43,267 research outputs found
The range of non-linear natural polynomials cannot be context-free
Suppose that some polynomial with rational coefficients takes only
natural values at natural numbers, i.e., . We show that the base- representation of is a
context-free language if and only if is linear, answering a question of
Shallit. The proof is based on a new criterion for context-freeness, which is a
combination of the Interchange lemma and a generalization of the Pumping lemma.Comment: This paper should be assigned to cs.FL, but I'm not endorsed over
ther
The range of non-linear natural polynomials cannot be context-free
summary:Suppose that some polynomial with rational coefficients takes only natural values at natural numbers, i. e., . We show that the base- representation of is a context-free language if and only if is linear, answering a question of Shallit. The proof is based on a new criterion for context-freeness, which is a combination of the Interchange lemma and a generalization of the Pumping lemma
Separations of Matroid Freeness Properties
Properties of Boolean functions on the hypercube invariant with respect to
linear transformations of the domain are among the most well-studied properties
in the context of property testing. In this paper, we study the fundamental
class of linear-invariant properties called matroid freeness properties. These
properties have been conjectured to essentially coincide with all testable
linear-invariant properties, and a recent sequence of works has established
testability for increasingly larger subclasses. One question left open,
however, is whether the infinitely many syntactically different properties
recently shown testable in fact correspond to new, semantically distinct ones.
This is a crucial issue since it has also been shown that there exist
subclasses of these properties for which an infinite set of syntactically
different representations collapse into one of a small, finite set of
properties, all previously known to be testable.
An important question is therefore to understand the semantics of matroid
freeness properties, and in particular when two syntactically different
properties are truly distinct. We shed light on this problem by developing a
method for determining the relation between two matroid freeness properties P
and Q. Furthermore, we show that there is a natural subclass of matroid
freeness properties such that for any two properties P and Q from this
subclass, a strong dichotomy must hold: either P is contained in Q or the two
properties are "well separated." As an application of this method, we exhibit
new, infinite hierarchies of testable matroid freeness properties such that at
each level of the hierarchy, there are functions that are far from all
functions lying in lower levels of the hierarchy. Our key technical tool is an
apparently new notion of maps between linear matroids, called matroid
homomorphisms, that might be of independent interest
Aspects of Non-commutative Function Theory
We discuss non commutative functions, which naturally arise when dealing with
functions of more than one matrix variable
Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited
Polynomial interpretations are a useful technique for proving termination of
term rewrite systems. They come in various flavors: polynomial interpretations
with real, rational and integer coefficients. As to their relationship with
respect to termination proving power, Lucas managed to prove in 2006 that there
are rewrite systems that can be shown polynomially terminating by polynomial
interpretations with real (algebraic) coefficients, but cannot be shown
polynomially terminating using polynomials with rational coefficients only. He
also proved the corresponding statement regarding the use of rational
coefficients versus integer coefficients. In this article we extend these
results, thereby giving the full picture of the relationship between the
aforementioned variants of polynomial interpretations. In particular, we show
that polynomial interpretations with real or rational coefficients do not
subsume polynomial interpretations with integer coefficients. Our results hold
also for incremental termination proofs with polynomial interpretations.Comment: 28 pages; special issue of RTA 201
Every locally characterized affine-invariant property is testable
Let F = F_p for any fixed prime p >= 2. An affine-invariant property is a
property of functions on F^n that is closed under taking affine transformations
of the domain. We prove that all affine-invariant property having local
characterizations are testable. In fact, we show a proximity-oblivious test for
any such property P, meaning that there is a test that, given an input function
f, makes a constant number of queries to f, always accepts if f satisfies P,
and rejects with positive probability if the distance between f and P is
nonzero. More generally, we show that any affine-invariant property that is
closed under taking restrictions to subspaces and has bounded complexity is
testable.
We also prove that any property that can be described as the property of
decomposing into a known structure of low-degree polynomials is locally
characterized and is, hence, testable. For example, whether a function is a
product of two degree-d polynomials, whether a function splits into a product
of d linear polynomials, and whether a function has low rank are all examples
of degree-structural properties and are therefore locally characterized.
Our results depend on a new Gowers inverse theorem by Tao and Ziegler for low
characteristic fields that decomposes any polynomial with large Gowers norm
into a function of low-degree non-classical polynomials. We establish a new
equidistribution result for high rank non-classical polynomials that drives the
proofs of both the testability results and the local characterization of
degree-structural properties
Tight polynomial worst-case bounds for loop programs
In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language - representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication - it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate. A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found. A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid
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