Phase transitions in axiomatic thought

Ein Aspekt dieser dissertation ist, einige bekannte Ordinalzahlbezeichnungssysteme für Peano Arithmetik zu untersuchen. Es wird gezeigt, dass Phasenübergänge in Bezug auf Beweisbarkeit und Unbeweisbarkeit beobachtet werden. Schließlich kommt man zu Vergleichen von Ordinalzahlbezeichnungssystemen. Diese Dissertation zeigt auch, wie man generell solche Phasenübergänge in Systemen, die stark genug im Sinne von Gödel sind, feststellen kann. Dabei spielt die Friedmansche Miniaturisierung die entscheidende Rolle. Ein anderer Punkt der Dissertation ist das Kanamori-McAloon-Prinzip mit Parameterfunktionen. Sie sind Varianten vom endlichen Ramseysatz und äquivalent zum Paris-Harrington-Prinzip. Es wird gezeigt, dass Phasenübergänge bezüglich der Beweisbarkeit des Kanamori-McAloon-Prinzips vorkommen, wenn die Parameterfunktionenen variieren. An aspect of the thesis is to investigate well-known ordinal notation systems for Peano arithmetic. It will be shown that the so-called phase transition phenomenon can be observed, i.e., there are thresholds between provability and unprovability. This investigation leads to a comparison of the ordinal notation systems. The thesis gives also a guide how one can generally establish such phase transitions in every logic system which is strong enough in the sense of Gödel. We shall see that Friedman style miniaturizations play the central role. Another point of the thesis is the parametrized version of the Kanamori-McAloon principle. This variants of the finite Ramsey theorem is equivalent to the Paris-Harrington principle. It will be shown that phase transitions occur with respect to the provability of the Kanamori-McAloon principle as the parameter function varies

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

Unprovability and phase transitions in Ramsey theory

The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions. Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas. Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions. Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem. Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability. Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families. Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details

Current research on G\"odel's incompleteness theorems

We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first incompleteness theorem, and the limit of the applicability of G\"{o}del's second incompleteness theorem.Comment: 54 pages, final accepted version, to appear in The Bulletin of Symbolic Logi

Phase transitions of iterated Higman-style well-partial-orderings

We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev's symmetric gap condition. For every d-times iterated wpo (SEQ(d), (sic)(d)) in question, d > 1, we fix a natural extension of Peano Arithmetic, T superset of PA, that proves the corresponding second-order sentence WPO (SEQ(d), (sic)(d)). Having this we consider the following parametrized first-order slow well-partial-ordering sentence SWP (SEQ(d), (sic)(d), r): (for all K > 0) (there exists M > 0) (for all x(0), . . . , x(M) is an element of SEQ(d)) ((for all i (there exists i < j <= M) (x(i) (sic)(d) x(j))) for a natural additive SEQ(d)-norm vertical bar.vertical bar and r ranging over EFA-provably computable positive reals, where EFA is an abbreviation for vertical bar Delta(0) + exp. We show that the following basic phase transition clauses hold with respect to T = Pi(0)(1)CA(<phi(d-1) (0)) and the threshold point 1. 1. If r < 1 then SWP (SEQ(d), (sic)(d), r) is provable in T. 2. If r > 1 then SWP (SEQ(d), (sic)(d), r) is not provable in T. Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA(0) and Delta(1)(1)CA, if d = 2 and d = 3, respectively. In the limit case d -> infinity we replace EFA-provably computable reals r by EFA-provably computable functions f : N -> R+ and prove analogous theorems. (In the sequel we denote by R+ the set of EFA-provably computable positive reals). In the basic case T = PA we strengthen the basic phase transition result by adding the following static threshold clause 3. SWP (SEQ(2), (sic)(2), 1) is still provable in T = PA (actually in EFA). Furthermore we prove the following dynamic threshold clauses which, loosely speaking are obtained by replacing the static threshold t by slowly growing functions 1(alpha) given by 1(alpha) (i) := 1 + 1/H-alpha(-1)(i), H-alpha being the familiar fast growing Hardy function and H-alpha(-1) (i) := min {j vertical bar H-alpha (j) >= i} the corresponding slowly growing inversion. 4. If alpha < epsilon(0), then SWP (SEQ(2), (sic)(2), 1(alpha)) is provable in T = PA. 5. SWP (SEQ(2), (sic)(2), 1(epsilon 0)) is not provable in T = PA. We conjecture that this pattern is characteristic for all T superset of PA under consideration and their proof-theoretical ordinals o (T), instead of epsilon(0)

Algorithmic Analysis of Infinite-State Systems

Many important software systems, including communication protocols and concurrent and distributed algorithms generate infinite state-spaces. Model-checking which is the most prominent algorithmic technique for the verification of concurrent systems is restricted to the analysis of finite-state models. Algorithmic analysis of infinite-state models is complicated--most interesting properties are undecidable for sufficiently expressive classes of infinite-state models. In this thesis, we focus on the development of algorithmic analysis techniques for two important classes of infinite-state models: FIFO Systems and Parameterized Systems. FIFO systems consisting of a set of finite-state machines that communicate via unbounded, perfect, FIFO channels arise naturally in the analysis of distributed protocols. We study the problem of computing the set of reachable states of a FIFO system composed of piecewise components. This problem is closely related to calculating the set of all possible channel contents, i.e. the limit language. We present new algorithms for calculating the limit language of a system with a single communication channel and important subclasses of multi-channel systems. We also discuss the complexity of these algorithms. Furthermore, we present a procedure that translates a piecewise FIFO system to an abridged structure, representing an expressive abstraction of the system. We show that we can analyze the infinite computations of the more concrete model by analyzing the computations of the finite, abridged model. Parameterized systems are a common model of computation for concurrent systems consisting of an arbitrary number of homogenous processes. We study the reachability problem in parameterized systems of infinite-state processes. We describe a framework that combines Abstract Interpretation with a backward-reachability algorithm. Our key idea is to create an abstract domain in which each element (a) represents the lower bound on the number of processes at a control location and (b) employs a numeric abstract domain to capture arithmetic relations among variables of the processes. We also provide an extrapolation operator for the domain to guarantee sound termination of the backward-reachability algorithm

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.