Randomisation and Derandomisation in Descriptive Complexity Theory

We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from PTIME. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in Cinf, finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures

Team Semantics and Recursive Enumerability

It is well known that dependence logic captures the complexity class NP, and it has recently been shown that inclusion logic captures P on ordered models. These results demonstrate that team semantics offers interesting new possibilities for descriptive complexity theory. In order to properly understand the connection between team semantics and descriptive complexity, we introduce an extension D* of dependence logic that can define exactly all recursively enumerable classes of finite models. Thus D* provides an approach to computation alternative to Turing machines. The essential novel feature in D* is an operator that can extend the domain of the considered model by a finite number of fresh elements. Due to the close relationship between generalized quantifiers and oracles, we also investigate generalized quantifiers in team semantics. We show that monotone quantifiers of type (1) can be canonically eliminated from quantifier extensions of first-order logic by introducing corresponding generalized dependence atoms

Introduction and Comparison of Dynamic Complexity Classes

This thesis gives some background and an introduction on dynamic complexity theory, a subfield of descriptive complexity theory in which queries on databases are maintained dynamically upon insertions and deletions to the database. The basic definitions of the dynamic complexity framework are given along with examples of queries maintainable with dynamic queries and a comparison of different dynamic complexity classes

The computational complexity of boundedly rational choice behavior

This research examines the computational complexity of two boundedly rational choice models that use multiple rationales to explain observed choice behavior. First, we show that the notion of rationalizability by K rationales as introduced by Kalai, Rubinstein, and Spiegler (2002) is NP-complete for K greater or equal to two. Second, we show that the question of sequential rationalizability by K rationales, introduced by Manzini and Mariotti (2007), is NP-complete for K greater or equal to three if choices are single valued, and that it is NP-complete for K greater or equal to one if we allow for multi-valued choice correspondences. Motivated by these results, we present two binary integer feasibility programs that characterize the two boundedly rational choice models and we compute their power. Finally, by using results from descriptive complexity theory, we explain why it has been so difficult to obtain `nice' characterizations for these models.

Capturing Polynomial Time using Modular Decomposition

The question of whether there is a logic that captures polynomial time is one of the main open problems in descriptive complexity theory and database theory. In 2010 Grohe showed that fixed point logic with counting captures polynomial time on all classes of graphs with excluded minors. We now consider classes of graphs with excluded induced subgraphs. For such graph classes, an effective graph decomposition, called modular decomposition, was introduced by Gallai in 1976. The graphs that are non-decomposable with respect to modular decomposition are called prime. We present a tool, the Modular Decomposition Theorem, that reduces (definable) canonization of a graph class C to (definable) canonization of the class of prime graphs of C that are colored with binary relations on a linearly ordered set. By an application of the Modular Decomposition Theorem, we show that fixed point logic with counting captures polynomial time on the class of permutation graphs. Within the proof of the Modular Decomposition Theorem, we show that the modular decomposition of a graph is definable in symmetric transitive closure logic with counting. We obtain that the modular decomposition tree is computable in logarithmic space. It follows that cograph recognition and cograph canonization is computable in logarithmic space.Comment: 38 pages, 10 Figures. A preliminary version of this article appeared in the Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS '17

Fixed-parameter tractability, definability, and model checking

In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For example, we establish the equivalence between the model-checking for existential first-order logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that model-checking for first-order formulas is fixed-parameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for every t >= 0 we prove an equivalence between model-checking for first-order formulas with t quantifier alternations and the parameterized halting problem for alternating Turing machines with t alternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows' W-hierarchy. On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixed-parameter tractable problems in terms of slicewise definability in finite variable least fixed-point logic, which is reminiscent of the Immerman-Vardi Theorem characterizing the class PTIME in terms of definability in least fixed-point logic.Comment: To appear in SIAM Journal on Computin