Abstract
In the previous chapter we saw that there is a decision procedure to determine if a linear sentence is true for a monoid presented by a finite, monadic and confluent string-rewriting system. By using linear sentences many decision problems can be solved for these systems (see Section 4.4). Here we will show that in general for finite, length-reducing and confluent string-rewriting systems the truth of linear sentences in the monoids so prescribed is undecidable; in fact, many decision problems like the extended word problem, that can be expressed through linear sentences, are undecidable in this setting. All these undecidability results will be derived from a presentation of recursively enumerable languages through finite, length-reducing, and confluent string-rewriting systems.
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