The final experiment concerns the chess endgame domain White King and Rook vs. Black King (KRK) [12]. The classification task is to distinguish between illegal and legal board positions. The 1BC representation employs a structural function board2whiteking to refer to the position of the White King (similarly for the other two pieces), and two structural functions pos2rank and pos2file to translate a position into rank and file. We have two propositional properties rankeq and fileeq equating rank/file with a number, and three relational properties adj, eq and lt to compare rank/files. The propositional elementary features in this domain are exemplified by board2whiteking(A,B),rankeq(B,1) and board2blackking(A,B),fileeq(B,8). The relational elementary features are of the form board2whiteking(A,B),board2whiterook(A,C),pos2rank(B,D), pos2rank(C,E),eq(E,D).
Following the results reported in [9], we used 5 training sets of 100 board positions each, and a test set of 5000 positions. Table 1 gives the accuracy over the training set, and the accuracy over the test set, averaged over the 5 training sets. 1BC 2/2 refers to features with no more than 2 literals and 2 variables, i.e. the propositional features referred to above (in this case pos2file and pos2rank are used as properties with a parameter as second argument). Similarly, 1BC 5/5 refers to features with up to 5 literals and 5 variables, which includes both the propositional and the relational features. 1BC FO refers to relational features only (the two propositional properties rankeq and fileeq were removed from the representation).
| System | Training accuracy | Test accuracy |
| Majority class | 64.0% sd. 3.0% | 66.3% |
| MLC++ | 79.0% sd. 3.1% | 57.0% sd. 2.6% |
| 1BC 2/2 | 79.0% sd. 3.5% | 56.2% sd. 1.4% |
| 1BC 5/5 | 91.2% sd. 2.5% | 84.3% sd. 5.2% |
| 1BC FO | 93.8% sd. 3.6% | 88.3% sd. 2.8% |
The results show that KRK-illegal is a difficult domain for a Bayesian classifier, since the best result reported in [9] was 98.1% on the test set, achieved by LINUS. Nevertheless, the experiment clearly demonstrates that the use of first-order features considerably improves the performance of the Bayesian classifier. With only propositional features, the result on the test set drops well below the majority class. This means that propositional features have actually negative information content [8]. To verify that this was not due to a bug in 1BC, we also ran the Bayesian Classifier in MLC++ on the same data, and got virtually the same results for the propositional features only.