# Questions tagged [finite-automata]

Questions about finite automata, an elementary automaton model with finite memory. It is equivalent to regular languages and the basis for many more complex models.

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### How to work thompson's construction algorithm?

Could someone explain to me, through an example, how the McNaughton–Yamada–Thompson algorithm works in a way that I can understand? Thank you in advance for your helpful response!
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### Is this correct Context free grammar(CFG) for these two languages?

Question 1: L = { 0^n 1^n | n > 0 } My answer = S -> 0 S 1 | 10 Question 2: L = { 101^n0^2n | n > 0 } My answer = S -> 101 S 00 | 100 Can anyone correct this if there is any issue with ...
1 vote
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### In regular language inference, how is the observation table kept consistent?

I am trying to understand the background literature on regular language inference in the TTT paper ("The TTT Algorithm: A Redundancy-Free Approach to Active Automata Learning" by Isberner, ...
1 vote
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### How to construct complement of NFA universality?

Given an input NFA, can one construct an NFA that is universal (that is, accepts all its inputs) if and only if, the input NFA isn't universal? I tried to use the fact that NFA-universality is PSPACE-...
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### Applications of ω-automaton in engineering

Finite automaton is a well-known topic which finds applications in software engineering (e.g. pattern matching or regular expressions in general). ω-automaton are another kind of automaton, defined ...
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### Determining the language of a DFA

I've tried quite a large amount of examples, starting from 1 bit all the way up to 5 bits yet I couldn't put my finger on any recurring pattern of words that are accepted. the DFA is as such: The ...
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1 vote
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### How to build a deterministic or non deterministic finite automat for a language

$L_1$ and $L_2$ are two languages that have a finite deterministic automaton. $Σ = \{0,1\}$ for the two languages. Let $L=\{x∈L_1\mid \text{there exists } y∈L_2,\text{such that } \#_0(x)=\#_0(y)\}$, ...
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### Proving that the intersection of two languages is regular or not?

Let $B$ and $C$ be two languages on $A = \{a,b\}$: $B = \{ w \mid w \text{ has the same number of }a\text{ and }b\text{ symbols}\}$ $C = \{ a^n b^m \mid n,m \ge 0\}$ Describe $B \cap C$ and determine ...
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### Intersection of a regular and a non-regular language can give a non-regular language?

Determine two languages L and L1 on the alphabet {a,b} with regular L, L∩L1 is non-regular and L1 does not contain L. I came up with this solution: L = { w | w ends with the string aba} L1 = {w | w = ...
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### Can the intersection of 2 non-regular languages ​be a regular and infinite language?

The outline of the exercise is this: Describe two non-regular L1 and L2 languages on A = {a,b} such that L1∩L2 is infinite and regular. The proposed solution to this exercise is: L1 = {a^n b^n | n >...
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### Proving a language is not regular using Pumping Lemma

So I am given the language qr where q is any combination of a's and b's. r is then the reverse of whatever q is. For example, abba is in the language because we can make a q = ab and r = ba I have to ...
1 vote
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### Is there a symbol in FA/FSM for everything else

I remember in a Computability Theory class building a state machine for an assignment that required many 'go-back' transitions on $\Sigma - symbols$ for all symbols not already handled. This was very ...
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### Can someone explain the following NFA?

I'm learning NFA and the following question is confusing me. Can Someone explain the answer. I have a feeling that it might be wrong. Thank you!
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### Create DFA such that every substring of w of length 3 contains two or three 0's with w={0,1}

I am attempting to draw the DFA of this problem but I'm kind of stuck. In another post I saw that the DFA should have 10 states. I'm finding it hard to even understand the question, is the input to ...
56 views

### 1 way 2 stack and 2 way 2 stack Pushdown Accepters that accepts $L=\{a^{(n^2)} \mid n \geq 1\}$

Using a 1 way 2 stack, and a 2 way 2 stack PDA, I want to check if the length of an input string is strictly a perfect square number. How can I do this in both approaches?
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### Why does lexer has O(n) time complexity?

