## Hierarchies in Transitive Closure Logic, Stratified Datalog and Infinitary Logic (1994)

Venue: | ANNALS OF PURE AND APPLIED LOGIC |

Citations: | 11 - 4 self |

### BibTeX

@INPROCEEDINGS{Grädel94hierarchiesin,

author = {Erich Grädel and Gregory L. McColm},

title = {Hierarchies in Transitive Closure Logic, Stratified Datalog and Infinitary Logic},

booktitle = {ANNALS OF PURE AND APPLIED LOGIC},

year = {1994},

pages = {167--176},

publisher = {}

}

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### OpenURL

### Abstract

We establish a general hierarchy theorem for quantifier classes in the infinitary logic L ! 1! on finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure. This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves a conjecture of Immerman. We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog.