@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}
}