According to my CS knowledge so far, a lexer uses DFA(which takes linear time) for 'each' token type to find the next token, so in the worst case, it should try 'all possible' token types of a ...
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### How to find prefixes and suffixes for infinite languages? (Automata)

L= {abc} prefix = {epsilon,a,ab,abc} suffix = {epsilon,c,bc,abc} It's easy to find suffixes and prefixes for finite Regular languages. But what will be the ...
310 views

### Turing machine for a^n b^m c^n d^m

The state diagram for the initial part of this turing machine given as: Here, we are basically traversing through the input tape, changing occurence of 'a' to X1, and 'c' to X2. After that we go back ...
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### What is the minimum length word accepted by the product of these simple loop automata?

Let $A_{n} = (aa|aaa|aaaa|\dots |a^{n-2})(a^{n})^*$ where $n \geq 4$ is some natural,and $A_2 = (a^2)^*, A_3 = (a^3)^*$. Clearly every transition is thus labeled by an $a$. From now on let $A_n$ ...
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### What is the Minimum length of a string that is accepted by a DFA that shows that the language accepted by that DFA is infinite?

What is the minimum length of a string that is accepted by a DFA, shows that the language accepted by that DFA is infinite? I checked this post How to determine if an automata (DFA) accepts an ...
230 views

### Construct a regular grammar that produces all possible strings of $\Sigma = \{a,b\}$ that do not contain substring 'abba'

I'm really stuck here and do not know what to do. So far, I've constructed a DFA and a regular expression that produces the aforementioned set of strings. Namely, the DFA looks like: After a lot of ...
1 vote
79 views

### DFA for even concatenation of strings from a language

If I have a deterministic finite automaton (DFA) with a language $W$, and I need to create another DFA that returns all the strings that are a concatenation of an even number of strings in $W$, how ...
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### After converting a CFG to a PDA, what is an example input string to the PDA?

For example, the following CFG: S$\rightarrow$aSb|$\epsilon$ A valid string within the language described by the CFG would be, e.g., "aabb". The converted PDA has the following form: The ...
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### Binary combinatorics with rank

I am looking at finding acceptable binary values with maximum 2 consecutive 1s and 0s, from a range of maximum 6 bits (2^6 values). Also, I want to rank and unrank these subset of values (in ...
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### DFA for "K" bit value with max "n" consecutive "0"s and "1"s

I had posted earlier a question regarding ranking max "n" consecutive "0"s and "1"s in a "K"-bit string - Order in a subset I got clarification regarding how ...
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### Minimum number of states in a DFA

Consider the language L given by the regular expression (a + b )*b(a + b) over the alphabet {a, b} . The smallest number of states needed in a deterministic finite-state automaton (DFA) accepting L is ...
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### How to convert a NFA to alternating finite automata AFA?

I am trying to construct an AFA from a NFA, how do I know if a state of NFA becoms existential or universal in the AFA?
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### Question about remainder in automata construction that checks divisibility

I'm trying to understand de construction of a DFA from the book "Introduction to Automata Theory, Languages, and Computation by John Hopcroft and Jeffrey Ullman 1st ed" It says: Where it ...
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### Is the set of all DFAs countable?

Let $\Sigma$ be a finite nonempty alphabet. Is the set of all DFAs over $\Sigma$ countable? I know the set of all regular languages is countable, however, it is impossible to build an injection from ...
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Let us assume, we have a FSM (to be precise, an epsilon-NFA) in which I've got redundant structures. I am looking for a framework that allows to factor out the redundant part into some kind of ...
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### Translating non-deterministic finite automata with counters to deterministic ones

I know there are algorithms for translating regular NFAs into their corresponding DFAs. Can these algorithms be applied to automata that employ transitions involving counters in a straightforward way, ...
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### How to convert NFA with multiple start states to NFA with single start state without epsilon transition?

I can easily convert NFA with multiple start states to NFA with single start state with epsilon transition. But I want to know how can I do so without epsilon transition Here's my idea Suppose s1 and ...
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### What can i say about L1 given that L2, L1L2 and L2L1 are regular?

