222,800 research outputs found
A Stronger LP Bound for Formula Size Lower Bounds via Clique Constraints
We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan (1995) and the theory of stable set polytope. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko (1971). Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give integrally matching upper and lower bounds of their formula size. We also show monotone formula size lower bounds of balanced recursive ternary majority functions improved from the quantum adversary bound of Laplante, Lee and Szegedy (2006)
Extreme Rays of AND-Measures in Circuit Complexity
This paper is motivated by the problem of proving lower bounds on the formula size of boolean functions, which leads to lower bounds on circuit depth. We know that formula size is bounded from below by all formal complexity measures. Thus, we study formula size by investigating AND-measures, which are weakened forms of formal complexity measures. The collection of all AND-measures is a pointed polyhedral cone; we study the extreme rays of this cone in order to better understand AND-measures. From the extreme rays, we attempt to discover useful properties of AND-measures that may help in proving new lower bounds on formula size and circuit depth. This paper focuses on describing some of the properties of AND-measures, especially those that are extreme rays. Furthermore, it describes some algorithhms for finding the extreme rays
The quantum adversary method and classical formula size lower bounds
We introduce two new complexity measures for Boolean functions, or more
generally for functions of the form f:S->T. We call these measures sumPI and
maxPI. The quantity sumPI has been emerging through a line of research on
quantum query complexity lower bounds via the so-called quantum adversary
method [Amb02, Amb03, BSS03, Zha04, LM04], culminating in [SS04] with the
realization that these many different formulations are in fact equivalent.
Given that sumPI turns out to be such a robust invariant of a function, we
begin to investigate this quantity in its own right and see that it also has
applications to classical complexity theory.
As a surprising application we show that sumPI^2(f) is a lower bound on the
formula size, and even, up to a constant multiplicative factor, the
probabilistic formula size of f. We show that several formula size lower bounds
in the literature, specifically Khrapchenko and its extensions [Khr71, Kou93],
including a key lemma of [Has98], are in fact special cases of our method.
The second quantity we introduce, maxPI(f), is always at least as large as
sumPI(f), and is derived from sumPI in such a way that maxPI^2(f) remains a
lower bound on formula size. While sumPI(f) is always a lower bound on the
quantum query complexity of f, this is not the case in general for maxPI(f). A
strong advantage of sumPI(f) is that it has both primal and dual
characterizations, and thus it is relatively easy to give both upper and lower
bounds on the sumPI complexity of functions. To demonstrate this, we look at a
few concrete examples, for three functions: recursive majority of three, a
function defined by Ambainis, and the collision problem.Comment: Appears in Conference on Computational Complexity 200
Resolution Lower Bounds for Refutation Statements
For any unsatisfiable CNF formula we give an exponential lower bound on the
size of resolution refutations of a propositional statement that the formula
has a resolution refutation. We describe three applications. (1) An open
question in (Atserias, M\"uller 2019) asks whether a certain natural
propositional encoding of the above statement is hard for Resolution. We answer
by giving an exponential size lower bound. (2) We show exponential resolution
size lower bounds for reflection principles, thereby improving a result in
(Atserias, Bonet 2004). (3) We provide new examples of CNFs that exponentially
separate Res(2) from Resolution (an exponential separation of these two proof
systems was originally proved in (Segerlind, Buss, Impagliazzo 2004))
Negative weights make adversaries stronger
The quantum adversary method is one of the most successful techniques for
proving lower bounds on quantum query complexity. It gives optimal lower bounds
for many problems, has application to classical complexity in formula size
lower bounds, and is versatile with equivalent formulations in terms of weight
schemes, eigenvalues, and Kolmogorov complexity. All these formulations rely on
the principle that if an algorithm successfully computes a function then, in
particular, it is able to distinguish between inputs which map to different
values.
We present a stronger version of the adversary method which goes beyond this
principle to make explicit use of the stronger condition that the algorithm
actually computes the function. This new method, which we call ADV+-, has all
the advantages of the old: it is a lower bound on bounded-error quantum query
complexity, its square is a lower bound on formula size, and it behaves well
with respect to function composition. Moreover ADV+- is always at least as
large as the adversary method ADV, and we show an example of a monotone
function for which ADV+-(f)=Omega(ADV(f)^1.098). We also give examples showing
that ADV+- does not face limitations of ADV like the certificate complexity
barrier and the property testing barrier.Comment: 29 pages, v2: added automorphism principle, extended to non-boolean
functions, simplified examples, added matching upper bound for AD
Lower Bounds for Monotone Span Programs
The model of span programs is a linear algebraic model of computation. Lower bounds for span programs imply lower bounds for contact schemes, symmetric branching programs and for formula size. Monotone span programs correspond also to linear secret-sharing schemes. We present a new technique for proving lower bounds for monotone span programs. The main result proved here yields quadratic lower bounds for the size of monotone span programs, improving on the largest previously known bounds for explicit functions. The bound is asymptotically tight for the function corresponding to a class of 4-cliques
Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds
Given a formula in quantifier-free Presburger arithmetic, if it has a
satisfying solution, there is one whose size, measured in bits, is polynomially
bounded in the size of the formula. In this paper, we consider a special class
of quantifier-free Presburger formulas in which most linear constraints are
difference (separation) constraints, and the non-difference constraints are
sparse. This class has been observed to commonly occur in software
verification. We derive a new solution bound in terms of parameters
characterizing the sparseness of linear constraints and the number of
non-difference constraints, in addition to traditional measures of formula
size. In particular, we show that the number of bits needed per integer
variable is linear in the number of non-difference constraints and logarithmic
in the number and size of non-zero coefficients in them, but is otherwise
independent of the total number of linear constraints in the formula. The
derived bound can be used in a decision procedure based on instantiating
integer variables over a finite domain and translating the input
quantifier-free Presburger formula to an equi-satisfiable Boolean formula,
which is then checked using a Boolean satisfiability solver. In addition to our
main theoretical result, we discuss several optimizations for deriving tighter
bounds in practice. Empirical evidence indicates that our decision procedure
can greatly outperform other decision procedures.Comment: 26 page
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