Questions tagged [formal-languages]
The study of formal languages (sets of strings or trees over an alphabet), rewriting systems and algorithms, recognition automata/algorithms, and related questions.
147
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Are 100% of statements undecidable, in Gödel's numbering? [duplicate]
Gödel's incompleteness theorem shows that there are undecidable statements, i.e., formal logical claims which neither have proofs nor disproofs.
In doing so, Gödel famously enumerated all well-formed ...
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4
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0
answers
93
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Are semilinear sets piecewise periodic?
I wanted to check my understanding of semilinear sets before I give a talk on them, and I haven't been able to find this exact perspective in any of the sources I've read through. Is it correct, and ...
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1
vote
0
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31
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The set of closed untyped $\lambda$-terms is not context-free?
The set of untyped $\lambda$-terms is obviously context-free. But, according to Barendregt's paper Discriminating coded lambda terms (six lines before Theorem 1.5), the set of closed untyped $\lambda$...
- 1,152
2
votes
0
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44
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A particular generalization of free partially commutative monoids
A trace monoid, or free partially commutative monoid, is one with the presentation $\langle \Sigma \mid a_1b_1 = b_1a_1, \dots, a_nb_n = b_na_n\rangle$. The theory of trace monoids has been well ...
- 21
4
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0
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146
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Corollaries of Kleene's Theorem (Regular Languages)
Kleene's theorem that finite automata (specifically, nondeterministic) are expressively equivalent to regular expressions seems to be a powerful and not immediately obvious tool for untangling the ...
- 399
4
votes
0
answers
91
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String rewrite system for algebraic knots/links?
$\newcommand\over{\vert}\newcommand\rot[1]{\mathopen<#1\mathclose>}$By its definition, an algebraic tangle, and by extension, its closure (knot or link) can be written as a string (of ...
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3
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0
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284
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Conversion of proofs between HoTT and ZFC
HoTT provides a foundation of math that remains mysterious for
many mathematicians including me. Hence this question.
There are several implementations of math based on ZFC, an
example being MetaMath. ...
- 4,570
2
votes
0
answers
53
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Polynomial-time algorithm for uniformly sampling the $n$-slice of a context-free language
Let $L\subset \Sigma^*$ be a context-free language. The $n$-slice is the intersection $L\cap \Sigma^n$ for a non-negative integer $n$.
Is there a polynomial-time algorithm for uniformly sampling from ...
- 21
2
votes
0
answers
91
views
Name for the theory of words with equal length, prefix, successors
I've worked with this theory for a while, but I've never been quite sure what to call it:
$$(\Sigma^*, =_{el}, \preceq, (S_a)_{a \in \Sigma})$$
Where
$\Sigma^*$ is the set of finite words on finite ...
- 399
0
votes
0
answers
94
views
Empty context-sensitive language independent of ZFC?
Is there a simple context-sensitive grammar $G$ such that $L(G)=\emptyset$ is independent of ZFC?
$L(G)$ is the formal language generated by $G$.
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5
votes
1
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Algorithms to factorize words into product of powers
I came across this problem, which I guess is well known to combinatorialists of words, so I write here to see if someone can help me with some references.
Let $A$ be a finite set of symbols, are there ...
- 477
2
votes
1
answer
125
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What is the cardinality of the set of Dyck natural numbers of semilength $k$?
In arXiv:2102.02777 ("Recursive Prime Factorizations: Dyck Words as Numbers"), I show that there is a 1:1 correspondence between $\mathbb{N} = \{0,1,2,3,4,\ldots\}$ and $\mathcal{D}_{r_{\...
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-1
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1
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Prove using Dyck naturals: for $n \in \mathbb{N}_{+}$ and big enough $k \in \mathbb{N}_{+}$, $p_{k-1} < \cdots < np_{k-a_{n}}$ (a is A073093)
While conducting research in connection with arXiv:2102.02777 ("Recursive Prime Factorizations: Dyck Words as Numbers"), I noticed certain interesting patterns, one of which inspired the ...
- 129
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1
answer
361
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Is there an equivalent of the incompleteness theorems/halting problem in category theory?
Taking the doctrine of computational trinitarianism ( https://ncatlab.org/nlab/show/computational+trinitarianism ), if one understands the incompleteness theorems as the "logic" version, and ...
- 141
4
votes
0
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A lemma from Jarden's and Lubotzky's paper 'Elementary equivalence of profinite groups'
I have a question about a reduction argument from
Jarden's and Lubotzky's paper 'Elementary equivalence of
profinite groups' in Lemma 1.1 on page 3:
Lemma 1.1: For each positive integer $n$ and each ...
