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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
2k 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 ...
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

Are classes of graphs represented by adjacency matrix ordered structures?

We know that FO[LFP] captures PTIME on the class of ordered structures. However, I have difficulties interpreting this result. From what I understand, it means that, given a constant, finite alphabet $...
34 votes
3 answers
3k views

Is Descriptive Complexity dead?

I recently started reading about Descriptive Complexity, the branch of Complexity Theory studying the logic languages needed to express complexity classes. The main milestone in the area seems to be ...
-3 votes
1 answer
107 views

Can one do descriptive complexity theory using abstract state machines?

I learned about ASM recently and was interested how it could used for descriptive complexity theory. Such link seems natural to me: you can give construction of algebraic model for formula as an ASM. ...
3 votes
0 answers
80 views

Extending fagin’s theorem for #P (for arbitary structure)

While i am reading Descriptive complexity of #P functions (Saluja) in theorem 1 he state that #FO coincides #P on ordered structures. This is a corollary from fagin’s theorem. I have read fagin’s ...
11 votes
1 answer
322 views

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 ...
1 vote
0 answers
89 views

Complexity class name for the class of languages that are $\Sigma^1_1$-definable over finite domains

Let ${\cal L}=\{Y_1,..., Y_k, X\}$ be a finite relational language such that $X$ is a unary relation name. Let $\phi(X,\bar{Y})\in{\cal L}$ be a first-order formula (the formula can have the equality ...
5 votes
1 answer
105 views

Nonterminal descriptional complexity of regular languages

Recently I became interested in grammar complexity of regular language. Prior to searching for literature, I tried to investigate it on my own, proving two lemmas from comment below. I am aware of an ...
6 votes
1 answer
263 views

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. ...
0 votes
0 answers
109 views

Does Descriptive Complexity techniques have the naturalisation barrier?

I wished to know if the proof attempts at separation of complexity classes via the methods outlined by Descriptive Complexity theorists naturalise? By naturalise I'm talking about the Idea of Natural ...
8 votes
1 answer
253 views

Second order logic = PH. Do even higher order logics correspond to anything on complexity side of things?

In the context of descriptive complexity PH is the class of languages expressible by statements of second-order logic. What classes of languages do higher order logics correspond to?
11 votes
0 answers
230 views

Descriptive Complexity characterzation of BPP

We know of descriptive complexity characterizations of classes such as P, and NP, which use First Order logic, and operators. Does BPP have a characterization under descriptive complexity, too(any ...
5 votes
1 answer
207 views

Reductions in Descriptive Complexity

Reducing one problem to another are well known in various settings, such as many-one, randomized, truth-table, logspace or a whole slew of other reductions. Descriptive complexity can alternately ...
2 votes
0 answers
69 views

Inexpressibility of Second order

In finite model theory, Ehrenfeucht-Fraïssé games gives us tools to prove inexpressibility results for FOL. Pebble games do the same for infinitary logic with finitely many variables. Do we have such ...
10 votes
1 answer
295 views

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, ...
10 votes
3 answers
6k 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 \...
1 vote
0 answers
148 views

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 ...
3 votes
1 answer
176 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?
3 votes
2 answers
146 views

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 $y \in L$ ...
7 votes
1 answer
472 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) := ...
0 votes
0 answers
159 views

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 ...
7 votes
3 answers
2k 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 ...
12 votes
1 answer
265 views

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 ...
1 vote
0 answers
155 views

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 ...
20 votes
0 answers
504 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 $MC(L,C)...
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

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^{...
7 votes
3 answers
340 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 ...
6 votes
1 answer
275 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 $...
21 votes
0 answers
281 views

Descriptive complexity characterization of TimeSpace classes

Are there descriptive complexity characterizations for TimeSpace complexity classes like $\mathsf{SC^i}= \mathsf{DTimeSpace}(n^{O(1)},O(\lg^i n))$?
9 votes
1 answer
429 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 ...
1 vote
1 answer
399 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 \...
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

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 $(...
11 votes
1 answer
473 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 ...
10 votes
2 answers
231 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, $\...
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 ...