8,312 research outputs found
Characterizing the Existence of Optimal Proof Systems and Complete Sets for Promise Classes.
In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language L? Q2: Do there exist complete problems for a given promise class C? For concrete languages L (such as TAUT or SAT) and concrete promise classes C (such as NP∩coNP, UP, BPP, disjoint NP-pairs etc.), these ques-tions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new character-izations for Q1 and Q2 that apply to almost all promise classes C and languages L, thus creating a unifying framework for the study of these practically relevant questions. While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount on advice is avail-able in the underlying machine model. This continues a recent line of research on proof systems with advice started by Cook and Kraj́ıček [6]
Seven characterizations of non-meager P-filters
We give several topological/combinatorial conditions that, for a filter on
, are equivalent to being a non-meager -filter. In
particular, we show that a filter is countable dense homogeneous if and only if
it is a non-meager -filter. Here, we identify a filter with a
subspace of through characteristic functions. Along the way, we
generalize to non-meager -filters a result of Miller about
-points, and we employ and give a new proof of results of
Marciszewski. We also employ a theorem of Hern\'andez-Guti\'errez and
Hru\v{s}\'ak, and answer two questions that they posed. Our result also
resolves several issues raised by Medini and Milovich, and proves false one
"theorem" of theirs. Furthermore, we show that the statement "Every non-meager
filter contains a non-meager -subfilter" is independent of
(more precisely, it is a consequence of
and its negation is a consequence of ).
It follows from results of Hru\v{s}\'ak and van Mill that, under
, a filter has less than types of
countable dense subsets if and only if it is a non-meager -filter.
In particular, under , there exists an ultrafilter
with types of countable dense subsets. We also show that such an
ultrafilter exists under .Comment: 17 page
Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
A total dominating set in a graph is a set of vertices such that every vertex
of the graph has a neighbor in the set. We introduce and study graphs that
admit non-negative real weights associated to their vertices such that a set of
vertices is a total dominating set if and only if the sum of the corresponding
weights exceeds a certain threshold. We show that these graphs, which we call
total domishold graphs, form a non-hereditary class of graphs properly
containing the classes of threshold graphs and the complements of domishold
graphs, and are closely related to threshold Boolean functions and threshold
hypergraphs. We present a polynomial time recognition algorithm of total
domishold graphs, and characterize graphs in which the above property holds in
a hereditary sense. Our characterization is obtained by studying a new family
of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of
independent interest.Comment: 19 pages, 1 figur
Nondeterministic functions and the existence of optimal proof systems
We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system
The Deduction Theorem for Strong Propositional Proof Systems
This paper focuses on the deduction theorem for propositional logic. We define and investigate different deduction properties and show that the presence of these deduction properties for strong proof systems is powerful enough to characterize the existence of optimal and even polynomially bounded proof systems. We also exhibit a similar, but apparently weaker condition that implies the existence of complete disjoint NPUnknown control sequence '\mathsf' -pairs. In particular, this yields a sufficient condition for the completeness of the canonical pair of Frege systems and provides a general framework for the search for complete NPUnknown control sequence '\mathsf' -pairs
A Categorical View on Algebraic Lattices in Formal Concept Analysis
Formal concept analysis has grown from a new branch of the mathematical field
of lattice theory to a widely recognized tool in Computer Science and
elsewhere. In order to fully benefit from this theory, we believe that it can
be enriched with notions such as approximation by computation or
representability. The latter are commonly studied in denotational semantics and
domain theory and captured most prominently by the notion of algebraicity, e.g.
of lattices. In this paper, we explore the notion of algebraicity in formal
concept analysis from a category-theoretical perspective. To this end, we build
on the the notion of approximable concept with a suitable category and show
that the latter is equivalent to the category of algebraic lattices. At the
same time, the paper provides a relatively comprehensive account of the
representation theory of algebraic lattices in the framework of Stone duality,
relating well-known structures such as Scott information systems with further
formalisms from logic, topology, domains and lattice theory.Comment: 36 page
Ultrafilter spaces on the semilattice of partitions
The Stone-Cech compactification of the natural numbers bN, or equivalently,
the space of ultrafilters on the subsets of omega, is a well-studied space with
interesting properties. If one replaces the subsets of omega by partitions of
omega, one can define corresponding, non-homeomorphic spaces of partition
ultrafilters. It will be shown that these spaces still have some of the nice
properties of bN, even though none is homeomorphic to bN. Further, in a
particular space, the minimal height of a tree pi-base and P-points are
investigated
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