- 4,994
5
votes
1
answer
359
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Computational complexity of proof verification
Let $\mathcal{L}$ be a recursive first-order theory, with a deductive system $\Xi$ (for instance, Hilbert-Ackerman proof system). Let $\phi$ be a formula and let $l=(\psi_1, \ldots, \psi_n=\phi)$ be a ...
- 1,895
5
votes
0
answers
112
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Self avoiding walks and context free languages
Let $G$ be an infinite, locally finite, connected graph whose arcs (oriented edges) are labelled by letters in a finite alphabet $\Sigma$ such that arcs starting in the same vertex are labelled by ...
- 2,461
1
vote
0
answers
41
views
Name for a class of languages closed under union, inverse generalised sequential machine mappings and intersection with regular languages
I asked this question on the TCS stackexchange but have so far received no answer:
Is there a name for classes of languages closed under finite union, inverse generalised sequential machine mappings ...
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3
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0
answers
536
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Can third-order arithmetic prove the consistency of second-order arithmetic?
I'm trying to get a deeper understanding of Buss's version of Gödel's speedup proof. In short, if we assume that $Z_0$ is first-order arithmetic, $Z_1$ is second-order arithmetic, and so on, then for $...
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8
votes
1
answer
320
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Is equality of formulas with floor rounding or integer division decidable?
As far as I know, formulae involving rationals and basic arithmetic ($+$, $-$, $\cdot$ and $/$) have decidable equality. Is this still the case if we add floor rounding (or integer division)?
Define ...
- 5,475
0
votes
1
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183
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Does the fixed point lemma / diagonalization require capturing or not?
Peter Smith's formulation of the diagonalization lemma is essentially as follows, from Theorem 47 of his (fantastic) online book:
If theory T extends Robinson Arithmetic, and P is an one-place open
...
- 31
1
vote
0
answers
198
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Is it possible to construct a formal language that allows to refer to specific real numbers that encode ordinals accidentally writable by an ITTM?
Let $A$ denote a particular (fixed) algorithm to encode ordinals as real numbers. The exact technical description of $A$ is irrelevant for this question: it can be any algorithm that is mathematically ...
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25
votes
5
answers
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Formalizations of the idea that something is a function of something else?
I'll state my questions upfront and attempt to motivate/explain them afterwards.
Q1: Is there a direct way of expressing the relation "$y$ is a function of $x$" inside set theory?
More ...
- 5,193
1
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1
answer
148
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Errors in Waksman's Solution to Cellular Automaton Firing Squad Problem?
Recently, a student and I have been working through Waksman's paper ``An Optimum Solution to the Firing Squad Synchronization Problem.'' The paper claims that for any value of $n$, the proposed ...
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2
votes
1
answer
65
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For synchronizing eulerian finite state machines every proper subset of states has some larger state set leads to this subset
Suppose we have a deterministic complete finite automaton which is synchronized, meaning we have a reset word, i.e. a word which resets the automaton to a definite state, regardless from which state ...
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16
votes
4
answers
832
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Representing mathematical statements as SAT instances
The following problem (call it THEOREMS) belongs to class NP.
Input: Mathematical statement $S$ (written in some formal system such as ZFC) and positive integer $n$ written in unary.
Output: "Yes" if ...
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0
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259
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How can I catalog these generalized Collatz problems?
The Collatz conjecture can be expressed in terms of a ruleset in the language $\{x,+,1,\rightarrow,;\}$:
$x + x + 1 \rightarrow x+x+x+1+1;$
$x + x \rightarrow x;$
Whenever a number matches the LHS ...
- 2,214
0
votes
0
answers
73
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Transformation or correspondence between language and real number
As we know, formal language can be regarded as a set of strings of alphabet, and real number can be regarded as sequence generated by set of integers, for example, denominators of the simple continued ...
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0
answers
228
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What does homomorphism between languages mean to the correspoding Turing Machines?
According to the article: every c.e.language over $\Sigma^*$can be formed by homomorphism from a Dyck language over $\Sigma^{'}$ intersection with a minimal linear language over $\Sigma^{'}$ to the ...
- 2,824
2
votes
1
answer
184
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Alternative notation for Kleene star
I am writing a paper which use two different operations on sets of works $X$, both of which I want to denote by a star, $X^{\ast}$. One of these operations is the Kleene star, and for whatever reason ...
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1
answer
382
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Can ETCC/ETCS talk about 'size issues'?
In material set theories (like ZFC), one can prove that there is no set of all sets. Can one prove a similar statement in ETCS? This exact statement "there is no set x such that y in x for every set y"...
