 A pair \(P=\langle \{{{\cal L}}_i\}_{i\in I}, \{f_i\}_{i\in I}\rangle\) as above is called a possible-translations frame for \({{\cal L}}\). We say that \(P\) is a possible-translations semantics for \({{\cal L}}\) if, for every \(\Gamma\cup\{\varphi\}\subseteq L(C)\),   \[\Gamma \vdash_{{\cal L}}\varphi \ \textrm{ iff } \ f_i(\Gamma) \vdash_{{{\cal L}}_i} f_i(\varphi), \ \textrm{ for every  } i \in I.\]   This means that checking derivability in \({{\cal L}}\) is equivalent to checking derivability in every factor logic \({{\cal L}}_i\) through the translations. In many cases, the factor logics \({{\cal L}}_i\) are presented by finite matrices. Since the length of a formula is finite, it is enough to test a finite number of translations in order to determine if a formula of \({{\cal L}}\) is valid in \({{\cal L}}\). Thus, checking the validity of a formula of \({{\cal L}}\) is equivalent to performing a finite number of finitary tests. This decidability property is of real advantage when the original logic \({{\cal L}}\) is not characterizable by finite matrices. For instance, in Carnielli 2000 (see also Marcos 1999) the well-know hierarchy \(\{{{\cal C}}_n\}_{n\in{\mathbb{N}}}\) of paraconsistent logics of N. da Costa, formed by logics which cannot be characterized by finite matrices, is represented by means of a PTS whose factors are presented through finite matrices; this grants a decision procedure for each logic \({{\cal C}}_n\).