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Questions tagged [fl.formal-languages]

formal languages, grammars, automata theory

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197 views

what is a model of computation, mathematically?

Where can I find a mathematical definition for "model of computation"? https://en.m.wikipedia.org/wiki/Model_of_computation doesn't provide a precise definition for "model of computation"--it doesn't ...
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0answers
34 views

Prove the Stack-Turing Machine is equivalent to the Classic TM

Consider a “Stack-Turing Machine” variant that operates with one infinite tape and one stack. At each step through the tape, the machine reads the input from the current tape location and from the top ...
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0answers
40 views

Incompleteness and term extraction

Is there a formalization, which from a proof that a system includes enough arithmetic extracts an arithmetic sentence in the language of PA, which is not provable in the given system? Imagine the ...
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0answers
56 views

Information theory for Mathematical Physics [duplicate]

What are some good introductory texts on information theory for someone who is classically trained in mathematical physics? Unfortunately my abilities in computer sciences and formal logic are next ...
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1answer
58 views

Chomsky-Schutzenberg Hierarchies explained for physicist (general) [closed]

I am classically trained in physics, however I have been interested in the use of information theory in studying some classical systems. As someone who is somewhat unfamiliar with the language of ...
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1answer
93 views

Ordered Grammar in THEORY OF COMPUTATION [closed]

What is ordered grammar in the theory of computation?
9
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1answer
151 views

What class of languages is recognized by finite-state automata with $k$ heads?

A DFA or NFA reads through an input string with a single head, moving left-to-right. It seems natural to wonder about finite-state machines that have multiple heads, each of which moves through the ...
1
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1answer
118 views

How are safety/liveness languages defined on the set of finite or infinite words?

Let $Σ$ be an alphabet (e.g., the powerset of atomic propositions coming from some Kripke structure, though such details are irrelevant here). For infinite words, a language $P\subseteq Σ^ω$ is ...
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1answer
120 views

Gödel-Numbering of the Context-Sensitive Languages

I would like to have a Gödel-numbering of the context-sensitive languages. Because there is no obvious syntactic distinction between LBAs and TMs, I cannot number the former in an immediate way. So I ...
5
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1answer
103 views

k-testable languages with non-constant k?

Let $p_t(w)$ and $s_t(w)$ denote the prefix and suffix of length $t$ of the word $w$, respectively. If $|w| < t$, then $p_t(w) = s_t(w) = w$. Furthermore, let $i_t(w)$ be the set of infixes of ...
3
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0answers
44 views

Extending the sequential calculus (logic over words) to allow a hierarchy of languages like the arithmetical hierarchy

Let $\Sigma$ be some finite alphabet. Then consider the logical language $\mathcal L = \{ R_a : a \in \Sigma \} \cup \{ <,= \}$ and first order formulas. For a given first order formula $\varphi$ a ...
2
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0answers
97 views

Notion of “quotient” or “inverse” for recognizable tree languages?

Related to my previous question but this time I have a better idea of what I'm actually asking. I'm looking at the following operation on recognizable tree languages (i.e. regular tree grammars, ...
2
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0answers
51 views

Regular Tree Languages are closed under quotient?

The Wikipedia page for Regular Tree Grammars notes that if $L_1$ and $L_2$ are regular tree languages, than $L_1 \setminus L_2$ is as well. However, it doesn't define this quotient operation for trees,...
17
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1answer
254 views

Is the Set of all Primitive Words a Prime Language?

A word $w$ is called primitive, if there is no word $v$ and $k > 1$ so that $w = v^k$. The set $Q$ of all primitive words over an alphabet $\Sigma$ is a well known language. WLOG we can choose $\...
8
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1answer
149 views

A conjecture related to the Cerny conjecture - counterexample/reference request

The Cerny conjecture is the statement that any synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$. The best current upper bound for the length of a ...
6
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1answer
151 views

Arranging letters to make a word in a regular language

Fix a regular language $L$ on the alphabet $\{a, b\}$, and consider the following problem. I am given as input: some number $m \in \mathbb{N}$ of copies of the letter $a$, and some number $n \in \...
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0answers
26 views

Is there an unambiguous grammar that has no left recursion or left factors, but is not in $LL(1)$?

I know that, for a grammar $G$ to belong to $LL(1)$, it is necessary that $G$ is not ambiguous; that is, every sentence has a unique parse tree in $G$. $G$ has no left recursion; that is, we can't ...
13
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1answer
237 views

Is { ww' | HamDist(w,w')>1 } context-free?

