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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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41 votes
6 answers
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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 ...
5 votes
3 answers
208 views

Why is order/choice an issue for a logic for PTIME

As I'm reading on the question of a logic for PTIME and in particular about CPT and its variants, whilst things make sense and I follow along, I came to realise that I don't fundamentally understand ...
4 votes
1 answer
171 views

Implicit characterization of sublogarithmic space

Let $SUBLOG = DSPACE(o(\log(n)) \setminus DSPACE(O(1))$ be the set of languages decidable with less than logarithmic space, but more than a constant amount of space, on a multi-tape Turing machine ...
1 vote
0 answers
45 views

19 votes
5 answers
1k views

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 ...
5 votes
1 answer
229 views

Expressiveness of Infinitary Logic

I'm trying to put together a general picture of the expressiveness of some logics: First-Order Logics, Fixed-Point Logics, (Finite Variable) Infinitary Logics and the respected versions with Counting. ...
2 votes
1 answer
89 views

4 votes
2 answers
849 views

Context-free Grammar for a Context-free Language Intersecting a Regular Language (get the Maximum Number of Rules)

It is well known that the intersection of $L \cap R$ of a context-free Language $L$ and a regular Language $R$ is context-free. Each proof I have seen constructs a automaton (a PDA) that accepts $L$ ...
9 votes
2 answers
1k views

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)...
2 votes
1 answer
337 views

What is a commutative transitive closure operator?

When reading about descriptive complexity theory, I have read about a "commutative transitive closure operator". I understand transitive closure operators, but what is a commutative transitive closure ...
15 votes
0 answers
322 views

Proof assistant formalizations of Finite Model Theory

I'm wondering if anyone knows of a formalization (even limited) of any part of finite model theory in any of the major proof assistants. (I'm most familiar with Coq, but Isabelle, Agda, etc. would ...
9 votes
2 answers
308 views

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 ...
4 votes
1 answer
211 views

5 votes
0 answers
203 views

Logic capturing automorphism-invariant $\mathsf{AC^0}$ properties

Q1. Is there a logic that is computable in polynomial-time which contains all order-invariant properties computable in smaller classes like $\mathsf{AC^0}$ (or $\mathsf{TC^0}$)? Motivation As you ...
7 votes
1 answer
185 views

Using MSOL for solving BIDS problem

From "Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width" (B. Courcelle et al) we know that any problem that can be written on MSOL (Monadic Second Order Logic) has a linear ...
10 votes
0 answers
363 views

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 ...
19 votes
0 answers
521 views

To what extent MSO = WS1S, when adding relations?

[This question has been asked on MathOverflow with no luck a month ago.] Let me first clarify my definitions. For a word $w \in \Sigma^*$, with $\Sigma =\{a_1, \ldots, a_n\}$, I define two ...
14 votes
2 answers
578 views

MSOL optimization problems on graphs of bounded cliquewidth, with cardinality predicates

CMSOL is Counting Monadic Second Order Logic, i.e. a logic of graphs where the domain is the set of vertices and edges, there are predicates for vertex-vertex adjacency and edge-vertex incidence, ...
5 votes
1 answer
330 views

What is the parameterized complexity of following model checking problem?

Input: Graph $G$ and formula $\varphi_1(\vec x),\varphi_2(\vec x)$ Parameter: $tw(G)+|\varphi_1|+|\varphi_2|$ Problem: Decide if $|\varphi_1(G)|=|\varphi_2(G)|$ where $tw(G)$ is the treewidth of \$...
2 votes
3 answers
155 views

Searching for matching queries

Suppose you have a large set of queries (could be in SQL form, but conceivably the same problem exists for search engine query strings or Lucene expressions, etc...) stored and you want to know which ...