Questions related to formal languages, grammars, and automata theory

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4
votes
2answers
89 views

Method for measuring the 'similarity' between FSA grammars?

I'm working with a pattern matching algorithm that generates an acyclic finite state automaton that accepts a given text string and all its substrings. The FSA algorithm is being run on a symbolic ...
2
votes
2answers
130 views

show that language $L'$ is regular (given $L$ regular)

I am working on the following question: $L$ is regular. Show that $L'=\{x|\exists y,z,\ xyz\in L\wedge |x|=|y|=|z|\} $ is also regular. Firstly I show my idea. When you accept it I will try to ...
10
votes
1answer
257 views

Algorithm to test whether a language is context-free

Is there an algorithm/systematic procedure to test whether a language is context-free? In other words, given a language specified in algebraic form (think of something like $L=\{a^n b^n a^n : n \in ...
7
votes
1answer
181 views

Is there a strictly non-deterministic one-counter language whose complement is one-counter?

Let $A= \{L \mid L \;\text{is one-counter and \(\bar{L}\) is also one-counter} \}$ Clearly, $\text{Deterministic one-counter} \subseteq A$ Is it the case that $ A = \text{Deterministic ...
6
votes
1answer
118 views

Using the Chomsky-Schutzenberger theorem to prove a language is not context-free?

The Chomsky-Schutzenberger representation theorem states that a language $L$ is context-free iff there is a homomorphism $h$, a regular language $R$, and a paired alphabet $\Sigma = T \cup ...
-1
votes
1answer
141 views

DFA for every run of a's=2 or 3

I am trying to create a dfa for L={w: every run of a's has length either two or three} this is my attempt at the solution..i feel like I am missing something..?
-2
votes
1answer
38 views

Prove that $(L^*M^*)^* = (L\cup M)^*$

I would like to find out how to prove this statement. Thank you. Well I think that I proved one part of the statement, but my proof doesn't really look elegant. My proof of $(L\cup M)^* \subset ...
20
votes
0answers
525 views

Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?

We define a regular tree language as in the book TATA: It is the set of trees accepted by a non-deterministic finite tree automaton (Chapter 1) or, equivalently, the set of trees generated by a ...
8
votes
0answers
60 views

Machines for context-free languages which gain no extra power from nondeterminism

When considering machine models of computation, the Chomsky hierarchy is normally characterised by (in order), finite automata, push-down automata, linear bound automata and Turing Machines. For the ...
7
votes
0answers
116 views

Test whether two languages are equal, when give in algebraic form

This sub-problem is motivated by Algorithm to test whether a language is regular. Suppose we have two languages $L_1,L_2$ that are expressed in "algebraic" form, as formalized below. I want to ...
6
votes
0answers
268 views

Can this CFG be written into an equivalent LL(1) grammar?

I have the following CFG which I suspect cannot be rewritten to one which is LL(1): $S \rightarrow \epsilon\ |\ aSbS\ |\ bSaS\ |\ cSdS\ |\ dScS$ I've thought about it for a while, and can't seem to ...
5
votes
0answers
57 views

Calculating with regexes

We use a regex engine (say, PCRE) that allows grouping subexpressions with parentheses and recalling the value they match in the search / replace expressions (backreferences, denoted by \i for ...
5
votes
0answers
154 views

How to disambiguate symbolic regular expressions

What I mean by a "symbolic regular expression" (if there already is a different name for this I'm not aware of it) is a regular expression that may include exponents that are symbolic arithmetic ...
4
votes
0answers
39 views

Languages recognized by finite state automata of polynomially growing size

In the course of my research (condensed matter physics stuff), I stumbled over the following concept: The class of regular languages can be defined via finite state machines (FSM): A language $L$ ...
3
votes
0answers
68 views

Prove or disprove that every $L$ in this class is a CFL iff $L$ is equivalent to a substitution

Let $L$ be a language with every string of the form $(w_i\#)^*$ with $w_i\in\{0,1\}^*$. Set $w'\sim w$ if there is a permutation $\pi_1$ such that $w_i=w'_{\pi_1(i)}$ for all $i$. If additionally ...
3
votes
0answers
375 views

Intuitive description for training of LSTM (with forget gate/peephole)?

I am a CS undergraduate (but I don't know much about AI though, did not take any courses on it, and definitely nothing about NN until recently) who is about to do a school project in AI, so I pick a ...
2
votes
0answers
67 views

Removing hidden ambiguity in grammar using left factoring

I am trying to reduce the grammar to LL(1) for a hypothetical language we created. I have removed most of the left factoring issues in the grammar, using the general rule of introducing new ...
2
votes
0answers
42 views

Tightest upper bound on length of distinguishing string in Hopcroft's algorithm

Hopcroft's algorithm is an algorithm for DFA minimization that produces a table identifying which pairs of states are distinguishable. What is the tightest possible upper bound (with proof) on the ...
2
votes
0answers
37 views

Tree Languages are Word Languages on an Infinite Alphabet of Contexts

I have been reading the book Tata (Tree Automata Techniques and Applications), and there is a sentence I have read thousands of times, yet still don't quite understand. In the beginning of Chapter 2, ...
2
votes
0answers
166 views

Good introductions to Formal Language Theory and Formal Grammars

Does anyone know any good introductions to Formal Language theory and Formal Grammar, that cover the mathematical basis of Syntax and things like context free grammars and pushdown automata. In ...
2
votes
0answers
46 views

What kind of structural features of strings can be described by regular grammars?

