Positive First-Order Logic on Words and Graphs

Denis Kuperberg (1, 2)

1 CNRS - Centre National de la Recherche Scientifique
2 LIP - Laboratoire de l'Informatique du Parallélisme

We study FO + , a fragment of first-order logic on finite words, where monadic predicates can only appear positively. We show that there is an FO-definable language that is monotone in monadic predicates but not definable in FO +. This provides a simple proof that Lyndon's preservation theorem fails on finite structures. We lift this example language to finite graphs, thereby providing a new result of independent interest for FO-definable graph classes: negation might be needed even when the class is closed under addition of edges. We finally show that given a regular language of finite words, it is undecidable whether it is definable in FO + .

Domaines: Logique en informatique [cs.LO]; Théorie et langage formel [cs.FL]; Logique [math.LO]

Denis Kuperberg. Positive First-Order Logic on Words and Graphs. Logical Methods in Computer Science, 19 (3), pp.7:1 - 7:35, 2023. ⟨hal-03865495v1⟩