Automata Theory, including abstract machines, grammars, parsing, grammatical inference, transducers, and finite-state techniques

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Algorith/ Equation to get the ith element in N x N

I am having a difficulty figuring out the equation to get the ith element in $\mathbb{N}\times \mathbb{N}$ ( crossing the set of natural numbers).We have $\mathbb{N}\times \mathbb{N} = \{ ...
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53 views

Prove the following context-free language is generated by this grammar.

I would like to prove the context-free language $$ \mathcal{A} = \{ w\#x ~:~ w^R \text{ is a substring of $x$ for } w,x \in \{0,1\}^* \}, $$ has the context free grammar \begin{align*} ...
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1answer
27 views

Language described by inverting accepting states of NFA

What is the formal language described by inverting accepting states of NFA? By inverting, I mean that rejecting states become accepting states and accepting states become rejecting states. Is there a ...
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141 views

Is the language “substrings of an even-lengthed regular language” also regular?

I want to prove that for a regular language $L$ where $\forall w \in L$ the length of $w$ is even, the language containing the first halves of the words of $L$ and the language containing the second ...
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91 views

NFA construction problem

I need to construct an automaton that recognizes the following language of strings over the alphabet $\{a,b\}$: The set of all strings over alphabet $\{a,b\}$ with the subsequence $abba$. (A ...
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1answer
108 views

Designing a Turing machine for $a^n$ $b^{2n}$ $a^n$

I am new to theory of automata, and i have a little knowledge of designing a turing machine, I am stuck in this question which is given to us as an assignment, $\{$$a^n$ $b^{2n}$ $a^n$ : ...
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57 views

Intersection of context-free language and its reversal

I know that intersection of two context-free languages is not always context-free and the following problem: Given two context-free languages A and B, is $A \bigcap B \neq \emptyset$ ? is ...
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52 views

Closure properties between 2 languages of different types

Whenever said - The intersection between a Context Free Language and a Regular Language is always Context Free, what is the best logical way to confirm the statement? I have this Chomsky hierarchy in ...
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42 views

Construct an automata for this language

Let $\mathcal{L}_1$ be the language over alphabet $\{0,1\}^*$. Define language $\mathcal{L}_2$, call even-$\mathcal{L}_1$, as: $$\mathcal{L}_2 = \{ w_2 w_4 \ldots w_{k} ~:~ w_1 w_2 w_3 w_4 \ldots ...
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43 views

Proving language as regular

Suppose that A and B are languages such that A o B is regular. Suppose that B is regular. Prove or disprove that A is regular. I am having a tough time with questions relating to proving a language ...
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91 views

Prove that $\{ww^R\#ww^R\}$ is not context free

I need to prove that $L = \{ww^R\#ww^R \; | \; w \text{ is in } \{a,b\}^*\}$ is not context free. I have tried using the pumping lemma for this. For $w=a^pb^pb^pa^p\#a^pb^pb^pa^p$. I have two cases ...
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63 views

Building a Deterministic Finite Automaton with the set {a}

Design a DFA for the set of strings in $\{a\}$* such that it's length is divisible by 3 or 5. The point that I am not understanding is, how many states do we need? Since we can't possibly represent ...
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30 views

question about determinstic pushdown automata

i need to show that there is no DPDA accepts the language $L=\{a^n*b^n \mid n>0\}\cup\{a^n*b^{2n} \mid n>0\}.$ i used the prefix property but i'm stuck showing that if $w,w' \in L$, $w$ is ...
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1answer
339 views

Context free grammar to NFA

I've been given an exercise to solve which goes as follows: generate an NFA from the given CFG, S -> AB | c A -> aAb | c B -> bBa | c Now correct me if I'm wrong, but if this language has an NFA it ...
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2answers
45 views

Did I find the right expression for the regular language for this FSA?

I have the following FSA, and the regular language that I found for it: Is this language correct? It doesn't match the solution in the book, but my teacher says there can be multiple equally ...
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2answers
68 views

How to break down a problem while constructing a CFG for a language?

