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I have personally enjoyed several Aha! moments from studying basic automata theory. NFAs and DFAs form a microcosm for theoretical computer science as a whole.
Does Non-determinism Lead to Efficiency? There are standard examples where the minimal deterministic automaton for a language is exponentially larger than a minimal non-deterministic automaton. ...
44
To complete the other answers: I think that Turing Machine are a better abstraction of what computers do than finite automata.
Indeed, the main difference between the two models is that with finite automata, we expect to treat data that is bigger than the state space, and Turing Machine are a model for the other way around (state space >> data) by making the ...
33
There are many good theoretical reasons to study N/DFAs. Two that immediately come to mind are:
Turing machines (we think) capture everything that's computable. However, we can ask: What parts of a Turing machine are "essential"? What happens when you limit a Turing machine in various ways? DFAs are a very severe and natural limitation (taking away ...
32
There is a known formula for minimum number of states for such a finite automaton. This depends on $n$ as well as the radix $R$ of the underlying positional representation.
If $n$ is coprime to $R$, then the minimal number of states is $n$. However, when $n$ shares a factor with the radix then the situation is rather complicated. See Mathematica Journal ...
32
There are two approaches when considering this question: historical that pertains to how concepts were discovered and technical which explains why certain concepts were adopted and others abandoned or even forgotten.
Historically, the Turing Machine is perhaps the most intuitive model of several developed trying to answer the Entscheidungsproblem. This is ...
31
To add one more perspective to the rest of the answers: because you can actually do stuff with finite automata, in contrast with Turing machines.
Just about any interesting property of Turing machines are undecidable. On the contrary, with finite automata, just about everything is decidable. Language equality, inclusion, emptiness and universality are all ...
28
Neither!
The best way to see this independence is to read the original papers.
Turing's 1936 paper introducing Turing machines does not refer to any simpler type of (abstract) finite automaton.
McCulloch and Pitts' 1943 paper introducing "nerve-nets", the precursors of modern-day finite-state machines, proposed them as simplified models of neural activity, ...
27
State. you need to learn that one can model the world (for certain problems) as a finite state space, and one can think about computation in this settings. This is a simple insight but extremely useful if you do any programming - you would encounter state again and again and again, and FA give you a way to think about them. I consider this to be a sufficient ...
25
I'm not quite sure which flavour of "why" you are looking for. One reason for the increase in power when allowing nondeterminism can be seen in the following example:
Let $L$ be the set of palindromes $w\bar{w}$ over some alphabet (of at least two symbols), where $\bar{w}$ is the reverse of $w$. An NPDA for this language can just keep pushing symbols onto ...
25
No, it's not. I know two major classes of techniques for avoiding inconsistency/Turing completeness.
The first line of attack is to set up the system so that syntax can be arithmetized, but Godel's fixed point theorem doesn't go through. Dan Willard has worked extensively on this and given consistent self-verifying logical systems. The trick is to eliminate ...
25
This is really a stubborn -- and well-studied -- problem. Regarding positive results, an exact algorithm by Kameda and Weiner, a heuristic approach by Polák, and a recent approach using SAT solvers by Geldenhuys et al. come to mind. But there seem to be far more negative results ruling out other possible approaches (e.g. approximation algorithms, special ...
24
Not necessarily. For instance, the two-dimensional block cellular automaton with two states, in which a cell becomes live only when its four predecessors have exactly two adjacent live cells, can simulate itself with a factor of two slowdown and a factor of two size blowup, but is not known to be Turing complete. See The B36/S125 “2x2” Life-Like Cellular ...
22
How about $L:=\{a^{n^2}\mid n\in\mathbb{N}\}$? It is easy to see that $L$ and its complement are not regular, and hence (as we are dealing with a unary alphabet) not context-free.
22
Simple answer: If there does exist a more efficient algorithm that runs in $O(n^{\delta})$ time for some $\delta < 2$, then the strong exponential time hypothesis would be refuted.
We will prove a stronger theorem and then the simple answer will follow.
Theorem: If we can solve the intersection non-emptiness problem for two DFA's in $O(n^{\delta})$ time,...
21
Although it is not really the reason they were originally studied, finite automata and the regular languages they recognize are tractable enough that they have been used as building blocks for more complicated mathematical theories. In this context see particularly automatic groups (groups in which the elements can be represented by strings in a regular ...
20
See Mike Domaratzki's paper, State complexity of proportional removals
http://dl.acm.org/citation.cfm?id=782471
http://www.cs.umanitoba.ca/~mdomarat/pubs/sc_jalc.ps
20
Consider password automata: for each $w\in\{0,1\}^n$, the DFA $M_w$ accepts the language $\{w\}$. In this case, a membership query is the same as an equivalence query --- and clearly, you'll need exponentially many of these to find the "needle in the haystack". (This is even if the learner knows in advance that the target automaton is of this form.) For a ...
