3-SAT → Variable-alphabet lettericity Complexity open

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Clone https://github.com/reduction-zoo/lettericity to investigate 3-SAT → Variable-alphabet lettericity. Read the fixed question and state. Use preparation notes, the test corpus, and verifier if present; otherwise complete Prepare before a reduction attempt. Follow the repository instructions and your AutoResearch pipeline.

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Why this question matters

This addresses recognition when the alphabet is an input parameter, beyond fixed-alphabet structural results.

Fixed-alphabet algorithms and supplied-partition formulations do not classify recognition when both the alphabet budget and the partition are part of the problem.

The reduction to construct

A source instance is an explicitly encoded Boolean formula with at most three literals per clause. Its outputs are satisfying Boolean assignments, or NO-SOLUTION when the formula is unsatisfiable.

Given a graph G and alphabet budget k, find a vertex order, a letter for every vertex, and a directed decoder on at most k letters such that each earlier/later vertex pair is adjacent exactly when its ordered letter pair is in the decoder.

Construct deterministic polynomial-time maps F and G: F sends every legal source instance to a legal target instance, and G(x,y) is a valid source output for every valid output y of F(x). Preserve the stated threshold, domain and promises. The requested complexity conclusion is NP-hardness for the stated target problem.

Both the order and the partition are free. A hard-looking encoding with a supplied partition need not prove the unrestricted result.

What a complete answer must include

Give explicit construction and recovery algorithms, a general proof for every legal input and every valid target output, and polynomial runtime and encoding-size bounds. Specify finite output encodings and handle NO-SOLUTION outputs when applicable. Tests compare recovered source outputs with independent source solutions.

References

Literature checked 2026-09-16. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.