From Patterns to Maze Structures: SMT-Based Path Synthesis and 2D/3D Construction

Jul 16, 2026

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A pipeline constructs maze structures from input patterns such as text or shapes by solving a central path-synthesis problem encoded in Satisfiability Modulo Theories. The SMT model imposes global constraints on adjacency, continuity, and pattern-constrained coverage, enabling fixed-bound instances to be solved via a single call. Synthesized paths become scaffolds for planar, self-avoiding routes or for layered traversals with prescribed over–under crossings, which guide both 2D maze completion and 3D realizations of woven mazes, extending prior Bridges work.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

Introduction

Problem scope and extensions

Example planar and woven outputs

Shengyi Wang

Three-step pipeline overview

Path Synthesis

Problem analysis

SMT encoding

Solver performance

arXiv :2607 .0978 1v 1 [ cs .AI] 8 Jul 2026 From Patterns to Maze Structures: SMT-Based Path Synthesis and 2D/3D Construction SHENGYI WANG, Shanghai Qi Zhi Institute, China We present a pipeline for constructing maze structures from input patterns such as text or shapes. The central path-synthesis problem is encoded in Satisfiability Modulo Theories as global constraints on adjacency, continuity, and pattern-constrained coverage, allowing each fixed-bound instance to be solved in one call. The resulting path is either a planar, self-avoiding route or a layered traversal with prescribed over–under crossings, and it serves as a scaffold for constructing planar mazes and three-dimensional realizations of woven mazes. This report extends the published Bridges 2026 conference paper with more representative SMT-LIB examples and a fuller account of how synthesized paths become concrete maze constructions in planar and three-dimensional form. 1 Introduction This report extends the Bridges 2026 conference paper [9], which introduced the SMT formulation for synthesizing maze solution paths from input patterns. The present version keeps that formulation but broadens the scope from path synthesis to maze construction. In particular, it gives more representative SMT-LIB fragments, describes how synthesized paths are completed into planar and woven maze structures, and develops the height information needed to realize layered crossings as three-dimensional maze geometry. We derive two distinct maze solution paths from the same input pattern: a planar path and a layered path with crossings. Each solution path then serves as a scaffold for maze construction. Figure 1 gives an example of the resulting planar and woven mazes, both with and without highlighted solution paths. (a) (b) (c) (d) Fig. 1. Planar and woven mazes derived from the same “∞” solution pattern: (a) Planar maze, (b) Planar maze with solution,(c) Woven maze, (d) Woven maze with solution. Author’s Contact Information: Shengyi Wang, Shanghai Qi Zhi Institute, Shanghai, China, [wangshengyi@sqz.ac.cn](wangshengyi@sqz.ac.cn). 2 Shengyi Wang The construction of the mazes considered in this paper can be viewed as a three-step process, as shown in Figure 2. First, a given text or image is converted into a set of grid positions. Second, a path is constructed on this grid, starting from a prescribed entry position and ending at a prescribed exit position, while traversing as many target grid positions as possible. Depending on the intended maze type, this path may be required to be planar or may admit layered crossings. Third, a maze is generated from the resulting path. Fig. 2. Three-step pipeline from an input pattern to a maze. Among these steps, the first is straightforward and can be handled using existing image processing tools, such as rasterization followed by binary thresholding to extract a target set of grid positions from the input pattern. The conference version focused primarily on the second step: finding a suitable path that satisfies global structural constraints while remaining faithful to the input pattern. This report retains that formulation and develops the third step in greater detail, including both planar maze completion and a three-dimensional realization of layered crossings. The third step builds on standard maze-generation ideas, for example those surveyed in [3] . From a graph-theoretic perspective, planar maze generation can be viewed as constructing a spanning tree of an underlying grid graph subject to the synthesized solution path. In the layered case, the crossing relation supplies additional topological information that must be turned into compatible height data before a three-dimensional maze geometry can be constructed. 2 Path Synthesis This section describes the synthesis of the solution path that serves as the combinatorial scaffold for the later maze construction. The path is modeled as a sequence of moves on g

FAQs

What is the main contribution of this report?

It extends the Bridges 2026 SMT formulation from synthesizing maze solution paths to also constructing maze structures, including planar completion and 3D realization of layered crossings with more representative SMT-LIB examples.

How does the pipeline convert input patterns into maze structures?

It rasterizes text or images into grid positions, synthesizes a path on the grid from a fixed entry to exit with coverage objectives (and crossing data when needed), then completes the resulting path into planar mazes and 3D woven-maze geometry.

How are planar and layered (woven) solutions generated from the same input?

The method derives two solution paths from the same pattern: a planar orthogonal self-avoiding route, and a layered route with crossings. Each path then serves as a scaffold for constructing the corresponding planar or woven maze, with layered crossings requiring height information for 3D geometry.

Jul 16, 2026

Upload by Eden