# The Polylog-Time Hierarchy Captured by Restricted Second-Order Logic

@article{Ferrarotti2018ThePH, title={The Polylog-Time Hierarchy Captured by Restricted Second-Order Logic}, author={Flavio Ferrarotti and Sen{\'e}n Gonz{\'a}lez and Klaus-Dieter Schewe and Jos{\'e} Maria Turull Torres}, journal={2018 20th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)}, year={2018}, pages={133-140} }

Let SO^plog denote the restriction of second-order logic, where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. In this article we investigate the problem, which Turing machine complexity class is captured by Boolean queries over ordered relational structures that can be expressed in this logic. For this we define a hierarchy of fragments Σ^plog_m (and Σ^plog_m) defined by formulae with alternating blocks of existential and…

## Topics from this paper

## 6 Citations

A Restricted Second-Order Logic for Non-deterministic Poly-Logarithmic Time

- Mathematics, Computer ScienceLog. J. IGPL
- 2020

A Fagin's style theorem is proved showing that the Boolean queries which can be expressed in the existential fragment of second-order logic corresponds exactly to the class of decision problems that can be computed by a non-deterministic Turing machine with random access to the input in time.

Descriptive complexity of deterministic polylogarithmic time and space

- Computer ScienceJ. Comput. Syst. Sci.
- 2021

A novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements and a variant of random-access Turing machines that can access the relations and functions of a structure directly are introduced.

Descriptive Complexity of Deterministic Polylogarithmic Time

- Computer ScienceWoLLIC
- 2019

A novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements and a variant of random-access Turing machines that can access the relations and functions of the structure directly are introduced.

Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems

- Computer Science, MathematicsFoIKS
- 2020

This paper shows that the descriptive complexity theory of polylogarithmic time is taken further showing that there are strict hierarchies inside each of the classes of the hierarchy.

Completeness in Polylogarithmic Time and Space

- Computer ScienceArXiv
- 2020

An alternative notion of completeness inspired by the concept of uniformity from circuit complexity is developed and proved and it is shown that complete problems can still play an important role in the study of the interrelationship between polylogarithmic and other classical complexity classes.

Logic, Language, Information, and Computation

- Computer ScienceLecture Notes in Computer Science
- 2018

Inhabitants of Intuitionistic Implicational Theorems discuss the role of language in the development of knowledge and the role that language plays in the acquisition of knowledge.

## References

SHOWING 1-10 OF 38 REFERENCES

Computing queries with higher-order logics

- Computer Science, MathematicsTheor. Comput. Sci.
- 2006

This article defines a logic which is called variable order logic (VO) which permits the use of untyped relation variables, i.e., variables of variable order, by allowing quantification over orders, and shows that this logic is complete, though even non-recursive queries can be expressed in VO.

Expressibility of Higher Order Logics

- Computer Science, MathematicsElectron. Notes Theor. Comput. Sci.
- 2003

A logic is defined which is called variable order logic (VO) which permits the use of untyped relation variables, i.e., variables of variable order, by allowing quantification over orders.

Truth Definitions and Higher Order Logics in Finite Models

- Mathematics
- 2005

The first part of the thesis concerns the expressive power of higher order logics in finite models. We introduce the notion of basic arity, a higher order analogue of the arity of a second order…

On Fragments of Higher Order Logics that on Finite Structures Collapse to Second Order

- Mathematics, Computer ScienceWoLLIC
- 2017

It turns out that there are many examples of properties of finite models (queries from the perspective of relational databases) which can be simply and elegantly defined by formulae of the higher-order fragments studied in this work.

A Second-Order Logic in Which Variables Range over Relations with Complete First-Order Types

- Computer Science2010 XXIX International Conference of the Chilean Computer Science Society
- 2010

The complexity class NP$^F$ is defined by using a variation of the relational machine of S. Abiteboul and V. Vianu and it is proved that this complexity class is captured by $\Sigma^{1,F}_1$.

On Higher Order Query Languages which on Relational Databases Collapse to Second Order Logic

- Mathematics, Computer ScienceArXiv
- 2016

This article defines a general schema of $\exists$TO formulas which consists of existentially quantifying a third-order linear digraph of polynomial length, that is, a sequence of structures that represents a computation, and gives a constructive proof of the fact that all $\exist$TO sub formulas of that schema can be translated into an equivalent SO formula.

Choiceless Polynomial Time

- Computer Science, MathematicsAnn. Pure Appl. Log.
- 1999

This work attempts to capture the choiceless fragment of PTime, a version of abstract state machines (formerly called evolving algebras) that is to replace arbitrary choice with parallel execution and is more expressive than other PTime logics in the literature.

Subclasses of Binary NP

- Mathematics, Computer ScienceJ. Log. Comput.
- 1998

It is shown that many of the semantical restrictions of binary NP have the same expressive power and establish a 4-level strict hierarchy, represented by sets, permutations, unary functions and arbitrary binary relations, respectively.

A Restricted Second Order Logic for Finite Structures

- Computer Science, MathematicsLCC
- 1994

We introduce a restricted version of second order logic SOω in which the second order quantifiers range over relations that are closed under the equivalence relation ≡k of k variable equivalence, for…

Expressing Properties in Second and Third Order Logic: Hypercube Graphs and SATQBF

- Mathematics, Computer ScienceLog. J. IGPL
- 2014

This work characterize in second-order logic the class of hypercube graphs and the classes SATQBF_k of satisfiable quantified Boolean formulae with k alternations of quantifiers and sketches a third- order logic sentence that defines the class SATQbf = \bigcup_{k \geq 1} SATQ BF_k.