675 research outputs found
Expressive power and complexity of a logic with quantifiers that count proportions of sets
We present a second-order logic of proportional quantifiers, SOLP, which is essentially a first-order language
extended with quantifiers that act upon second-order variables of a given arity r and count the fraction of elements in
a subset of r-tuples of a model that satisfy a formula. Our logic is capable of expressing proportional versions of
different problems of complexity up to NP-hard as, for example, the problem of deciding if at least a fraction 1/n of
the set of vertices of a graph form a clique; and fragments within our logic capture complexity classes as NL and P,
with auxiliary ordering relation. When restricted to monadic second-order variables, our logic of proportional
quantifiers admits a semantic approximation based on almost linear orders, which is not as weak as other known logics with counting quantifiers (restricted to almost orders), for it does not have the bounded number
of degrees property. Moreover, we show that, in this almost-ordered setting, different fragments of this logic vary in their expressive power, and show the existence of an infinite hierarchy inside our monadic language.
We extend our inexpressibility result of almost-ordered structure to a fragment of SOLP, which in the presence of full order captures P. To obtain all our inexpressibility results, we developed combinatorial games appropriate
for these logics, whose application could go beyond the almost-ordered models and hence are interesting by themselves.Peer ReviewedPreprin
Logics of Finite Hankel Rank
We discuss the Feferman-Vaught Theorem in the setting of abstract model
theory for finite structures. We look at sum-like and product-like binary
operations on finite structures and their Hankel matrices. We show the
connection between Hankel matrices and the Feferman-Vaught Theorem. The largest
logic known to satisfy a Feferman-Vaught Theorem for product-like operations is
CFOL, first order logic with modular counting quantifiers. For sum-like
operations it is CMSOL, the corresponding monadic second order logic. We
discuss whether there are maximal logics satisfying Feferman-Vaught Theorems
for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th
birthday. The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-23534-9_1
Queries with Guarded Negation (full version)
A well-established and fundamental insight in database theory is that
negation (also known as complementation) tends to make queries difficult to
process and difficult to reason about. Many basic problems are decidable and
admit practical algorithms in the case of unions of conjunctive queries, but
become difficult or even undecidable when queries are allowed to contain
negation. Inspired by recent results in finite model theory, we consider a
restricted form of negation, guarded negation. We introduce a fragment of SQL,
called GN-SQL, as well as a fragment of Datalog with stratified negation,
called GN-Datalog, that allow only guarded negation, and we show that these
query languages are computationally well behaved, in terms of testing query
containment, query evaluation, open-world query answering, and boundedness.
GN-SQL and GN-Datalog subsume a number of well known query languages and
constraint languages, such as unions of conjunctive queries, monadic Datalog,
and frontier-guarded tgds. In addition, an analysis of standard benchmark
workloads shows that most usage of negation in SQL in practice is guarded
negation
Rewritability in Monadic Disjunctive Datalog, MMSNP, and Expressive Description Logics
We study rewritability of monadic disjunctive Datalog programs, (the
complements of) MMSNP sentences, and ontology-mediated queries (OMQs) based on
expressive description logics of the ALC family and on conjunctive queries. We
show that rewritability into FO and into monadic Datalog (MDLog) are decidable,
and that rewritability into Datalog is decidable when the original query
satisfies a certain condition related to equality. We establish
2NExpTime-completeness for all studied problems except rewritability into MDLog
for which there remains a gap between 2NExpTime and 3ExpTime. We also analyze
the shape of rewritings, which in the MMSNP case correspond to obstructions,
and give a new construction of canonical Datalog programs that is more
elementary than existing ones and also applies to formulas with free variables
Approximate formulae for a logic that capture classes of computational complexity
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Logic Journal of IGPL following peer review. The definitive publisher-authenticated version Arratia, Argimiro; Ortiz, Carlos E. Approximate formulae for a logic that capture classes of computational complexity. Logic Journal of IGPL, 2009, vol. 17, p. 131-154 is available online at: http://jigpal.oxfordjournals.org/cgi/reprint/17/1/131?maxtoshow=&hits=10&RESULTFORMAT=&fulltext=Approximate+formulae+for+a+logic+that+capture+classes+of+computational+complexity&searchid=1&FIRSTINDEX=0&resourcetype=HWCITThis paper presents a syntax of approximate formulae suited for the logic with counting quantifiers SOLP. This logic was formalised by us in [1] where, among other properties, we showed the following facts: (i) In the presence of a built–in (linear) order, SOLP can describe NP–complete problems and some of its fragments capture the classes P and NL; (ii) weakening the ordering relation to
an almost order we can separate meaningful fragments, using a combinatorial tool
adapted to these languages.
The purpose of our approximate formulae is to provide a syntactic approximation
to the logic SOLP, enhanced with a built-in order, that should be complementary of the semantic approximation based on almost orders, by means of producing logics where problems are syntactically described within a small counting error. We introduce a concept of strong expressibility based on approximate formulae, and show that for many fragments of SOLP with built-in order, including ones that capture P and NL, expressibility and strong expressibility are
equivalent. We state and prove a Bridge Theorem that links expressibility in
fragments of SOLP over almost-ordered structures to strong expressibility with
respect to approximate formulae for the corresponding fragments over ordered
structures. A consequence of these results is that proving inexpressibility over
fragments of SOLP with built-in order could be done by proving inexpressibility
over the corresponding fragments with built-in almost order, where separation
proofs are allegedly easier.Peer ReviewedPostprint (author’s final draft
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