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

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What are the states of this NFA?

I have to realize an NFA that recognizes the language of strings on the alphabet {a, b} ending with: bb, ba, baa. I thought that there must be the following states: $q_0$: the string ends with bb. ...
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215 views

Nondeterministic finite automaton proof

I am having a really hard time working the problem below out. I am not sure I am even on the right direction with this logic . Swapping the accept and reject states alone is not sufficient to accept ...
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53 views

Is this NFA correct for the language {w | w ends with an a and no a occurs between any occurrences of b}?

Language: {w | w ends with an a and no a occurs between any occurrences of b} The NFA must have exactly 3 states.
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719 views

Language that is recursively enumerable, but not recursive

I have a problem with this task: Show that this language is recursive enumerable, but not recursive: $L = \{ w \in \{0,1\}^* | M_w(x)\; \text{converges for some input}\; x \}$ (where $M$ is turing ...
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1answer
47 views

Show Two f.s.a. machines accept different input

I'm trying to solve it for two hours already. I know it somehow related to the pumping lemma Let $M_1 = \langle Q_1,S,f_1,s_1,F_1\rangle$ and $M_2 = \langle Q_2,S,f_2,s_2,F_2\rangle$ be two machines, ...
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1answer
676 views

Show two finite state machines are equivalent

Suppose $M_1 = \langle Q_1,S,R,f_1,g_1\rangle$ and $M_2 = \langle Q_2,S,R,f_2,g_2\rangle$ are two strongly connected machines. I need to show that $M_1 \equiv M_2$ iff there exist a state $p \in Q_1$ ...
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1answer
54 views

Proving $\{ll^{R}l|l\in\{a,b\}^{*}\}$ is not context free using the pumping lemma

How can I prove, using the pumping lemma for context free languages, that $\{ll^{R}l|l\in\{a,b\}^{*}\}$is not a context free language ? I tried to put $n$ as the pumping lemma constant and chose ...
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1answer
150 views

finite state machine is strongly connected

Let M be an n-state reduced strongly connected finite state machine. prove there exists an input string $w$, where $|w|\le n(n-1)/2$, s.t. M assumes each of its states at least once in response to ...
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1answer
198 views

Examples of epsilon transitions

I understand the meaning of epsilon transitions, but could someone give example where epsilon transition becomes handy?
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1answer
127 views

Random cellular automaton with three colors.

Does exist a Cellular Automata Rule that is RANDOM (like rule 30) and has 3 colors? I mean, as Wolfram says in his book, rule 30 shows a random behavior with some limits. But this happens using 2 ...
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1answer
112 views

What is this operation between sets {a, b}{c, d} do?

Not sure what this operation does, which is why i'm on here. It's not the cartesian product and no idea what it's called. I need to know to prove: For any language L, (Null set)L = L(null set) = ...
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1answer
711 views

Proving regular expressions to be equivalent

I'm trying to prove that two regular expressions are equivalent. I mean prove in the rigorous sense of the word (i.e. this is a legit proof). The process is to show that R1 is a subset of R2, and ...
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32 views

How to construct NFAs that recognize the following languages.

I am new to this computation theory and I am trying to answer the following question. Can you please check if I am on the right track? If there is any material that I can study for problems like ...
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2answers
56 views

Formal Languages - Regular Expression

I've been battling the following two questions for more than a day today. Write a regular expression (comprised of {a, b}) that contain at least two b and do not contain abb. Write a regular ...
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2answers
46 views

Draw a finite state machine which will accept the regular expression $(a^2)^* + (b^3)^*$

Draw a finite state machine which will accept the regular expression: $(a^2)^* + (b^3)^*$ In particular, I am confused by the $+$ sign, what does it exactly mean? Most literature I could find about ...
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3answers
29 views

Reducing a Decidability Problem to the Halting Problem

Let $L = \{(M, n): M$ halts on less than $n$ elements from a set S $\}$ I'm trying to come up with a generalization on how to solve these types of problems so I have not defined what S is. Since the ...
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1answer
39 views

Can someone explain me the proof?

I don't really understand the proof and I don't understand why it is M2 that "leads" to a reject state instead of M1.
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28 views

Minimal DFA for a given regular expression

How can I construct a minimal DFA from the following definition? $L=\{w \in \{a,b,c\}^* $: if the second-to-last letter from w is an $a$, then the number of $c$'s $\le 1\}$ I've already made a ...
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1answer
40 views

Finite state machine

I am doing discrete math, and we are studying Finite State Machines. But i am a little confuse on how to do this. Here is a question, Write a regular expression for the language, and define a finite ...
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1answer
19 views

Is it possible to make a PDA for $\{ ww : w \in \{ 0,1 \}^* \}$?

Consider the language $L = \{ ww : w \in \{ 1,0 \}^* \}$. I know it's easy to make a PDA for $\{ w w^\text{R} : w \in \{ 0,1 \}^* \}$ where $w^{\text{R}}$ is the reverse of $w$, but I can't think of ...
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1answer
46 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 ...
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18 views

Can the NFA constructs be minimized?

When converting a regular expression to a NFA you need to use certain constructs. My question is can these constructs be minimized? We have one with 4 states, I want to use the one with 2 states ...
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67 views

How do we prove that a particular DFA is minimal?

