@phdthesis{Travers2007,
  author    = {Travers, Stephen},
  title     = {Structural Properties of NP-Hard Sets and Uniform Characterisations of Complexity Classes},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-27124},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2007},
  abstract  = {This thesis is devoted to the study of computational complexity theory, a branch of theoretical computer science. Computational complexity theory investigates the inherent difficulty in designing efficient algorithms for computational problems. By doing so, it analyses the scalability of computational problems and algorithms and places practical limits on what computers can actually accomplish. Computational problems are categorised into complexity classes. Among the most important complexity classes are the class NP and the subclass of NP-complete problems, which comprises many important optimisation problems in the field of operations research. Moreover, with the P-NP-problem, the class NP represents the most important unsolved question in computer science. The first part of this thesis is devoted to the study of NP-complete-, and more generally, NP-hard problems. It aims at improving our understanding of this important complexity class by systematically studying how altering NP-hard sets affects their NP-hardness. This research is related to longstanding open questions concerning the complexity of unions of disjoint NP-complete sets, and the existence of sparse NP-hard sets. The second part of the thesis is also dedicated to complexity classes but takes a different perspective: In a sense, after investigating the interior of complexity classes in the first part, the focus shifts to the description of complexity classes and thereby to the exterior in the second part. It deals with the description of complexity classes through leaf languages, a uniform framework which allows us to characterise a great variety of important complexity classes. The known concepts are complemented by a new leaf-language model. To a certain extent, this new approach combines the advantages of the known models. The presented results give evidence that the connection between the theory of formal languages and computational complexity theory might be closer than formerly known.},
  subject      = {Berechnungskomplexit{\"a}t},
  language  = {en}
}
@phdthesis{Dose2021,
  author    = {Dose, Titus},
  title     = {Balance Problems for Integer Circuits and Separations of Relativized Conjectures on Incompleteness in Promise Classes},
  doi       = {10.25972/OPUS-22220},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-222209},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2021},
  abstract  = {This thesis is divided into two parts. In the first part we contribute to a working program initiated by Pudl{\´a}k (2017) who lists several major complexity theoretic conjectures relevant to proof complexity and asks for oracles that separate pairs of corresponding relativized conjectures. Among these conjectures are: - \(\mathsf{CON}\) and \(\mathsf{SAT}\): coNP (resp., NP) does not contain complete sets that have P-optimal proof systems. - \(\mathsf{CON}^{\mathsf{N}}\): coNP does not contain complete sets that have optimal proof systems. - \(\mathsf{TFNP}\): there do not exist complete total polynomial search problems (also known as total NP search problems). - \(\mathsf{DisjNP}\) and \(\mathsf{DisjCoNP}\): There do not exist complete disjoint NP pairs (coNP pairs). - \(\mathsf{UP}\): UP does not contain complete problems. - \(\mathsf{NP}\cap\mathsf{coNP}\): \(\mathrm{NP}\cap\mathrm{coNP}\) does not contain complete problems. - \(\mathrm{P}\ne\mathrm{NP}\). We construct several of the oracles that Pudl{\´a}k asks for. In the second part we investigate the computational complexity of balance problems for \(\{-,\cdot\}\)-circuits computing finite sets of natural numbers (note that \(-\) denotes the set difference). These problems naturally build on problems for integer expressions and integer circuits studied by Stockmeyer and Meyer (1973), McKenzie and Wagner (2007), and Glaßer et al. (2010). Our work shows that the balance problem for \(\{-,\cdot\}\)-circuits is undecidable which is the first natural problem for integer circuits or related constraint satisfaction problems that admits only one arithmetic operation and is proven to be undecidable. Starting from this result we precisely characterize the complexity of balance problems for proper subsets of \(\{-,\cdot\}\). These problems turn out to be complete for one of the classes L, NL, and NP.},
  subject      = {NP-vollst{\"a}ndiges Problem},
  language  = {en}
}
@unpublished{Dandekar2008,
  author    = {Dandekar, Thomas},
  title     = {Why are nature´s constants so fine-tuned? The case for an escalating complex universe},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-34488},
  year      = {2008},
  abstract  = {Why is our universe so fine-tuned? In this preprint we discuss that this is not a strange accident but that fine-tuned universes can be considered to be exceedingly large if one counts the number of observable different states (i.e. one aspect of the more general preprint http://www.opus-bayern.de/uni-wuerzburg/volltexte/2009/3353/). Looking at parameter variation for the same set of physical laws simple and complex processes (including life) and worlds in a multiverse are compared in simple examples. Next the anthropocentric principle is extended as many conditions which are generally interpreted anthropocentric only ensure a large space of different system states. In particular, the observed over-tuning beyond the level for our existence is explainable by these system considerations. More formally, the state space for different systems becomes measurable and comparable looking at their output behaviour. We show that highly interacting processes are more complex then Chaitin complexity, the latter denotes processes not compressible by shorter descriptions (Kolomogorov complexity). The complexity considerations help to better study and compare different processes (programs, living cells, environments and worlds) including dynamic behaviour and can be used for model selection in theoretical physics. Moreover, the large size (in terms of different states) of a world allowing complex processes including life can in a model calculation be determined applying discrete histories from quantum spin-loop theory. Nevertheless there remains a lot to be done - hopefully the preprint stimulates further efforts in this area.},
  subject      = {Natur},
  language  = {en}
}