After reading the recent question "Is the complement of $\{ www \mid ...\}$ context-free?"; I remembered a similar problem I wasn't able to disprove: Is $L = \{ ww' \mid w,w' \in \{0,1\}^* \land |w|...
9
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3answers
389 views

Continuous mathematics and formal language theory

Whether there are some results on solving formal languages problems using mathematical analysis, continuous mathematics. For example, solving the intersection non-emptiness problem for a context-free ...
5
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1answer
158 views

For which $R$ is $\{0^a10^b10^c\mid R(a,b,c)\}$ context-free?

Unless I'm mistaken, a language of the form $\{0^a10^b\mid R(a,b)\}$ is context-free if and only if $R$ is a finite union of linear (in)equalities involving integer constants and the variables $a$ and ...
10
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3answers
323 views

Is the complement of { www | … } context-free?

It is well-known that the complement of $\{ ww \mid w\in \Sigma^*\}$ is context-free. But what about the complement of $\{ www \mid w\in \Sigma^*\}$?
6
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1answer
226 views

Size of complement of context-free language

Let $L$ be a context-free language, $\bar L$ be its complement and $\bar L_n$ be the length $n$ words in $\bar L_n$. What is known about $|\bar L_n|$? Note that it is known that $|L_n|$ is either ...
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1answer
81 views

Why can't a left-recursive, non-deterministic, or ambiguous grammar be LL(1)?

I've learned from several sources that an LL(1) grammar is: unambiguous, not left-recursive, and, deterministic (left-factorized). What I can't fully understand is why the above is true for any LL(1)...
11
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2answers
265 views

Does a given regular language contain an infinite prefix-free subset?

A set of words over a finite alphabet is prefix-free if there are no two distinct words where one is a prefix of the other. The question is: What is the complexity of checking whether a regular ...
9
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0answers
98 views

Are there cascade decompositions of machines that are more general than finite automata?

The idea of decomposing automata and their associated semi-groups into irreducible sub-components is due to Krohn & Rhodes and has been explored relatively thoroughly. Krohn & Rhodes gave an ...
9
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1answer
88 views

Is there a method for proving non-regularity of string transformations?

There are a number of different models for defining transformations between languages. Finite state transducers and MSO-definable graph transformations over string graphs are the two that I am best ...
20
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1answer
1k views

Number of words of length n in a context-free language

Denote by $w_n$ the number of words of length $n$ in a (possibly ambiguous) context-free language. What is known about $w_n$? I'm sure this has been studied a lot, but I couldn't find anything at ...
14
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4answers
528 views

Base-k representations of the co-domain of a polynomial - is it context-free?

In chapter 4 of Jeffrey Shallit's A Second Course in Automata Theory the following problem is listed as open: Let $p(n)$ be a polynomial with rational coefficients such that $p(n) \in \mathbb{N}$ for ...
6
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0answers
74 views

Reference request: transforming a grammar to Greibach normal form preserves the number of parse trees

I believe that most "natural" ways of transforming a grammar to the GNF should preserve the number of parse trees for each string. For example, Urbanek's construction from the paper "On Greibach ...
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1answer
193 views

What is the practical importance of making or using a Turing complete language? [closed]

I get what a Turing machine is and what language is a Turing-complete language but when someone introduces me to a new programming language (like Solidity) and says it is Turing complete, what am I ...
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0answers
59 views

Rational power series over $\mathbb N \cup \{\infty\}$, rationality of singular part

Let $\Sigma$ be a finite alphabet, and consider the formel power series over $\Sigma$ considered as non-commuting variables with coefficients in the semiring $\mathcal N := \mathbb N \cup \{\infty\}$ ...
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0answers
50 views

What is the interpretation of an infinite formal context-free grammar?

Let $L$ be a language as follows: $$ \begin{align*} L &::= a\ |\ L^{*}\\ \end{align*} $$ Now, suppose I apply some sort of transformation $T : N \rightarrow N$ where $N$ is the set of non-...
3
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3answers
109 views

Example of monoid $M$ such that $\operatorname{RAT}(M) \not\subseteq \operatorname{REC}(M)$

Let $M$ be a monoid, the family of rational sets $\operatorname{RAT}(M)$ is defined as the smallest set containing the finite subsets, and closed under union, concatentaion and the star operation. The ...
9
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1answer
137 views

Generalisation of the statement that a monoid recognizes language iff syntactic monoid divides monoid

Let $A$ be a finite alphabet. For a given language $L \subseteq A^{\ast}$ the syntactic monoid $M(L)$ is a well-known notion in formal language theory. Furthermore, a monoid $M$ recognizes a language $...
4
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1answer
371 views

