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

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L is a context free language over {0, 1}, prove, disprove:

cont... L is a context free language over {0, 1}, prove, disprove: L1 is a CFL over {a, b}, where L1 is the language of all words from L, that 0 is converted to a and 1 is converted to bba. Thanks ...
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2answers
692 views

How to prove by pumping lemma these languages are not regular?

$L_1 = (a^k * b^r \mid k \neq r^2)$ $L_2 = (a ^{\sum_i ^n t} \mid n > 0 )$
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0answers
163 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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1answer
139 views

Number of states required for deterministic finite automata

Lemma in text: Let $c$ be a constant and $L = \{1^c\}$ (the singleton language containing the string of $c$ many 1's). Then no DFA with < $c$ states can accept $L$. The given proof assumes ...
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1answer
506 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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1answer
92 views

Prove min(L) = all words in L that they don't have any prefix of themselves in L

We define the minimal words language of $L, \min(L)$, to be the language of all words in $L$ that don't have any prefix in $L$. Assume $L$ is regular language. I need to prove by building an ...
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1answer
361 views

Show that every finite automaton is a one-state pushdown automaton

I'm reading a book that states: Every finite automaton is a one-state push-down automaton How can I go about proving it?
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2answers
588 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 $n=200$.
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3answers
111 views

showing that a regular language is regular after taking a letter off or after adding letters

I'll be happy to recieve help with this one: Given the regular language $L$ defined over alphabet $\{a,b\}$, show that the following languages are also regular: $\{xy\mid xay\in L\}$ ...
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1answer
810 views

Formal proof of the concatenation of two regular languages automaton

During an exercise for college, given two NFA's, $A_1\text{ and }A_2$ that accept the languages $L_1\text{ and }L_2$, I've built a NFA, $M$ that accepts the language $L_1*L_2$ (concatenation). The ...
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1answer
63 views

Is regularity is preserved under reversal?

When talking about languages and regular languages. Can I say that reversal preserved regularity since if the language L is regular, we can generate it by right linear grammar. Therefore, the ...
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2answers
110 views

Solve right linear set equations

Trying to solve this set of equations. I'm feeling like I'm making it so complicated. Of course + is union. Am I on the right track? A = 0B + 1D B = 0C + 1A C = 0A + 1B + λ D = OD + 1C + λ A ...
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2answers
401 views

Proof Two Grammars Generate the Same Language

I have two right linear grammars and I need to prove they both generate the same language. What is the right way to do it? L1: $S \rightarrow 0A$ $S \rightarrow 1B$ $A \rightarrow 0A$ $A ...
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1answer
79 views

Is this NFA correct?

For the language L = $\{\Sigma^*. 0 .\Sigma^5 . 1. \Sigma^*\}$ The NFA must have 8 states. Also, what would be the upper bound on the number of states of a DFA recognizing L.
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1answer
75 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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3answers
80 views

Is this DFA correct for the language $\{w \mid w \in \{ab\}^*\}$?

Language: $\{w \mid w \in \{ab\}^*\}$
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1answer
949 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
1k 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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2answers
529 views

Can _any_ NFA be converted to a DFA?

I was wondering if for every NFA there exists an equivalent DFA? I think the answer is yes. How would one prove it? Since I'm just starting up learning about Automata I'm not confused about this and ...
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1answer
412 views

Creating a minimal dfa from a regular expression

Having a bit of difficult with the following question: Create a minimal dfa for the language $L(r)$ where $r = a^*\bigl((ab+b)^*\bigr)$?
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2answers
437 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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2answers
77 views

What is the difference between the automatas for the regular expressions (a + b)* and (a* + b*)?

I know that the automaton for the regular expression (a + b)* will just have one state, where the initial state = the accepting state and there is one edge going into that state labelled a,b. Sorry, ...
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1answer
125 views

Proving what are the equivalence classes of the relation $R_{L}$

I tried writing a solution for a question that defined a language $L$ and asked to find the equivalence classes of the relation $R_{L}$ The TA wrote that "this is not what you should prove" (or "this ...
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0answers
105 views

A question about the language of a non-deterministic FSA defined by a non-deterministic FSA with epsilon moves

I have tried to solve the following problem and my claims were marked as wrong, I would appreciate it if someone could point out my mistake Let $A=(Q,\Sigma,q_{o},\delta,F)$ be a finite state ...
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0answers
116 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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1answer
107 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 ...
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1answer
58 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
53 views

Legal actions on a PDA and terminology

I am unsure about the following so I would like to verify if my statements are true: We can remove at most a single character ($Z\in\Gamma$) from the PDA (top ?) of the stack with one step of the ...
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1answer
152 views

Proving that for every context-free language there exist a pushdown automata $M$ s.t. $L=L_{e}(M)$

