Definability and Transformations for Cost Logics and Automatic Structures

We provide new characterizations of the class of regular cost functions (Colcombet 2009) in terms of first-order logic. This extends a classical result stating that each regular language can be defined by a first-order formula over the infinite tree of finite words with a predicate testing words for equal length. Furthermore, we study interpretations for cost logics and use them to provide different characterizations of the class of resource automatic structures, a quantitative version of automatic structures. In particular, we identify a complete resource automatic structure for first-order interpretations.

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