From 824e880b82073c2ddd3de580b0b6f6b3bff7d9d9 Mon Sep 17 00:00:00 2001 From: Thomas Walker Lynch Date: Sun, 12 Jan 2025 10:43:28 +0000 Subject: [PATCH] close to getting Example_SRTM_Diagonal --- .../diagonal_traversal.txt" | 53 ++++++ .../CountingNumber$Topo_Infinite.class | Bin 0 -> 1474 bytes .../CountingNumber$Topo_Null.class | Bin 0 -> 1452 bytes .../CountingNumber$Topo_Rightmost.class | Bin 0 -> 1468 bytes .../CountingNumber$Topo_Segment.class | Bin 0 -> 1787 bytes .../CountingNumber/CountingNumber.class | Bin 0 -> 1976 bytes .../CountingNumber/Example_CountingNumber_0 | 2 + .../Example_CountingNumber_0.class | Bin 0 -> 2157 bytes developer/example/CountingNumber/temp.java | 44 ----- developer/example/IndexTree/Example_4x4 | 2 - developer/example/IndexTree/Example_4x4.class | Bin 1817 -> 0 bytes .../example/IndexTree/Example_SRTM_Diagonal | 2 - .../IndexTree/Example_SRTM_Diagonal.class | Bin 1799 -> 0 bytes .../IndexTree/Example_SRTM_Diagonal.java | 51 +++--- developer/example/IndexTree/Graph.class | Bin 808 -> 0 bytes developer/example/IndexTree/Label.class | Bin 2227 -> 0 bytes developer/example/IndexTree/Label.java | 29 +++- developer/example/IndexTree/Node.class | Bin 856 -> 0 bytes .../example/IndexTree/SRTM_Child$1.class | Bin 1317 -> 0 bytes .../example/IndexTree/SRTM_Child$2.class | Bin 1256 -> 0 bytes .../example/IndexTree/SRTM_Child$3.class | Bin 1339 -> 0 bytes developer/example/IndexTree/SRTM_Child.class | Bin 1549 -> 0 bytes developer/example/IndexTree/SRTM_Child.java | 23 ++- .../example/IndexTree/SRTM_Diagonal.java | 159 +++++++++--------- developer/example/SRTM/Example_SRTMI_Array | 2 + .../example/SRTM/Example_SRTMI_Array.class | Bin 0 -> 2046 bytes developer/example/SRTM/Example_SRTM_List | 2 + .../example/SRTM/Example_SRTM_List.class | Bin 0 -> 2105 bytes developer/example/SRTM/Example_SRTM_List.java | 2 +- .../javac\360\237\226\211/Ariadne_Label.java" | 4 +- .../Ariadne_Label_String.java" | 29 ++-- .../Ariadne_SRTM_List.java" | 49 ++++-- "developer/tool\360\237\226\211/make" | 2 +- "developer/tool\360\237\226\211/make_example" | 2 +- 34 files changed, 258 insertions(+), 199 deletions(-) create mode 100644 "developer/document\360\237\226\211/diagonal_traversal.txt" create mode 100644 developer/example/CountingNumber/CountingNumber$Topo_Infinite.class create mode 100644 developer/example/CountingNumber/CountingNumber$Topo_Null.class create mode 100644 developer/example/CountingNumber/CountingNumber$Topo_Rightmost.class create mode 100644 developer/example/CountingNumber/CountingNumber$Topo_Segment.class create mode 100644 developer/example/CountingNumber/CountingNumber.class create mode 100755 developer/example/CountingNumber/Example_CountingNumber_0 create mode 100644 developer/example/CountingNumber/Example_CountingNumber_0.class delete mode 100644 developer/example/CountingNumber/temp.java delete mode 100755 developer/example/IndexTree/Example_4x4 delete mode 100644 developer/example/IndexTree/Example_4x4.class delete mode 100755 developer/example/IndexTree/Example_SRTM_Diagonal delete mode 100644 developer/example/IndexTree/Example_SRTM_Diagonal.class delete mode 100644 developer/example/IndexTree/Graph.class delete mode 100644 developer/example/IndexTree/Label.class delete mode 100644 developer/example/IndexTree/Node.class delete mode 100644 developer/example/IndexTree/SRTM_Child$1.class delete mode 100644 developer/example/IndexTree/SRTM_Child$2.class delete mode 100644 developer/example/IndexTree/SRTM_Child$3.class delete mode 100644 developer/example/IndexTree/SRTM_Child.class create mode 100755 developer/example/SRTM/Example_SRTMI_Array create mode 100644 developer/example/SRTM/Example_SRTMI_Array.class create mode 100755 developer/example/SRTM/Example_SRTM_List create mode 100644 developer/example/SRTM/Example_SRTM_List.class diff --git "a/developer/document\360\237\226\211/diagonal_traversal.txt" "b/developer/document\360\237\226\211/diagonal_traversal.txt" new file mode 100644 index 0000000..b10710f --- /dev/null +++ "b/developer/document\360\237\226\211/diagonal_traversal.txt" @@ -0,0 +1,53 @@ +Does there exist a node in the tree, such that if that node were marked before the diagonal generating algorithm is run, that that node will never be reached no matter how long the diagonal generating algorithm runs? Or perhaps you can't tell. +ChatGPT said: +ChatGPT +This is an interesting and profound question that touches on the completeness of the diagonal traversal algorithm in enumerating all nodes in an infinite tree. Let’s analyze it step by step. + +Key Observations About the Algorithm +Breadth and Depth Expansion: + +The diagonal algorithm explicitly alternates between "expanding down" to include children of nodes on the current diagonal and "expanding across" to include siblings of nodes already encountered. +Infinite Iterations: + +The algorithm iteratively generates new diagonals, and given infinite time, it will traverse every node that can be reached by a finite number of inc_down and inc_across operations. +Traversal Guarantee: + +The algorithm systematically enumerates every node along both depth and breadth. No node can be skipped permanently unless: +It cannot be reached via any sequence of inc_down and inc_across operations starting from the root. +The marking somehow prevents traversal. +Impact of Marking a Node +If a node is marked before the algorithm starts: + +Does Marking Influence the Algorithm? + +In the current implementation, the diagonal algorithm operates solely on labels and their incremental modifications (inc_down and inc_across). The marking state of a node is not factored into the traversal logic. Therefore: +Marking a node has no direct effect on whether the algorithm traverses it. +The algorithm will still reach every node, regardless of whether it is marked. +Can a Marked Node Be Skipped? + +No, the diagonal algorithm’s design ensures that every node in the infinite tree will eventually be reached, even if it is marked, because: +Marking is a separate property and does not alter the traversal mechanics. +The algorithm is exhaustive in both depth (inc_down) and breadth (inc_across). +Does Any Node Remain Unreachable? +To ensure there is no unreachable node: + +Structural Guarantees: + +The index tree is connected, meaning every node is reachable from the root via a finite sequence of operations. +Algorithm Completeness: + +The diagonal algorithm guarantees eventual traversal of every node by alternating between expanding depth and breadth. +No Skipped Nodes: + +Since the algorithm generates all possible sequences of inc_down and inc_across, there cannot exist a node in the tree that remains permanently unreachable. +Conclusion +No, there does not exist a node in the infinite index tree that, if marked beforehand, would never be reached by the diagonal generating algorithm. + +The diagonal algorithm traverses every node eventually, regardless of marking. +Marking only adds an attribute to nodes but does not interfere with the traversal mechanism. +Let me know if you'd like a deeper mathematical formalism for this or additional edge cases analyzed! 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