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

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1answer
124 views

Pumping Lemma for regular languages proof template

http://www.cs.uiuc.edu/class/fa06/cs273/Lectures/pumping-lemma/pumping-lemma.html So, I went to that site and it says: $w = xyz$ $|xy| \leq p$ $|y| \geq 1$ for all $i$, $xy^iz$ is in ...
2
votes
1answer
72 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 ...
2
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0answers
52 views

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 ...
2
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0answers
81 views

Blanks in the Tape of a Turing Machine

I used to have a lot of trouble with Turing Machines, primarily because I thought that in-between input symbols on the tape, one could have an arbitrary number of blanks, so every input would need to ...
2
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1answer
79 views

Build regular expression from language

I have the following language: L = {w $\in$ {a,b}* | aa is not part of w}. I have to construct a regular grammar from this language and I thought about first finding the regular expression from the ...
2
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0answers
64 views

Turing Machines

Suppose that $\Sigma$ is a finite set and that $L_1$, $L_2$ and $L_3$ are Turing acceptable subsets of $\Sigma^*$ that satisfy the following properties: $L_1 \cup L_2 \cup L_3 = \Sigma^*$; $L_1 \cap ...
2
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1answer
267 views

Converting CFG to CNF

I have the following problem of converting CFG to CNF: $$ \begin{aligned} S \Rightarrow\,& bA \mid aB\\ A \Rightarrow\,& bAA \mid as \mid a\\ B \Rightarrow\,& BB\mid bs\mid b ...
2
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1answer
27 views

Properties of “fail-safe” languages

I'm wondering if anyone has any experience with the concept of a "fail-safe" language. And, if so, where could I find more information on the subject. To explain what a "fail-safe" language is: Let ...
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0answers
31 views

Is there a 2D 3-colorstate mobile automaton that grows like $ln^{0,5}(t)$?

Define an integer function $f(t)$ for an integer $t>25$ such that $|f(f(t)) - ln(t)| < \sqrt {ln(t)}+2$. Define $L(X(t))$ as the number of nonwhite states at iteration $t$ of mobile automaton ...
2
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0answers
60 views

Expressiveness of finite memory programs

Assume we have a simple programming language with while, if, := (assignment), ...
2
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0answers
53 views

Quality of Reduction of finite automata

I am looking for an example, which corresponds to what I've learned in my Applied Automata Theory Class: Given a NFA $\mathcal{A}$, a $\approx _\mathcal{A}$ quotient automaton can be bigger then a ...
2
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2answers
151 views

An NFA with $\Sigma = \{1\}$ with $x^2$ accepting runs on strings $1^x$ for all $x \geq 0$ - how to construct?

One of my homework assignments requires us to construct an NFA over the alphabet $\{1\}$ which has exactly $x^2 + 3$ accepting runs over the input string 1^x for all $x \in \mathbb{N}$. Now, the +3 ...
2
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0answers
127 views

Mathematical formulation of 'Indra's net' [closed]

Quoting Wikipedia: "Imagine a multidimensional spider's web in the early morning covered with dew drops. And every dew drop contains the reflection of all the other dew drops. And, in each ...
1
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2answers
311 views

Converting from NFA to a regular expression.

This is a NFA, I have been working to covert it to a regular expression. After I'am done, I arrive at an expression as follows $$ \left(((a\cup b)a^*b) (ba^*b)^*a\right)^* \left(((a\cup b)a^*b) ...
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4answers
171 views

Prove this language is not regular [closed]

How do I prove that this language = {1^k | k is a perfect square} is not regular by showing that no DFA can accept the language?
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2answers
95 views

If $L_2L_1$ is accepted by a DFA, is $L_1$ too?

Given that $L_2, L_2L_1$ are accepted by a DFA, is $L_1$ accepted by a DFA too? What is the general approach to such question? What if instead of $\cdot$ we are given that $L_2 \cup L_1$ is ...
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2answers
165 views

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 ...
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2answers
195 views

regular expression

I would like to write the regular expression for the set of all binary strings where there are no three consecutive 0's. The following strings are part of the language: ...
1
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4answers
161 views

Non-Deterministic Turing Machine Algorithm

I'm having trouble with this question: Write a simple program/algorithm for a nondeterministic Turing machine that accepts the language: $$ L = \left\{\left. xw w^R y \right| x,y,w \in \{a,b\}^+, ...
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3answers
671 views

Is indistinguishability an equivalence relation?

Let x and y be strings and let L be any language. We say that x and y are distinguishable by L if some string z exists whereby exactly one of the strings xz and yz is a member of L; otherwise, ...
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2answers
68 views

Why this lemma is true?

Let $\Sigma$ be an alphabet of size $|\Sigma|=k$. Let $w\in\Sigma^*$ be a word over $\Sigma$. If $|w| > 2^k$, then $w$ contains an infix $y$ with $|y|\ge 2$, such that every letter occurring in y ...
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2answers
274 views

Is this DFA correct

I'm supposed to construct a DFA which accepts { w | w is a word except 'aa' and 'aaa' } Is this the correct solution? The thick line state is supposed to be the end state. EDIT Sry, somehow ...
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2answers
95 views

Convert NFA with only accepting states to regular expression?

