On the Expressive Power of Monadic Least Fixed Point Logic Nicole Schweikardt ⋆ Institut f¨ ur Informatik, Humboldt-Universit¨ at Berlin Unter den Linden 6, D-10099 Berlin, Germany Email: schweika@informatik.hu-berlin.de Url: http://www.informatik.hu-berlin.de/˜schweika Abstract. Monadic least fixed point logic MLFP is a natural logic whose expres- siveness lies between that of first-order logic FO and monadic second-order logic MSO. In this paper we take a closer look at the expressive power of MLFP. Our results are 1. MLFP can describe graph properties beyond any fixed level of the monadic second-order quantifier alternation hierarchy. 2. On strings with built-in addition, MLFP can describe at least all languages that belong to the linear time complexity class DLIN. 3. Settling the question whether addition-invariant MLFP ? = addition-invariant MSO on finite strings would solve open problems in complexity theory: “=” would imply that PH = PTIME whereas “=” would imply that DLIN = LINH. 1 Introduction A central topic in Finite Model Theory has always been the comparison of the expres- sive power of different logics on fi nite structures. One of the main motivations for such studies is an interest in the expressive power of query languages for relational databases or for semi-structured data such as XML-documents. Relational databases can be mod- eled as fi nite relational structures, whereas XML-documents can be modeled as fi nite labeled trees. Since first-order logic FO itself is too weak for expressing many interest- ing queries, various extensions of FO have been considered as query languages. When restricting attention to strings and labeled trees, monadic second-order logic MSO seems to be “just right”: it has been proposedas a yardstick for expressiveness of XML query languages [7] and, due to its connection to fi nite automata (cf., e.g., [25]), the model-checking problem for (Boolean and unary) MSO-queries on strings and labeled trees can be solved with polynomial time data complexity (cf., e.g., [6]). On fi nite relational structures in general, however, MSO can express complete problems for all levels of the polynomial time hierarchy [1], i.e., MSO can express queries that are believed to be far too diffi cult to allow effi cient model-checking. ⋆ Parts of this work were done while the author was supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD) in order to visit the Laboratory for Foundations of Computer Science, University of Edinburgh, Scotland, U.K.