A problem I came across was: Design a CFG for the language $\{a^ib^jc^k\,|\,i=j+k \}$ The solution I came up with : $S\rightarrow aSc\,|\,S_1$ $S\rightarrow aS_1b\,|\,\epsilon$ It took ...
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2k views

Help understanding a 'reversing a string' Turing Machine

I am having a bit of a confusion understanding some transitions in a Turing Machine. Its an example from Introduction to Languages and the Theory of Computation by John C. Martin. I've attached the ...
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461 views

Prove that $L=\{a^nb^nc^md^m \mid m,n >=0\}$ is context free language

I'm trying to write the grammar of this language, in order to prove that it is CFL but I'm stuck because m or n could be 0. The language is: $L=\{a^nb^nc^md^m \mid m,n >=0\}$ . If they were ...
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1answer
78 views

Proving languages are undecidable

Let L5 be the language { {G,D} | G is a CFG, D is a DFA, and L(D) ⊆ L(G) } Show that L5 is undecidable. Is L5 R.E.? Is it co-R.E.? I am not quite sure where exactly to start with this. Could ...
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3answers
101 views

regular language proof, a language partial to a regular one

So, I've been trying to solve a question I got and I think I'm correct but I'm not positive. Is the language {w| www belongs to L' and L' is regular} regular? I couldnt find any way to prove it isnt ...
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27 views

Nondeterministic final automat - example

Show a possibly nondeterministic FA to accept the following language: $$\left\{w\in\{a,b\}^*:w\text{ contains at least one instance of }aaba,bbb,\text{ or }ababa\right\}$$
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32 views

Language of Grammar

Let $G = (V,T,S,P)$ be the phrase structure grammar with $V = \{0,1,A,S\}$, $T=\{0,1\}$, and a set of productions $P$ consisting of: $S \to 1S$ $S \to 00A$ $A \to 0A$ $A \to 0$ What is the ...
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31 views

Designing a Pushdown Automation to accept a language

Im a novice trying to understand the theory of computation.Im trtying to learn about PDA.I understand that it is a machine counterpart of CFG.Im basically referring to Introduction to Automata Theory ...
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41 views

Designing a turing Machine belonging to a language

Im trying to learn the concept of turing machines.I have understood the basic stuff like its a simple mathematical model of a computer and its parts.Now im asked to create a turing machine. ...
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1answer
105 views

Can 2 items be added/taken away from a stack in push down automata at once?

Here is a language and 2 ways (I hope) of representing it with a PDA. Can I use the notation (b,a $\to$ ee) or anything of the like, to take away 2 items from the top of a list at once? Such as I ...
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603 views

Construct a PDA to accept the language

construct a PDA that accepts the language: a) $L_1 = \{ a^k b^k c^i \mid k,i \ge 0 \}$ my answer is : $$\begin{align*} &S\to AA\\ &A\to abc \mid ab \mid c \mid \lambda \end{align*}$$ b) ...
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165 views

How to design a Context-Free Grammar and Pushdown Automaton for the following language:

How would you design a context-free grammar for the following language? $\{p^n \ r^m \ p \ \ b^{m+n} \ \ r^2 ∣ m,n\geq 0\}$ Derive a Pushdown Automaton that accepts the same language as the CFG. ...
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1answer
37 views

Definition of “equivalence classes”

I am studying finite state automata and learning how to prove a machine uses the minimum number of states. I have come across the Myhill-Nerode theorem and one of the corollaries states the following ...
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60 views

What is the highest state in the context of finite state automata?

I am doing an assignment for my Theory of Computation course. We are writing a function and I am having a hard time understanding what "highest" state means in the following context: ...
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1answer
115 views

Automata Language regularity proof by construction.

I've been trying to prove or disprove a question that popped during our last session in Uni, we've been using automaton constructing to prove regularity for a while now and I really do have the handle ...
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66 views

For $\sum = \{ 0,1 \}$, $A$ has strings which contain a $1$ in their middle third, and a $B$ which contain two $1$'s in their middle third.

Language $A$ can also be represented as, $$A = \{ uvw \mid u,w \in \sum^*\text{ and, }v \in \sum^* 1 \sum^*\text{ and, }|u| = |w| \ge |v| \}$$ Language $B$ can also be represented as, $$B = \{ uvw ...
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324 views

Help with proving a language is regular (Sipser problem 1.49a)

I am working through Sipser's Introduction to the theory of computation on my own, so I don't have access to a teacher. Hopefully you can help me! Giving hints is highly appreciated. This question is ...
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230 views

Divisibility problem using DFA

Original problem: Create a DFA for every positive integer $k$, so that when DFA takes a binary string (reading from most significant bit), decides whether the number is divisible by $k$. A DFA for a ...
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71 views

Pumping Lemma for Context Free Languages: Is this language CFL?

