10,231 research outputs found
On the Complexity of Existential Positive Queries
We systematically investigate the complexity of model checking the
existential positive fragment of first-order logic. In particular, for a set of
existential positive sentences, we consider model checking where the sentence
is restricted to fall into the set; a natural question is then to classify
which sentence sets are tractable and which are intractable. With respect to
fixed-parameter tractability, we give a general theorem that reduces this
classification question to the corresponding question for primitive positive
logic, for a variety of representations of structures. This general theorem
allows us to deduce that an existential positive sentence set having bounded
arity is fixed-parameter tractable if and only if each sentence is equivalent
to one in bounded-variable logic. We then use the lens of classical complexity
to study these fixed-parameter tractable sentence sets. We show that such a set
can be NP-complete, and consider the length needed by a translation from
sentences in such a set to bounded-variable logic; we prove superpolynomial
lower bounds on this length using the theory of compilability, obtaining an
interesting type of formula size lower bound. Overall, the tools, concepts, and
results of this article set the stage for the future consideration of the
complexity of model checking on more expressive logics
A Theory of Formal Synthesis via Inductive Learning
Formal synthesis is the process of generating a program satisfying a
high-level formal specification. In recent times, effective formal synthesis
methods have been proposed based on the use of inductive learning. We refer to
this class of methods that learn programs from examples as formal inductive
synthesis. In this paper, we present a theoretical framework for formal
inductive synthesis. We discuss how formal inductive synthesis differs from
traditional machine learning. We then describe oracle-guided inductive
synthesis (OGIS), a framework that captures a family of synthesizers that
operate by iteratively querying an oracle. An instance of OGIS that has had
much practical impact is counterexample-guided inductive synthesis (CEGIS). We
present a theoretical characterization of CEGIS for learning any program that
computes a recursive language. In particular, we analyze the relative power of
CEGIS variants where the types of counterexamples generated by the oracle
varies. We also consider the impact of bounded versus unbounded memory
available to the learning algorithm. In the special case where the universe of
candidate programs is finite, we relate the speed of convergence to the notion
of teaching dimension studied in machine learning theory. Altogether, the
results of the paper take a first step towards a theoretical foundation for the
emerging field of formal inductive synthesis
The Value of Help Bits in Randomized and Average-Case Complexity
"Help bits" are some limited trusted information about an instance or
instances of a computational problem that may reduce the computational
complexity of solving that instance or instances. In this paper, we study the
value of help bits in the settings of randomized and average-case complexity.
Amir, Beigel, and Gasarch (1990) show that for constant , if instances
of a decision problem can be efficiently solved using less than bits of
help, then the problem is in P/poly. We extend this result to the setting of
randomized computation: We show that the decision problem is in P/poly if using
help bits, instances of the problem can be efficiently solved with
probability greater than . The same result holds if using less than
help bits (where is the binary entropy function),
we can efficiently solve fraction of the instances correctly with
non-vanishing probability. We also extend these two results to non-constant but
logarithmic . In this case however, instead of showing that the problem is
in P/poly we show that it satisfies "-membership comparability," a notion
known to be related to solving instances using less than bits of help.
Next we consider the setting of average-case complexity: Assume that we can
solve instances of a decision problem using some help bits whose entropy is
less than when the instances are drawn independently from a particular
distribution. Then we can efficiently solve an instance drawn from that
distribution with probability better than .
Finally, we show that in the case where is super-logarithmic, assuming
-membership comparability of a decision problem, one cannot prove that the
problem is in P/poly by a "black-box proof.
Querying the Guarded Fragment
Evaluating a Boolean conjunctive query Q against a guarded first-order theory
F is equivalent to checking whether "F and not Q" is unsatisfiable. This
problem is relevant to the areas of database theory and description logic.
Since Q may not be guarded, well known results about the decidability,
complexity, and finite-model property of the guarded fragment do not obviously
carry over to conjunctive query answering over guarded theories, and had been
left open in general. By investigating finite guarded bisimilar covers of
hypergraphs and relational structures, and by substantially generalising
Rosati's finite chase, we prove for guarded theories F and (unions of)
conjunctive queries Q that (i) Q is true in each model of F iff Q is true in
each finite model of F and (ii) determining whether F implies Q is
2EXPTIME-complete. We further show the following results: (iii) the existence
of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof
of the finite model property of the clique-guarded fragment; (v) the small
model property of the guarded fragment with optimal bounds; (vi) a
polynomial-time solution to the canonisation problem modulo guarded
bisimulation, which yields (vii) a capturing result for guarded bisimulation
invariant PTIME.Comment: This is an improved and extended version of the paper of the same
title presented at LICS 201
- …