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

  title={A Restricted Second-Order Logic for Non-deterministic Poly-Logarithmic Time},
  author={Flavio Ferrarotti and Sen{\'e}n Gonz{\'a}lez and Klaus-Dieter Schewe and Jos{\'e} Maria Turull Torres},
We introduce a restricted second-order logic $\mathrm{SO}^{\mathit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin's style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\mathrm{SO}^{\mathit{plog}}$ corresponds exactly to… Expand
Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems
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. Expand
Completeness in Polylogarithmic Time and Space
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. Expand


The Polylog-Time Hierarchy Captured by Restricted Second-Order Logic
The problem, which Turing machine complexity class is captured by Boolean queries over ordered relational structures that can be expressed in second-order logic, is investigated and the relevance of this logic and complexity class by several problems in database theory is demonstrated. Expand
A Second-Order Logic in Which Variables Range over Relations with Complete First-Order Types
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 $\Sigma^{1,F}_1$. Expand
Second-Order Logic over Strings: Regular and Non-regular Fragments
An exhaustive classification of the regular and nonregular prefix classes of general second-order logic, and derive of complexity results for the corresponding model checking problems. Expand
Existential second-order logic over graphs: charting the tractability frontier
A dichotomy holds, i.e., each prefix class of existential second-order logic either contains sentences that can express NP-complete problems or each of its sentences expresses a polynomial-time solvable problem. Expand
Choiceless Polynomial Time
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. Expand
Capturing Complexity Classes by Fragments of Second-Order Logic
  • E. Grädel
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1992
It is shown that all these logics collapse to their existential fragments and are strictly weaker than previously known logics for these classes and fail to express some very simple properties. Expand
Existential second-order logic over graphs: Charting the tractability frontier
This article completely characterize the computational complexity of prefix classes of existential second-order logic in three different contexts: (1) over directed graphs, (2) over undirected graphs with self-loops and (3) over undirected graphs without self-Loops. Expand
Existential second-order logic over strings
ESO and ESO are the maximal standard ESO-prefix classes contained in MSO, thus expressing only regular languages, and the following dichotomy theorem is proved: An ESO prefix-class either expresses onlyregular languages (and is thus in ESO), or it expresses some NP-complete languages. Expand
The complexity of theorem-proving procedures
  • S. Cook
  • Computer Science, Mathematics
  • STOC
  • 1971
It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be “reduced” to the problem of determining whether a given propositional formula is aExpand
A restricted second order logic for finite structures
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, forExpand