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I also have the impression that Descriptive Complexity is a less active area of research nowadays. Nevertheless, there are some topics in which people are still active: Rank logics: Rank Logic is Dead, Long Live Rank Logic! by Grädel and Pakusa Symmetric Circuits for Rank Logic by Dawar and Wilsenach Separating Rank Logic from Polynomial Time by Lichter ...


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Definitely still active in the area of Weisfeiler-Leman-style algorithms for isomorphism problems such as Graph Isomorphism. The connection with logic was first (I believe) made in Immerman-Lander and Cai-Fürer-Immerman. Very recently adapted to Group Isomorphism [Brachter-Schweitzer]. Certain logics are equivalent in their power to distinguish graphs (resp....


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Lemma 1. Consider $L = (ab)^* + (ba)^*$: There exists CFG with two variables which generates $L$ There exist no CFG with one variable which generates $L$ For proving (1) we may just consider the following CFG $$ S \rightarrow bAa \mid A , \, A \rightarrow abA \mid \varepsilon$$ The second proposition is a bit more tricky. Suppose that there exists CFG $G = ...


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