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

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Regular language restricted to smaller alphabet is regular

Let $L$ be a regular language on some alphabet $\Sigma$, and let $\Sigma_1 \subset \Sigma$ be a smaller alphabet. Consider $L_1$ the subset of $L$ whose elements are made up only of symbols from ...
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212 views

Combining two DFAs into an NFA to recognize concatenation

Suppose we have two separate DFAs that each recognize their own language. What is the most efficient way to combine these two DFAs into one NFA that recognizes the concatenation of both languages?
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1answer
84 views

How to construct a grammar G such that L(G)={x^ny^mx^my^n/m,n>1}?

construct a grammar $G$ such that $L(G)=\{x^ny^mx^my^n/m,n>1\}$? I don't have much idea how to approach this one. Could some help me to understand how to approach these kinds of problem?
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344 views

Regular expression and DFA/NFA questions

If a language L is generated by a regular expression, then L is recognized by a DFA. I think this is true, because regular expressions describe regular languages, those of which are exactly ...
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2answers
233 views

Why is a 3-PDA no more powerful than a 2-PDA?

A k-PDA is a pushdown automata with k stacks. My textbook on Computation Theory has an exercise that asks to prove that 3-PDA is no more powerful than a 2-PDA. Now consider the language that I made ...
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1answer
45 views

Proof of a Known Claim About Languages

I would like to know how to prove that there is no non-trivial language $L$ that satisfies the following condition: $${\large \left(\overline{L}\right)^* = \overline{L^*}}$$ "Non-trivial" is ...
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1answer
24 views

Reversed Language of a Non Regular Language

Is the following saying true or false? In any case why? Thanks!
2
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1answer
63 views

Context free grammar for language

I'm learning how to generate context-free grammar for a language. $L=\{{a}^i {b}^j {c}^k\, |\,i=j\lor j=k$ Here is how I tried ...
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2answers
41 views

What method should you follow to show what a DFA does?

I'm specifically looking for help analyzing the following DFA. What steps would one follow to show what language this particular DFA accepts? To me it seems quite random, and I can't figure out a ...
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2answers
56 views

Reduction between two languages and a common one

My question is as following : Let $A$ and $B$ be some languages, there exist a language $C$ such that $A\le C$ and $B\le C$, where "$\le$" means "reducible to", so $A\le C$ means there is a mapping ...
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2answers
38 views

Determining a language with a Turing Machine

How can I build a Turing Machine that determines the following language? $$L_{E - DFA} = \{\langle A \rangle | \text{$A$ is a $DFA$ and $L(A) = \varnothing$}\}$$ Thanks alot
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78 views

Why isn't this a sufficient proof?

So basically, we have a question that asks us to prove that given a particular Deterministic finite automaton (DFA), there is a symbol for which we can get to a state $q$ from a state $p$ given a ...
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26 views

If $s \geq 3$, $3$ divides $s$, and $t = s/3$, then $t+1 < s$.

I am using the pumping lemma to prove a language is not regular, and would like to assert what I have stated in the title of the question to complete my proof. That is, if $s \geq 3$, $3$ divides $s$, ...
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3answers
123 views

NFA for $(ab|a)^{*}$ using only 2 states

In Introduction to the Theory of Computation by Michael Sipser, there's an example which shows how to convert the regular expression $ (ab|a)^{*}$ into an NFA. The "standard" method results in 8 ...
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0answers
59 views

Is the PDA I drew correct?

Could you tell me if the language $L=\{a^{m}b^{n}:m \neq n,m>0,n>0 \} $ is accepted by the following pushdown automaton, where the alphabet of the stack is $\{a,z\}$ and $z$ is the initial ...
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49 views

Theory of Automata concepts

I just started taking Theory of Automata and I'm having a hard time understanding some of the concepts. It's been only a week and the following questions are my homework. I'm not asking you to do my ...
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1answer
63 views

Prove that RE is closed under reduction

Prove that the class RE is closed under reduction. Definitions: A language $ A \subseteq \Sigma^*$ is called reducible to $ B \subseteq \Gamma^*$ ( denoted by $A \leq B$) if there is a computable ...
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1answer
39 views

Languages that are not comparable in $R$

I want to know if there are $2$ languages $A,B\in{R}$ such that there's no reduction between them. Namely, $2$ languages $A$ and $B$ $\in$ $R$ such that $A\not\le B$ and $B\not\le A$ Thanks a lot!
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63 views

Determining whether a language belongs to R or RE (Turing Machines)

Does the language L belong to R, RE or neither? In each case, why? Thanks a lot!
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1answer
59 views

is this language regular or not?

