In this lecture we discuss descriptive complexity in some more detail. This subject is concerned with understanding the relationship between logics, whose formulas generate symmetric algorithms, and complexity classes conventionally defined in terms models with no such symmetry condition (e.g. Turing machines). We discuss the foundational result in this field, Fagin’s theorem, which establishes that those problems in NP are precisely those that can be defined in existential second-order logic. This leads naturally to the central open question in the field: Is there a similar characterisation for P?

We introduce various logics that have arisen naturally in the field, often motivated by this central question, including fixed-point logic (FP) . We discuss the Immerman-Vardi theorem, which establishes that FP characterises P over ordered structures, and how understanding the definability of order is central to understanding the relationship between logic and computation.

We discuss finite-variable fragments of logics and game-based characterisations of these logics, as well how these may be used to establish inexpressibility results for fixed-point logic and its extensions. First, we discuss finite-variable fragments of first-order logic and show that their expressive power may be characterised using a pebble game. We note that each formula of FP may be translated to a family of first-order formulas with a constant bound on the number of variables used in each formula. We show how this relationship enables the use of pebble games for establishing inexpressibility results for FP. We discussion an FP with a counting operator, a logic we call fixed-point logic with counting (FPC). This logic is often said to express almost all problems that are obviously in P and is of central importance in descriptive complexity. More formally, FPC has been shown to capture P over all proper minor closed classes of graphs. We note, however, that FPC provably doesn’t capture P.

We discuss counting quantifiers and extensions of finite-variable fragments with these quantifiers. We recall the equivalence of these finite-variable counting logics and the Weisfeiler-Lehman algorithms for distinguishing graphs. We also discuss various other characterisations of these logics from a number of different fields and two (equivalent) bijection-game characterisations of their expressive power. In the same way as for FP, FPC formulas may be translated to families first-order formulas with counting quantifiers with a constant bound on the number of variables. This allows us to use the bijection-game characterisation to prove inexpressibility results for FPC.

We lastly discuss a family of graph constructions named for Cai, Fürer, and Immerman, called the CFI constructions. We observe, but do not yet prove, that these graphs may be used to construct a query in P not inexpressible in FPC. The proof of this result, which uses bijection games, is discussed in the next lecture.

Part 1

Part 2

Part 3

Further Details

The lecture slides are available here. The notes for the discussion sessions are to be found here.