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Exploring Goal Nodes in Recursive Functions: Bound Analysis

This document investigates the analysis of goal nodes in recursive functions and the patterns of their behavior based on various bounds. Each function, such as f(n), h(n), and g(n), reveals different pathways and shifts when manipulated with boundaries set at 4, 5, and 7. The goal is to understand how these configurations impact the function calculations and determine the minimum fringe necessary for reinitializing the bounds. By studying these patterns, we aim to shed light on the nuances of recursive functions in computational theory.

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Exploring Goal Nodes in Recursive Functions: Bound Analysis

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  1. root f(n) 12 g(n), h(n) 0,4 goal

  2. 12 18 7 10 12 12

  3. 4 bound=4 0,4 12 18 7 7 12 12

  4. 4 bound=4 0,4 4 1,3 12 18 7 7 12 12

  5. 4 bound=4 0,4 4 1,3 5 2,3 12 18 7 7 12 12

  6. 4 bound=4 0,4 4 1,3 5 2,3 12 18 7 7 12 12

  7. 4 bound=4 0,4 4 1,3 5 4 2,3 2,2 12 18 7 7 12 12

  8. 4 bound=4 0,4 4 1,3 5 4 2,3 2,2 12 8 6,2 18 7 7 12 12

  9. 4 bound=4 0,4 4 1,3 5 4 2,3 2,2 12 8 6,2 18 7 7 12 12

  10. 4 bound=4 0,4 4 8 1,3 5,3 5 4 2,3 2,2 12 8 6,2 18 7 7 12 12

  11. 4 bound=4 0,4 4 8 1,3 5,3 5 4 2,3 2,2 12 8 6,2 18 7 7 12 12

  12. 4 bound=4 0,4 4 8 1,3 5,3 5 4 2,3 2,2 12 8 6,2 18 7 take minimum fringe f(n) as next bound and start again… 7 12 12

  13. 4 bound=5 0,4 12 18 7 7 12 12

  14. 4 bound=5 0,4 4 1,3 12 18 7 7 12 12

  15. 4 bound=5 0,4 4 1,3 5 2,3 12 18 7 7 12 12

  16. 4 bound=5 0,4 4 1,3 5 2,3 12 7 5,2 18 7 7 12 12

  17. 4 bound=5 0,4 4 1,3 5 2,3 12 7 5,2 18 7 7 12 12

  18. 4 bound=5 0,4 4 1,3 5 4 2,3 2,2 12 7 5,2 18 7 7 12 12

  19. 4 bound=5 0,4 4 1,3 5 4 2,3 2,2 12 7 8 5,2 6,2 18 7 7 12 12

  20. 4 bound=5 0,4 4 1,3 5 4 2,3 2,2 12 7 8 5,2 6,2 18 7 7 12 12

  21. 4 bound=5 0,4 4 8 1,3 5,3 5 4 2,3 2,2 12 7 8 5,2 6,2 18 7 7 12 12

  22. 4 bound=5 0,4 4 8 1,3 5,3 5 4 2,3 2,2 12 7 8 5,2 6,2 18 7 take minimum fringe f(n) as next bound and start again… 7 12 12

  23. 4 bound=7 0,4 12 18 7 7 12 12

  24. 4 bound=7 0,4 4 1,3 12 18 7 7 12 12

  25. 4 bound=7 0,4 4 1,3 5 2,3 12 18 7 7 12 12

  26. 4 bound=7 0,4 4 1,3 5 2,3 12 7 5,2 18 7 7 12 12

  27. 4 bound=7 0,4 4 1,3 5 2,3 12 7 5,2 18 7 7,0 7 12 12

  28. 4 bound=7 0,4 4 1,3 5 2,3 12 7 5,2 18 7 7,0 7 12 12 we found a goal node

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