Backus–Naur form grammar that generates classifiers 

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... episodes were learnt as in a classical XCS to produce a maximally general accurate population, termed Optimum population set (OPS). This OPS was then the environment where each generalised rule was a message passed to the S-Expression based XCS, termed S-XCS. This could be contrasted with a single population version (without OPS) that directly learnt from the raw environmental messages, again utilising the S-Expression alphabet, in order to discover any benefits of abstraction. Following the S-Expression paradigm [7][ 8] the classifiers’ building blocks consists of either non-terminals (i.e. functions) or terminals (i.e. variables or constants), see figure 3. These can be tailored to the domain, e.g. VALUEAT which returns the bit value of a position within the classifier and ADDROF which returns the integer counterpart of a string of bits. Tailored functions rely upon domain knowledge, which reduces the applicability of the technique unless used with other general functions. AND, OR, NOT – Binary functions; the first two receive two arguments and the last one only one. Arguments can be either binary values directly from the leaf nodes of the condition trees, or integers (values >1 are treated as the binary ‘true’ or ‘1’) if the functions are located in the middle of the trees. • PLUS, MINUS, MULTIPLY, DIVIDE, POWEROF – Arithmetic functions; they all receive two arguments apart from the final one that receives only one argument. These arguments can be either binary values or integers as above. The POWEROF function treats its argument as an exponent of base two. • VALUEAT, ADDROF – Domain specific functions; VALUEAT receives one argument (pointer to an address, which is capped to the string length) and returns a value (environmental ‘value’ at the referenced address). ADDROF receives two arguments symbolising the beginning and end (as positions in the environmental string) for specifying the binary string to be converted to an integer. There is no ordering as the lowest index is considered the least significant bit and the highest index as the most significant bit. There is no match set using S-Expressions since the classifiers represent a complete solution by using variables and not predetermined bit strings with fixed actions. Thus the actions are computed (similar to piece-wise linear approximators). It is noted that VALUEAT points to a value and that it is impossible to consistently point to an incorrect value, which does not favour the low prediction classifiers that occur in a classical XCS. Other methods were also appropriately altered, e.g. Classifier Covering or completely removed, e.g. Subsumption Deletion. Coverage was triggered when the average fitness was below 0.5 in an action set and created a random valid tree. In initial tests numerosity tended to be small due to the diversity of the S- Expressions, which tended to distort fitness measurements. Thus, numerosity was removed and absolute accuracy was used as fitness instead of the relative accuracy. The update function, including accuracy-based fitness and action selection in exploit trials are the same as in classical XCS. The first system created was the S_XCS which uses only three Binary Functions (AND, OR, NOT). The Covering procedure created one classifier per invocation. The second system was the S_XCS1 which was created in order to investigate the effect of tailored building blocks (e.g. ADDROF, VALUEAT) and to determine if an increased variety of functions causes bloating or prevents learning. System parameters are β =0.2, initial error ε =0.5, accuracy parameters α =0.1, ε 0 =10, ν =5, GA threshold θ GA =25, deletion probabilities θ del =20, mutation rate μ =0.01, τ =0.2, crossover possibility χ =1, population size N=400 [2]. S_XCS discovers the Disjunctive Normal Form of the 3-MUX problem (see figure 4 & table 1), but fails to scale. However, S_XCS1 does scale, see figures 5 & 6. Provided sufficient time is allowed the system will discover the most compact functions suited to the domain, see table 2. This time may be greatly reduced by the use abstraction, where generalisation is accomplished using the ternary alphabet (where possible) and then semantic knowledge is gained by training on the optimum set, see figure 7. The OPS can not be learnt practically for >135 MUX and thus the known ternary solution was used in figure 7 and table 3, which only shows the potential scalability of S-Expressions. The best discovered solutions are shown in tables 2 and 3 (highest fitness in bold). Unnecessary functions, such as POWEROF, are more likely to appear without abstraction. The results of using S_XCS1 with a limited set of the most suited functions (discovered though abstraction) did not provide significant advantage, comparing figure 8 with figure 4. It is only the hypothetical case of figure 7, which demonstrates optimum performance is possible as the domain scales. XCS scales well on the MUX problems as the learning is polynomial complexity [9]. However, XCS still struggles to solve the 135-MUX problem whereas humans are capable of understanding the concept behind all MUX problems. The S_XCS system does exhibit potential in developing solutions that resemble the Disjunctive or the Conjunctive Normal Form of the k-MUX problem, but it did not scale well. The reason for this was that no partially correct classifiers entered the population rendering the Genetic Algorithm redundant and only Covering could create a correct classifier. The problem with competent classifiers not entering the population was that the environmental variables changed at every epoch and the functions (AND, OR, ...