On the Halting Problem of Finite-State Programs

The undecidability of the halting problem is a well-known research result of theoretical computer science, dating back to Turing’s work in 1936. Nevertheless, it is commonly known that the halting problem on finite-state computer systems is decidable. Thus, any undecidability proof given for the halting problem must imply that it does not apply to finite-state computer systems. The aim of this paper is to deepen the understanding of why the undecidability proofs of the halting problem cannot be instantiated as finite-state programs. To bridge the gap between theory and practice, the arguments are based on simple mathematics rather than an extensive use of abstract formalisms.