## 5 Results  
The results of the first experimental phase, in which we looked for the emergence of the mentioned concepts, are reported in Figures 5 and 6. We trained, for each concept and for 10 runs, a linear SVC in separating the activations yielded by all the 24 layers on balanced samples of 2000 functions, where 1000 represent the concept and the other 1000 do not represent it. It can be observed that the two models exhibit a similar pattern for all the considered concepts, but that different concepts define problems of different difficulties: for instance, the average accuracy for the cyclomatic complexity concept is around 91% in all the layers, while for the cast to integer we move from 80% to 90%. Some sub-concepts (i.e., cyclomatic complexity) have a constant average accuracy along the layers, while others (e.g., cast to integer) present some increasing or decreasing trends. Note also that, as expected, the accuracy of the SVCs trained on random concepts (Figure 6) is always around 50%, thus confirming the validity of the results obtained on the well-defined concepts. After having observed the _absolute_ emergence of the concepts, we measured their impact on the original task. We performed two analyses. The results of the first one, obtained by considering only vulnerable instances and that are directly derived from the original TCAV [2], are reported in Table 1, and they give interesting insights on how different concepts have a different influence on the decision. For instance, it is clear that the presence of a cast to integer has a much stronger positive effect than the many-to-many I/O relationship. Additionally, notice that the influence of the randomly defined concept is irrelevant, since for both the models, it is about 50%. For the second measure of sensitivity we propose, based on the classification of instances which are generated by the DSGE evolutionary algorithm, we performed 16 experiments, each corresponding to a combination of choices for concept, CWE, and maximization or minimization of fitness, respectively, on safe and on vulnerable code. As a result, each experiment generated an overall population of around 2200 unique individuals, whose fitness ranges were polarized toward positive or negative values by the chosen optimization. Nonetheless we checked the values ranges, and they always included both positive and negative values, apart from experiments aiming at minimizing for CWE-789 the presence of \"cast\" elements in the input, where no individual with negative fitness value was generated (bottom-left box in Figure 7). We recall that having fitness values spanning both positive and negative values means that each experiment generated both individuals with the strong presence of a concept together with individuals with little presence of the same concept. The accuracy of our fine-tuned transformers on each (sub)population is presented in Table 2. Each cell corresponds to a given choice for concept, CWE, and ground truth of individuals (we maximized the concept for safe instances, i.e., with ground truth 0, and minimized it for vulnerable ones, i.e., with ground truth 1). In each cell, the upper part reports the classifier accuracy on instances with strong presence of the concept, and conversely, the lower part shows the accuracy on those with little concept signal. The numbers show quantitatively how input with more, or less, signal for a concept (i.e., the value of the decision function of the corresponding SVC, on activations induced by the instance) affects the accuracy of the classification for safe and for vulnerable inputs. As we can see, the two networks have a different behavior on the evolved instances, and are also different with respect to each concept. For instance, we can state that the classifier for CWE-789, when fed with the instances we generated outside the training set, has better accuracy, and it seems less impacted by the presence, or absence, of our concepts. On the other hand, the transformer for CWE-369 has generally worse accuracy, on such instances.