Questions tagged [fl.formal-languages]

formal languages, grammars, automata theory

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611 views

Why is the Pumping Lemma sometimes called Bar-Hillel's Lemma?

There are several papers in the literature that refer to the Pumping Lemma for context free languages as Bar-Hillel's Lemma (for example, here, here, and on the Wikipedia page). However, the first ...
13
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0answers
2k views

minimizing size of regular expression

Suppose we have a regular language specified by a regex, for example, (ab|ac)* and we wish to find an equivalent regex with the minimal number of symbols, (a(b|c))*. Is there any efficient way to do ...
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287 views

Reference request: exponential growth rates of subsequence-closed languages are integers

This question is migrated from MathOverflow, where it did not receive any answers a year ago. For a language $L$ over the finite alphabet $\Sigma$, let $L_n$ denote the set of words in $L$ of length $...
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989 views

Is CFL strictly contained in NL?

We know that $\mathsf{REG}=\mathsf{NSPACE}(O(1))$ and $\mathsf{CSL}=\mathsf{NSPACE}(O(n))$. What is the relation of $\mathsf{CFL}$ and $\mathsf{NSPACE}(O(\log n))=\mathsf{NL}$? Is $\mathsf{CFL}$ a ...
12
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297 views

Survey on infinite alphabet automata?

The paper "Symbolic Finite State Transducers, Algorithms and Applications" by Bjorner et al (to appear at POPL 2012) describes one type of finite-state, infinite-alphabet automata/transducers by using ...
11
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0answers
301 views

Can we approximate the number of words accepted by an NFA?

Let $M$ be an acyclic NFA. Since $M$ is acyclic, $L(M)$ is finite. In a related question, it was suggested that exact counting of the number of words accepted by $M$ is $\#P$-Complete. The second ...
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111 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 ...
10
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0answers
145 views

A language outside the Boolean closure of stochastic languages

Stochastic languages, that is, those accepted by probabilistic automata, are known to not be closed under intersection, union, concatenation, and morphism, even on unary languages. I have two ...
10
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1answer
414 views

Reversible Turing tarpits?

This question is about whether there are there any known reversible Turing tarpits, where "reversible" means in the sense of Axelsen and Glück, and "tarpit" is a much more informal concept (and might ...
9
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120 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 $WP(G)$ ...
9
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0answers
334 views

Deciding if a language induced by a Presburger formula is context-free

Is the following problem decidable? Given $n$ and a Presburger arithmetic formula $\phi(x_1,x_2,\dots,x_n)$, determine whether the language $\{a_1^{i_1} \dots a_n^{i_n}:\phi(i_1,i_2,\dots,i_n)\}$ ...
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196 views

Simplifying the disjoint union of wildcard strings

Setting: patterns with "don't care" symbols, binary alphabet. For example, pattern $x = 001?$ represents the set $L(x) = \{0010, 0011\}$. We are given a set $P$ of disjoint patterns: $L(x) \cap L(y) ...
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1k views

Is there an ambiguity test for CFGs faster than trying all strings?

It is well known that testing whether a grammar is ambiguous is undecidable. It is however trivially decidable for any $G$ whether $L_n(G) := \{ w | w \in L(G) \wedge |w| \leq n \}$ for any $n \in \...
7
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143 views

Has a result of Book and Greibach on Quasi-Realtime languages been improved?

Quasi-realtime languages are defined as languages accepted by nondeterministic multitape Turing machines in quasi-real time. Ronald Book and Sheila Greibach have shown in their 1970 paper that every ...
7
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740 views

Names for the left- and right-hand sides of a grammar production?

Problem I'm writing a document where I have to describe some of the properties of a type system as they relate to a particular formal grammar. I was trying to refer to the right-hand-sides of the ...
7
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337 views

Examples of non-CSLs not created through diagonalization

Hopcroft & Ullman 1979, Intro to Automata Theory, Languages, & Computation states (p. 224) that "almost any language one can think of is CSL; the only known proofs that certain languages are ...
7
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334 views

The semantics of Parsing Expression Grammars

Is there a simple and intuitive explanation for the fact that the following parsing expression (where S is the starting symbol, $...
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0answers
82 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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527 views

Does PEG contain CFG?

