655 research outputs found
Stability of Boolean function classes with respect to clones of linear functions
We consider classes of Boolean functions stable under compositions both from
the right and from the left with clones. Motivated by the question how many
properties of Boolean functions can be defined by means of linear equations, we
focus on stability under compositions with the clone of linear idempotent
functions. It follows from a result by Sparks that there are countably many
such linearly definable classes of Boolean functions. In this paper, we refine
this result by completely describing these classes. This work is tightly
related with the theory of function minors, stable classes, clonoids, and
hereditary classes, topics that have been widely investigated in recent years
by several authors including Maurice Pouzet and his coauthors.Comment: 44 page
Stability of Boolean function classes with respect to clones of linear functions
We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linear equations, we focus on stability under compositions with the clone of linear idempotent functions. It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related with the theory of function minors, stable classes, clonoids, and hereditary classes, topics that have been widely investigated in recent years by several authors including Maurice Pouzet and his coauthors
Function classes and relational constraints stable under compositions with clones
The general Galois theory for functions and relational constraints over
arbitrary sets described in the authors' previous paper is refined by imposing
algebraic conditions on relations
Functional equations, constraints, definability of function classes, and functions of Boolean variables
The paper deals with classes of functions of several variables defined on an arbitrary set A and taking values in a possibly different set B. Definability of function classes by functional equations is shown to be equivalent to definability by relational constraints, generalizing a fact established by Pippenger in the case A = B = {0,1}. Conditions for a class of functions to be definable by constraints of a particular type are given in terms of stability under certain functional compositions. This leads to a correspondence between functional equations with particular algebraic syntax and relational constraints with certain invariance properties with respect to clones of operations on a given set. When A = {0,1} and B is a commutative ring, such B-valued functions of n variables are represented by multilinear polynomials in n indeterminates in B[X1,..., Xn], Functional equations are given to describe classes of field-valued functions of a specified bounded degree. Classes of Boolean and pseudo-Boolean functions are covered as particular cases
The Complexity of Reasoning for Fragments of Autoepistemic Logic
Autoepistemic logic extends propositional logic by the modal operator L. A
formula that is preceded by an L is said to be "believed". The logic was
introduced by Moore 1985 for modeling an ideally rational agent's behavior and
reasoning about his own beliefs. In this paper we analyze all Boolean fragments
of autoepistemic logic with respect to the computational complexity of the
three most common decision problems expansion existence, brave reasoning and
cautious reasoning. As a second contribution we classify the computational
complexity of counting the number of stable expansions of a given knowledge
base. To the best of our knowledge this is the first paper analyzing the
counting problem for autoepistemic logic
Lectures on Designing Screening Experiments
Designing Screening Experiments (DSE) is a class of information - theoretical
models for multiple - access channels (MAC). We discuss the combinatorial model
of DSE called a disjunct channel model. This model is the most important for
applications and closely connected with the superimposed code concept. We give
a detailed survey of lower and upper bounds on the rate of superimposed codes.
The best known constructions of superimposed codes are considered in paper. We
also discuss the development of these codes (non-adaptive pooling designs)
intended for the clone - library screening problem. We obtain lower and upper
bounds on the rate of binary codes for the combinatorial model of DSE called an
adder channel model. We also consider the concept of universal decoding for the
probabilistic DSE model called a symmetric model of DSE.Comment: 66 page
A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits
We study the size blow-up that is necessary to convert an algebraic circuit
of product-depth to one of product-depth in the multilinear
setting.
We show that for every positive
there is an explicit multilinear polynomial on variables
that can be computed by a multilinear formula of product-depth and
size , but not by any multilinear circuit of product-depth and
size less than . This result is tight up to the
constant implicit in the double exponent for all
This strengthens a result of Raz and Yehudayoff (Computational Complexity
2009) who prove a quasipolynomial separation for constant-depth multilinear
circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an
exponential separation in the case
Our separating examples may be viewed as algebraic analogues of variants of
the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan
(STOC 2016), who used them to prove lower bounds for constant-depth Boolean
circuits
Generalized Satisfiability Problems via Operator Assignments
Schaefer introduced a framework for generalized satisfiability problems on
the Boolean domain and characterized the computational complexity of such
problems. We investigate an algebraization of Schaefer's framework in which the
Fourier transform is used to represent constraints by multilinear polynomials
in a unique way. The polynomial representation of constraints gives rise to a
relaxation of the notion of satisfiability in which the values to variables are
linear operators on some Hilbert space. For the case of constraints given by a
system of linear equations over the two-element field, this relaxation has
received considerable attention in the foundations of quantum mechanics, where
such constructions as the Mermin-Peres magic square show that there are systems
that have no solutions in the Boolean domain, but have solutions via operator
assignments on some finite-dimensional Hilbert space. We obtain a complete
characterization of the classes of Boolean relations for which there is a gap
between satisfiability in the Boolean domain and the relaxation of
satisfiability via operator assignments. To establish our main result, we adapt
the notion of primitive-positive definability (pp-definability) to our setting,
a notion that has been used extensively in the study of constraint satisfaction
problems. Here, we show that pp-definability gives rise to gadget reductions
that preserve satisfiability gaps. We also present several additional
applications of this method. In particular and perhaps surprisingly, we show
that the relaxed notion of pp-definability in which the quantified variables
are allowed to range over operator assignments gives no additional expressive
power in defining Boolean relations
- …