737 research outputs found
Exponential Lower Bounds and Separation for Query Rewriting
We establish connections between the size of circuits and formulas computing
monotone Boolean functions and the size of first-order and nonrecursive Datalog
rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower
bounds and separation results from circuit complexity to prove similar results
for the size of rewritings that do not use non-signature constants. For
example, we show that, in the worst case, positive existential and nonrecursive
Datalog rewritings are exponentially longer than the original queries;
nonrecursive Datalog rewritings are in general exponentially more succinct than
positive existential rewritings; while first-order rewritings can be
superpolynomially more succinct than positive existential rewritings
Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy
It is shown that for any fixed , the -fragment of
Presburger arithmetic, i.e., its restriction to quantifier alternations
beginning with an existential quantifier, is complete for
, the -th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
and . This result completes the
computational complexity landscape for Presburger arithmetic, a line of
research which dates back to the seminal work by Fischer & Rabin in 1974.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
-fragment of Presburger arithmetic: given a -formula
, it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure
Complexity of ITL model checking: some well-behaved fragments of the interval logic HS
Model checking has been successfully used in many computer science fields,
including artificial intelligence, theoretical computer science, and databases.
Most of the proposed solutions make use of classical, point-based temporal
logics, while little work has been done in the interval temporal logic setting.
Recently, a non-elementary model checking algorithm for Halpern and Shoham's
modal logic of time intervals HS over finite Kripke structures (under the
homogeneity assumption) and an EXPSPACE model checking procedure for two
meaningful fragments of it have been proposed. In this paper, we show that more
efficient model checking procedures can be developed for some expressive enough
fragments of HS
Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation
We present an efficient proof system for Multipoint Arithmetic Circuit
Evaluation: for every arithmetic circuit of size and
degree over a field , and any inputs ,
the Prover sends the Verifier the values and a proof of length, and
the Verifier tosses coins and can check the proof in about time, with probability of error less than .
For small degree , this "Merlin-Arthur" proof system (a.k.a. MA-proof
system) runs in nearly-linear time, and has many applications. For example, we
obtain MA-proof systems that run in time (for various ) for the
Permanent, Circuit-SAT for all sublinear-depth circuits, counting
Hamiltonian cycles, and infeasibility of - linear programs. In general,
the value of any polynomial in Valiant's class can be certified
faster than "exhaustive summation" over all possible assignments. These results
strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed
by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas
running in time, nearly-linear time MA-proof systems for
counting orthogonal vectors in a collection and finding Closest Pairs in the
Hamming metric, and a MA-proof system running in -time for
counting -cliques in graphs.
We point to some potential future directions for refuting the
Nondeterministic Strong ETH.Comment: 17 page
Invariant Generation through Strategy Iteration in Succinctly Represented Control Flow Graphs
We consider the problem of computing numerical invariants of programs, for
instance bounds on the values of numerical program variables. More
specifically, we study the problem of performing static analysis by abstract
interpretation using template linear constraint domains. Such invariants can be
obtained by Kleene iterations that are, in order to guarantee termination,
accelerated by widening operators. In many cases, however, applying this form
of extrapolation leads to invariants that are weaker than the strongest
inductive invariant that can be expressed within the abstract domain in use.
