Classifies problems based on how hard it is to express the problem in some logical formalism.
Questions tagged [descriptive-complexity]
37 questions
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3 answers
Extension of SQL capturing $\mathsf{P}$
According to Immerman, the complexity class associated with SQL queries is exactly the class of safe queries in $\mathsf{Q(FO(COUNT))}$ (first-order queries plus counting operator): SQL captures safe queries. (In other words, all SQL queries have a…
Kaveh
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Can joins be parallelized?
Suppose we want to join two relations on a predicate. Is this in NC?
I realize that a proof of it not being in NC would amount to a proof that $P\not=NC$, so I'd accept evidence of it being an open problem as an answer.
I'm interested in the general…
Xodarap
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For what kinds of languages is min |NFA| = Ω(min |DFA|)?
Consider a regular language $L$. Let $D(L)$ be a minimal DFA for $L$ and $N(L)$ be a minimal NFA for $L$ (minimal in the sense of the smallest possible number of states for an automaton that recognizes the given language). Write $|A|$ for the size…
Karthik C. S.
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3 answers
How to calculate the number of states in designing a Turing machine?
I would like to ask how to determine the number of states when designing a Turing machine from the description for a language? For example:
$\qquad \displaystyle L = \{wcw \mid w \in \{0,1\}^*\}.$
I mean how to know how many states are there in the…
Chau
5
votes
1 answer
When does the function mapping a string to its prefix-free Kolmogorov complexity halt?
In Algorithmic Randomness and Complexity from Downey and Hirschfeldt, it is stated on page 129 that
$\qquad \displaystyle \sum_{K(\sigma)\downarrow} 2^{-K(\sigma)} \leq 1$,
where $K(\sigma)\downarrow$ means that $K$ halts on $\sigma$, $\sigma$…
pintoch
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4
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First Order interpretation of arbitrary structures as a graph
I am currently trying to get some intuition on the concept of First Order reductions, and have come across this exercise question by Immerman, dubbed "Everything is a Graph".
Given some arbitrary relational structure $S$ of some vocabulary $\sigma$,…
Syzygy
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Can a transcendental number like $e$ or $\pi$ be compressed as not algorithmically random?
The related and interesting fields of Information Theory, Turing Computability, Kolmogorov Complexity and Algorithmic Information Theory, give definitions of algorithmically random numbers.
An algorithmically random number is a number (in some…
Nikos M.
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Why doesn't descriptive complexity theory solve P = NP?
According to the Wikipedia page on Descriptive complexity theory:
In the presence of linear order, first-order logic with a least fixed point operator gives P, the problems solvable in deterministic polynomial time.
Existential second-order logic…
Jesus is Lord
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3
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2 answers
Lower bound on size of proof that a list of integers is sorted
Suppose we have a list of unbounded integers, written in binary, and we want to write a (formal) proof that the list is sorted in ascending order.
Such a proof might look (informally) like: "2 < 3, and 3 < 5, and ... and 71 < 79, so the list is…
Chris Pressey
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3
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1 answer
State complexity of homomorphisms of regular languages
Given a DFA $A = (Q, \Sigma, \delta, q_0, F)$ with $n$ states and a homomorphism $h: \Sigma \to \Gamma^*$. It is easy to see that the family of regular languages is closed under homomorphisms using regular expressions or by construction of a…
PeterMcCoy
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3
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How to describe Deterministic Transitive Closure in FOL?
In "Finite Model Theory and Its Applications", page 152, it is said that Deterministic Transitive Closure, on ordered finite structures, captures LOGSPACE.
Hence, taking into account that FOL captures LOGSPACE, the following should be…
441Juggler
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Computability of Kolmogorov complexity in total languages
It is well known that the Kolmogorov complexity is uncomputable in Turing-complete programming languages. However, what about total programming languages?
For example, is the Kolmogorov complexity of natural numbers computable if we work in Gödel's…
xuq01
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Do undecidable problems have no HO query? If so, could I have an example?
In descriptive complexity, HO corresponds to ELEMENTARY.
ELEMENTARY is a subset of R, so therefore all HO queries are decidable.
Then undecidable problems have no corresponding HO query. Is my understanding correct? If so, can I have an example…
Jesus is Lord
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2
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1 answer
NFA state complexity for the complement of EPAL restricted to a fixed length
I've been having trouble proving the next statement:
Let $L_n=\{ww, |w|=n\}$ (the set of equal-length palindromes (EPAL) restricted to length $2n$). Prove that $L^c_n$ can be accepted by an NFA with at most $O(n^2)$ states.
What I've been trying…
christian jones
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Completeness and first order logic with Least fixed point operator (LFP)
Is there any result about the extension of first order logic with least fixed point operator, being complete (as logic in general on infinite structures too) or not? In other words does the Goedel completeness theorem of first order logic extent to…
DrkCostas
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