214,577 research outputs found
On equations over sets of integers
Systems of equations with sets of integers as unknowns are considered. It is
shown that the class of sets representable by unique solutions of equations
using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in
T} and with ultimately periodic constants is exactly the class of
hyper-arithmetical sets. Equations using addition only can represent every
hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can
also be represented by equations over sets of natural numbers equipped with
union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in
T, \: m \geqslant n}. Testing whether a given system has a solution is
-complete for each model. These results, in particular, settle the
expressive power of the most general types of language equations, as well as
equations over subsets of free groups.Comment: 12 apges, 0 figure
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Complex Algebras of Arithmetic
An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a
sequence of arithmetic and logical operations to be performed on sets of
natural numbers. Arithmetic circuits can also be viewed as the elements of the
smallest subalgebra of the complex algebra of the semiring of natural numbers.
In the present paper, we investigate the algebraic structure of complex
algebras of natural numbers, and make some observations regarding the
complexity of various theories of such algebras
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
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