Fixed-Point Definability and Polynomial Time

My talk will be a survey of recent results about the quest for a logic capturing polynomial time. In a fundamental study of database query languages, Chandra and Harel [4] first raised the question of whether there exists a logic that captures polynomial time. Actually, Chandra and Harel phrased the question in a somewhat disguised form; the version that we use today goes back to Gurevich [15]. Briefly, but slightly imprecisely,1 a logic L captures a complexity class K if exactly those properties of finite structures that are decidable in K are definable in L. The existence of a logic capturing PTIME is still wide open, and it is viewed as one of the main open problems in finite model theory and database theory. One reason the question is interesting is that we know from Fagin's Theorem [9] that existential second-order logic captures NP, and we also know that there are logics capturing most natural complexity classes above NP. Gurevich conjectured that there is no logic capturing...

• Type: Conference
• Year: 2009
• Where: CSL
• Authors: Martin Grohe
• Keywords: Automated Reasoning, CSL 2009, Logic, Logic Capturing, Logic Capturing Ptime