explicit task configuration.The reported effects include altered action-sampling probabilities and merged states with different terminal outcomes.Source: §VI · p. 6