Suppose you wanted to find a Regular Expression that defines the language accepted by the folowing Finite State Automaton. Where S1 and S2 are both accepting states. How would I go on doing this?
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3answers
95 views

CFG for language l

please solve this question.thanks Consider the language L expressed by (a+b)*a defined over Σ = {a, b}. Draw FA and construct the CFG corresponding to the language L.
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2answers
360 views

Constructing a finite automata from a subset of its language

I am attempting to solve the following problem: Let $M=(Q,\Sigma,\delta,q_0,F)$ be a deterministic finite automata which accepts $L(M)$, and let $E$ be the subset of $L(M)$ consisting of all words of ...
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1answer
37 views

Write the regular expression of the language that the DFA accepts.

I am given a DFA and I have tried to write the regular expression of the language that it accepts. This is the DFA that I am given: I have found some words that the DFA accepts: ...
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2answers
49 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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2answers
29 views

Regular expression translation.

Given a set {1,2,...9} how can I construct a regular expression starts with a 3 has no 8's and has even number of 6's? Here's what I tried: $$$$ Define a new set no8 = {1,2,3,4,5,6,7,9} ...
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2answers
200 views

The halting problem for tapes that are or are not completely blank

Is it possible to construct a Turing machine that halts only if the tape is not completely blank? Also, is it possible to construct one to halt if the tape is completely blank? Intuitively, I think ...
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3answers
152 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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1answer
71 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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2answers
32 views

Construct a deterministic finite automation

The question asks to: construct a DFA which accepts exactly $\frac{n(n-1)(n-2)}{6} + \frac{n(n-1)}{2}+1$ many members of $\{0, 1\}^n$ for every n. I have no idea where to start to constructing the ...
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2answers
1k views

Concatenation of 2 finite Automata

I have some problems understanding the algorithm of concatenation of two NFAs. For example: How to concatenate A1 and A2? A1: ...
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2answers
204 views

Prove that any finite set of words is regular. How many states is sufficient for a single word $w_1…w_m$?

DFA Prove that any finite set of words is regular. How many states is sufficient for a single word $w_1...w_m$? For part 2, wouldn't it require M states if the word length is M?
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1answer
121 views

Is the language $\{i|f(i)=1\}$ recursive, function $f$ is described further inside. [duplicate]

Possible Duplicate: Show $f$ is primitive recursive, where $f(n) = 1$ if the decimal expansion of $\pi$ contains $n$ consecutive $5$'s $$L = \{i\mid f(i)=1\}$$ $f(i)$ equals $1$ if ...
1
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1answer
381 views

A regular expression for the words that don't contain the sequence $ab$ over $\{a,b,c\}$

The following is an exercise in a book I am reading: Let $\Sigma=\{a,b,c\}$, define $L$ to be the language of all words over $\Sigma$ that do not contain $ab$ as a sub-word. Find a regular ...
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2answers
44 views

Why |w|>=m in pumping lemma?

If L is a regular language, then there exists a constant n (which depends on L) such that ...
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1answer
45 views

Regular expressions

I have this assignment and I have to prove that $$ (b+aa^* b)+(b+aa^* b)(a+ba^* b)^* (a+ba^* b) = a^* b(a+ba^* a)b^* $$ How do I prove this? What I have is this: $$\begin{align} \text{LHS} ...
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3answers
50 views

Show that a language is regular

Show that language $B$ is regular: $$B = \left\{1^ky\mid y\in \{0,1\}^*\right\} $$ $y$ contains $\ge k$ symbols $1$ So I try in following way - I'll draw DFA: What about my solution? Is it good?
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2answers
31 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
30 views

FSM to be regular with atleast two 0's and at most two 1's

Is it possible to construct a FSM to prove that the set $X$ is regular, where $$ X = \{s \in \{0,1\}^* \mid \text{$s$ contains at least two $0$'s and at most two $1$'s}\}\ ? $$
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1answer
42 views

Can someone explain to me how this is a proof?

I honestly don't understand how this proves anything.
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2answers
43 views

Closed regular languages

Are regular languages closed under the following construction? $f(L) = \{w \mid w \in L$ and for all prefixes $x$ of $w$ it holds that $x \notin L$ $\}$
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1answer
25 views

Subset of regular languages

Assume $L_1$ and $L_2$ two regular languages, and $L_1\subseteq L\subseteq L_2$. Does this imply that $L$ is a regular language? Thanks in advance.
1
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1answer
89 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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2answers
71 views

New automaton that runs two others step by step?

If I have two automatons for two languages ($M_1, M_2, L_1,L_2$ respectively), what would be the procedure to mix them by defining a new automaton such that the new automaton would accept the words ...
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1answer
169 views

Is $\epsilon$ in every alphabet?

Given a $\Sigma$ an alphabet, is $\epsilon$ in it logically? For example, if I have a function $ f : \Sigma \to \Sigma $, can I define it for example $ f(\sigma) = \epsilon$? even if my alphabet is ...
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2answers
162 views

Whether $L=\{(a^m,a^n)\}^*$ is regular or not?

I am condidering the automatic structure for Baumslag-Solitar semigroups. And I have a question. For any $m,n \in Z$, whether the set $L=\{(a^m,a^n)\}^*$ is regular or not. Here a set is regular means ...
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1answer
639 views

How to prove CYK algorithm has $O(n^3)$ running time

I have a final coming up in few days, and the professor mentioned the CYK algorithm. I want to be prepared for the final. I'm trying to find out how to prove the algorithm has worst case running time ...
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2answers
728 views

Showing that a language is regular - Pushdown Automaton

So what I have to prove is that $L$ is regular given that the stack of PDA for $L$ never grows beyond $n$ entries on any input, and in this case $n=200$.