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Theoretical and practical aspects of the verification of infinite-state systems have attracted a lot of interest in the verification community throughout the last 30 years. One goal is to identify classes of infinite-state systems that admit decidable decision problems on the one hand, and which are sufficiently general to model systems, programs or protocols with unbounded data or recursion depth on the other hand. The first part of this thesis is concerned with the computational complexity of verifying counter automata, which are a fundamental and widely studied class of infinite-state systems. Counter automata consist of a finite-state controller manipulating a finite number of counters ranging over the naturals. A classic result by Minsky states that reachability in counter automata is undecidable already for two counters. One restriction that makes reachability decidable and that this thesis primarily focuses on is the restriction to one counter. A main result of this thesis is to show that reachability in one-counter automata with counter updates encoded in binary is NP-complete, which solves a problem left open by Rosier and Yen in 1986. We also consider parametric one-counter automata, in which counter updates can be parameters ranging over the naturals. Reachability for this class asks whether there are values of the parameters such that a target configuration can be reached from an initial configuration. This problem is also shown to be NP-complete. Subsequently, we establish decidability and complexity results of model checking problems for one- counter automata with and without parameters for specifications written in EF, CTL and LTL. The second part of this thesis is about the verification of programs with pointers and linked lists in the framework of separation logic. We consider the fragment of separation logic introduced by Berdine, Calcagno and O'Hearn in 2004 and the problem of deciding entailment between formulae of this fragment. We improve the known coNP upper bound and show that this problem can actually be solved in polynomial time. This result is based on a novel approach in which we represent separation logic formulae as graphs and decide entailment between them by checking for the existence of a graph homomorphism. We complement this result by considering various natural extensions of this fragment which make entailment coNP-hard.

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Higher-Order Pushdown Automata (HOPDA) are abstract machines equipped with a nested stacks of stacks ... of stacks of stacks. Collapsible pushdown automata (CPDA) enhance these stacks with the addition of ‘links’ emanating from atomic elements to the higher-order stacks below. For trees CPDA are equi-expressive with recursion schemes, which can be viewed as simply-typed λY terms. With vanilla HOPDA, one can only capture schemes satisfying a syntactic constraint called safety. This dissertation begins with some results concerning the significance of links in terms of recursion schemes. We introduce a fine-grained notion of safety that allows us to correlate the need for links of a given order with the imposition of safety on variables of a corresponding order. This generalises some joint work with William Blum that shows we can dispense with homogeneous types when characterising safety. We complement this result with a demonstration that homogeneity by itself does not constrain the expressivity of otherwise unrestricted recursion schemes. The main results of the dissertation, however, concern the configuration graphs of CPDA. Whilst the configuration graphs of HOPDA are well understood and have decidable MSO theories (they coincide with the Caucal hierarchy), relatively little is known about the transition graphs of CPDA. It is known that they already have undecidable MSO theories at order-2, but Kartzow recently showed that 2-CPDA graphs are tree automatic and hence first-order logic is decidable at order-2. We provide a characterisation of the decidability of first-order logic on CPDA graphs in terms of quantifier-alternation and the order of CPDA stacks and the links contained within. Whilst this characterisation is fairly comprehensive, we do leave open the question of decidability for some sub-classes of CPDA. It turns out that decidability can be highly sensitive to the order of links in a stack relative to the order of the stack itself. In addition to some strong and surprising undecidability results, we also develop further Kartzow’s work on 2-CPDA. We introduce prefix-rewrite systems for nested-words that characterise the configuration graphs of both 2-CPDA and 2-HOPDA, capturing the power of collapse precisely in terms outside of the language of CPDA. It also formalises and demonstrates the inherent asymmetry of the collapse operation. This generalises the rational prefix-rewriting systems characterising conventional pushdown graphs and we believe establishes the 2-CPDA graphs as an interesting and robust class.

In the 1980s and 1990s, Joyal and Street developed a graphical notation for various flavours of monoidal category using graphs drawn in the plane, commonly known as string diagrams. In particular, their work comprised a rigorous topological foundation of the notation. In 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notions they called ⊸-schedules, ⊗-schedules and heaps. Schedules described interleavings of plays in games formed using ⊸ and ⊗, and heaps provided pointers used for backtracking. Their definitions were combinatorial in nature, but researchers often draw certain pictures when working in practice. In this thesis, we extend the framework of Joyal and Street to give a formal account of the graphical methods already informally employed by researchers in game semantics. We give a geometric formulation of ⊸-schedules and ⊗-schedules, and prove that the games they describe are isomorphic to those described in Harmer et al.’s terms, and also those given by a more general graphical representation of interleaving across games of multiple components. We further illustrate the value of the geometric methods by demonstrating that several proofs of key properties (such as that the composition of ⊸-schedules is associative) can be made straightforward, reflecting the geometry of the plane, and overstepping some of the cumbersome combinatorial detail of proofs in Harmer et al.’s terms. We further extend the framework of formal plane diagrams to account for the heaps and pointer structures used in the backtracking functors for O and P.

The specification of enterprise information systems using formal specification languages enables the formal verification of these systems. Reasoning about the properties of a formal specification is a tedious task that can be facilitated much through the use of an automated reasoner. However, set theory is a corner stone of many formal specification languages and poses demanding challenges to automated reasoners. To this end a number of heuristics has been developed to aid the Otter theorem prover in finding short proofs for set-theoretic problems. This dissertation investigates the applicability of these heuristics to next generation theorem provers. / Computing / M.Sc. (Computer Science)