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Degrees of Unsolvability

Jahanjou, Hamid Reza | 2010

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 41456 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Ardeshir, Mohammad
  7. Abstract:
  8. One of the the most fundamental questions in computability (recursion) theory is the exact structure of D and in particular that of R. Over the years, various efforts have eventuated in Degree ?eory, an interesting subfield of computability theory, which is concerned with various properties of these algebraic structures which have turned out to be extremely complex. Beyond any doubt, the most useful tool for proving theorems related toRis the Priority Method which is first introduced to solve the Post’s problem. Over the years, increasingly more sophisticated versions of the method have been used to prove complex statements about R. Roughly one may classify the priority arguments into three camps: Finite Injury Priority Method (0' arguments), Infinite Injury Priority Method (0'' arguments) and the 0''' argument (a.k.a. the monster priority method!) In the 70’s, the Tree Method, was first introduced to analyze 0''' arguments, turned out to be a very useful framework for priority arguments in general and has been studied to this day. In this MSc thesis, we perform a detailed survey of the various priority arguments. In chapter 1, fundamental definitions and theorems of computability theory are presented. Next, in chapter 2, we focus on Degree ?eory; in particular, we introduce the jump operator and prove that R is an upper semi-la?ice. Finally, in chapter 3, we discuss priority arguments in detail; in particular, finite injury, infinite injury arguments and the tree method are analyzed
  9. Keywords:
  10. Computability ; Degree Theory ; Priority Method ; Mathematical Logic

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