Undecidable Properties on Length-Two String Rewriting Systems

Length-two string rewriting systems are length preserving string rewriting systems that consist of length-two rules. This paper shows that confluence, termination, left-most termination and right-most termination are undecidable properties for length-two string rewriting systems. This results mean that these properties are undecidable for the class of linear term rewriting systems in which depth-two variables are allowed in both-hand sides of rules.