A Gödel logic descriptor for chains of regular languages

Fortresses constitute a class of language descriptors completely based only on two fundamental concepts of propositional logic: the notion of logical consequence and the notion of substitution. Fortresses based on classical propositional logic precisely recognise the class of regular languages. In this paper, we characterise the formal languages obtained by replacing, in fortresses, the notion of logical consequence in classical logic with the one in Gödel infinitely-valued logic. We prove that fortresses in Gödel logic exactly recognise finite chains of inclusions of regular languages. To prove this, we make use of a Stone’s type categorical duality between the algebraic semantics of Gödel propositional logic and the category of finite forests and open maps.