Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [descriptive-complexity]

Classifies problems based on how hard it is to express the problem in some logical formalism.

1
vote
0answers
9 views

Structure of numbers in descriptive complexity

Descriptive complexity is a useful way to free yourself from computational considerations when studying complexity. One of those considerations is the encoding of the structures you are working on. ...
2
votes
0answers
24 views

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 ...
0
votes
0answers
28 views

What descriptive complexity SO-Horn query is equivalent to determining whether or not a graph is bipartite?

I am told "SO-Horn = P". Determining whether or not a graph is bipartite is in P. Therefore there should be a SO-Horn query for it. I'm still new to all this, but here's a SO query I came up with ...
0
votes
1answer
168 views

First Order Logic, First Order Logic + Recurrence and SQL

we know that SQL standard is equivalent to First Order Logic (FOL). I've seen at my lecture that graph connectivity cannot be expressed by FOL, so in SQL as well. But we know that we can easily solve ...
2
votes
1answer
102 views

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 ...
1
vote
1answer
34 views

For a turing machine which computes $y$ with argument $x$, how much does the descriptional length of a machine producing $y$ without input increases

Let $f$ be some function in some programming language (like C), and we need $n$ bits to store this function. Suppose we have some fixed value $v$ for the argument, then let g() { f(v) } be the ...
4
votes
1answer
153 views

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 ...
0
votes
1answer
65 views

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 ...
1
vote
1answer
70 views

Reference proof for Second-Order Logic captures Polynomial-Time Hierarchy

I'm looking for a complete proof of $\mathrm{PH=SO}$. The (admittedly few) textbooks and papers i've looked at all either state that it's a corollary from Fagin's Theorem, or leave it as an exercise ...
0
votes
2answers
123 views

Descriptive complexity: 3-colorability example

So in Neil Immerman's book http://books.google.co.kr/books?id=kWSZ0OWnupkC&pg=PA113&lpg=PA113#v=onepage&q&f=false, 3-colorability problem in descriptive complexity fashion is expressed ...
0
votes
3answers
58 views

A graph in descriptive complexity - is $x$ already a vertex?

So suppose that there is an undirected graph with edge connections known. Now in first-order logic there is quantifier $\forall x$. Then does this automatically refer to vertexes, or can we use ...
4
votes
1answer
89 views

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 ...
2
votes
0answers
40 views

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 ...
4
votes
3answers
619 views

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 ...
2
votes
1answer
55 views

About proofs in descriptive complexity

In descriptive complexity, we have theorems that look like $\mathrm{ESO} = \mathrm{NP}$ or "on linearly ordered structures, $FO(LFP) = P$", but I don't really understand the proofs of those. For the ...
0
votes
1answer
152 views

NFA and DFA storage cost

In some paper I read, A theoretical worst case study shows that a single regular expression of length $n$ can be expressed as an NFA with $O(n)$ states. When the NFA is converted into a DFA, ...
7
votes
1answer
223 views

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 ...
1
vote
1answer
176 views

Expressing complexity class P using first-order logic with LFP

Can anyone show how to express complexity class P using first-order logic with LFP? (descriptive complexity)
1
vote
1answer
59 views

Operators in descriptive complexity

When we talk about operators in descriptive complexity, are they something like this: for example, if transitive closure operator $TR$ is available, we can use variable $y$ that we define as $TR(x)$ ...
3
votes
2answers
110 views

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,...
5
votes
3answers
1k views

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 ...
4
votes
1answer
202 views

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)\...
9
votes
1answer
240 views

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 ...
19
votes
3answers
283 views

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 ...