A Linear-Size Block-Partition Fibonacci Encoding for Gödel Numbering
This provides a linear-size encoding for Gödel numbering, addressing efficiency in theoretical computer science and formal logic, though it is incremental as it builds on known concepts like Fibonacci encoding and Zeckendorf's theorem.
The paper tackles the problem of encoding finite strings as natural numbers efficiently, achieving a worst-case digit count growth of Θ(m) for strings of length m, which matches the information-theoretic lower bound up to a constant factor, while showing that an existing method (Rosko's binary carryless pairing) has exponential Θ(2^m) growth.
We construct an encoding of finite strings over a fixed finite alphabet as natural numbers, based on a block partition of the Fibonacci sequence. Each position in the string selects one Fibonacci number from a dedicated block, with unused indices between blocks guaranteeing non-adjacency. The encoded number is the sum of the selected Fibonacci numbers, and Zeckendorf's theorem guarantees that this sum uniquely determines the selection. The encoding is injective, the string length is recoverable from the code, and the worst-case digit count of the encoded number grows as $Î(m)$ for strings of length $m$, matching the information-theoretic lower bound up to a constant factor. We also prove that the natural right-nested use of Rosko's (2025) binary carryless pairing for sequence encoding has worst-case $Î(2^m)$ digit growth, an exponential blowup that the block-partition construction avoids entirely.