I have problem with this language $$L = \{ a^n b^m : \text{$n+m$ is odd} \}$$ is it regular or not My Solution I used pumping lemma, w = a^2p b^2p+1 (the same for a^2p+1 b^2m ) ...
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0answers
49 views

Extended PDA vs TM

We studied in class that PDA is less powerful than TM. My question is: Extended PDA : for every $\alpha,\beta \in \Gamma \cup \{\epsilon\}$, $\sigma \in \Sigma \cup \{\epsilon\}$, $q,r \in Q$, $w ...
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1answer
80 views

Finding regular expressions

I'm given the DFA shown below and need to find regular expressions for the following languages: $L_{1,2}^0, L_{2,1}^6, L_{2,5}^4, L_{2,3}^5, L_{1,3}^5$. The language $L_{p,q}^r$ is defined as ...
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84 views

Showing that 2 languages are context free

I have these 2 languages: $$L_1 = \left\{a^ib^jc^k: k\ge i+j\right\}\\ L_2 = \left\{w_1cw_2 : w_1,w_2\in\{a,b\}^\ast\land |w_1|_a = |w_2|_a\right\}$$ How can I determine that they are context free ...
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91 views

Push down automata for context free grammar

I'm having trouble finding the PDA for this language $L = \{x^{3i} y^j z^k\ |\ i \ge 0 \land k \gt 2j \gt 0\}$ The ...
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45 views

Language requiring a DFA with a certain number of states to implement

For any function $f\colon\{0,1\}^n\to\{0,1\}$, define a language $S_f = \{(b_1,b_2,\ldots ,b_n)\in\{0,1\}^n : f(b_1,b_2,\ldots ,b_n) = 1\}$. So all words in the langugage has same length $n$. I have ...
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1answer
67 views

(Transformation) Semigroups, the semigroup $\mathbf D_n$ and the wreath product

I have some trouble understanding the following proof, were I can't even figure out how some terms are defined. But first I state some definitions and preliminary lemmas. A transformation semigroup ...
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1answer
168 views

Prove that a PDA with accept states accepts all context-free languages

Or in other words that $\forall L: L \in DCFL => L \in CFL$. First of all, does this statement even require a proof? My idea was to let L be an arbitrary language, such that $L \in DCFL$, this ...
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1answer
51 views

build finite automaton for language minimize states

I want to build a finite automaton that accepts $a^nb^n, n \gt 0, m \ge 0$. I can't do it unless the FA has two final states, i.e.: $delta(q0, a) = q1 delta(q1, a) = q1 delta(q1, b) = q2 delta(q2, ...
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1answer
216 views

DFA for Boolean Formula

Let $ f\left( b_{1}, \dots , b_{n} \right)$ be a boolean function. Define $S_{f} = \{\left( b_{1}, \dots , b_{n} \right): f\left( b_{1}, \dots , b_{n} \right)=1; b_{i} \in \{0,1\}, 1\leq i \leq n \}$ ...
4
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1answer
384 views

The shortest word in context free language

Let $G=(\Sigma,N,R,S)$ be a context-free grammar. For every production rule A --> w, we say that its length is $r$ if $|w|=r$. In addition $n = |N|$, and $k =$ the maximal length of a production rule ...
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1answer
31 views

Help Specify possible definitions for this Boolean Function

My math is rusty, but I need some guidance here. Problem I wish to design a stochastic, boolean procedure $f(state)$, that picks a winner, $f(state_{win})\to 1$ or loser, $f(state_{loss})\to 0$. I ...
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0answers
95 views

Oracle Turing machine - $E_{\text{TM}}$ and $PCP$.

$$E_{\text{TM}}=\{\langle M\rangle|M\text{ is a TM and $L(M)=\emptyset$}\}.$$ $E_{\text{TM}}$ is undecidable $$PCP=\{\langle P\rangle|P\text{ is an instance of the Post Correspondence Problem with a ...
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1answer
37 views

Irregular $a^nb^n$

We studied in class that regular languages closed under intersection. My question is : if we take the irregular language $L =$ {$a^nb^n : n\geq 0$} and the regular finite language $L' = \{a^3 ...
2
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1answer
75 views

