Inner-Learning: From Hand-Coded Beliefs to Declarative Knowledge in Trick-Taking Games

We present a declarative approach to belief-state modeling in trick-taking card games. Starting from a state-of-the-art Skat player, we automatically extract its hand-coded inference rules into a domain-specific language built on the primitive $\neg\mathsf{holds}(P,C)$, denoting that player $P$ cannot hold any card in set $C$. Round-trip code generation verifies that the extracted rules reproduce the original engine's intended behavior, up to three audited bugs in the original that the declarative engine corrects. We then grow the rule base through symmetric derivation and annotation of an expert strategy document---unifying code, symmetry, and expertise in one representation. We extend the knowledge representation with cardinality constraints: at-most-$k$ and at-least-$k$ bounds on the cards a player holds from a named group, expressed in the same declarative language and enforced at world-generation time. Confidence parameters connect symbolic rules to probabilistic belief updates. Self-play evaluation shows the declarative engine measurably improving winning percentage and score over the original hand-coded engine, with the extracted rules, the human-annotated rules, and the cardinality constraints each contributing to the effect.