Another well-known source of imprecision of traditional abstract interpretation
techniques stems from their use of join operators at merge nodes in the control
flow graph. The mentioned weaknesses may prevent these methods from proving
safety properties. The technique we develop in this article addresses both of
these issues: contrary to Kleene iterations accelerated by widening operators,
it is guaranteed to yield the strongest inductive invariant that can be
expressed within the template linear constraint domain in use. It also eschews
join operators by distinguishing all paths of loop-free code segments. Formally
speaking, our technique computes the least fixpoint within a given template
linear constraint domain of a transition relation that is succinctly expressed
as an existentially quantified linear real arithmetic formula. In contrast to
previously published techniques that rely on quantifier elimination, our
algorithm is proved to have optimal complexity: we prove that the decision
problem associated with our fixpoint problem is in the second level of the
polynomial-time hierarchy.Comment: 35 pages, conference version published at ESOP 2011, this version is
a CoRR version of our submission to Logical Methods in Computer Scienc
Existentially Restricted Quantified Constraint Satisfaction
The quantified constraint satisfaction problem (QCSP) is a powerful framework
for modelling computational problems. The general intractability of the QCSP
has motivated the pursuit of restricted cases that avoid its maximal
complexity. In this paper, we introduce and study a new model for investigating
QCSP complexity in which the types of constraints given by the existentially
quantified variables, is restricted. Our primary technical contribution is the
development and application of a general technology for proving positive
results on parameterizations of the model, of inclusion in the complexity class
coNP
Path Checking for MTL and TPTL over Data Words
Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are
quantitative extensions of linear temporal logic, which are prominent and
widely used in the verification of real-timed systems. It was recently shown
that the path checking problem for MTL, when evaluated over finite timed words,
is in the parallel complexity class NC. In this paper, we derive precise
complexity results for the path-checking problem for MTL and TPTL when
evaluated over infinite data words over the non-negative integers. Such words
may be seen as the behaviours of one-counter machines. For this setting, we
give a complete analysis of the complexity of the path-checking problem
depending on the number of register variables and the encoding of constraint
numbers (unary or binary). As the two main results, we prove that the
path-checking problem for MTL is P-complete, whereas the path-checking problem
for TPTL is PSPACE-complete. The results yield the precise complexity of model
checking deterministic one-counter machines against formulae of MTL and TPTL
On the Succinctness of Query Rewriting over OWL 2 QL Ontologies with Shallow Chases
We investigate the size of first-order rewritings of conjunctive queries over
OWL 2 QL ontologies of depth 1 and 2 by means of hypergraph programs computing
Boolean functions. Both positive and negative results are obtained. Conjunctive
queries over ontologies of depth 1 have polynomial-size nonrecursive datalog
rewritings; tree-shaped queries have polynomial positive existential
rewritings; however, in the worst case, positive existential rewritings can
only be of superpolynomial size. Positive existential and nonrecursive datalog
rewritings of queries over ontologies of depth 2 suffer an exponential blowup
in the worst case, while first-order rewritings are superpolynomial unless
. We also analyse rewritings of
tree-shaped queries over arbitrary ontologies and observe that the query
entailment problem for such queries is fixed-parameter tractable
Ontology-based data access with databases: a short course
Ontology-based data access (OBDA) is regarded as a key ingredient of the new generation of information systems. In the OBDA paradigm, an ontology defines a high-level global schema of (already existing) data sources and provides a vocabulary for user queries. An OBDA system rewrites such queries and ontologies into the vocabulary of the data sources and then delegates the actual query evaluation to a suitable query answering system such as a relational database management system or a datalog engine. In this chapter, we mainly focus on OBDA with the ontology language OWL 2QL, one of the three profiles of the W3C standard Web Ontology Language OWL 2, and relational databases, although other possible languages will also be discussed. We consider different types of conjunctive query rewriting and their succinctness, different architectures of OBDA systems, and give an overview of the OBDA system Ontop
Revisiting Synthesis for One-Counter Automata
We study the (parameter) synthesis problem for one-counter automata with
parameters. One-counter automata are obtained by extending classical
finite-state automata with a counter whose value can range over non-negative
integers and be tested for zero. The updates and tests applicable to the
counter can further be made parametric by introducing a set of integer-valued
variables called parameters. The synthesis problem for such automata asks
whether there exists a valuation of the parameters such that all infinite runs
of the automaton satisfy some omega-regular property. Lechner showed that (the
complement of) the problem can be encoded in a restricted one-alternation
fragment of Presburger arithmetic with divisibility. In this work (i) we argue
that said fragment, called AERPADPLUS, is unfortunately undecidable.
Nevertheless, by a careful re-encoding of the problem into a decidable
restriction of AERPADPLUS, (ii) we prove that the synthesis problem is
decidable in general and in N2EXP for several fixed omega-regular properties.
Finally, (iii) we give a polynomial-space algorithm for the special case of the
problem where parameters can only be used in tests, and not updates, of the
counter
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