Model checking is a fully-automatic formal verification method that has been extremely successful in validating and verifying safety-critical systems in the past three decades. In the past fifteen years, there has been a lot of work in extending many model checking algorithms over finite-state systems to finitely representable infinitestate systems. Unlike in the case of finite systems, decidability can easily become a problem in the case of infinite-state model checking. In this thesis, we present generic and specific techniques that can be used to derive decidability with near-optimal computational complexity for various model checking problems over infinite-state systems. Generic techniques and specific techniques primarily differ in the way in which a decidability result is derived. Generic techniques is a “top-down” approach wherein we start with a Turing-powerful formalismfor infinitestate systems (in the sense of being able to generate the computation graphs of Turing machines up to isomorphisms), and then impose semantic restrictions whereby the desired model checking problem becomes decidable. In other words, to show that a subclass of the infinite-state systems that is generated by this formalism is decidable with respect to the model checking problem under consideration, we will simply have to prove that this subclass satisfies the semantic restriction. On the other hand, specific techniques is a “bottom-up” approach in the sense that we restrict to a non-Turing powerful formalism of infinite-state systems at the outset. The main benefit of generic techniques is that they can be used as algorithmic metatheorems, i.e., they can give unified proofs of decidability of various model checking problems over infinite-state systems. Specific techniques are more flexible in the sense they can be used to derive decidability or optimal complexity when generic techniques fail. In the first part of the thesis, we adopt word/tree automatic transition systems as a generic formalism of infinite-state systems. Such formalisms can be used to generate many interesting classes of infinite-state systems that have been considered in the literature, e.g., the computation graphs of counter systems, Turing machines, pushdown systems, prefix-recognizable systems, regular ground-tree rewrite systems, PAprocesses, order-2 collapsible pushdown systems. Although the generality of these formalisms make most interesting model checking problems (even safety) undecidable, they are known to have nice closure and algorithmic properties. We use these nice properties to obtain several algorithmic metatheorems over word/tree automatic systems, e.g., for deriving decidability of various model checking problems including recurrent reachability, and Linear Temporal Logic (LTL) with complex fairness constraints. These algorithmic metatheorems can be used to uniformly prove decidability with optimal (or near-optimal) complexity of various model checking problems over many classes of infinite-state systems that have been considered in the literature. In fact, many of these decidability/complexity results were not previously known in the literature. In the second part of the thesis, we study various model checking problems over subclasses of counter systems that were already known to be decidable. In particular, we consider reversal-bounded counter systems (and their extensions with discrete clocks), one-counter processes, and networks of one-counter processes. We shall derive optimal complexity of various model checking problems including: model checking LTL, EF-logic, and first-order logic with reachability relations (and restrictions thereof). In most cases, we obtain a single/double exponential reduction in the previously known upper bounds on the complexity of the problems.

In this thesis we study sparse classes of graphs and their properties usable for design of algorithms and data structures. Our specific focus is on the con- cepts of bounded expansion and tree-depth, developed in recent years mainly by J. Nešetřil and P. Ossona de Mendez. We first give a brief introduction to the theory as whole and survey tools and results from related areas of parametrised complexity and algorithmic model theory. The main part of the thesis, application of the theory, presents two new dynamic data structures. The first is for keeping a tree-depth decomposition of a graph, the second counts appearances of fixed subgraphs in a given graph. The time and space complexity of operations of both structures is guaranteed to be low when used for sparse graphs. 1