Questions tagged [automata]

Questions about mathematical devices that read an input stream symbol by symbol and use a state transition map to produce an output stream, maybe using secondary storage.

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61
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7answers
26k views

Is a Turing Machine "by definition" the most powerful machine?

I agree that a Turing Machine can do "all possible mathematical problems". But that is because it is just a machine representation of an algorithm: first do this, then do that, finally output that. ...
51
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8answers
103k views

How to prove a language is regular?

There are many methods to prove that a language is not regular, but what do I need to do to prove that some language is regular? For instance, if I am given that $L$ is regular, how can I prove that ...
50
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2answers
39k views

Is a push-down automaton with two stacks equivalent to a turing machine?

In this answer it is mentioned A regular language can be recognized by a finite automaton. A context-free language requires a stack, and a context sensitive language requires two stacks (which is ...
44
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2answers
2k views

Determining capabilities of a min-heap (or other exotic) state machines

See the end of this post for some clarification on the definition(s) of min-heap automata. One can imagine using a variety of data structures for storing information for use by state machines. For ...
39
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2answers
6k views

Are there inherently ambiguous and deterministic context-free languages?

Let us call a context-free language deterministic if and only if it can be accepted by a deterministic push-down automaton, and nondeterministic otherwise. Let us call a context-free language ...
36
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8answers
9k views

Are there any non-finite automata?

In automata theory, we all read automata as finite automata, from the very beginning. What I want to know is, why are automata finite? To be clear, what is it in an automaton that is finite - the ...
34
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9answers
4k views

What is the significance of context-sensitive (Type 1) languages?

Seeing that in the Chomsky Hierarchy Type 3 languages can be recognised by a state machine with no external memory (i.e., a finite automaton), Type 2 by a state machine with a single stack (i.e. a ...
32
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2answers
1k views

Equivalence of Büchi automata and linear $\mu$-calculus

It's a known fact that every LTL formula can be expressed by a Büchi $\omega$-automaton. But, apparently, Büchi automata are a more powerful, expressive model. I've heard somewhere that Büchi automata ...
30
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4answers
5k views

How to simulate backreferences, lookaheads, and lookbehinds in finite state automata?

I created a simple regular expression lexer and parser to take a regular expression and generate its parse tree. Creating a non-deterministic finite state automaton from this parse tree is relatively ...
27
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4answers
86k views

How to show that a "reversed" regular language is regular

I'm stuck on the following question: "Regular languages are precisely those accepted by finite automata. Given this fact, show that if the language $L$ is accepted by some finite automaton, then $L^{...
26
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10answers
11k views

Why is non-determinism a useful concept?

An automaton is an abstract model of a digital computer. Digital computers are completely deterministic; their state at any time is uniquely predictable from the input and the initial state. When we ...
26
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3answers
2k views

What are the conditions for a NFA for its equivalent DFA to be maximal in size?

We know that DFAs are equivalent to NFAs in expressiveness power; there is also a known algorithm for converting NFAs to DFAs (unfortunately I do now know the inventor of that algorithm), which in ...
25
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2answers
9k views

How is the rule 110 Turing complete?

I've read the wikipedia page for rule 110 in cellular automata, and I more or less know how they work (a set of rules decides where to draw the next 1 or 0). I've just read they're Turing complete, ...
23
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2answers
1k views

Is there a "natural" undecidable language?

Is there any "natural" language which is undecidable? by "natural" I mean a language defined directly by properties of strings, and not via machines and their equivalent. In other words, if the ...
23
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1answer
3k views

Decide whether a context-free languages can be accepted by a deterministic pushdown automaton

Given a context-free grammar G, there exists a Nondeterministic Pushdown Automaton N that accepts exactly the language G accepts. (and visa versa) There may also exist a Deterministic Pushdown ...
22
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3answers
6k views

How to prove that DFAs from NFAs can have exponential number of states?

All non-deterministic finite automata can be turned into equivalent deterministic finite automata. However, a deterministic finite automata only allows a single arrow per symbol pointing from a state. ...
20
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9answers
4k views

Would it be wrong to say that the processor (and hardware) is the implementation of an interpreter for machine language?

