I thought that the $L\_{\alpha}$ notation denotes a set of reals that can occur on the output tape at stage $\beta$, where $\beta$ is any ordinal less than $\alpha$ and the input is an arbitrary real. 

Similarly, the $L\_{\alpha}[x\_1, x\_2, \ldots, x\_{n-1}, x\_n]$ notation would denote a set of reals that can occur on the output tape at stage $\beta$, where $\beta$ is any ordinal less than $\alpha$, assuming that an Infinite Time Turing Machine interprets the input as $n$ separate reals and $x\_i$ denotes an $i$-th real. 

But then I read that the recognizable closure of Infinite Time Turing Machines is equal to $L\_{\sigma}$ (see Lemma 3.2 in the paper [“Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”](https://arxiv.org/abs/1512.06101)), where $\sigma$ is a particular countable ordinal. 

And here my understanding seems to contradict the intended meaning of the notation: assume that the input is a real that encodes any well-order $W$ of order-type greater than $\sigma$ and consider an Infinite Time Turing Machine that checks if $W$ is, in fact, a well-order. This machine will halt at stage $\tau$, where $\tau$ is greater than $\sigma$. 

What is the meaning of $L\_{\alpha}$ and $L\_{\alpha}[x\_1, x\_2, \ldots, x\_{n-1}, x\_n]$ notations in relation to Infinite Time Turing Machines?