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Descriptive Complexity for Counting Problems

Yaghoubi, Omid | 2023

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 56247 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Ebrahimi Boroojeni, Javad
  7. Abstract:
  8. Descriptive complexity refers to the difficulty of expressing a problem in a formal language, allowing for a complete and precise description of the problem, including output specifications for function problems. Despite the success of the descriptive approach in computational complexity, this approach has been predominantly used for decision problems. In contrast to decision problems, a counting problem can be considered as a function with the range of natural numbers. This thesis focuses on the examination of counting problems from a descriptive perspective, exploring the connection between their descriptive and computational versions. Toward this goal, weighted logic is utilized to describe this type of problem, where each formula from this logic is interpreted with an element from a given semiring, diverging from the semantics of classical formal languages. The two-layered grammar in this descriptive language enables us to leverage the classical results in descriptive complexity for effectively describing counting problems. In this study, it is shown that by restricting this language and modifying its underlying logic, important classes of counting problems can be captured within computational complexity
  9. Keywords:
  10. Computational Complexity ; Descriptive Complenity ; Counting Problems ; Finite Model Theory

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