Multiple-Model Induction in Eleusis

lbe card game Elcllsis [1.51 poscs an induct:on problem in which no single model-bascd gcncralization Sp.1CC capturcs all of the desircd gcncf:lliz.1tions of the input data. A progr:lr:l has becn dcvcloped which solves this problem by se.1(":hing scveral ditTerem genNali7..1tion spaces for pl.111Siblc Elcusis rules. Multiple generalization spaces require multiple data transformation mcthods "for transforming thc data into 'a fonn suitable fOf gcncrali7.ation. Examplcs of thc program's opcralion arc givcn and suggestions are made for further research. ' Introduction Simon and Lea [121 have described induction as a search process which uses two spaces: aspace of instances and a ):pace of rules. This characterizatior: of indllclion is very apt alld has recently been extended by Mitchell to take ad... antage of the gcnerai-to'spccific ordcring of sentences in the gcneralization space [1O). One of tile major goals for future research in induction and problem'solYins is the <lbility to solve problems whose fOlmlilation is imprecise-whose generalization spaces have not been provided in advance to the progr~lm. The work on the card game Elellsis described herein is a very small step in this direction. Elcllsis provides an induction problem fur which no single generalization space is sufficient. Instead, several different model-b.lScd gener~1ization spaces were identilicd and ( implemented. The Eleusio; program searches this "meta-space" of models and withill cach model. conducts a more traditional scarch for plausible gencr:llizations of the input data. This paper describes the card game E\cusis and the methods used to perfonn induction in multiple gcneml!zation spaces. A Program. for Eleusis F,1cusis (developed by Robert Abbott It 5D is a card game in which players attempt to induce a secret nile invented by the dealer. The secret rule describes a linear sequence of cards, In their turns, the players attcmpt to extend this sequence by playing additional cards from their hands. 111e dealer gives no information aside from indicating whether or not each play is correcL Players arc penalized for incorrect plays by having additional C:lrds added to their hands. The game ends when a player empties his hand. A record of the play is maintained ac; a layout (Figure 1) in which the top row, or mail/line, contains all of the correctly-played cards in sequence. Incorrect cards arc placed in side lines below the main tine card which ~hey follow. mainline: 3H 9S 4C JD 2C 100 8H 7H 2C 5H sidelines: JD AH AS 10H 50 8H lOS QD Rule 1: "If the last card is odd. pby black, if the last card is c\'en, play red:· Figure 1. S;mlplc Elcusis Layout (after (lD. l11e scoring in Eleusis encourages the dealer to choose niles of intermediate difficulty. The de:tler's score is dClennined by the dilf~rcl1cc between the highest and lowcsl scores of the p1<IYCrs. Thus, a good nile is one which is easy lor some playcrs and hard fur others. This research sought to dcvelop a program which could serve as an intelligent assistant to a hu:nan Eleusis player, The program needed to be ahle to: ( ~ discover niles which plausibly describe the layout. ~ accept niles typed by the user and test them agJinst the layout. ~ extend the layout by suggesting cards to be played from the playcr's hand. The LC'Jrning Problem This paper rocusses on the first goal of the E1cusis assistant: the induction of nIles which plausibly describ\! the layout In the terminology of Simon and Lea. the purpose of the program is to find niles in sOllie generali/ation space which pl,;lIsibly descnbe the layout (which is an ckment of the instance sp,lce). The remainder of this paper diS<;lIsses what is meant by the tenns pl(!/lsibly dcscribe and lome gencralization spac~. then describes me search process used to search generalization space, . and finally presents an extended example . • DescrilltiollS A rule describes the layout if ~ the rule could be used to nondeterministically generate the layout. and ~ none of the incorrect plays could be generated using the rule. In general. a rule describes a whole set of possible layouts just as a grammar describes a set of sentences (a language), ror example, Rule 1 describes allY layout in which all black: cards are preceded hy odd cards and all red cards arc preceded by even cards (and in which 4111 black cards played incorrectly were preceded by even cards and all red cards played incorrectly were preceded by odd cards). ". ( We stress that the layout generation is nondeterministic becausc of the worle done on sequence extrapolation [11] where deterministic descriptions arc soughL In 8eusis there are usually several ways to continue the main line. Plausibility The conditions for plausibility can be expressed by the following heuristics: 1. Prefer rules with intermediate degree of complexity. In Eleusis, Occam's Razor docs not always apply. The dealer is unlikely to choose a rule which is extremely simple becausc it would be too easy to discover. Very complex rules will not be discovered by anyone so the dealer is not likely to choose them either. 2. Prefcr rules with intermediate degree of generality. The degree of generality-the number of possible layouts which satisfy a mle-is the degree of nondeterrninism in the rule. Eleusis rules arc typically nondetemtinistic. Furthermore, mles with low degree of generality lead to many incorrect plays thus rendering them easy to discover. Rules of high generality lead to few. incorrect plays and arc ~herefore diflicult to discover. 3. Look for symmetric disjunctions. Rule 1 is an excellent example of a symmetric disjunctive rule. Must often in Elcusis, the terms of a disjunction define mutlmlly exclusive cases which have sume symmetric relationship to each other. '!1le dealer is very likely to choose such rules because tbey are not tou hmd-nor too easy-to discover. lbesc plausibility heuristics are easy to formulate in Eleusis because we have insight into the dealer's goals and behavior. I n real w(lrld problems. we arc rarely so fortunate, but plausibility heuristies call usually still be fonnul<lted. Heuristic 3 (symmetry) is huilt into lhe model-fitting algorithms described below. The other heuristics are used to evaluate and disc.:'Jfd implausible rules once they have been discovered.