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, 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 PTIME. If this conjecture was true, this would not only imply that PTIME 6= NP, but it would also show that NP and the complexity classes above NP have a fundamentally different structure than the class PTIME and presumably most natural complexity classes below PTIME. (This aspect is highlighted by a result due to Dawar [6], also see [13].) On the positive side, Immerman [18] and Vardi [23] proved that least fixedpoint logic FP captures polynomial time on the class of all ordered finite structures. Here we say that a logic L captures a complexity class K on a class C of finite structures if exactly those properties of structures in C decidable in K are definable in L. It is easy to prove that FP does not capture PTIME on the class of all finite structures. Immerman [19] proposed the extension FP + C of fixed-point logic by counting operators as a candidate for a logic capturing PTIME. It is not easy to prove, but true nevertheless, that FP + C does not capture PTIME. This was shown by Cai, Fürer, and Immerman in 1992 [3]. Fixed-point definability on graphs with excluded minors Even though the logic FP + C does not capture PTIME on the class of all finite structures, it does capture PTIME on many natural classes of structures. Immerman and Lander [20] proved that FP + C captures PTIME on the class of all trees. In 1998, I proved that FP + C captures PTIME on the class of all planar 1 For a precise definition of a logic capturing PTIME, I refer the reader to Grädel’s excellent survey [10] on descriptive complexity theory. graphs [11] and around the same time, Julian Mariño and I proved that FP + C captures PTIME on all classes of structures of bounded tree width [14]. In [12], I proved the same result for the class of all K5-free graphs, that is the class of all graphs that have no complete graph on five vertices as a minor. A minor of graph G is a graph H that can be obtained from a subgraph of G by contracting edges. By (the easy direction of) Kuratowski’s Theorem, the class of all K5-free graphs contains all planar graphs. We say that a class C of graphs excludes a minor if there is a graph H that is not a minor of any graph in C. Very recently, I proved the following theorem, which generalises all these previous results, because all classes of graphs appearing in these results exclude minors. Theorem FP + C captures PTIME on all classes of graphs that exclude a minor. The main part of my talk will be devoted to this theorem.