66,231 research outputs found
Computabilities of Validity and Satisfiability in Probability Logics over Finite and Countable Models
The -logic (which is called E-logic in this paper) of
Kuyper and Terwijn is a variant of first order logic with the same syntax, in
which the models are equipped with probability measures and in which the
quantifier is interpreted as "there exists a set of measure
such that for each , ...." Previously, Kuyper and
Terwijn proved that the general satisfiability and validity problems for this
logic are, i) for rational , respectively
-complete and -hard, and ii) for ,
respectively decidable and -complete. The adjective "general" here
means "uniformly over all languages."
We extend these results in the scenario of finite models. In particular, we
show that the problems of satisfiability by and validity over finite models in
E-logic are, i) for rational , respectively
- and -complete, and ii) for , respectively
decidable and -complete. Although partial results toward the countable
case are also achieved, the computability of E-logic over countable
models still remains largely unsolved. In addition, most of the results, of
this paper and of Kuyper and Terwijn, do not apply to individual languages with
a finite number of unary predicates. Reducing this requirement continues to be
a major point of research.
On the positive side, we derive the decidability of the corresponding
problems for monadic relational languages --- equality- and function-free
languages with finitely many unary and zero other predicates. This result holds
for all three of the unrestricted, the countable, and the finite model cases.
Applications in computational learning theory, weighted graphs, and neural
networks are discussed in the context of these decidability and undecidability
results.Comment: 47 pages, 4 tables. Comments welcome. Fixed errors found by Rutger
Kuype
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
Classifying the Arithmetical Complexity of Teaching Models
This paper classifies the complexity of various teaching models by their
position in the arithmetical hierarchy. In particular, we determine the
arithmetical complexity of the index sets of the following classes: (1) the
class of uniformly r.e. families with finite teaching dimension, and (2) the
class of uniformly r.e. families with finite positive recursive teaching
dimension witnessed by a uniformly r.e. teaching sequence. We also derive the
arithmetical complexity of several other decision problems in teaching, such as
the problem of deciding, given an effective coding of all uniformly r.e. families, any such that
, any and , whether or not the
teaching dimension of with respect to is upper bounded
by .Comment: 15 pages in International Conference on Algorithmic Learning Theory,
201
Recognizing well-parenthesized expressions in the streaming model
Motivated by a concrete problem and with the goal of understanding the sense
in which the complexity of streaming algorithms is related to the complexity of
formal languages, we investigate the problem Dyck(s) of checking matching
parentheses, with different types of parenthesis.
We present a one-pass randomized streaming algorithm for Dyck(2) with space
\Order(\sqrt{n}\log n), time per letter \polylog (n), and one-sided error.
We prove that this one-pass algorithm is optimal, up to a \polylog n factor,
even when two-sided error is allowed. For the lower bound, we prove a direct
sum result on hard instances by following the "information cost" approach, but
with a few twists. Indeed, we play a subtle game between public and private
coins. This mixture between public and private coins results from a balancing
act between the direct sum result and a combinatorial lower bound for the base
case.
Surprisingly, the space requirement shrinks drastically if we have access to
the input stream in reverse. We present a two-pass randomized streaming
algorithm for Dyck(2) with space \Order((\log n)^2), time \polylog (n) and
one-sided error, where the second pass is in the reverse direction. Both
algorithms can be extended to Dyck(s) since this problem is reducible to
Dyck(2) for a suitable notion of reduction in the streaming model.Comment: 20 pages, 5 figure
Hard Properties with (Very) Short PCPPs and Their Applications
We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed ?, we construct a property P^(?)? {0,1}^n satisfying the following: Any testing algorithm for P^(?) requires ?(n) many queries, and yet P^(?) has a constant query PCPP whose proof size is O(n?log^(?)n), where log^(?) denotes the ? times iterated log function (e.g., log^(2)n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n?polylog(n)).
As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed ?, we construct a property that has a constant-query tester, but requires ?(n/log^(?)(n)) queries for every tolerant or erasure-resilient tester
Hyper-polynomial hierarchies and the polynomial jump
AbstractAssuming that the polynomial hierarchy (PH) does not collapse, we show the existence of ascending sequences of ptime Turing degrees of length ω1CK in PSPACE such that successors are polynomial jumps of their predecessors. Moreover these ptime degrees are all uniformly hard for PH. This is analogous to the hyperarithmetic hierarchy, which is defined similarly but with the (computable) Turing degrees. The lack of uniform least upper bounds for ascending sequences of ptime degrees causes the limit levels of our hyper-polynomial hierarchy to be inherently non-canonical. This problem is investigated in depth, and various possible structures for hyper-polynomial hierarchies are explicated, as are properties of the polynomial jump operator on the languages which are in PSPACE but not in PH
Uniform Random Sampling of Traces in Very Large Models
This paper presents some first results on how to perform uniform random walks
(where every trace has the same probability to occur) in very large models. The
models considered here are described in a succinct way as a set of
communicating reactive modules. The method relies upon techniques for counting
and drawing uniformly at random words in regular languages. Each module is
considered as an automaton defining such a language. It is shown how it is
possible to combine local uniform drawings of traces, and to obtain some global
uniform random sampling, without construction of the global model
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