- 81
3
votes
2
answers
435
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Topology induced by context-free language
Is there any way to reasonably define a topology on a context-free-language language? In other words, given a context-free grammar (or perhaps a grammar from an interesting subclass of context-free ...
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2
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What exactly is a judgement?
Before formulating my question, let me briefly sum up what I know about the topic (feel free to correct me if something I claimed is false!). This is for you good to see what my state of knowledge is, ...
user99916
5
votes
1
answer
386
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What do we call this quantifier ("binder")?
There's a quantifier ("binder", whatever), call it $\alpha$, defined as follows: $\alpha x.\tau$ is the (usually infinite) expression obtained by applying the substitution $\{x \mapsto \tau\}$ to the ...
- 3,613
5
votes
2
answers
479
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Neighbourhood of a word and Levenshtein distance
The Levenshtein distance or Edit distance $$ lev(U,V) $$ between two strings $U$ and $V$ over a finite alphabet $\Sigma$ of size $ \left| \Sigma \right| = \sigma ,$ is the minimal number of insertions,...
- 61
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0
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387
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Computing the ordinal of a rational language well-partially-ordered by the subword relation
Let $\Sigma$ be a finite set or "alphabet", $\Sigma^*$ the free monoid on $\Sigma$ or set of "words". If $w,w'\in \Sigma^*$, write $w\leq w'$ when $w$ is a "subword" of $w'$, i.e., can be obtained by ...
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13
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0
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299
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Reference request: exponential growth rates of subword-closed languages are integers
For a language $L$ over the finite alphabet $\Sigma$, let $L_n$ denote the set of words in $L$ of length $n$. The word $u$ is a subword of $w$ if $u$ can be obtained from $w$ by deleting letters (...
- 2,281
9
votes
1
answer
599
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Coherence and rewriting
In category theory there are numerous coherence theorems (https://ncatlab.org/nlab/show/coherence+theorem). One example is the Mac Lane's coherence theorem for monoidal categories. This and probably ...
- 2,272
3
votes
1
answer
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How to get $\omega$-regular expression from buchi automaton
Is there an algorithm or a trick on how to get $\omega$-regular expressions from Buchi automatons? If yes, is there also some way to do create minimal such regular expressions?
It is extremely ...
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1
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Deterministic Finite Automata question [closed]
I am very new to finite automata, and I came across an issue in my professors lecture slides which I think is wrong, and I'd wonder if any of you could confirm:
Alphabet: {1}
Automata
Surely the ...
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0
votes
0
answers
192
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Is the positive existential theory undecidable?
Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?
How can we prove the (...
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6
votes
1
answer
466
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Show that the positive existential theory is undecidable
To show that the positive existential theory of $\mathbb{C}[t, e^{\lambda t} \mid \lambda \in \mathbb{C}]$ in the language $\{+, \cdot , ' , 0 , 1, t\}$ is undecidable we have to prove the following: $...
- 299
2
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0
answers
45
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Relation between indexed languages (OI-macro or context-free tree) and scattered context languages
I'm not sure about the relation between indexed languages (generated by indexed grammars--Aho) and scattered context languages (generated by
scattered context grammars--J Hopcroft).
I think that ...
- 21
1
vote
1
answer
149
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Terminology for set of infinite strings with a certain prefix
Let $\mathcal{A}$ be a finite alphabet, and let $C$ be the Cantor space $\mathcal{A}^\omega$ under the product topology.
Given a finite string $s \in \mathcal{A}^*$, let $C(s)$ be the set of all ...
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1
vote
1
answer
241
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The automorphism groups of smallest grammars of a language string are isomorphic
Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = \...
- 182
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votes
0
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211
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Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?
Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"?
Or is there something else that states ...
- 1,853
7
votes
2
answers
204
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Is this variant of the balanced bracket language context free?
Consider the language generated by the following context free grammar:
$$
S \to SS \quad S \to () \quad S \to (S) \quad S \to [] \quad S \to [S]
$$
There is a one-to-one correspondence between this ...
- 3,630
0
votes
0
answers
101
views
Counting path generating sentences in a specific formal language
Given a formal grammar of a language or an Turing machine of the language, can we count the path that generating sentences of the language?
For example, we know that if the grammar is context-free ...
- 2,824
6
votes
2
answers
208
views
Formal languages with non-unique interpretations of terms
In mathematical logic and model theory, one considers interpretations of syntactic expressions: terms without free variables are interpreted as elements of some structure, formulas without free ...
- 1,390
3
votes
3
answers
204
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Are there any results on well-quasi-ordering of languages?
There are a number of papers that I can find about well-quasi-orders in formal lnaguage theory, by Kunc, de Luca, D'Alessandro, and Varricchio, among others. I am interested, however, in well-quasi ...
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