# What is the motivation behind defining Deterministic Looping Automata?

I was wondering about what could possibly be the motivation behind defining the deterministic looping automata? What puzzles me is that they accept a word iff they have a run on it!

I believe they are strongly related to DBWs. And i would like to see how.

My intuition: since we can assume W.l.o.g that each non-accepting state $q$ in a DBW $A$ has a loop of non-accepting states containing q. Maybe one can look at the $strongly$ $connected$ $components$ of $A_{rej}$, Where $A_{rej}$ denotes $A$ restricted to non-accepting states, as looping automata.

Relevant papers could help.

Note: i am new to research. In particular i have few knowledge about DBWs. Its more a reference than a question.

thanks.

• What's a DBW? Which definition of looping automata are you using? – András Salamon Sep 29 '17 at 15:30
• DBW $A=(\Sigma, Q, q_0, \delta, F)$ is deteriministic büchi automaton where $\delta$ is a partial function from $Q \times \Sigma$ to $Q$. And a deterministic looping automaton $A$ is defined exactly as the deterministic büchi automaton but all states are accepting. i.e., $F=Q$. – Bader Abu Radi Sep 29 '17 at 15:53

In a looping automaton, the transition function is not assumed to be total. That is, for a state $q$ and a letter $\sigma$, it could be the case that $\delta(q,\sigma)$ is undefined. Intuitively, this is the same as having a transition to a rejecting sink.
The reason for studying these automata is that they correspond to safety properties. A safety property is a language $L$ such that for every word $w\notin L$, there exists some prefix $x$ of $w$ such that for every suffix $y$, $x\cdot y\notin L$. That is, once you read $x$, there is no way to get in the language. These properties are useful in practice, as they correspond to "system failure", in a way. That is, once something really bad happens, you can't recover.