On Fixed-Point Logic With Counting

One of the fundamental results of descriptive complexity theory, due to Immerman [12] and Vardi [17], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered finite structures by a logic in a similar way. The most obvious shortcoming of fixed-point logic itself on unordered structures is that it cannot count. Immerman [13] responded to this by adding counting constructs to fixed-point logic. Although it has been proved by Cai, Fürer, and Immerman [1] that the resulting fixed-point logic with counting, denoted by IFP+C, still does not capture all of polynomial time, it does capture polynomial time on several important classes of structures (on trees, planar graphs, structures of bounded tree-width [14, 8, 9]). The main motivation for such capturing results is that they may give a better understanding of polynomial time. But of course this requires that the logical side is well understood. We hope that our analysis of IFP+C-formulas will help to understand better the expressive power of IFP+C; in particular, we derive a normal form. Moreover, we obtain a problem complete for IFP+C under first-order reductions. IFP+C-formulas work with two kinds of objects. The elements of the structure they are talking about (referred to as points) and the numbers obtained by counting. The numbers can be seen as an additional, naturally ordered, domain. Analyzing the proofs of the capturing results for IFP+C mentioned above, we observe that the formulas describing polynomial time problems there work as follows: