Descriptive complexity classifies problems based on how hard it is to express the problem in some logical formalism.
3
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162 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 $L$ ...
2
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1answer
170 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 ...
7
votes
2answers
659 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 ...
1
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0answers
85 views
Least fixed point logic is efficiently $\operatorname{P}$-bounded for $\operatorname{P} \Leftrightarrow L_\leq$ is a logic for $\operatorname{P}$
A least-fixed point (LFP) formula is $\leq m$-invariant iff f.a. structrues $\mathcal{A}$ with $|A| \leq m$ and all orderings $<_1,<_2$ on $A$
$$(\mathcal{A},<_1) \models_{LFP} \varphi ...
1
vote
1answer
80 views
Consequences of a $p$-optimal proof system for $\operatorname{TAUT}$
I'm reading a paper which shows the result:
$(1)$ There is a $p$-optimal proof system for $\operatorname{TAUT}$. $\Leftrightarrow$
$(2)$ $L_{\leq}$ is a $P$-bounded logic for $P$.
Both $(1)$ and ...
5
votes
2answers
352 views
Understanding least-fixed point logic
To better understand a paper I'm trying to get a brief understanding of least-fixed point logic. There are a few points where I am stuck.
If $G = (V,E)$ is a graph and
$$ \Phi(P) = \{(a,b) \mid G ...
5
votes
2answers
177 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 ...
12
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0answers
158 views
Descriptive complexity characterization of TimeSpace classes
Are there descriptive complexity characterizations for TimeSpace complexity classes like $\mathsf{SC^i}= \mathsf{DTimeSpace}(n^{O(1)},O(\lg^i n))$?
8
votes
1answer
163 views
FO-uniform AC0 with some predicate
My question is about finite model theory/descriptive complexity, so $FO(R)$ will mean "first order over finite binary words, using predicates Rs and a unary predicate P true on the position of the 1 ...
11
votes
0answers
155 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 ...
10
votes
2answers
175 views
When does an FO property kill off NL-hardness?
Context: We consider only digraphs. Let CYCLE be the language of graphs with a cycle; it is an NL-complete problem. Let HASEDGE be the language of graphs with at least one edge. Then trivially, ...
4
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0answers
136 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 ...
7
votes
1answer
127 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 ...
16
votes
3answers
819 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 ...
5
votes
1answer
272 views
What is First-Order Rewritable (and FO-Query)?
I just wonder what FO Rewritable is, put an example to make it clearer for me. Also, I heard that a language that is FO Rewritable is very good, in what sense?
It is said as follow:
A class C of ...
9
votes
2answers
257 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
1answer
277 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 ...
2
votes
3answers
133 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 ...
10
votes
1answer
316 views
Could a descriptive complexity version of Rice's theorem be used to separate AC0 and PSPACE?
In this question, it was mentioned that there are descriptive complexity versions of Rice's theorem. I found a proof of the following theorem:
Given a complexity class C, nontrivial properties of ...
8
votes
1answer
134 views
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, ...
14
votes
2answers
427 views
Are there descriptive complexity representations of quantum complexity classes?
The title more or less says it all, but I guess I could add a bit of background and some specific examples I'm interested in.
Descriptive complexity theorists, such as Immerman and Fagin, have ...
6
votes
1answer
249 views
SAT in finite model theory without order
It is well known in finite model theory that without an order on the input, the expressivity is very limited. For example it is known that $FO(<,\textit{PFP})$ is equal to PSPACE, and ...
9
votes
1answer
313 views
Finite One-Way Permutation with Infinite Domain
Let $\pi \colon \{0,1\}^* \to \{0,1\}^*$ be a permutation. Note that while $\pi$ acts on an infinite domain, its description might be finite. By description, I mean a program that describes $\pi$'s ...
16
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0answers
370 views
Model-checking for three-variable logics and restricted structures
Denote the $k$-variable fragment of logic $L$ by $L^{(k)}$. The model-checking problem for a logic $L$ with respect to a class of structures $C$, denoted $MC(L,C)$, is the decision problem
...
6
votes
1answer
145 views
Non-interesting numbers via resource-bounded properties?
There is an old joke about the smallest non-interesting number being interesting in itself (I have heard it attributed to Richard Hamming). This is then used to justify the argument that every number ...
12
votes
3answers
189 views
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 ...
10
votes
1answer
341 views
Ehrenfeucht-Fraïssé games (Ajtai-Fagin in fact) for regular languages.
Immerman (Descriptive Complexity, 1999) presents the EF games for existential monadic second order (Ajtai-Fagin games) on page 127. As $\exists$MSO on words is equivalent to regular languages, the ...
30
votes
4answers
818 views
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 ...
10
votes
0answers
286 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 ...
15
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0answers
367 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 ...