I found this question in one of our past exams, and I'm not to sure about the correct answer. I have a language L1 (which i don't know anything about) and another language L2, which is regular, the ...
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### For each of the following languages, give a regular expression over {a, b}

$\{a^{2n}b^{n+k+1}a^k ∈ \{a, b\}^∗ \mid n \ge 0, k \ge 0\}$
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### How to create a DFA with memory

How can I create a DFA that has memory such that I could write a rule for this DFA that states ...
108 views

### What strings are accepted by the described DFA?

Could someone clarify what strings are accepted by the described DFA? Draw a DFA for the set of all strings with two consecutive 0’s followed by two consecutive 1’s over Ʃ= {0,1}
1 vote
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### Does graph traversal require stack machine?

I'm writing a graph traversal function to be used in a garbage collector. To avoid stack overflow, I used a finite state machine. Roughly, it descends into child nodes recursively to mark objects, and ...
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### nfa of the Language L={w belongs to (a,b)*/w starts with aa or ends with aa} with or being not exclusive

I have a question I need to give the NFA of the following language: L={w belongs to (a,b)*/w starts with aa or ends with aa} with or being not exclusive meaning I can have a word that starts with aa ...
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### Does there exist terminology to discern NFA states and NFA-transformed-into-DFA state?

I can't find this from surface search in the literature When one observes/simulates NFA after processing several characters, one has to consider both "internal states of NFA" (individual ...
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1 vote
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### Meaning/veracity of "each state has [..] or two outgoing ϵ transitions" in Thompson's construction

The dragon book lists properties of an NFA N(r) created using Thompson's construction, in particular: Each state of N(r) other than the accepting state has either one outgoing transition on a symbol ...
1 vote
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### Regex to DFA - How do I implement parsing preferences in regex search?

I've tried a to implement a Regex to DFA converter, and it works, so far, but I don't understand how to implement "parsing preferences" in the DFA. A classic example is $a^*$. This regex is ...
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### Real-life applications of pure Mealy machines

I'm currently studying formal methods in software engineering related to state machines, specifically Mealy machines. This made me wonder how relevant Mealy machines really are for practical ...
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### Finite State Machine without recursion

I have found utility in state machines that do not have the complexity of recursion: no self or ancestral transitions. Is there a name for this sub-category of FSMs? Are they commonly used or ...
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### Theory of computation

I am trying to look answer for this question of toc please help me find the answer. The question is : Construct epsilon NFA(Non deterministic finite automata) for regular expression (0+1)*1(0+1)
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### Creating a deterministic finite automaton for strings of 2k ones and 3q zeros or a general language

I was requested to draw the graph of a finite state machine whose language is $$L = \{ w \in \{0, 1\}: w \text{ has 2k ones and 3q zeros }\}$$ In other words, the number of ones must be even and ...
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1 vote
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### Counting States in the trim automaton for $\cup_{i=1}^{p} L_i \circ L'_i$

Preliminaries. Let $n,m,i,j,p,c \in \mathbb{N}$ with $n,m,i,j,p,c \geq 1$. Let our alphabet be $\{0,1\}$, with non-empty languages $L_i \subseteq \Sigma^n$ and $L'_i \subseteq \Sigma^m$. The other ...
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### ALL_{NFA} is PSPACE-complete

Show that $ALL_{NFA}$ = {$\langle M\rangle : M$ is $NFA$ and $L(M) = \Sigma^*$} is $\text{PSPACE-complete}$. I've manged to prove that the langauge is in $\text{PSPACE}$. Indeed, it is easy to see ...
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### Does my finite state automaton accept a string iff it contains the given string as a substring?

I am trying to write down the generalized form of the finite automata which accept strings which contain as a substring an arbitrary string. Here is what I have come up with — I was hoping someone ...
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### Counting States in the trim automaton for $(L_1 \cup L_2 \cup \ldots \cup L_p) \circ L'$

Preliminaries. Let $n,m,p \in \mathbb{N}$ with $n,m,p > 1$. We allow that $p$ could be large but still bounded by a function of $n$: $p = O(2^n)$. Let our alphabet be $\Sigma = \{0,1\}$, with non-...
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### Counting States in the trim automaton for $L\circ L'$
Preliminaries. Let $n,m \in \mathbb{N}$. Let our alphabet be $\Sigma = \{0,1\}$, with non-empty languages $L \subseteq \Sigma^n$ and $L' \subseteq \Sigma^m$. We follow the standard definition for ...