# Is it possible to convert LBA into DFA?

Today I learned about an abstract class of machines called linear bounded automata.

It is intended to model real-world computers that have a limited amount of memory. I have always thought that real computers are DFAs due to the finite memory (but the DFA is a terribly poor abstraction).

Is it possible to convert LBA into an equivalent DFA by making every possible tape configuration into a state of the DFA?

## 2 Answers

No, you can't do this. A single LBA can process inputs of any possible finite length. For any individual input, there are only finitely many possible tape configurations but, over all possible inputs or all possible lengths, there are there are infinitely many possible tape configurations. A finite automaton can't have one state for each of these configurations.

In particular, LBAs can accept non-regular languages, such as $$\{a^nb^n\mid n\geq 0\}$$ so there are LBAs that can't be converted to a single DFA.

What you could do is produce an infinite family of DFAs, one for each possible input length. Each of these would have a finite number of states. However, such an infinite family could recognize any language – even undecidable ones! – because every language contains only finitely many strings of each length and every finite language is regular.

Yes, you just modelled the LBA by a finite state device. You have to decide what are the actions of that model, probably the instructions of the LBA ("in state $$q$$ on reading $$a$$ on the tape do ...").

But there is a catch. You have modelled the LBA together with its input. That means you will get a different finite state automaton for each input of the LBA. Which is drastically different from DFA: they can accept strings of arbitrary length.

I will not start the discussion whether real computers are finite state automata. Oh boy. That question has been asked: "Are real computers finite state machines?" (and was closed as "unclear what you're asking"), and several times before: "Does our PC work as Turing Machine?". Check the responses there.