18
State complexity is really about concise description of an object (in this case, a regular language), not about computational complexity. The general topic is called "descriptional complexity" in the literature and draws its inspiration, in part, from the classic 1971 paper of Meyer and Fischer entitled "Economy of Expression by Automata, Grammars, and ...
18
You are asking (at least) two different questions: (a) What parts of theory build on finite automata nowadays? (b) Why were finite automata developed in the first place? I think the best way to address the latter is to look at the old papers, such as:
Rabin, Scott, Finite Automata and Their Decision Problems, 1959
Here are the first two paragraphs:
...
18
Perhaps I found some relevant information in:
Jean-Michel Autebert, Jean Berstel, Luc Boasson; Context-Free Languages and Pushdown Automata; Handbook of Formal Languages; 1997, pp 111-174
DPDAs without $\epsilon$-transitions are known as realtime deterministic pushdown automata.
They are less powerful than DPDAs, for example
$L = \{ a^n b^p c a^n \mid ...
18
In my paper with Domaratzki and Kisman, "On the number of distinct languages accepted by finite automata with n states" published in J. Automata, Languages, and Combinatorics 7 (2002) we proved that if $G_k (n)$ is the number of distinct languages accepted by NFA's with $n$ states over a $k$-letter alphabet, and $g_k (n)$ is similarly the number of distinct ...
18
First, the name of the conjecture is "Hartmanis-Stearns", not "Hartmanis-Stearn".
Second, the Hartmanis-Stearns conjecture concerns those real numbers computable by a multi-tape Turing machine in real time; in other words, the TM must compute the n'th digit in n time.
Third, the result of Adamczewski et al. is only about finite automata and deterministic ...
17
Two more constructions: Brzozowski-McCluskey aka state elimination [1], and Gaussian elimination in a system of equations using Arden's Lemma. The best source on these is probably Jacques Sakarovitch's book [2].
[1] J. Brzozowski, E. McCluskey Jr., Signal flow graph techniques for sequential circuit state diagrams, IEEE Transactions on Electronic
Computers ...
17
This question is addressed in Section 2 of [1], which shows (Theorem 2.6) that the problem is
in P if $L(\alpha)$ is finite;
coNP-complete if $L(\alpha)$ is infinite but bounded (i.e. $L(\alpha)\subseteq w_1^*w_2^*\ldots w_k^*$ for some $w_1,\ldots, w_k$);
PSPACE-complete otherwise.
[1] Harry B. Hunt, Daniel J. Rosenkrantz, Thomas G. Szymanski, On the ...
16
I thought about this problem again, and I think I have a full proof. It is a bit more tricky than what I anticipated. Comments are very welcome! Update: I submitted this proof on arXiv, in case this is useful to someone: http://arxiv.org/abs/1207.2819
$\DeclareMathOperator{\fp}{fp}$
$\DeclareMathOperator{\lp}{lp}$
$\newcommand{\fpp}[1]{\widehat{\fp{#1}}}$
$\...
16
Another reason is that they're relatively practical theoretical models. A Turing machine, apart from the impossibility of the infinite tape, is kind of an awkward fit for what it's like to program a computer (note that this is not a good analogy to begin with!). PDAs and DFAs however are quite amenable to being models of actual programs in the sense that a ...
16
According to Garey and Johnson (p. 174), REGULAR EXPRESSION NON-UNIVERSALITY is PSPACE-complete. This is the problem of deciding whether a regular expression over $\{0,1\}$ does not generate all strings. So your problem is also PSPACE-complete.
Here is one way to see that the OP's problem is in PSPACE. Given a DFA $A$ and a regular expression $r$, construct ...
16
There is even a stronger result than your request:
There are exponentially-ambiguous NFAs for which the minimal polynomially-ambiguous NFAs are exponentially larger, and in particular the minimal UFAs.
Check this paper by Hing Leung.
16
Let me provide you with an algorithm for recursively constructing an infinite state machine to decide any language $L \subseteq \{0,1\}^\ast$ that you like.
Make the initial state accept if the empty string is in the language.
Create two states for the strings 0 and 1, which the initial state branches to depending on whether the first symbol is 0 or 1. ...
15
Here is another example:
$ \mathtt{L} = \{ x\#y \mid x \in \mathtt{EQ},y \in \overline{\mathtt{EQ}} \} $,
where
$ \mathtt{EQ}=\{ a^nb^nc^n \mid n \geq 0 \} $ and $ \overline{\mathtt{EQ}} $ is the complement of $\mathtt{EQ}$.
It is a well known fact that $\mathtt{EQ}$ is not in $ \mathsf{CFL} $.
Assume that $ \mathtt{L} $ is recognized by a PDA $...
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