1,680 research outputs found
Locally monotone Boolean and pseudo-Boolean functions
We propose local versions of monotonicity for Boolean and pseudo-Boolean
functions: say that a pseudo-Boolean (Boolean) function is p-locally monotone
if none of its partial derivatives changes in sign on tuples which differ in
less than p positions. As it turns out, this parameterized notion provides a
hierarchy of monotonicities for pseudo-Boolean (Boolean) functions. Local
monotonicities are shown to be tightly related to lattice counterparts of
classical partial derivatives via the notion of permutable derivatives. More
precisely, p-locally monotone functions are shown to have p-permutable lattice
derivatives and, in the case of symmetric functions, these two notions
coincide. We provide further results relating these two notions, and present a
classification of p-locally monotone functions, as well as of functions having
p-permutable derivatives, in terms of certain forbidden "sections", i.e.,
functions which can be obtained by substituting constants for variables. This
description is made explicit in the special case when p=2
Pivotal decompositions of functions
We extend the well-known Shannon decomposition of Boolean functions to more
general classes of functions. Such decompositions, which we call pivotal
decompositions, express the fact that every unary section of a function only
depends upon its values at two given elements. Pivotal decompositions appear to
hold for various function classes, such as the class of lattice polynomial
functions or the class of multilinear polynomial functions. We also define
function classes characterized by pivotal decompositions and function classes
characterized by their unary members and investigate links between these two
concepts
Pseudorandom Generators for Width-3 Branching Programs
We construct pseudorandom generators of seed length that -fool ordered read-once branching programs
(ROBPs) of width and length . For unordered ROBPs, we construct
pseudorandom generators with seed length . This is the first improvement for pseudorandom
generators fooling width ROBPs since the work of Nisan [Combinatorica,
1992].
Our constructions are based on the `iterated milder restrictions' approach of
Gopalan et al. [FOCS, 2012] (which further extends the Ajtai-Wigderson
framework [FOCS, 1985]), combined with the INW-generator [STOC, 1994] at the
last step (as analyzed by Braverman et al. [SICOMP, 2014]). For the unordered
case, we combine iterated milder restrictions with the generator of
Chattopadhyay et al. [CCC, 2018].
Two conceptual ideas that play an important role in our analysis are: (1) A
relabeling technique allowing us to analyze a relabeled version of the given
branching program, which turns out to be much easier. (2) Treating the number
of colliding layers in a branching program as a progress measure and showing
that it reduces significantly under pseudorandom restrictions.
In addition, we achieve nearly optimal seed-length
for the classes of: (1) read-once polynomials on
variables, (2) locally-monotone ROBPs of length and width
(generalizing read-once CNFs and DNFs), and (3) constant-width ROBPs of length
having a layer of width in every consecutive
layers.Comment: 51 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
Representation of States on Effect-Tribes and Effect Algebras by Integrals
We describe -additive states on effect-tribes by integrals.
Effect-tribes are monotone -complete effect algebras of functions where
operations are defined by points. Then we show that every state on an effect
algebra is an integral through a Borel regular probability measure. Finally, we
show that every -convex combination of extremal states on a monotone
-complete effect algebra is a Jauch-Piron state.Comment: 20 page
Some applications of the ultrapower theorem to the theory of compacta
The ultrapower theorem of Keisler-Shelah allows such model-theoretic notions
as elementary equivalence, elementary embedding and existential embedding to be
couched in the language of categories (limits, morphism diagrams). This in turn
allows analogs of these (and related) notions to be transported into unusual
settings, chiefly those of Banach spaces and of compacta. Our interest here is
the enrichment of the theory of compacta, especially the theory of continua,
brought about by the immigration of model-theoretic ideas and techniques
Type-Decomposition of a Pseudo-Effect Algebra
The theory of direct decomposition of a centrally orthocomplete effect
algebra into direct summands of various types utilizes the notion of a
type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly)
noncommutative version of an effect algebra. In this article we develop the
basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set
to PEAs, and show that TD sets induce decompositions of centrally orthocomplete
PEAs into direct summands.Comment: 18 page
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