a subfield of computational complexity. Instead of creating a program, logical operators, like quantifiers and least fixed point, are used to categorize problems
Questions tagged [descriptive-complexity]
27 questions
4
votes
1 answer
Computing the Sum of Log n Bits in First Order Logic
I have read the so-called Bit-Sum Lemma from
Neil Immerman. "Descriptive Complexity" (Lemma 1.18) and from
Barrington, Immerman and Straubing. "On uniformity within $NC^1$" (Lemma 7.2)
but I am unable to understand a small part of the proof.
The…
4
votes
2 answers
Any examples of exact calculation of Kolmogorov Complexity??
First question: It is known that Kolmogorov Complexity (KC) is not computable (systematically). I would like to know if there are any "real-world" examples-applications where the KC has been computed exactly and not approximately through practical…
Robert_Lewis
- 43
3
votes
0 answers
Inexpressibility results in $FO+LFP$
When we want to prove that some property is not expressible in certain logic, we often use EF-game or it's variants to show this result in finite model theory.
for example, we can show that connectivity is not FO-expressible by classic EF-game on…
deopen
- 33
3
votes
1 answer
Kolmogorov Complexity and Compression Schemes
My question concerns strings with low Kolmogorov Complexities and if there is a single compression scheme that can be used to compress them
I have been introduced to Kolmogorov Complexity through Introduction to Theory of Computation, According to…
Anwar
- 256
3
votes
1 answer
Applications of Model Theory and Category Theory
Do Model Theory and Category Theory have applications in solving Complexity and Game Theory problems in computer science?
I am looking for an example of these...(If there is any...)
user850424
- 869
3
votes
0 answers
Kolmogorov complexity of substring if string is divided according to rule
Denote the plain Kolmogorov complexity of a string $u$ by $C(u)$. Now let $u$ be a string of length $n$ with $C(u) \ge n - O(1)$ and suppose $u = u_1 \cdots u_{\log n}$, a subdivision of the string. Further suppose for each $1 \le i \le \log n$ we…
StefanH
- 18,586
2
votes
1 answer
A question on Kolmogorov Complexity
Is it true that for all strings of a given length at least one has its Kolmogorov complexity equal to its length ?
Is there a proof if the answer is in affirmative?
(For any alphabet with more than symbol)
ARi
- 454
2
votes
0 answers
Coarsenings of the topology on $2^\omega$ with $F_\sigma$ (sub)base
Motivation: I am interested in computational representations of topological spaces which are particularly "explicit", in the somewhat vague sense that we can specify everything we care about using only binary strings. I'll try to illustrate what I'm…
Robin Saunders
- 956
2
votes
1 answer
Complexity of formulae involving projective sets in Polish spaces
I am no expert in descriptive set theory, but for some reason want to estimate the complexity of a certain formula.
The framework. Let be $F$ a projective set in the Polish space $X=\mathbb R^{\mathbb N}$, say it is in the family $\mathbf \Pi^1_n$.…
Tomasz Kania
- 16,996
2
votes
1 answer
How many different graphs of order $n$ are there?
I'm interested in all four combinations: directed and undirected, cyclic and acyclic.
I'm having trouble calculating how big the complexity gets as you add more nodes to a graph. Clearly, the number of possible graphs goes up with adding…
2
votes
1 answer
The problem $K(x) \le K(y)$ is not decidable for Kolmogorov complexity $K$
Let $X$ be some finite alphabet. Given $(x,y) \in X^{\ast}\times X^{\ast}$, how to show that $K(x) \le K(y)$ is not decidable?
I know that $K(x) \le k$ for some fixed $k$ is not decidable, so I tried to reduce the problem. I also know that $K(1^n)…
StefanH
- 18,586
2
votes
1 answer
LFP - shortest path problem
Curious question:
Can anyone show me how to describe shortest path problem using LFP + first order logic?
I am just getting lost on how to describe the problem, though I know that LFP + first-order logic matches to the complexity class $P$.
Thanks.
user27515
- 905
2
votes
0 answers
Each recursive approximating sequence for Kolmogorov complexity is not uniform
Denote the plain Kolmogorov complexity by $C(x)$.
Let $\phi(t,x)$ be a recursive function and $\lim_{t\to\infty} \phi(t,x) = C(x)$ for all $x$. For each $t$ define $\psi_t(x) := \phi(t,x)$ for all $x$. Then $C$ is the limit of the sequence of…
StefanH
- 18,586
1
vote
1 answer
What to study to learn descriptive complexity?
I have an assignment to study the descriptive complexity of a given device that is described with some algebra and informal statements.
I have a background in computer engineering but I haven't deeply studied this field. I've been trying to read a…
ivarec
- 133
1
vote
0 answers
Basic questions about descriptive complexity
I'm trying to learn descriptive complexity, and I'm having trouble on a basic level wrapping my head around what it means for a logical formula to define a computational language. I've tried and failed to find a gentle introduction to the field…
GMB
- 4,256