Despite their considerable expressive power, all PEGs can be parsed in linear time using a tabular or memoizing parser (8). These properties strongly suggest that CFGs and PEGs define incomparable ...
6
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273 views

Language of stack configurations of a pushdown automaton

Consider a pushdown automaton $A$ with stack alphabet $\Gamma$. Let $L$ be the language on $\Gamma$ of the stack configurations encountered during accepting runs of $A$. Is $L$ a context-free language?...
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114 views

Languages whose Parikh image is recognizable

Let $\Sigma$ be some alphabet, and $p : \Sigma^* \to \mathbb N_0^{|\Sigma|}$ the Parikh map. A formal language $L \subseteq \Sigma^*$ is called a slip-language, if $p(L)$ is a semilinear set. By ...
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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 $A\in N$, $t\in T(N\cup T)$, $t_x\in T(N\cup T\cup \{x_1,\dots,x_n\})$, $T(Z)$ ...
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0answers
97 views

Salomaa's axiomatisation of regular languages and the use of regular expression in it

I am reading the classical article of A. Salomaa where he gives two axiom systems for regular sets and proofs consistency and completeness. As I have understood it, an axiomatic system in some logic (...
5
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0answers
143 views

Mastery-based grading for Theory of Computation

I would be interested to learn of anyone's experience using mastery-based (or "mastery-level") grading in a Theory of Computation course. Usually this requires—at a minimum— a detailed ...
5
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0answers
72 views

How much smaller can universal Turing machines get if they only need to be universal for a subclass?

Say that a Turing machine $U$ is universal for a class $\mathcal{C}$ of languages if for any language $L \in \mathcal{C}$, there is a word $w_L$ with: $$(\forall w)\quad w \in L \Leftrightarrow U(w_L, ...
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165 views

The regularity of Markov chains with a threshold

(This question has been asked on math.se, with no response.) I am studying Paz's "Introduction to Probabilistic Automata" and there is an exercise I cannot solve: Ex. 11, p. 170: Let $\Sigma = \{a\...
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390 views

Is there a model theory for Haskell type classes?

I am trying to understand the semantics of Haskell’s type classes (TCs) from a model-theory point of view. It might difficult to give precise model theoretic semantics to type classes (see 1, and 2). ...
4
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688 views

Variant of a proof using Ogden's lemma

I am trying to understand better the proof that the language $K=\{a^{i}b^{j}c^{k} ~|~ i \neq j, i \neq k, j \neq k$} is not context-free. (see It only looks like a homework problem…), and the use of ...
4
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0answers
531 views

Minimal context-free Grammar for a special one-letter Language

For natural numbers $n \geq 5$, $m \geq 2^{n-2} + 1$ the following context-free language is given: $$ L_{n,m} = \{ a^i | 2 \leq i \leq m \} \setminus \{a^{2^i}|2 \leq i \leq n-2\} $$ Find and ...
4
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231 views

Intersection between register automata and pushdown automata over infinite alphabet

I'm not an expert in automata theory, this is a reference request. As far as I have understood it is known in the automata comunity that register automata by Kaminski are closed by intersection with ...
4
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0answers
318 views

Is $\{a^n b^m : 1 \le m \le 2^{2^n}\}$ a permitting-context language?

