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Unification of Higher-Order Patterns for Dependent Types

Ketab Ghale Haji Ali, Milad | 2014

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 46109 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Ardeshir Bahrestaghi, Mohammad
  7. Abstract:
  8. Immensity and profundity of mathematical logic have provided an opportunity for its interconnection with other branches of science, one of which is computer science where theory of dependent types and lambda calculus lounges amongst the area of their overlap. The former has been engendered for the sake of deepening the connection between mathematical logic and programming. On the other side, lambda calculus which is a formal system in mathematical logic and computer science, developed as a tool for formal computations, could also be counted as a programming discipline. It is the objective of this thsis to study the two aforementioned fields with emphasis on unification of higher-order patterns which is held common in between these realms. As it is known, higher-order unification is not decidable. However, it has some fragments which admit practical algorithms used extensionally in logical frameworks. For exmple, a fragment of highe-order unification in λ→ and λΠ called pattern fragment has been proved to be decidable. Despite the decidability, the aformentioned fragment is too restrictive. In theorem provers like Twelf, this restriction sometimes forces the program to solve sub-problems which fall into pattern fragment and postpone those which do not, untill enough information is added to the constraints. Moreover, some systems implement λΠΣ calculus. Therefore the given algorithms do not cover numerous unification problems falling into Pattern fragment of λΠΣ calculus. This thesis is devoted to discussing an algorithm given for unification of higher-order patterns in λΠΣ calculus.
  9. Keywords:
  10. Lambda Calculus ; Unification ; Dependent Types ; Higher-Order Patterns

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