Related papers: Arithmetic with Limited Exponentiation
Complexity theory offers a variety of concise computational models for computing boolean functions - branching programs, circuits, decision trees and ordered binary decision diagrams to name a few. A natural question that arises in this…
We give a generalization of Kung's theorem on critical exponents of linear codes over a finite field, in terms of sums of extended weight polynomials of linear codes. For all i=k+1,...,n, we give an upper bound on the smallest integer m…
We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite…
Consider the problem of computing quantized linear functions with only a few queries. Formally, given $\mathbf{x}\in \mathbb{R}^k$, our goal is to encode $\mathbf{x}$ as $\mathbf{c} \in \mathbb{R}^n$, for $n > k$, so that for any…
We show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig's Interpolation Theorem holds, but a strong form of this theorem does not hold. Translating these…
This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication…
We show in Bishop's constructive mathematics---in particular, using countable choice---that weak K\"{o}nig's lemma implies the uniform continuity theorem.
Given a first-order theory $T$ formulated in the usual language of first-order arithmetic, we say that $T$ is of *restricted complexity* if there is some natural number $n$ and some set $\mathcal A$ of $\Sigma_n$-sentences such that $T$ can…
We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our…
We use a very simple version of the optimized (linear) $\delta $ - expansion by scaling the free part of the Lagrangian with a variational parameter. This method is well suited to calculate the renormalized coupling constant in terms of the…
We study the expressive power of fragments of inclusion logic under the so-called lax team semantics. The fragments are defined either by restricting the number of universal quantifiers or the arity of inclusion atoms in formulae. In case…
A set of integers $A$ is computably encodable if every infinite set of integers has an infinite subset computing $A$. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this paper, we extend this…
Logics with team semantics provide alternative means for logical characterization of complexity classes. Both dependence and independence logic are known to capture non-deterministic polynomial time, and the frontiers of tractability in…
Studying Birkhoff sums of non-integrable functions involves the challenge of large observations depending on the sampled orbit, which prevents pointwise limit theorems. To address this issue, the largest observations are removed, this…
These lectures describe the most important theoretical methods in b-physics. We discuss the formalism of effective weak Hamiltonians, heavy quark effective theory, the heavy quark expansion for inclusive decays of b-hadrons and, finally,…
We explore finite-field frameworks for quantum theory and quantum computation. The simplest theory, defined over unrestricted finite fields, is unnaturally strong. A second framework employs only finite fields with no solution to x^2+1=0,…
We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning,…
We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak Konig's Lemma within the…
The replacement (or collection or choice) axiom scheme asserts bounded quantifier exchange. We prove the independence of this scheme from various weak theories of arithmetic, sometimes under a complexity assumption.
By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…