343

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 ...


34

Yes, there is. Define a context-free expression to be a term generated by the following grammar: $$ \begin{array}{lcll} g & ::= & \epsilon & \mbox{Empty string}\\ & | & c & \mbox{Character $c$ in alphabet $\Sigma$} \\ & | & g \cdot g & \mbox{Concatenation} \\ & | & \bot & \mbox{...


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 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 ...


27

Every context-free language has either polynomial growth or exponential growth. In the notation of the question poser: Either there is a polynomial $p$ so that $w_n\le p(n)$ for all $n$ Or there exists a $c>1$, so that $w_n\ge c^n$ for infinitely many $n$. This has been shown for instance in: Roberto Incitti: "The growth function of context-free ...


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 ...


23

Unambiguous context-free parsing is in $O(n^2)$ using Earley's algorithm. Whether there exists a parsing algorithm working in linear-time on all the unambiguous context-free grammars is an open problem. One of the most advanced statements of this kind is due to Leo [1991], who showed that a variant of Earley parsing works in linear time for all LRR ...


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


19

About Q1: Both the ambiguity problem (given a CFG, whether it is ambiguous) and the inherent ambiguity problem (given a CFG, whether its language is inherently ambiguous, i.e. whether any equivalent CFG is ambiguous) are undecidable. Here are the original references: The undecidability of ambiguity was proved by Cantor (1962), Floyd (1962), and Chomsky and ...


19

There is the notion of primality of a language. It asks whether L can be written as $L_1 \cdot L_2$ where neither factor contains the empty word. A language is prime if it cannot be written in this form. For a given regular language, represented by a DFA, it is shown in [MNS] that it is PSPACE-complete to decide primality. [MNS] Wim Martens, Matthias ...


18

There has been a lot done applying category theory to regular languages and automata. One starting point is the recent papers: Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages by Bart Jacobs A Bialgebraic Approach to Automata and Formal Language Theory by James Worthington. In the first of these papers, the structure of ...


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: ...


17

Lower bounds for algebraic circuits In the setting of algebraic circuits, where a lower bound on circuit size is analogous to a lower bound on time, many results are known, but there are only a few core techniques in the more modern results. I know you asked for time lower bounds, but I think in many cases the hope is that the algebraic lower bounds will ...


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

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

It's discussed in one of the very first papers about strings and complexity, namely, Dana Angluin, Finding patterns common to a set of strings, J. Comput. System Sci. 21 (1980), 46-62. Look at Theorem 3.6. The problem is NP-complete. It's also in A. Ehrenfeucht, G. Rozenberg, Finding a homomorphism between two words is NP-complete, Inform. Process. Lett. ...


16

They are typically called AND-functions. (I'm not joking.) Indeed, this concept has been considered before, and that's what people call them. See, for example, the book by Kobler, Schoning, and Toran on Graph Iso, where they talk about AND- and OR-functions for GI. And, by the way, there is an OR-function for GI (ibid.). The question of an AND-function for ...


15

Every unary context-free language is regular. (e.g. a direct consequence of Parikh's theorem) If every iterative/pumping pair of a context-free language L is degenerated, then L is regular, i.e. L is regular if, for all words x,u,y,v,z it satisfies: $$xu^nyv^nz \in L, \text{for all } n \geq 0 \implies xu^iyv^jz \in L, \text{ for all }i,j \geq 0.$$This was ...


15

Visibly pushdown automata (or nested word automata, if you prefer working with nested words instead of finite words) extend the expressive power of deterministic finite automata: the class of regular languages is strictly contained within the class of visibly pushdown languages. For deterministic visibly pushdown automata, the language inclusion problem can ...


15

Take $S_5$ as alphabet and $$L= \{ \sigma_1\cdots \sigma_n \in S_5^*\mid \sigma_1\circ\cdots\circ\sigma_n = \text{Id}\}$$ Barrington proved in [2] that $L$ is $\textrm{NC}^1$-complete for $\textrm{AC}^0$ reduction (and even with a more restrictive reduction actually). In particular this shows that regular languages are not in $\textrm{TC}^0$ if $\textrm{...


15

Another paper to look at: Kai Salomaa, "Language Decompositions, Primality, and Trajectory-Based Operations", 2008.


14

Finite automata in which the initial state is also the unique accepting state have the form $r^∗$, where $r$ is some regular expression. However, as J.-E. Pin points out below, the converse is not true: there are languages of the form $r^*$ which are not accepted by a DFA with a unique accepting state. Intuitively, given a sequence of states $q_0, \ldots, ...


14

An $\omega$-regular language is actually quite low in the Borel hierarchy (inside $\Delta_3$), a result due to R. McNaughton, Testing and generating infinite sequences by a finite automaton, Information and Control 9 (1966), 521-530. For a proof and more details, you can look at Chapter 3 of the following book D. Perrin et J.-É. Pin, Infinite words, ...


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