Capture Complexity by Partition

We show in this paper a special extended logic, partition logic based on so called partition quantifiers, is able to capture some important complexity classes NP, P and NL by its natural fragments. The Fagin’s Theorem and Immerman-Vardi’s Theorem are rephrased and strengthened into a uniform partition logic setting. Also the dual operators for the partition quantifiers are introduced to expose some of their important model-theoretic properties. In particular they enable us to show a 0-1 law for the partition logic, even when finite variable infinitary logic is adjunct to it. As a consequence, partition logic cannot count without built-in ordering on structures. Considering its better theoretical properties and tools than those of second order logic, partition logic may provide us with an alternative, yet uniform insight for descriptive complexity.