The question is basically in the title. I know that a computers hardware is of course some physical object, where as the interpreter is some abstract thing that does something abstract with the code ...
20
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4answers
33k views

What is the difference between finite automata and finite state machines?

I have used FSM in Digital sequential Circuit designs. But I am unfamiliar with Finite Automata. Can somebody help me in understanding 'basic' difference between the two ?
20
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4answers
2k views

Defining the halting problem for non-deterministic automata

The primary definition of Turing machine (TM), at least in my own reference textbook (Hopcroft+Ullman 1979) is deterministic. Hence my own understanding of the halting problem is primarily for ...
20
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1answer
16k views

How do I write a proof using induction on the length of the input string?

In my Computing Theory course, a lot of our problems involve using induction on the length of the input string to prove statements about finite automata. I understand mathematical induction, however ...
19
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3answers
407 views

Is this language defined using twin primes regular?

Let $\qquad L = \{a^n \mid \exists_{p \geq n}\ p\,,\ p+2 \text{ are prime}\}.$ Is $L$ regular? This question looked suspicious at the first glance and I've realized that it is connected with the ...
19
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2answers
2k views

Can there be 'dead states' in a context-free grammar?

Can a context-free grammar include "dead states" from an automaton, such as $$G = \big(\{a, b, c\}, \{A, B, C\}, \{A\to aB, B\to b, B\to C, C\to cC\}, A\big)\,?$$ The production rules $B\to C$ and $...
17
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1answer
5k views

Can a language have more than one DFA?

For Example when we consider a DFA that allows the strings having neither sub strings 00 or 11 I can produce the following two ...
17
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2answers
2k views

Is a Turing machine without the ability to write on blank cells less powerful than standard Turing?

Is a Turing machine without the ability to write on blank cells less powerful than standard Turing? I think the answer is yes but i'm unable to find a computation that standard Turing machine can do ...
17
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1answer
10k views

Construct a PDA for the complement of $a^nb^nc^n$

I am wondering if this is even possible, since $\{a^n b^n c^n \mid n \geq 0\} \not\in \mathrm{CFL}$. Therefore a PDA that can distinguish a word $w\in\{a^n b^n c^n \mid n \geq 0\}$ from the rest of $...
16
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2answers
6k views

Can a deterministic finite automaton ever go into an infinite loop?

As the title suggests, is this possible? Or does it halt execution when it hits the trap state? Thanks to anyone who can clear this up.
16
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3answers
6k views

Why is the halting problem decidable for LBA?

I have read in Wikipedia and some other texts that The halting problem is [...] decidable for linear bounded automata (LBAs) [and] deterministic machines with finite memory. But earlier it is ...
16
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2answers
538 views

Languages accepted by modified versions of finite automata

A deterministic finite automaton (DFA) is a state machine model capable of accepting all and only regular languages. DFAs can be (and usually are) defined in such a way that each state must provide ...
16
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1answer
573 views

Proving closure under reversal of languages accepted by min-heap automata

This is a follow-up question of this one. In a previous question about exotic state machines, Alex ten Brink and Raphael addressed the computational capabilities of a peculiar kind of state machine: ...
15
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5answers
2k views

Are there undecidable properties of non-turing-complete automata?

Are there undecidable properties of linear bounded automata (avoiding the empty set language trick)? What about for a deterministic finite automaton? (put aside intractability). I would like to get ...
15
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3answers
22k views

Why is this true: “There are countably many Turing Machines” [duplicate]

It is said that there are uncountably many languages but only countably many Turing Machines. Could someone make this clear to me? And this doesn't mean that the set of TM is finite, yes?
15
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2answers
2k views

What is the origin of λ for empty string?

I usually use the symbol $\varepsilon$ for empty string (empty word or null string). But I know some people use $\lambda$ instead of $\varepsilon$. I think $\varepsilon$ is derived from the word "...
15
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2answers
4k views

Which languages are recognized by one-counter machines?

