Past exam of the computer science course of the University of Oxford / 2015 / Models of computation / 3 / d / ii

The problem is undecidable. Reduce the halting problem to it. Given a Turing machine $M$ and input word $w$, construct a Turing machine $T$ over $\{a,b,c\}$ with the following behavior on input $x$:

This is an effective construction: $M,w$ are fixed data stored in the description of $T$. Its recognized formal language isTherefore $T$ accepts exactly one input word if and only if $M$ halts on $w$. A supposed decision procedure for the singleton-language problem for Turing machines, applied to the constructed $T$, would decide the halting problem. The standard undecidability of the halting problem excludes that possibility.