I am learning for the first time the Pumping Lemma for CFL, and I thought I understood how it works until I came across this example: "Show that $L = \{a^m b^m c^n \mid m \leq n\}$ is not a CFL." My ...
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1answer
51 views

Turing machine that goes left on first symbol

I have a turing machine with transitions given by the following table I'm inputting the string aaaa. So if I look at the first symbol "a" in state A, it says to replace it with an X, go into state ...
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1answer
67 views

How to prove that a language `L` is not a regular language?

Given the following question: Prove that the following language is not a regular language: A language L in alphabet $\Sigma = \{a, b\}$ where every word ...
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179 views

State diagram of DFA

I'm trying to understand how to create state diagram of DFA. I found following example. On the first diagram I dont understand why we need fourth state when third state is final and there is no ...
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how to create transition system/automata for modulo 4

I don't know how to think when to build a transition system/automata to calculate modulo 4 a of binary numbers. I know that the last two binary digits gives the rest but I need to go through hole ...
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1answer
1k views

Create DFA that accept language where number of 0's is even and after every 1 goes 0

Alphabet ${} = \{0,1\}$. Language $L = \{ w \in \{0,1\}^* \mid \text{ number of $0$'s in $w$ is even and after every $1$ goes $0$} \}$. I'm trying to create DFA that accepts language $L$. ...
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Size of automata or regular expressions avoiding cross patterns

Let $\Sigma$ be an alphabet of finite size $k$, and $n$ some integer. I am interested in the language of words of size $n$ that do not contain $abab$ as a subword, for any pair $(a,b) \in \Sigma$ (I ...
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1answer
64 views

Is it possible to build an Pushdown Automata for an Ambiguous Context-Free Grammar?

Say I have the following grammar: $$S \to \epsilon \mid [S] \mid (S) \mid SS$$ This grammar is ambiguous as both the following parse trees yield the empty string $$S \to \epsilon$$ $$S \to SS \to ...
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44 views

Nondeterministic finite automaton understanding problem

It is probably a silly question but I have problem understanding it. Let's say I have to design a nondeterministic finite automaton that accepts the language consisting of words containing a string of ...
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1answer
54 views

Pushdown automaton design

I have to design a PDA that recognizes the language: $$L=\{w \mid \#(a,w) - 3\#(b,w) = 2\} $$ where $\#(a,w)$ means the number of letters $a$ in $w$ My idea is to count $a$'s and $b$'s. I have to ...
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55 views

Show a Turing Machine accepts all DFAs

You have a Turing Machine $T = \{ \{ A \} \mid A$ is a DFA and $L(A) = \Sigma^* \}$ i.e. the DFA that accepts all languages. Show it's decidable. Can you use the complement of this, the DFA that ...
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1answer
64 views

Deterministic finite automaton algorithm

I am having a little trouble understanding this question. For a DFA M = (Q, Σ, δ, q0, F), we say that a state q ∈ Q is reachable if there exists some string w ∈ Σ∗ such that q = δ∗(q0, w). Give an ...
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52 views

Pumping lemma to prove that a language is not context free

We've got $L = 0^{x^{2}}$. So we let $w = 0^{p^{2}}$, and we know that we can split w into $w = u\cdot v\cdot w\cdot x\cdot y$ , according to the pumping lemma for CFGs. I'd like to know how to ...
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284 views

CFG and PDA for w1#w2

Looking for a Context Free Grammar and Push Down Automata to describe a language made of two words, separated by a #, where the first words is not equal to the second word. For this example, we can ...
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Is this correct way to prove that { $uv$ $\mid$ $u$ and $v$ are strings over $(0,1)$ & $|u| = |v|$ } language is not regular?

I proved this using Pumping Lemma in the following way, Taking $p$ as pumping length then let us take a string $S$ where, $|u| = |v| = p$ Now according to pumping lemma $S$ can be split into three ...
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82 views

How to prove that $ \{ 0^n 1^{5n} : n \ge 10000 \} $ is not a regular language?

I proved that $$ \{0^n 1^{5n} : n \ge 0\} $$ is not a regular language using Pumping Lemma by following way. Solve by contradiction that $ L = \{ 0^n 1^{5n} : n >= 0 \}$ is regular language. ...
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Converting the NFA produced from the language $a^nb^n : n\geq 0$ to a DFA to show its regular? Leading to question about pumping lemma.

I am reading about the pumping lemma, and having a hard time understanding it. I noticed that it is used to prove a language is not regular by contradiction. So you must first prove that a language in ...