10 research outputs found
On the Complexity of Nonrecursive XQuery and Functional Query Languages on Complex Values
This paper studies the complexity of evaluating functional query languages
for complex values such as monad algebra and the recursion-free fragment of
XQuery.
We show that monad algebra with equality restricted to atomic values is
complete for the class TA[2^{O(n)}, O(n)] of problems solvable in linear
exponential time with a linear number of alternations. The monotone fragment of
monad algebra with atomic value equality but without negation is complete for
nondeterministic exponential time. For monad algebra with deep equality, we
establish TA[2^{O(n)}, O(n)] lower and exponential-space upper bounds.
Then we study a fragment of XQuery, Core XQuery, that seems to incorporate
all the features of a query language on complex values that are traditionally
deemed essential. A close connection between monad algebra on lists and Core
XQuery (with ``child'' as the only axis) is exhibited, and it is shown that
these languages are expressively equivalent up to representation issues. We
show that Core XQuery is just as hard as monad algebra w.r.t. combined
complexity, and that it is in TC0 if the query is assumed fixed.Comment: Long version of PODS 2005 pape
The Descriptive Complexity of the Deterministic Exponential Time Hierarchy
AbstractIn Descriptive Complexity, we investigate the use of logics to characterize computational complexity classes. Since 1974, when Fagin proved that the class NP is captured by existential second-order logic, considered the first result in this area, other relations between logics and complexity classes have been established. Well-known results usually involve first-order logic and its extensions, and complexity classes in polynomial time or space. Some examples are that the first-order logic extended by the least fixed-point operator captures the class P and the second-order logic extended by the transitive closure operator captures the class PSPACE. In this paper, we will analyze the combined use of higher-order logics of order i, HOi, for i⩾2, extended by the least fixed-point operator, and we will prove that each level of this hierarchy captures each level of the deterministic exponential time hierarchy. As a corollary, we will prove that the hierarchy of HOi(LFP), for i⩾2, does not collapse, that is, HOi(LFP)⊂HOi+1(LFP)
Complexity Hierarchies and Higher-order Cons-free Term Rewriting
Constructor rewriting systems are said to be cons-free if, roughly,
constructor terms in the right-hand sides of rules are subterms of the
left-hand sides; the computational intuition is that rules cannot build new
data structures. In programming language research, cons-free languages have
been used to characterize hierarchies of computational complexity classes; in
term rewriting, cons-free first-order TRSs have been used to characterize the
class PTIME.
We investigate cons-free higher-order term rewriting systems, the complexity
classes they characterize, and how these depend on the type order of the
systems. We prove that, for every K 1, left-linear cons-free systems
with type order K characterize ETIME if unrestricted evaluation is used
(i.e., the system does not have a fixed reduction strategy).
The main difference with prior work in implicit complexity is that (i) our
results hold for non-orthogonal term rewriting systems with no assumptions on
reduction strategy, (ii) we consequently obtain much larger classes for each
type order (ETIME versus EXPTIME), and (iii) results for cons-free
term rewriting systems have previously only been obtained for K = 1, and with
additional syntactic restrictions besides cons-freeness and left-linearity.
Our results are among the first implicit characterizations of the hierarchy E
= ETIME ETIME ... Our work confirms prior
results that having full non-determinism (via overlapping rules) does not
directly allow for characterization of non-deterministic complexity classes
like NE. We also show that non-determinism makes the classes characterized
highly sensitive to minor syntactic changes like admitting product types or
non-left-linear rules.Comment: extended version of a paper submitted to FSCD 2016. arXiv admin note:
substantial text overlap with arXiv:1604.0893
A comparison between algebraic query languages for flat and nested databases
AbstractRecently, much attention has been paid to query languages for nested relations. In the present paper, we consider the nested algebra and the powerset algebra, and compare them both mutually as well as to the traditional flat algebra. We show that either nest or difference can be removed as a primitive operator in the powerset algebra. While the redundancy of the nest operator might have been expected, the same cannot be said of the difference. Basically, this result shows that the presence of one nonmonotonic operator suffices in the powerset algebra. As an interesting consequence of this result, the nested algebra without the difference remains complete in the sense of Bancilhon and Paredaens. Finally, we show there are both similarities and fundamental differences between the expressiveness of query languages for nested relations and that of their counterparts for flat relations
Towards Tractable Algebras for Bags
AbstractBags, i.e., sets with duplicates, are often used to implement relations in database systems. In this paper, we study the expressive power of algebras for manipulating bags. The algebra we present is a simple extension of the nested relation algebra. Our aim is to investigate how the use of bags in the language extends its expressive power and increases its complexity. We consider two main issues, namely (i) the impact of the depth of bag nesting on the expressive power and (ii) the complexity and the expressive power induced by the algebraic operations. We show that the bag algebra is more expressive than the nested relation algebra (at all levels of nesting), and that the difference may be subtle. We establish a hierarchy based on the structure of algebra expressions. This hierarchy is shown to be highly related to the properties of the powerset operator
The most nonelementary theory (a direct lower bound proof)
We give a direct proof by generic reduction that a decidable rudimentary theory of finite typed sets [Henkin 63, Meyer 74, Statman 79, Mairson 92] requires space exceeding infinitely often an exponentially growing stack of twos. This gives the highest currently known lower bound for a decidable logical theory and affirmatively answers to Problem 10.13 of [Compton & Henson 90]: Is there a `natural' decidable theory with a lower bound of the form , where is not linearly bounded? The highest previously known lower and upper bounds for `natural' decidable theories, like WS1S, S2S, are `just' linearly growing stacks of twos
On the Complexity of Queries in the Logical Data Model
We investigate the complexity of query processing in the logical data model (LDM). We use two measures: data complexity, which is complexity with respect to the size of the data, and expression complexity, which is complexity with respect to the size of the expressions denoting the queries. Our investigation shows that while the operations of product and union are essentially first-order operations, the power set operation is inherently a higher-order operation and is exponentially expensive. We define a hierarchy of queries based on the depth of nesting of power set operations and show that this hierarchy corresponds to a natural hierarchy of Turing machines that run in multiply exponential time