The book I am reading have proof for the statement Every context-free language there exist a pushdown automata $M$ s.t. $L=L_{e}(M)$ For the case $\epsilon\not\in L$. The proof uses greibach ...
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1answer
181 views

Proof related to a finite state machine

I have this confusion related to a finite state machine M such that if the number of states n>=2, then there exits i $ \overset{i}\equiv{}= {\overset{i+1}\equiv{}}$ I mean the $i^{th}$ equivalence ...
2
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1answer
99 views

Definition of a deterministic Pushdown automaton

According to my book the definition of a deterministic Pushdown automaton allows for $\delta(q,\epsilon,Z)$ to be non-empty if $$\forall\sigma\in\Sigma:\,\delta(q,\sigma,Z)\neq\emptyset$$ Can someone ...
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2answers
100 views

Pushdown automata - definition and definition of $\vdash$

I am reading about pushdown automata and I don't understand the definition of $\vdash$. My book writes that $$(q,aw,Z\alpha)\vdash(p,w,\beta\alpha)$$ if $$(p,\beta)\in\delta(q,a,Z)$$ Can someone ...
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1answer
110 views

Finding if two machines can be equivalent

I have this problem: Consider the following machines M1 and M2. M1 has initial state A and the initial state of M2 is unspecified. Can the machines be made equivalent by the correct choice of ...
2
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2answers
3k views

Pushdown automata for the language $\{ w001w^\text{R} : w = \{0,1\}^* \}$

I'm trying to make a PDA that accepts the language $\{ w001w^\text{R} : w = \{0,1\}^* \}$ by empty stack. (Here $w^\text{R}$ denotes the reverse of the string $w$.) Our stack symbol s $\#$. I've come ...
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2answers
766 views

Existence of NFA of a reverse of a language

Hi, I'm really stuck on how to prove the following: Given that $L$ is a language and $L'$ is a set such that $L' = \{w \mid w' \in L\}$ where $w$ is the reverse of $w'$, e.g., if $w = a a b$ then $w' ...
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1answer
193 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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0answers
196 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
47 views

Given two NFAs, is there a way to figure out if there exists a language that works for both of them?

Given two non-deterministic finite automaton, is there a way to determine if there exists a single language that satisfies them both?
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3answers
258 views

Deciding equivalence of regular languages

Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows: build two finite automata $M_R$ and $M_S$ such that $L(R) = L(M_R)$ and $L(S) ...
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2answers
52 views

Derivation of a property of a language

I am confused how this relation is derived for a language on alphabet V A,B The relation is $$ (A\cup B)^*=(A^*B^*)^* $$ I am confused how this is derived. Any pointers?
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38 views

Confusion related to a property of languages on some alphabet V

I came across this relation betwen tww sets of languages formed from the alphabet V. A,B The relation is $$ A^*\cup B^* =((A\cup B)^*)^* $$ I am confused how this is derived. Any pointer?
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1answer
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Existence of NFA for this language

I'm given a task to find (and prove) such language $L$ in the alphabet $\Sigma = \{a,b\}$ with all words less than $1000$ in length, for which any DFA/NFA will have more than $10^{10}$ of states. For ...
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1answer
218 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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3answers
130 views

Showing $L=\{uw \mid \exists v:uv\in L_{1},vw\in L_{2}\}$ is regular

Let $L_{1,}L_{2}$ be regular languages and define $L:=\{uw \mid \exists v\in\Sigma^{*}:uv\in L_{1},vw\in L_{2}\}$. I wish to prove that $L$ is regular using only closure properties (such as ...
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1answer
71 views

Is $f$ is computable by a finite automaton, the dual of $f$ is thus computable also?

et $A$ be a finite alphabet. Let $A^*$ denote the language of all words in $A$, and $\epsilon$ the empty word. Let $\rho : A^* \to A^*$ denote the "reverting" map, that transforms $a_1a_2\ldots a_n$ ...
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1answer
343 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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1answer
75 views

Regular expression arithmetics

What are the rules of regular expression arithmetics ? For example: Let $\Sigma=\{0,1\}$ $1. 1+01=(\epsilon+0)1$. $2. (\epsilon+00)^*=(00)^*$
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124 views

A question about the regular languages being closed under Boolean operation (how to generalize)

I know that if $L_{1},L_{2}$ are regular languages then so is $L_{1}\cap L_{2},L_{1}\cup L_{2}$ are regular languages, I also know that $L$ is regular $\implies L^{c}$ is regular . It is easy to ...
3
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1answer
255 views

Proving Turing Completeness by Simulating Rule 110

Something I've heard often is that Rule 110 is the `simplest' Turing-complete formalism. As a programming exercise in a language I am new to, I implemented a function that computes from an initial ...