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Questions tagged [descriptive-complexity]

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

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40 votes
6 answers

Is there a logic without induction that captures much of P?

The Immerman-Vardi theorem states that PTIME (or P) is precisely the class of languages that can be described by a sentence of First-Order Logic together with a fixed-point operator, over the class of ...
András Salamon's user avatar
11 votes
1 answer

reference request: deciding validity of higher-order quantified boolean formulas is not Kalmar-elementary

$\newcommand\iddots{⋰}$In "A simple proof of a theorem of Statman" (TCS 1992), Harry Mairson gives a simple proof of Statman's result that deciding $\beta\eta$-equality of terms in simply typed lambda ...
Noam Zeilberger's user avatar
9 votes
2 answers

Intuition behind proof systems

I'm trying to under stand the paper On p-Optimal Proof Systems and Logic for PTIME. There is a notion called proof systems in the paper and I do not get the intution: $\Sigma = \{0,1\}$ ... We ...
Joachim's user avatar
  • 513
19 votes
5 answers

Why do relational databases work at all, given the theoretical exponential complexity of answer finding (in the size of the query)?

It seems to be known that to find an answer to a query $Q$ over a relational database $D$, one needs time $|D|^{|Q|}$, and one cannot get rid of the exponent $|Q|$. As $D$ can be very large, we wonder ...
imz -- Ivan Zakharyaschev's user avatar
12 votes
3 answers

Is there a natural restriction of VO logic which captures P or NP?

The paper Lauri Hella and José María Turull-Torres, Computing queries with higher-order logics, TCS 355 197–214, 2006. doi: 10.1016/j.tcs.2006.01.009 proposes logic VO, variable-order logic. This ...
András Salamon's user avatar
10 votes
0 answers

Collapsing of exptime and alternation bounded turing machine

This question was already asked on math overflow, but I did not find any answer to my question (or let say the answer was that up to the knowledge of those people, no answer were known) Let C be a ...
Arthur MILCHIOR's user avatar
9 votes
1 answer

Is testing two SO-Horn queries for equivalence decidable?

It follows from Rice's theorem that you cannot determine whether or not two Turing machines decide the same language. My question is: Does this also apply in descriptive complexity settings, ...
user avatar
9 votes
2 answers

How can we express "$P=PSPACE$" as a first-order formula? [closed]

How can we express "$P=PSPACE$" as a first-order formula? Which level of the arithmetic hierarchy contains this formula (and what is the currently known minimum level of the hierarchy that contains it)...
mars's user avatar
  • 101
6 votes
1 answer

What is FO(REGULAR)? (The descriptive complexity equivalent of NC1)

According to Immerman's Descriptive Complexity diagram, there is a logic called $\mathsf{FO(REGULAR)}$ which captures $\mathsf{NC}^1$. However, I can't find the reference where this logic is defined. ...
Caleb Stanford's user avatar