Counter machines with two or more counters are typically shown to be equivalent to Turing machines in courses on the theory of computation. However, I have not seen a formal analysis of which ...
15
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1answer
2k views

Computational power of deterministic versus nondeterministic min-heap automata

This is a follow-up question of this one. In a previous question about exotic state machines, Alex ten Brink and Raphael addressed the computational capabilities of a peculiar kind of state machine: ...
14
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5answers
34k views

In a DFA, does every state have a transition on every symbol of the alphabet?

If not, then what does it mean when for some state $q$ and some symbol $a$, $\delta(q, a)$ does not exist?
14
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4answers
1k views

Bridge theorems for group theory and formal languages

Is there some natural or notable way to relate or link math groups and CS formal languages or some other core CS concept e.g. Turing machines? I am looking for references/applications. However note ...
14
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3answers
11k views

Why is the Kleene star operator also called the Kleene 'closure' operator?

I've found that if I don't understand the etymology behind a cs/programming term, it usually means that I've missed or misunderstood some important underlying concept. I don't understand why the ...
14
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2answers
2k views

How did the word "production" end up being a synonym with the word "rule" in the context of Computer Science?

I am studying formal languages and production bases systems (rule-bases systems) and I am a little confused about why do these two word "production" and "rule" mean the same thing in so many context ...
14
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1answer
14k views

Difference between regular expression and grammar in automata

I am new to automata, and I have been given a brief introduction to regular expressions only yesterday. I have read the various rules which to define a regular expression. But I am unable to ...
14
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3answers
23k views

How to create DFA from regular expression without using NFA?

Objective is to create DFA from a regular expression and using "Regular exp>NFA>DFA conversion" is not an option. How should one go about doing that? I asked this question to our professor but he ...
14
votes
4answers
830 views

Push Down Automatons "guess" - what does that mean?

I realize non-deterministic pushdown automata can be an improvement over deterministic ones as they can "choose" among several states and there are some context-free languages which cannot be accepted ...
14
votes
3answers
33k views

How does an NFA use epsilon transitions?

In the picture below, I'm trying to figure out what exactly this NFA is accepting. What's confusing me is the $\epsilon$ jump at $q_0$. If a $0$ is entered, does the system move to both $q_0$ and $...
14
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1answer
1k views

Monadic Second Order Logic for Dummies

I am programmer with a grip on automata, but not on logic. I read in papers that the two are very tightly related. Deterministic Finite Automata (DFA), Tree Automata and Visibly Pushdown Automata are ...
14
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1answer
8k views

Single-tape Turing Machines with write-protected input recognize only Regular Languages

Here is the problem: Prove that single-tape Turing Machines that cannot write on the portion of the tape containing the input string recognize only regular languages. My idea is to prove that this ...
13
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3answers
3k views

Is it mandatory to define transitions on every possible alphabet in Deterministic Finite Automata?

Tomorrow is my presentation and I want to clear my concepts… I've read that in DFA, "For each state, transition on all possible symbols (alphabet) should be defined." Is for each state, defining ...
13
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4answers
3k views

Star free language vs. regular language

I was wondering, since $a^*$ is itself a star-free language, is there a regular language that is not a star-free language? Could you give an example? (from wikipdia) Lawson defines star-free ...
13
votes
1answer
348 views

Computing the intersection of two NPDA where it is possible

Apropois to Raphael's suggestion on Intersection of two NPDAs: Let $A_1$ and $A_2$ NPDA for context-free languages $L_1$ and $L_2$, respectively. Assuming that we know that $L = L_1 \cap L_2$ is ...
12
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2answers
5k views

Is the equality of two DFAs a decidable problem?

So given two DFAs, is the problem of finding if they generate the same language a Decidable problem? I already know that Equality of two CFL is not Decidable but what about Equality of two DFAs? ...
12
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3answers
2k views

Is there an efficient test for if an NFA accepts a subset of another NFA?

So, I know that testing if a regular language $R$ is a subset of regular language $S$ is decidable, since we can convert them both to DFAs, compute $R \cap \bar{S}$, and then test if this language is ...
12
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3answers
779 views

Why nondeterminism?

Theory of computation often involves nondeterministic models of computation. Some examples include nondeterministic finite automata (NFAs), nondeterministic pushdown automata (PDAs), and ...

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