Representation Scheme

The scheme we used to represent knowledge about the game's state space was similar to the well known "logic box" model. At the most basic level we use a disjunctive clause to represent the possible cards that a player can have:

$$d(i) = c_1 \vee c_2 \vee \dots \vee c_n$$

Each disjunction is also subject to the restriction that $c \leq length(d(i)) \leq n$. In this sense $d(i)$ is not actually a true disjunction, but this does not affect the correctness of our approach. In our scheme, we consider $P+1$ players, since the hidden cards can be seen as belonging to an additional player that never participates in the game. For this reason we define the function $cards(i) = C$ when $1 \leq i \leq p$, and $cards(P+1) = k$.

The one basic operation in this scheme is an assertion that a player does not have a certain card. All other assertions are based on this basic operation. For example, to assert that a player has a certain card, we simply assert that all other players do not have the card.

The iterative removal of variables from a disjunction simulates the operation of the Elimination Algorithm. The Elimination Algorithm is an online algorithm for learning monotone disjunction that begins with the assumption that all attributes are included in the disjunction. As new evidence is observed attributes are slowly removed from the hypothesis until only the relevant attributes (the cards that the player has) remain.

We also use some secondary data structures to aid or augment our basic scheme. First we dichotomize every player's card disjunction into two sets $K$ and $U$:

• $K$ (known cards): these are the cards in a players disjunction that no other player can have. It must be the case that $\vert K\vert \leq cards(i)$. $K$ can also be seen as a conjunction. Every card in $K$ must belong to the player for $K$ to be true.

• $U$ (uncertain cards): these are the cards in a players disjunction that are also part of the disjunction for some other player. If $\vert U\vert < cards(i)$, then $U$ can be viewed as an ordinary disjunction.

This separation is useful for making various inferences.

The disadvantage of this basic scheme is that only one disjunction is stored, when in fact throughout the game, as other players make interrogations, we would like to introduce new disjunctions over the "asked" cards (assuming the interrogated player responded with a card). To solve this problem, we augment our representational model by assigning each player a list of clauses. A new clause is added every time that player is interrogated and he or she responds with a card. These clauses are exactly like the disjunctions mentioned above. The only difference is that they are not required to have a minimum length of $cards(i)$. Clauses are removed when they are redundant with respect some other clause, or with respect to $K$.