Context-free grammars, as well as other types of grammars, can naturally associate structure with the strings of the defined language, for example tree structures in the case of context-free language. ...
2
votes
0answers
126 views

Example of execution fragment of multi-process transition system

Here is a simple transition system of beverage vending machine: The exemplary execution fragments can look like this: Now, imagine we have multi-process TS where processes are identical and ...
2
votes
0answers
88 views

The grammar of the GeoQuery language

GeoQuery is a dataset used for benchmarking semantic parsers. It contains 880 queries about USA geography. The queries are in Prolog format, for example: ...
2
votes
0answers
67 views

Computational power of nondeterministic type-2 min-heap automata

I have asked a series of questions concerning capabilities of a certain class of exotic automata which I have called min-heap automata; the original question, and links to others, can be found here. ...
1
vote
0answers
60 views

Are DCFLs closed under concatenation with a regular language?

I have found various opinions saying they are (a link to one is given in D.W.'s comment). However, a proof that DCFLs themselves are not closed under concatenation found here on StackExchange seems to ...
1
vote
0answers
46 views

How to convert CFG with Kleene Star, Kleene Plus, and Question Mark to Chomsky Normal Form?

I am fairly new to formal language theory but understand how to convert simple CFGs into both Chomsky normal form and Greibach normal form. However, I have not seen any examples of how to do that when ...
1
vote
0answers
26 views

How many restricted length strings are there without significant repetitions

Let us fix an alphabet $\Sigma$ of size $c$, then we have the finite language $\Sigma^n$ which is the set of all $n$ length words. For each $N,M$ how many words are there in $\Sigma^n$ such that no ...
1
vote
0answers
65 views

How to draw a clearly arranged DFA of a language with modulo rules?

I know how to draw a DFA, but I have problems with this specific one: ${L = \{ w \in \{a,b,c\}^* \mid \ |w|_a \equiv |w|_b - 2|w|_c \mod \ 5 \} }$ This language is regular and there has to exist a ...
1
vote
0answers
161 views

Is the complement of this language Context-Free $\{(a^nb^n)^m \mid n>0,m>0\}$?

I've been asked to decide whether a given language is a Context-Free Language (CFL). If yes, I should find the grammar that creates her, and if not, I need to prove it (with the pumping lemma). The ...
1
vote
0answers
118 views

How to describe the language generated by S → a | S + S | S S | S * | ( S )

I am trying to solve the following problem from Aho, et al., Compilers: Principles, Techniques, & Tools (2nd ed.), exercise 2.2.2e: What language is generated by the following grammar? ...
1
vote
0answers
38 views

Is this theorem about left-factored grammars correct?

I am working on CFG grammars, LL grammars in particular and I encountered the following theorem in the slides of presentations written by my professor: A CFG grammar cannot be left-factored if all ...
1
vote
0answers
109 views

Correct approach to Mapping Reduction from $E_{TM}$

as the title states, I am trying to figure out if my approach to solving mapping reduction from $E_{TM}$ to some other language is correct. As you surely know, $E_{TM} = \left \{ < M> \mid M \ ...
0
votes
0answers
20 views

Rules language / DSL expressivity measure

Languages to express domain rules are quite diverse from very simple and inexpressive to Turing-complete programming languages. If we consider developing some DSL (domain-specific language), is there ...
0
votes
0answers
58 views

The pumping lemma - Proving that this language is NOT context free

I would like to find out if this language is context free or not: $\qquad L=\{a^{i}b^{j}c^{k} \mid i<j,i+2j+3<k\}$. I've tried to apply the pumping lemma taking out $w=a^n b^{n+1}c^{3n+6}$ ...
0
votes
0answers
103 views

What is the limit for Turing machines with 2 states and 3 symbols that halt?

I read here that a proof has been offered that a Turing Machine with 2 states and 3 symbols can be universal (in that it is capable of arbitrary finite computations). Even if this proof is accepted, ...
0
votes
0answers
55 views

regular expression of star-height 1

Is there a regular expression of star-height 1 (i.e. without two nested Kleene stars) for the following language : $a^*(bb^*aa^*ba^*)^*$ ?
0
votes
0answers
89 views

How to convert the following grammar to LL(1)?

The following grammar is given: \begin{align*} M &\rightarrow d M d \\ M &\rightarrow e M e \\ M &\rightarrow f M f \\ M &\rightarrow \varepsilon \end{align*} I've checked it with ...
0
votes
0answers
58 views

Union, Intersection, Difference, etc. of different types of languages

I am preparing for a competitive exam (GATE) in which questions are asked in Automata about operations among different types of languages. For example, If $L_1$ is recursive & $L_2$ is ...
0
votes
0answers
52 views

Designing CFG for sequences of words of which two arbitrary ones are reversals

Let $L$ = {$x_1\#x_2\#...\#x_k$ : $k\;\ge\;1$, each $x_i\;\in\;\{0,1\}^*$ and $\exists i,j$ such that $i < j$ and $x_i$ = $x^R_J$}. For example, $001001\#0010\#100100\#00001$ is in $L$ because ...