Needed for [this](https://arxiv.org/abs/1204.6506) paper:


Here is a possibly more clear version of my question. A Turing machine (with $1$ tape) has sets of tape letters $Y$, state letters $Q$, two symbols $\alpha$ and $\omega$ that mark the ends of the tape and a set of commands $\Theta$. A configuration is any word of the form $\alpha uqv\omega$ where $u,v$ are words in $Y$, $q\in Q$. Usually we distinguish the stop state $q\_0$ and the input state $q\_1$. The  *input* configuration is any configuration of the form $\alpha uq\_1\omega$. A command of the Turing machine is a substitution $aqb\to a'q'b'$ where $a,a', b, b'$ are either empty or letters or symbols $\alpha,\omega$ (with natural restriction: if, say, $a=\alpha$, then $a'=\alpha$, etc.).The command is applicable to a configuration $\alpha uqv\omega$ if the configuration contains a subword equal to $aqb$.


The machine can start working with any configuration $\alpha u q v\omega$. If it starts working with an input configuration $\alpha u q\_1\omega$ and ends with a configuration containing $q\_0$ we say that the machine accepts $u$.The machine can stop without accepting by arriving to a configuration where no command from $\Theta$ is applicable.


The language of all words accepted by a Turing machine $M$ is denoted by $L(M)$. Any recursive set $L$ of words is accepted by a (deterministic) Turing machine which stops on any input word (but accepts only words from $L$). That machine may never stop when started with some non-input configuration. For every $L$ it is possible to construct a Turing machine with $L=L(M)$ and which stops starting with every configuration (not necessarily input).


* Question: Where in the literature can I find a construction of such a Turing machine (for every recursive $L$)?



Turing machines that always halt
Jean-Camille Birget answered my question. These are called universally halting Turing machines. 
The oldest reference is: 


Martin Davis (1956). A note on universal Turing machines. In Shannon,
 C. E., McCarthy, J., eds, Automata Studies, pp. 167-175. Princeton
 University Press.


Birget proved a complexity version of this:
 Every deterministic Turing machine with time complexity $T(n)$ is equivalent to a deterministic Turing machine which halts after $O(T(n))$ steps, no matter what configuration of size $n$ this machine starts in [J.C. Birget, Infinite String Rewrite Systems and Complexity, J. Symbolic Computation (1998) 25, 759-793.]


 **Update** Friedrich Otto sent the following two more references: 


Herman, G.T.,
 Strong computability and variants of the uniform
 halting problem,
Zeitschrift fuer mathematische Logik und
 Grundlagen der Mathematik,
 17,
 1971,
 115--131


Shepherdson, J.C.,
 Machine configuration and word problems of given
 degree of unsolvability,
 Zeitschrift fuer mathematische Logik und
 Grundlagen der Mathematik,
 11,
 1965,
 149--175


