The structure of graphs and new logics for the characterization of Polynomial Time

Descriptive complexity theory is the area of theoretical computer science that is concerned with the abstract characterization of computations by means of mathematical logics. It shares the approach of complexity theory to classify problems on the basis of the resources that are needed to solve them. Instead of computational resources, however, descriptive complexity theory considers the expressiveness necessary for a logic to define classes of problems. Its primary goal is to provide alternative characterizations of computational complexity that give us a way to reason about the strengths and limitations of computational procedures without reference to the underlying machine model. This thesis is making contributions to three strands of descriptive complexity theory. First, we consider an incongruence between machine computations, which receive ordered strings as their input, and logics, whose abstract view on input structures omits the ordering. We show that a combinatorial, singly exponential-time graph canonization algorithm of Corneil and Goldberg (1984) can be extended to edge-colored directed graphs. The algorithm’s input invariance allows us to implement it in the logic Choiceless Polynomial Time with Counting (C̃PT+C) if we restrict our attention to logarithmic-sized fragments of such graphs. This means that on structures whose relations are of arity at most 2, the polynomial-time (PTIME) properties of logarithmic-sized fragments are precisely characterized by C̃PT+C. The second contribution investigates the descriptive complexity of PTIME computations on restricted classes of graphs. We present a novel canonical form for the class of interval graphs which is definable in fixed-point logic with counting (FP+C), resulting in FP+C capturing PTIME on this graph class. The methods developed for this result may serve as the foundation for a systematic study of the interrelation of logics and complexity classes on the basis of modular decompositions of graphs. Then we introduce the notion of non-capturing reductions that lead us to a wide variety of graph classes on which computational problems are equally hard to characterize as on the class of all graphs. We finally mold our methods into a canonical labeling algorithm for interval graphs which is computable in logarithmic space (LOGSPACE) and we draw the conclusion that interval graph isomorphism is complete for LOGSPACE. In this way, our methods developed for the logical domain make a contribution to classical complexity theory. In the final part of this thesis, we take aim at the open question whether there exists a logic which generally captures polynomial-time computations. We introduce a variety of rank logics with the ability to compute the ranks of matrices over (finite) prime fields. We argue that this introduction of linear algebra makes for a robust notion of a logic and that it increases the expressiveness of FP+C and many other logics considered in the context of capturing PTIME. By adapting a construction of Hella (1996), we establish that rank logics strictly gain in expressiveness when we increase the number of variables that index the matrices we consider. Rank computations are the first natural operation for which this property is shown in the presence of recursion. Our particular choice of rank computations is justified by showing that most classic problems in linear algebra can either be defined by rank logics or are already definable in FP+C. Then we establish a direct connection to standard complexity theory by showing that in the presence of orders, a variety of complexity classes between LOGSPACE and PTIME can be characterized by suitably defined rank logics. Our exposition provides evidence that rank logics are a natural object to study and establishes the most expressive of our rank logics as a viable candidate for capturing PTIME, suggesting that rank logics need to be better understood if progress is to be made towards a logic for polynomial time.