We study local consequence relations in modal extensions of product logic over Kripke models with either valued (fuzzy) or crisp accessibility relations. In both settings, we consider semantics over the full class of product algebras as well as over the standard product algebra on $[0,1]$. Our main result is a constructive reduction of these modal logics to propositional product logic. As consequences, we prove that all the resulting systems are decidable and standard complete, i.e., the local consequence relation over all product algebras coincides with the one induced by the standard product algebra. In the valued-accessibility case, our methods strengthen previous results on decidability by extending them from theoremhood to arbitrary local consequence relations, and covering standard completeness. In the crisp case, the techniques are substantially different and yield, to the best of our knowledge, the first decidability and standard completeness results for local modal product logics with crisp accessibility relations.

翻译：我们研究在具有取值（模糊）或清晰可达关系的Kripke模型上，模态扩展乘积逻辑中的局部推论关系。在这两种设定下，我们考虑了基于乘积代数全类以及基于$[0,1]$上标准乘积代数的语义。我们的主要结果是将这些模态逻辑构造性地归约为命题乘积逻辑。作为推论，我们证明所有得到的系统都是可判定的且标准完备的，即所有乘积代数上的局部推论关系与标准乘积代数所诱导的局部推论关系一致。在取值可达情形下，我们的方法将之前关于可判定性的结果从定理证明扩展至任意局部推论关系，并涵盖了标准完备性。在清晰可达情形下，相关技术本质上不同，并且据我们所知，首次得到关于具有清晰可达关系的局部模态乘积逻辑的可判定性和标准完备性结果。