## Paper: The boundedness problem for monadic universal first-order logic (at LICS 2006)

**Martin Otto**

### Abstract

We consider the monadic boundedness problem for least fixed points over FO formulae as a decision problem: Given a formula (X, x), positive in X, decide whether there is a uniform finite bound on the least fixed point recursion based on . Few fragments of FO are known to have a decidable boundedness problem; boundedness is known to be undecidable for many fragments. We here show that monadic boundedness is decidable for purely universal FO formulae without equality in which each non-recursive predicate occurs in just one polarity (e.g., only negatively). The restrictions are shown to be essential: waving either the polarity constraint or allowing positive occurrences of equality, the monadic boundedness problem for universal formulae becomes undecidable. The main result is based on a model theoretic analysis involving ideas from modal and guarded logics and

### BibTeX

@InProceedings{Otto-Theboundednessprobl, author = {Martin Otto}, title = {The boundedness problem for monadic universal first-order logic}, booktitle = {Proceedings of the Twenty-First Annual IEEE Symposium on Logic in Computer Science (LICS 2006)}, year = {2006}, month = {August}, pages = {37--46}, location = {Seattle, Washington, USA}, publisher = {IEEE Computer Society Press} }