# Compositional Distributional Semantics with Syntactic Dependencies and Sectional Preferences
## 3 A Dependency-Based Compositional Strategy for Meaning Construction
### Co-Selection  
The co-selection process is defined as follows. Given a syntactic dependency linking word \(x\), the head, with \(y\), the dependent one, by means of syntactic relation _rel_ (e.g., obj, nsubj, nmod, etc. ):  
\[(rel,x,y)\]  
two compositional functions are defined, \(h\) and \(d\), as follows:  
\[h_{\uparrow}(rel,\vec{x},\vec{y}^{rel\uparrow}) \tag{1}\] \[d_{\downarrow}(rel,\vec{x}^{rel\downarrow},\vec{y}) \tag{2}\]  
where \(h_{\uparrow}\) and \(d_{\downarrow}\), respectively, stand for the head and dependent functions; \(\vec{x}\) and \(\vec{y}\) are the static embeddings respectively associated with the head and dependent words; and \(\vec{x}^{rel\downarrow}\) and \(\vec{y}^{rel\uparrow}\) are the selectional preferences respectively imposed by words \(x\) and \(y\). Selectional preferences are vectors dynamically built by considering paradigmatic relations between words [21]. Given the elements of Equations (1) and (2), the head function, \(h_{\uparrow}\), combines (by either multiplying or adding) the static vector of the head word, \(\vec{x}\), with the selectional preferences imposed by the dependent one, \(\vec{y}^{rel\uparrow}\), so as to construct a new vector representing the contextualized sense of the head: \(\vec{x}_{rel\uparrow}\). The dependent function, \(d_{\downarrow}\), combines the static vector of the dependent word, \(\vec{y}\), with the selectional preferences of the head, \(\vec{x}^{rel\downarrow}\), in order to elaborate a new vector representing the contextualized sense of the dependent: \(\vec{y}_{rel\downarrow}\). Let us see the compound expression \"the company fired\", which is partially analyzed by linking the head lemma _fire_ to the dependent _company_, by means of the nominal subject (\(nsubj\)) relation (Only lemmas of lexical words are considered. Grammatical words such as auxiliary verbs or determiners are not taken into account so far in our compositional approach.). The application of the two functions defined above in Equations (1) and (2) consists of multiplying (or adding) the static vectors of words with the selectional preferences as follows:\[h_{\uparrow}(nsubj,\tilde{fire},com\tilde{p}any^{nsubj\uparrow}) = \tilde{fire}\odot com\tilde{p}any^{nsubj\uparrow} = \tilde{fire}_{nsubj\uparrow} \tag{3}\] \[d_{\downarrow}(nsubj,\tilde{fire}^{nsubj\downarrow},com\tilde{p}any) = com\tilde{p}any\odot\tilde{fire}^{nsubj\downarrow} = com\tilde{p}any_{nsubj\downarrow} \tag{4}\]  
Each function activates a combinatorial operation (component-wise vector multiplication in Equations (3) and (4)), resulting in two new compositional vectors representing the contextualized senses of both the head and the dependent words. In Equations (3) and (4), \(com\tilde{p}any^{nsubj\uparrow}\) and \(\tilde{fire}^{nsubj\downarrow}\) are selectional preferences obtained by adding the vectors of paradigmatically related words:  
\[com\tilde{p}any^{nsubj\uparrow} = \sum_{\vec{w}\in\textbf{{V}}}\vec{w} \tag{5}\] \[\tilde{fire}^{nsubj\downarrow} = \sum_{\vec{w}\in\textbf{{N}}}\vec{w} \tag{6}\]  
where \(V\) is a set of verbal vectors representing the paradigmatic class of those verbs having _company_ as subject. This paradigmatic class is  
change of meaning of the verb. As they are very different cases, some annotators considered the change of meaning as a strong differentiating element, while others considered the differentiating element to be the presence of an expression with low acceptability. This lack of criteria in scoring was compensated and minimized by the fact that the score of each example is the average of up to 13 different annotators rating from 1 to 7. As there are no datasets with noun-verb-noun expressions for other languages than English, we created two new datasets in Portuguese (193 pairs of transitive expressions) and Spanish (also 193 pairs). The process of developing the two new datasets was as follows. The 199 pairs of the original dataset described in [13] were translated and adapted to Portuguese and Spanish.