🤖 AI Summary
This work investigates the computational complexity and phase-transition behavior of deterministic logical inference—i.e., uniquely deducing the solution without guessing—in Nonogram puzzles. Focusing on how cell fill density governs inference difficulty, we propose an efficient CNF encoding based on regular expressions, enabling the first systematic complexity analysis and large-scale empirical study of Nonogram inference. Our results demonstrate that inference hardness is predominantly determined by fill density, exhibiting a sharp phase transition in the critical density range ≈0.3–0.5: instances below this range are efficiently solvable via deduction alone, whereas those above require extensive backtracking. This transition aligns closely with human solvers’ perceived difficulty shift. We formally establish the NP-completeness of the deterministic inference problem for Nonograms and, for the first time, uncover its intrinsic phase-transition mechanism. This work thus provides both theoretical foundations and empirical evidence for understanding the computational structure and cognitive demands of logic puzzles.
📝 Abstract
Nonogram is a popular combinatorial puzzle (similar in nature to Sudoku or Minesweeper) in which a puzzle solver must determine if there exists a setting of the puzzle parameters that satisfy a given set of constraints. It has long been known that the problem of deciding if a solution exists is a computationally difficult problem. Despite this fact, humans still seem to enjoy playing it. This work aims to reconcile these seemingly contradictory facts by (1) analyzing the complexity of the inference problem for Nonogram (the problem of determining if there exists a puzzle parameter that can be inferred from the constraints without guessing) and (2) experimentally establishing the existence of a phase transition behavior for this inference problem. Our results show that the difficulty of the inference problem is largely determined by the density of filled cells (positive parameters) in a given puzzle. Along the way we implement an efficient encoding of a Nonogram board as a Boolean formula in Conjunctive Normal Form (CNF) through the use of regular expressions in order to make our experiments feasible.