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.

Filter by
Sorted by
Tagged with
51
votes
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 ...
11
votes
4answers
6k views

NFA with exponential number of states when determinized

How can I build an example of a regular language where the minimal DFA has $2^n$ states and the minimal NFA has $n$ states? Obviously the DFA's state-set should contain all subsets of the the NFA's ...
44
votes
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 ...
14
votes
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
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 ...
14
votes
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?
2
votes
2answers
10k views

Convert regular expression to DFA

How do you construct a DFA from a language that has a + sign? e.g. $L = \{(a+b)\}*$
50
votes
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 ...
30
votes
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 ...
20
votes
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 ...
25
votes
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, ...
17
votes
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 $...
15
votes
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 ...
12
votes
4answers
4k views

Can an FSA count?

This may be a silly question. It seem clear that an FSA, since it is finite, can only count the number of symbols in its input string up to a number bounded by the number of its states. But now ...
4
votes
2answers
3k views

Infinite Intersection/Union of regular languages

Hello I'm having trouble understanding how an intersection/union of regular languages can be regular and in other case non-regular. Can someone please give me some good examples?
27
votes
4answers
85k 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^{...
3
votes
4answers
110k views

Regular Expression for even-odd language of string

I am new to Automata theory and would to make a regular expression for "even-odd" strings, defined over $\Sigma = \{a,b\}$, which is the set of strings with even numbers of $b$'s and odd number of $a$'...
39
votes
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 ...
3
votes
2answers
7k views

Prove Queue Automaton is equivalent to Turing Machine

A deterministic queue automaton (DQA) is like a PDA except the stack is replaced by a queue. A queue is a tape allowing symbols to be written (push) on the left-end and read (pull) on the right-end. ...
22
votes
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. ...
6
votes
2answers
28k views

Complement of Non deterministic Finite Automata

It's known that the complement of a DFA can be easily formed. That is, given a machine $M$, we can construct $M'$ such that $L(M') = \Sigma^* \setminus L(M)$. Is it possible to construct such a ...
3
votes
1answer
2k views

Is $L_{half} = \{w : \text{for some } z \in L, x \in \Sigma^*, z = wx \wedge |w| = |x| \} $ regular? [duplicate]

Suppose we have some regular language $L$, then can we say that $$L_{half} = \{w : \text{for some } z \in L, x \in \Sigma^*, z = wx \wedge |w| = |x| \} $$ is also regular? I have a 'feeling' that ...
8
votes
3answers
2k views

Why isn't it simple to count the number of words in a regular language?

Given a DFA, A, let L(A) denote the number of words A accepts. I think it's easy to calculate L(A): Translate the encoding of A into a regular expression. If the Kleene star appears anywhere in the ...
11
votes
2answers
2k views

Smallest DFA that accepts given strings and rejects other given strings

Given two sets $A,B$ of strings over alphabet $\Sigma$, can we compute the smallest deterministic finite-state automaton (DFA) $M$ such that $A \subseteq L(M)$ and $L(M) \subseteq \Sigma^*\setminus B$?...
6
votes
1answer
5k views

DFA for a strings whose every subsequence of length five has at least two zeroes

I have a regular language consisting of such {0,1}^k sequences, in which every subsequence of length 5 has at least two 0's in ...
10
votes
1answer
833 views

How to prove that ε-loops are not necessary in PDAs?

In the context of our investigation of heap automata, I would like to prove that a particular variant can not accept non-context-sensitive languages. As we have no equivalent grammar model, I need a ...
5
votes
4answers
9k views

Decide whether a DFA accepts the empty language

Let $X = \{\langle M \rangle\ |\ M\text{ is a finite state machine and }L(M) = \emptyset\}$ where $\langle M \rangle$ is an encoding of the machine $M$. Is $X$ Turing decidable? Why or why not?
61
votes
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. ...
26
votes
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 ...
9
votes
4answers
11k views

Does our PC work as Turing Machine?

Does our PC work as Turing Machine? The model of a Turing Machine consists of infinite memory tape, which means infinite states. But suppose if our PC has 128 MB memory and 30GB disk it would have ...
6
votes
1answer
1k views

Is a LBA with stack more powerful than a LBA without?