I guess we need to prove that there is no redundant state, so can we use state elimination and prove that the regular language is minimal? We could prove that the regular language is minimal by ...
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47 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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48 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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46 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
36 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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42 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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41 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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148 views

DFA for Boolean Formula

Let $ f\left( b_{1}, \dots , b_{n} \right)$ be a boolean function. Define $S_{f} = \{\left( b_{1}, \dots , b_{n} \right): f\left( b_{1}, \dots , b_{n} \right)=1; b_{i} \in \{0,1\}, 1\leq i \leq n \}$ ...
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80 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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47 views

CERNY CONJECTURE ,reset sequence

given a reset/synchronizing sequence $w\in \Sigma^*$ ( w is a word in the input alphabet of the DFA which sends any state of the DFA to one and the same state), prove for an automata with k states ...
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1answer
48 views

One counter automata

A one-counter automaton $M = (Q, S, \Gamma, t, s, A)$ is a pushdown automaton where the stack alphabet $\Gamma$ contains just two symbols $\#$ and $g$. The symbol $\#$ is initially written on the ...
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45 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 ...
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45 views

$DFA/NFA$ for $L(OPPOSITE)=\{uv:vu\in L\}$

I'm trying to prove that: $L(OPPOSITE)=\{uv:vu\in L\} \in L_{FA}$ given that: $L \in L_{FA}$ . I'm trying to construct a finite automata that accepts $L(OPPOSITE)$ in order to prove it but I got ...
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1answer
55 views

Working with the word w⋅y, while given the word y⋅w

$L$ is a regular language. I am given $F(L)$ such that $$F(L)= \{wy \mid yw\in L\}$$ I need to prove that if $L$ belongs to $L_\text{dfa}$, $F(L)$ also belongs to $L_\text{dfa}$. I am having a hard ...
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73 views

Constructing an NFA accepted by a grammar

Contruct an NFA of the language accepted by the grammar below. $$G=(\{S,A,B\}, \{a,b,c\},S,P)$$ $P: S\rightarrow abaS\ \ | \ cA\\ \ \ \ \ \ \ A\rightarrow bA\ \ | \ cB \ \ | \ aa\\ \ \ \ \ \ ...
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82 views

interpreting language for finite state machine

Can someone explain to me what this means in clear english and maybe give me a hint for how to make a NDFSM (non-deterministic finite state machine) that accepts this language? I understand that the 3 ...
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2answers
77 views

Finite automata and languages question

I have attempted these few simple questions, can someone let me know if this is correct please? If not please provide the answer as I learn better that way and if possible explain. i) FA1 Start = ...
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18 views

Given two minimal FSMs with one accepting a subset of the other, must a simulation exist?

As part of an example, Abstract and Concrete Categories, section 3.35, claims: For every two minimal [by number of states] $\Sigma$-acceptors $A$ and $A'$, there exists at most one simulation $A ...
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135 views

Draw the state diagrams for the PDAs

Give informal English descriptions of PDAs for the languages and draw the state diagrams for the PDAs. The complement of the language $ [a^nb^n| n ≥ 0] $ My informal description: The PDA uses it's ...
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1answer
390 views

Proof that equal-length-concatenation is a context-free language?

If A and B are languages, define A⋄B={xy | x ∈ A and y ∈ B and |x|=|y|}. For example, if A = {00, 101, 111} and B= {1, 11, 00110}, we would have A⋄B={0011}. Show that if A and B are regular, ...
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171 views

Proving Language accepted by DFA

I am stuck with a problem, as my proving skills aren't good(trying to improve). Prob: Given a State Table of DFA, decribe what language is accepted, and prove by induction it accepts that language, ...
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129 views

PDA state diagram with an inifinite languge but with no looping states

For class I'm supposed to create a PDA state diagram that is capable of generating an infinite language with no state q such that q is reachable from the start state, there is no cycle within the ...
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2answers
266 views

For the regular expression, (a* + b*) . (a.b)* , does the following automaton recognise the language it describes?

I constructed the automaton below using the assumption that the language described by the regular expression above only accepted the following strings: Empty, aabab, babab, aaaabab, bbbabab etc ...
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102 views

Mathematical formulation of 'Indra's net'

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 ...
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161 views

Confusion related to state equivalence of finite state machines

I have this confusion if there are two states of a machine p and q. Let x be an input string such that length of x = k, g be the output function and let g(p,x) and g(q,x) be the output when the input ...
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1answer
77 views

How to formally describe this Uppaal automata?

I have the following simple automata: What I'm looking for is a formal description of this based on the definition here $A=(\Sigma,\Gamma,S,s_0,\delta,\omega, F)$ How to declare all the ...
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509 views

NFA to DFA conversion, half the power set

Is there a way to tell when a NFA will use at least half the power set when converted to a DFA. I tried to create a few examples, but i just can't see a pattern that would say whether an NFA will use ...
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1answer
128 views

Difference between $(a|b)^\ast$ and $a^\ast b^\ast$?

What is the difference between $(a|b)^\ast$ and $a^\ast b^\ast$? Can you show more examples of Kleene star and patterns and explain a little bit? I've searched so many sites in Google, but it returns ...