Size-Change Termination, Monotonicity Constraints and Ranking Functions

Size-Change Termination (SCT) is a method of proving program termination based on the impossibility of infinite descent. To this end we may use a program abstraction in which transitions are described by monotonicity constraints over (abstract) variables. When only constraints of the form x>y' and x>=y' are allowed, we have size-change graphs. Both theory and practice are now more evolved in this restricted framework then in the general framework of monotonicity constraints. This paper shows that it is possible to extend and adapt some theory from the domain of size-change graphs to the general case, thus complementing previous work on monotonicity constraints. In particular, we present precise decision procedures for termination; and we provide a procedure to construct explicit global ranking functions from monotonicity constraints in singly-exponential time, which is better than what has been published so far even for size-change graphs.Comment: revised version of September 2

Complexity Hierarchies Beyond Elementary

We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a non-elementary complexity, which occur naturally in logic, combinatorics, formal languages, verification, etc., with complexities ranging from simple towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep updating the catalogue of problems from Section 6 in future revision

An order-theoretic characterization of the Howard-Bachmann-hierarchy

In this article we provide an intrinsic characterization of the famous Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees with respect to a homeomorphic embeddability relation. We use our calculations to draw some conclusions about some corresponding subsystems of second order arithmetic. All these subsystems deal with versions of light-face Π₁¹-comprehension

Connecting the two worlds: well-partial-orders and ordinal notation systems

Kruskal claims in his now-classical 1972 paper [47] that well-partial-orders are among the most frequently rediscovered mathematical objects. Well partial-orders have applications in many fields outside the theory of orders: computer science, proof theory, reverse mathematics, algebra, combinatorics, etc. The maximal order type of a well-partial-order characterizes that order’s strength. Moreover, in many natural cases, a well-partial-order’s maximal order type can be represented by an ordinal notation system. However, there are a number of natural well-partial-orders whose maximal order types and corresponding ordinal notation systems remain unknown. Prominent examples are Friedman’s well-partial-orders of trees with the gap-embeddability relation [76]. The main goal of this dissertation is to investigate a conjecture of Weiermann [86], thereby addressing the problem of the unknown maximal order types and corresponding ordinal notation systems for Friedman’s well-partial orders [76]. Weiermann’s conjecture concerns a class of structures, a typical member of which is denoted by T (W ), each are ordered by a certain gapembeddability relation. The conjecture indicates a possible approach towards determining the maximal order types of the structures T (W ). Specifically, Weiermann conjectures that the collapsing functions #i correspond to maximal linear extensions of these well-partial-orders T (W ), hence also that these collapsing functions correspond to maximal linear extensions of Friedman’s famous well-partial-orders