The firing squad synchronization problem for squares with holes

TL;DR

This paper extends the firing squad synchronization problem to square grids with holes, providing minimal-time solutions for the case of one hole and analyzing minimum firing times for two-hole configurations, advancing automata theory.

Contribution

Introduces the FSSP variation for squares with holes, proving minimal-time solutions for one hole and analyzing minimum firing times for two holes, a challenging extension of classical results.

Findings

SH[1] has a minimal-time solution.

Minimum firing time for SH[2] configurations is determined.

Finding solutions for multiple holes remains a complex challenge.

Abstract

The firing squad synchronization problem (FSSP, for short) is a problem in automata theory introduced in 1957 by John Myhill. Its goal is to design a finite automaton A such that, if copies of A are placed in a line and connected and are started at time 0 with their leftmost copy in a special triggering state, then at some time (the "firing time") all copies enter a special "firing state" simultaneously for the first time. FSSP has many variations and for many of them we know minimal-time solutions (solutions having shortest firing time). One of such variations is the FSSP for squares (denoted by SQ) in which copies are placed in a square. In this paper we introduce a variation which we call the FSSP for squares with k holes and denote by SH[k] by slightly modifying SQ (k >= 1). In the variation, copies of a finite automaton are placed in a square but there are k positions ("holes") in the square where no copies are placed. We show that SH[1] has a minimal-time solution. Moreover, for each problem instance (a placement of copies in a square) C of SH[2], we determine the minimum firing time of C (the minimum value of firing times of C by A where A ranges over all solutions of SH[2]). The variation SQ was introduced and its minimal-time solutions were found in 1970's. However, to find minimal-time solutions of SH[k], a very simple modification of SQ, seems to be a very difficult and challenging problem for k >= 2.