Reducing A$_\text{TM}$ to REGULAR$_\text{TM}$

We can solve A$_\text{TM}$ problem using REGULAR$_\text{TM}$. Assume $R$ is a Turing machine that decides REGULAR$_\text{TM}$. We construct $S$ to decide A$_\text{TM}$ as follows On input ...
2
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1answer
56 views

Language concatenation

We learned in class that the regular languages are closed under concatenation (e.g $L_1L_2 =\{ w_1w_2 : w_1 \in L_1,w_2 \in L_2\}$ is a regular language if $L_1$ and $L_2$ are also regular ...
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2answers
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Proving $L=\{0^n \mid \text{n is a perfect square}\}$ is not a Regular Language without the Pumping Lemma

Is this a valid way of going about proving the proposition? Let $L = \{0^n \mid \text{n is a perfect square}\}$. The regular languages are closed under concatenation. So if $x \in L, y \in L$, then ...
2
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1answer
158 views

DFA Rejection State

I'm being asked to construct a DFA for the language over $\{0,1\}$ such that each string of five consecutive symbols contain at least two zeroes. In my construction, it seems to me that it would make ...
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1answer
20 views

proving regular language

let $L$ be a language over the alphabet $\{a,b\}$ that maintains that for each $w \in L$ ,the difference in absolute between the number of apearences of the letter $a$ and the number of apearences ...
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1answer
74 views

Grammar outside the Chomsky Hierarchy

This grammar describes a language that may fall outside the Chomsky Hierarchy (CH): \begin{array}{l} S \to abAbba \\ A \to abA \mid bbaB \\ B \to aab \\ \lambda \to Aab \mid aB \\ \end{array} Going ...
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1answer
44 views

Consider this grammar

Consider this grammar: \begin{array}{l} S \to aabBba \mid aAb \mid aab \\ bBb \to bCa \mid aaa\\ aA \to aC \mid bba\\ C \to aab \mid Cb \end{array} This is clearly context-sensitive (CS). It's not ...
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42 views

Context-sensitive grammar for this language

In order to write a context-sensitive grammar for: $L = \{ a^{n} b^{n} c^{n} d^{n} : n \ge 1 \}$ One possible set of productions is: $S \rightarrow aBCd | abcd $ $aB \rightarrow aaBb | ab | ...
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48 views

Assign Integer to Each Turing Machine

I have the following problem: suppose that we have an infinite set of symbols, $A = \{a_1, a_2, ...\}$ from which all Turing Machine input alphabets are chosen. Show how we could assign an integer to ...
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pumping lemma for regular languages and DFA with k states

The Formal statement of the pumping lemma for regular languages: Let $L$ be a regular language. Then there exists an integer $p \ge 1$ depending only on $L$ such that every string $w$ in $L$ of ...
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672 views

Routing Automaton

Is there a formal proof for the following question? For a DFA $M= (Q,\Sigma,\delta,s,A)$, we extend the function $\delta : Q \times \Sigma^* \to Q$, such that every $w \in \Sigma^* $, ...
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1answer
70 views

Infinite regular languages

There is a formal proof for the following sentence? For every 2 languages $A,B$, we write A@B if A subset of B and B\A infinite. Prove that if $A,B$ regular languages and A@b, than exists regular ...
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1answer
87 views

Is there a DFA with $k+2$ states which its reverse has $2^k$ states

I am trying to figure out if there exists a DFA $M$ with $k+2$ states (for every $k\in \mathbb{N}$ ) so that every automaton which accepts $L(M)^R$ has at least $2^k$ states. I am trying to find an ...
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1answer
43 views

Prove the existence of $C\in L_{regular}$ so that: $A \prec C \prec B $

Given $A,B$ regular languages. Prove the existence of $C\in L_{regular}$ so that: $A \prec C \prec B $ Whereas $A\prec B$ stands for: $A\subset B $ and $B\setminus A $ is infinite regular language. I ...
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Turing decidable/undecidable

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 $M$. can you prove whether $X$ is Turing ...
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0answers
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halting problem

Prove that it is undecidable for the halting problem of a deterministic Turing machine which accepts nonrecursive language or in-other-words: let's say we have a deterministic Turing machine which ...
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130 views

What is the language generated by G and how to draw the finite state automaton that recognizes this language?

G = (V, T, S, P) V = (0, 1, A, B, S) T = {0, 1} S is start S -> 0A S -> 1A A -> 0B B -> 1A B -> 1 For the drawing, I am confused about the last ...