arXiv subjects

Senén González

Publications and source records attributed to Senén González.

3 recordsLinked to original sources

arXiv2019

Descriptive Complexity of Deterministic Polylogarithmic Time and Space

We propose logical characterizations of problems solvable in deterministic polylogarithmic time (PolylogTime) and polylogarithmic space (PolylogSpace). We introduce a novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements. We prove that the inflationary and partial fixed point vartiants of this logic capture PolylogTime and PolylogSpace, respectively. In the course of proving that our logic indeed captures PolylogTime on finite ordered structures, we introduce a variant of random-access Turing machines that can access the relations and functions of a structure directly. We investigate whether an explicit predicate for the ordering of the domain is needed in our PolylogTime logic. Finally, we present the open problem of finding an exact characterization of order-invariant queries in PolylogTime.

cs.LO↗

arXiv2019

Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems

The polylogarithmic time hierarchy structures sub-linear time complexity. In recent work it was shown that all classes $\tildeΣ_{m}^{\mathit{plog}}$ or $\tildeΠ_{m}^{\mathit{plog}}$ ($m \in \mathbb{N}$) in this hierarchy can be captured by semantically restricted fragments of second-order logic. In this paper the descriptive complexity theory of polylogarithmic time is taken further showing that there are strict hierarchies inside each of the classes of the hierarchy. A straightforward consequence of this result is that there are no complete problems for these complexity classes, not even under polynomial time reductions. As another consequence we show that the polylogarithmic time hierarchy itself is strict.

cs.CC↗

arXiv2018

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

Let $\mathrm{SO}^{\mathit{plog}}$ 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 $Σ^{\mathit{plog}}_m$ (and $Π^{\mathit{plog}}_m$) defined by formulae with alternating blocks of existential and universal second-order quantifiers in quantifier-prenex normal form. We first show that the existential fragment $Σ^{\mathit{plog}}_1$ captures NPolyLogTime, i.e. the class of Boolean queries that can be accepted by a non-deterministic Turing machine with random access to the input in time $O((\log n)^k)$ for some $k \ge 0$. Using alternating Turing machines with random access input allows us to characterise also the fragments $Σ^{\mathit{plog}}_m$ (and $Π^{\mathit{plog}}_m$) as those Boolean queries with at most $m$ alternating blocks of second-order quantifiers that are accepted by an alternating Turing machine. Consequently, $\mathrm{SO}^{\mathit{plog}}$ captures the whole poly-logarithmic time hierarchy. We demonstrate the relevance of this logic and complexity class by several problems in database theory.

cs.LO↗