2-Head 2D Returning Finite Automata

2026-06-25

AI Summary

This study investigates the computational power of two-head finite automata in two-dimensional picture language recognition and their relationship with context-free matrix grammars (CFMGs) and returning pushdown automata (RPDAs). It introduces two novel models of two-head returning finite automata operating on rectangular pictures: the 2-HRFA, which allows backward moves, and the B2-HRFA, which enforces synchronous head movement—a constraint newly introduced in this work. Through formal language-theoretic analysis and closure properties, the paper establishes that the class of languages recognized by 2-HRFA is a proper subset of those recognized by RPDAs and incomparable with CFMGs. Furthermore, it demonstrates that B2-HRFA strictly lies between RFA and 2-HRFA in recognition power, thereby establishing a strict hierarchy among these three models.

Abstract

We introduce and study a family of two-head finite automata called two head returning finite automata (2-HRFA) operating on rectangular arrays of picture languages, in which both heads move in opposite directions. We show that the class of picture languages accepted by 2-HRFA is incomparable with the class of languages generated by context-free matrix grammars (CFMG), while it forms a proper subset of the class of languages accepted by returning pushdown automata (RPDA). In addition, we define a constrained variant, both head stepping two head returning finite automata (B2-HRFA), in which both heads are required to move in a synchronized, stepwise fashion. We prove that the class of languages accepted by returning finite automata (RFA) is a proper subset of the class of languages accepted by B2-HRFA, which in turn is a proper subset of the class of languages accepted by 2-HRFA. Closure properties for both the families of languages accepted by 2-HRFA and B2-HRFA are also investigated.

Problem

• picture languages
• two-head finite automata
• returning automata
• context-free matrix grammars
• closure properties

Innovation

• two-head finite automata
• picture languages
• returning automata
• synchronized movement
• closure properties