Even so a linear bounded automata (LBA) is strictly more powerful than a pushdown automata (PDA), adding a stack to a LBA might make it more powerful. A LBA with stack should not be Turing complete, ...
3
votes
3answers
14k views

Construct Turing Machine which accepts the language $ww$

I try to construct a TM that accepts the language $\{ ww \mid w \in \{a,b\}^* \}$. Between the words $w$ is no delimeter, so I don't know, how my TM can know where the first $w$ ends and the second $...
2
votes
1answer
7k views

Pushdown automaton for complement of { ww | ... }

I want to be able to describe the idea behind the pushdown automaton (no tables or diagrams). So, I already know that $L = \{ ww \mid w \text{ in } (0,1)^*\}$ is not context free. Since CFL are not ...
20
votes
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 ...
7
votes
3answers
3k views

Prove that A** = A*, where A is a language over Σ*

Let $\mathcal A$ be an arbitrary language over $\Sigma^*$ Proof. To prove, $\mathcal A^{**} = \mathcal A^* $ $\mathcal A^{**} = \left( \mathcal A^0 \cup \mathcal A^1 \cup {...} \cup \mathcal A^n \...
3
votes
1answer
3k views

How can one simulate a PDA with a FIFO queue PDA?

I'm trying to figure out how a pushdown automata (PDA), which we know uses a stack (LIFO) can be simulated by a queue (FIFO). I understand that in a regular PDA, we only have access to the top most ...
1
vote
2answers
3k views

When generating a PDA from a CFG do I have a receiving state?

Thw Wikipedia article on Pushdown automata doesn't explain what the receiving state is for the generated PDA it just states that there is but one state.
0
votes
1answer
2k views

How to construct a DFA for this?

Let $C = shuffle(A, B)$ denote the shuffle $C$ of two languages $A$ and $B$, it consists of all strings $w$ of the form $w = a_1b_1a_2b_2....a_kb_k$, for $k > 0$, with $a_1a_2 ··· a_k \in A$ and $...
7
votes
2answers
1k views

Do NPDA work in parallel?

Assume my language is $$ L= ww^{r}\ $$ Now when we use NPDA for this,we will guess middle every time. It may be actual middle or it may not, so a new branch is created every time as I have a choice ...
5
votes
2answers
2k views

Designing a PDA w/o $\epsilon$-moves and $\leq 2$ states to accept an $\epsilon$-free CFL by final state

I understand that any CFL can be accepted by a PDA by final state or empty store but I have been rather stumped by this question. The question states that the PDA has at most 2 states. Clearly 1 will ...
5
votes
3answers
11k views

Write a regular expression - Contains an equal number of 01 and 10 subtrings

I'm trying to write a regular expression for the language $L\subseteq\{0,1\}^*$ of strings that begin with $0$ and contain an equal number of occurrences of the substrings $01$ and $10$. ...
4
votes
1answer
407 views

Proof that PDA's with different definitions have same expressive power

Let $P$ be a push down automaton $(Q,\Sigma,\Gamma,\delta,q_s,F)$, where $Q$ is the set of states, $\Sigma$ is the input alphabet $\Gamma$ is the stack alphabet $\delta$ is the transition function ...
3
votes
2answers
178 views

Using pumping lemma to show a language is not context free(Complicated)

How can i show that the following long language is not context free using the pumping lemma? $L=\left\{abc^{i_1}bc^{i_2}...bc^{i_{2m}}def^{j_1}ef^{j_2}..ef^{j_{2n}}ghq^{k_1}hq^{k_2}...hq^{k_o}\right\}...
0
votes
2answers
3k views

Are supersets of non-regular languages also non-regular?

I have to proof that if $L_1 \subset L_2$ and $L_1$ is not regular then $L_2$ it not regular. This is my proof. Is it valid? Since $L_1$ is not regular, there does not exists a finite automata $M_1$ ...
15
votes
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?
32
votes
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 ...
20
votes
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 ?
4
votes
3answers
12k views

Minimum number of states in DFA accepting strings where the numbers of a and b are divisible by X and Y respectively?

While studying automata theory a typical problem that I face is of the following type: Constructing a DFA with minimum number of states for all strings over $\{a,b\}$ which have number of $a$’s ...
16
votes
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 ...
5
votes
3answers
5k views

Testing two DFAs generate the same language by trying all strings upto a certain length

Given the language $EQ_{\mathrm{dfa}} =$ $\{\langle A, B\rangle\mid A$ and $B$ are two DFAs and $L(A) = L(B)$ $\}$ Prove that $EQ_{\mathrm{dfa}}$ is decidable by testing the two DFAs on all strings ...

1
2 3 4 5