A random permitting-context grammar is a context-free grammar $(N, \Sigma, P, S)$ equipped with a function $p : P \rightarrow 2^N$. The rule $A \rightarrow x$ can be applied to $uAw \Rightarrow uxw$ ...
4
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1answer
344 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 $DCFL$?
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84 views

The number of words of length $n$ in a context-sensitive language

Let $L$ be a context-sensitive language, $s_{L}(n)$ is denoted by the number of words of length $n$ in $L$. What is known about $s_{L}(n)$? Note that it is known that $s_{L}(n)$ is either polynomial,...
3
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0answers
45 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 ...
3
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0answers
124 views

Justifying the state of virtual memory as a vector space

First, I'm mostly experienced with Math, which I hope won't be too inconvenient. I saw Operational Calculus on Programming Spaces by Sajovic and Vuk, which seemed very interesting to me (for a "short ...
3
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0answers
89 views

Grammar with “dead” derivation chains

This question is inspired by the great answer given by Jeffrey Shallit on my question about proper CSG for $a^n b^n c^n$ language. Disclaimer:: I'm not arguing about correctness of this grammar. This ...
3
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0answers
168 views

Context-Sensitive Grammar characteristic properties

This question can look like some kind of puzzle, but it is actually part of more complex applied problem. Let's consider subspace of Context-Sensitive Grammars, which contains grammars which can not ...
3
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0answers
82 views

A question on the introduction of the Wagner hierarchy from K. Wagner's original paper

My question is related to the seminal paper On $\omega$-regular sets by K. Wagner, which introduced a hierarchy which is now know as the Wagner- (or Wadge-) hierarchy of $\omega$-regular sets. In ...
3
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0answers
201 views

Mildly dependently-typed metalanguage for mildly context-sensitive object languages

This is almost certainly not a new idea, but I haven't seen it elaborated or discussed elsewhere. A very natural way to represent the abstract syntax of an object language in a typeful metalanguage is ...
3
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0answers
105 views

Is there research on “minimal” Turing-universal Markov algorithms?

The Markov algorithm is a simple model of computation. For other models of computation, such as Turing machines, cellular automata, tag systems, etc., there is research on the "minimal" instances of ...
3
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0answers
144 views

Restricted-Input Automaton

In the classic setting, an automaton for a language $L$ is required to accept all words in $L$ and reject/get stuck on every word in $\Sigma^*\setminus L$. All of the related concepts are then ...
3
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0answers
171 views

Deciding whether a binary multiplicity automaton has empty language

Multiplicity automatons (see here) is an interesting model. They have the (almost) same syntax as a non-deterministic finite automatons, but instead of deciding whether a word belongs to a language, ...
3
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0answers
111 views

Transfering properties from subsets of $X^*$ to subsets of $X^{\omega}$ by using the topology induces by Cantor space

A language $L \subseteq X^*$ is non-counting of order $n > 0$ iff for all $u,v, w \in X^*$ $$ uv^nw \in L \Leftrightarrow uv^{n+1} w \in L. $$ A $\omega$-language (set of infinite sequences) $L \...
3
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0answers
218 views

Learning about Nested Stack Automata

I want to learn about nested stack automata. However my efforts to find a suitable learning resource have so far been abortive: The Wikipedia article on nested stack automata is a stub. Alfred Aho's ...
3
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0answers
233 views

Providence of pumping lemmas for regular languages

I'm looking to track down who discovered the following pumping lemmas for regular languages. (where $p$ is the pumping constant.) Reg($L) \rightarrow \exists p\forall w(\in L) \forall u_1u_2v(\in \...
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0answers
67 views

Proof: Why are MM-1QFA strictly more powerful than MO-1QFA? // Quantum automata

While dealing with quantum finite automata (QFA), I repeatedly come across the statement that measure-many QFA (MM-1QFA, KW97) are strictly more powerful than measure-once QFA (MO-1QFA, MC97). More ...
2
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0answers
101 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
55 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,...
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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0answers
80 views

Is there a DCSL that cannot be recognized in O(n^2) steps by a deterministic LBA?

Is there a context sensitive language $L$ so that $L$ cannot be recognized by a deterministic linear bounded turing machine in $O(n^2)$ steps, but still can be recognized by a deterministic LBA? The ...