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.
158 questions
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Determine equivalences in the generated collection of subgroups and quotients
Let $A$ be an abelian group, and $B_1, B_2, \dots, B_m$ be subgroups of $A$. Define the family of subgroups $\mathcal{D}_0 = \{ \{0\}, A, B_1, B_2, \dots, B_m \}$.
Let $\mathcal{C}_1$ be the ...
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Rewriting systems for finite groups [closed]
This is a question about rewriting systems & languages for finite groups. I'm sure everything must be in the literature somewhere, but I find it hard to navigate the references I have (for example ...
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What is the max number of self-segregating words of length n?
A set of words S is called self-segregating if you don't need whitespaces to read them. It means that for any two words from S no new words from S arise between them.
For example the set ab, bc, ac, ...
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Proof of dynamic programming calculation of Levenshtein distance
Let s1 and s2 are 2 arbitrary strings with lengths l1 and ...
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Normal form for terms in language with two ring structures
Suppose I have two different ring structures on the same domain $\langle R,+,\cdot,0,1\rangle$, $\langle R,\oplus,\otimes,\bar 0,\bar 1\rangle$ and I throw the structures together into a single common ...
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An overview of mathematical-logical approaches in formalizing natural languages
Crossposted on Mathematics SE
I am an undergraduate mathematics student with a keen interest in pursuing research in the formalization of natural languages (from a more mathematical-logical approach),...
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Language equivalence between deterministic and non-deterministic counter net
One-Counter Nets (OCNs) are finite-state machines equipped with an integer counter that
cannot decrease below zero and cannot be explicitly tested for zero.
An OCN $A$ over alphabet $\sum$ accepts a ...
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Filling in some missing squares for classes of power series
This question concerns various important classes of formal power series. For concreteness and convenience, let us work with power series $F(x) = \sum_{n\geq 0}c_n x^n \in \mathbb{C}[[x]]$, i.e., with ...
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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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Counter net decidability [closed]
Let one Deterministic Counter Net ($\mathrm{1DCN}$), which is a finite-state automata where every state is complete means all states has transition of all input symbols and their respective weight ...
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Are there more true statements than false ones?
It is a nontrivial fact that half the primes are $\equiv 1 \pmod{4}$ and the other half are $\equiv 3\pmod{4}$. The Chebyshev bias suggests, however, that the latter class of primes is winning the ...
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Propositional logic without rules of inference and assumptions (except MP) [closed]
I was wondering whether it would be possible to do propositional logic without any rules of inference and assumptions (except modus ponens).
I have the following axioms:
$ p \to (q \to p) $
$ (p \to (...
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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 ...
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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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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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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 ...
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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 $\lambda$...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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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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 ...
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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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Predicates of infinite arity
Infinitary logic considers languages being infinite by infinite conjunctions and disjunctions.
I wonder why it not considers languages being infinite by relations and functions of infinite arity.
...
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Which Turing machines accept the language of trivial words in a finitely presented group?
Let $G$ be a finitely presented group with generators $g_1, g_1^{-1},\ldots, g_n, g_n^{-1}$. Let $L(G)$ be the language of all those words in $g_1, \ldots, g_n$ which represent the trivial element of $...
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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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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 ...
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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 ...
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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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Regular languages and the pumping lemma
Let's say that I want to prove that a language is not regular.
The only general technique I know for doing this is the so-called "pumping lemma", which says that if $L$ is a regular language, then ...
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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 ...
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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 ...
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Is there a natural family of languages whose generating functions are holonomic (i.e. D-finite)?
Let $L$ be a language on a finite alphabet and let $L_n$ be the number of words of length $n$. Let $f_L(x) = \sum_{n \ge 0} L_n x^n$. The following are well-known:
If $L$ is regular, then $f_L$ is ...
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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 ...
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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 ...
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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
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What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?
Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that'...
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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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When may function (meromorphic) be expanded as power series with coefficients of integers
Let $F$ be meromorphic function, with what properties may it be expanded as power series with coefficients of integers in such a form:
$$F=\sum_0^{\infty}a_i x^i,a_i\in \mathbb{N} \bigcup 0,\exists M \...
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Inherent ambiguity of the context-sensitive language $L = {a^ib^ic^id^je^jf^j \bigcap a^ib^jc^id^je^if^j} $ or $a^nb^nc^nd^ne^nf^n$
What is the definition of ambiguity of context-sensitive
grammar?This is relevant to the definition of inherent ambiguity of
context-sensitive language.And any proof for the inherent ambiguity of ...
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Theory interpreted in non-set domain of discourse may be consistent?
Following the blow. I will try to ask question in order to check if I well understand what was pointed. I decide to ask another question, because mathoverflow is not projected to be good environment ...
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Semantics of Higher-Order Logics
I've been trying to get to grips with the various semantics commonly discussed in formal logic. Specifically, the nature and role of interpretations of first and higher-order logics is slightly ...
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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 ...
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Context free grammar + functions = ?
If you start with the rules for building a context-free grammar and extend them by allowing left-hand nonterminals to be functions of one or more arguments, does that go beyond the definition of a ...
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Satisfiability problem for FOL[<,R]
Let FOL[<,R] be the fragment of first-order logic enriched with two relational symbols < and R and the first-order axioms that say:
< is a strict partial order and R is an irreflexive and ...
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Status of an open problem about semilinear sets
In his book "The Mathematical Theory of Context-Free Languages" (1966), Ginsburg mentioned the following open problem:
Find a decision procedure for determining if an arbitrary semilinear set
is a ...
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Why do I find Category Theory mostly just a way to make simple things difficult?
I have a basic working knowledge of category thoery since I do research in programming languages and typed lambda-calculus. Indeed, I have refereed many papers in my area based on category theory.
...
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Automatic groups - recent progress
Epstein's (et al.) "Word Processing in Groups" is a quite comprehensive monograph on automatic groups, finite automata in geometric group theory, specific examples like braid groups, fundamental ...
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