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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P and Descriptive Complexity
In the Complexity Zoo, it says [1] that, in descriptive complexity, $P$ can be defined by three different kind of formulae, $FO(LFP)$ which is also $FO(n^{O(1)})$, and also as $SO(HORN)$.
However, ...
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Proof of SAT is complete for NP via first-order reductions
So I have been reading this: https://people.cs.umass.edu/~immerman/book/ch7.pdf
I do not understand the proof of theorem 7.16, which says that SAT is complete for NP via first-order reductions. My ...
4
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1answer
111 views
Descriptive model theory classification of Counting hierarchy
Descriptive model theory uses logic to characterize complexity classes
How to model
Counting Hierarchy
PSPACE
in descriptive model theory?
7
votes
1answer
311 views
Where does the “intuitive” understanding of Kolmogorov complexity fails
Often, the Kolmogorov complexity of some string $x$ is defined as the length of the shortest program producing $x$, for example on wikipedia.
So to give this more formal meaning, define
$$
K'(x) := ...
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2answers
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Construct proof systems for common algorithmic task, like equivalence of regular expressions
A propositional proof system according to Cook and Reckhow for a language $L \subseteq \Sigma^{\ast}$ is a deterministic polynomial time function $f : \Sigma^{\ast} \to L$ that is onto.
For $y \in L$ ...
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Questions about the definition of the Quantum Turing Machine
I am trying to have a better understanding of the definition of the Quantum Turing Machine.
My questions:
If the output of a quantum program is the eigenvalue of the ground state of a Hamiltonian ...
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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 ...
10
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What do dichotomy theorems feed on?
It is well known that certain classes of NP-problems have dichotomy theorems, which guarantee that every task in the class is either NP-complete or is in P. The best known such result is
Schaefer's ...
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Do problems have to be statable in $\Pi_1$ to use Levin's universal search to find short proofs if P=NP
In If P=NP, could we obtain proofs of Goldbach's Conjecture etc.? it talks about the hypothetical world where P=NP and using the proof of it to prove a problem/theorem assuming that it has a short ...
5
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1answer
176 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. ...
7
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227 views
Completeness of the extension of first order logic with least fixed point order operator
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 ...
2
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Infinitary Counting Logics: 1-sorted vs. 2-sorted framework
There are two ways to extend infinitary logic with counting:
Grädel's way (cf. p. 11):
We extend $L_{\infty\omega}$ by introducing a counting existential quantifier:
$$ \mathcal{A} \models \exists^{...
6
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1answer
182 views
Is infinitary logic a logic in the sense of Gurevich?
Gurevich provides an exact definition of what Logic capturing PTIME is.
An abstract logic $L$ consists of
a set of $L[\tau]$-sentences for each vocabulary $\tau$,
and a mapping that maps a property $...
8
votes
1answer
364 views
What is wrong with this $\mathsf{L} \subseteq \mathsf{L}-$uniform $\mathsf{NC}^1$ argument?
The following is not believed to be true:
$\mathsf{L} \subseteq \mathsf{L}-\mbox{uniform } \mathsf{NC}^1$
Can you help me see where the argument breaks down?
The directed reachability problem is ...
3
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2answers
597 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
281 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 ...
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899 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)...
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1answer
289 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 $A$
$$(\mathcal{A},<_1) \models_{LFP} \varphi \...
2
votes
1answer
142 views
Consequences of a $p$-optimal proof system for $\operatorname{TAUT}$
I'm reading a paper which shows the result:
$(1)$ There is a $p$-optimal proof system for $\operatorname{TAUT}$. $\Leftrightarrow$
$(2)$ $L_{\leq}$ is a $P$-bounded logic for $P$.
Both $(1)$ and $(...
9
votes
3answers
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Understanding least-fixed point logic
To better understand a paper I'm trying to get a brief understanding of least-fixed point logic. There are a few points where I am stuck.
If $G = (V,E)$ is a graph and
$$ \Phi(P) = \{(a,b) \mid G \...
9
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2answers
258 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 ...
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Descriptive complexity characterization of TimeSpace classes
Are there descriptive complexity characterizations for TimeSpace complexity classes like $\mathsf{SC^i}= \mathsf{DTimeSpace}(n^{O(1)},O(\lg^i n))$?
9
votes
1answer
280 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 ...
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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
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205 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, $\...
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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
1answer
151 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 ...
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5answers
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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 ...
6
votes
3answers
822 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 ...
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2answers
417 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
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1answer
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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
3answers
145 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
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1answer
380 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
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1answer
153 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, ...
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2answers
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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 ...
8
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1answer
324 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 $FO(\textit{...
10
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1answer
432 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 ...
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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
$MC(L,C)...
6
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1answer
151 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 ...
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3answers
230 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
478 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 ...
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5answers
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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 ...
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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 ...
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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 ...