How to start learning formal language theory

I sincerely apologize if this is not appropriate in this stack Q&A, though it seemed the most fitting. I want to learn formal language theory, as well as generating grammars etc. The purpose is ...
7
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1answer
192 views

Finding a minimal DFA whose language has a desired intersection with another

Suppose I have regular languages $B \subseteq A$, with corresponding (known) minimal deterministic finite automata $M_A, M_B$. I would like to find another regular language $C$ such that $B = A \cap ...
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0answers
31 views

Inductive definition of language operators like the set of all permutations of a word came from the shuffle operator

Let $X$ be a finite alphabet. Given two words $u, v \in X^{\ast}$ the shuffle operator is defined to be $$ u || v := \{ u_1 v_1 u_2 v_2 \ldots u_n v_n : 1 \le i \le n, u_i, v_i \in X^{\ast}, u = u_1 \...
1
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1answer
68 views

Relation between REG and NLOGTIME?

What is known about the relation between the class of regular lanuages and NLOGTIME? Is any class contained in another one? You have some choice of how to define NLOGTIME to get one or both ...
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0answers
94 views

Possibly small circuit complexity class containing REG?

What is the smallest well-known Boolean-circuit complexity class containing all the regular languages over the binary alphabet {0,1}? If we believe Theorem 2 in Koucký, Circuit Complexity of ...
3
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1answer
254 views

Converting Kuroda normal form rules to the Penttonen normal form

Let us say we have some abstract context-sensitive grammar in the Kuroda normal form, which is where all production rules are of the form: $AB\rightarrow CD$ or $A\rightarrow BC$ or $A\rightarrow B$...
4
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1answer
336 views

What is the complexity of counting parse trees?

A Counting Problem Given a CFG $G$ and a string $s$, how many distinct parse trees are there for the string $s$? An Example Instance Let's consider an example instance consisting of a CFG $G$ with ...
4
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1answer
283 views

Is it known if $CFL \subseteq NSPACE(o(log^2(n)))$?

$CFL$ is the class of context-free languages. Question Is $CFL$ known to be solvable in $o(log^{2}(n))$ non-deterministic space? What about $DCFL$?
9
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0answers
117 views

Deterministic context-free languages that can be represented as the word problem of a group

Consider a group $G$. We call $G$ virtually free is it contains a free subgroup of finite index. If $G$ is finitely generated by some set $X \subseteq G$ one can consider the word problem $WP(G)$ ...
3
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1answer
69 views

Looking for a particular normal form for Context-sensitive grammar

I am wondering if there is a described normal form for Context-sensitive grammar, which is something similar to Kuroda normal form and Greibach normal form. That is to say, each rule in such form ...
3
votes
1answer
96 views

Generalizing Parsing Expression Grammar for Context Sensitive Grammars

One of the things I like about parsing expression grammars is that they're automatically unambiguous, and unambiguity is a very important property to have. However, context-free grammars are somewhat ...
0
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1answer
166 views

Visualizing the parse structure of a range concatenation grammar

The above is a good visualization of a derivation for a specific sentence in a context free language. You can find many more on Google Images by searching "context free grammar." Let's consider the ...
6
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1answer
120 views

Is this generalization of context free grammars known and strict?

Let $\Sigma$ be an finite alphabet and $(N, \circ)$ a semigroup. The semigroup operation on $N$ can be extended to $\mathscr{P}(N)$: $N_1 \circ N_2 := \{ \; n_1 \circ n_2 \; | \; n_1 \in N_1, \; n_2 \...
2
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0answers
36 views

A class of languages admitted by a class of grammars equivalent to $\mathbf{PR}$?

Is there a class of languages $L(G)$ admitted by a class of phrase structure grammars $G$ equivalent to $\mathbf{PR}$? (the class of primitive recursive languages = $\mathbf{LOOP}$)? In greater ...
2
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1answer
83 views

Is there a name for this property of a term rewriting system?

Given TRS let's call it top-reducible or left-reducible if no rule's right hand side is contained in any rule's left hand side non-trivially. A term A is contained in an other one B trivially if ...
5
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0answers
160 views

“Context” understanding in tree grammars

The Context-Free tree grammar has rules of the form: $A\rightarrow t$ or $A(x_1,\dots,x_n)\rightarrow t_x$, where $A\in N$, $t\in T(N\cup T)$, $t_x\in T(N\cup T\cup \{x_1,\dots,x_n\})$, $T(Z)$ ...