A Complete Finite Equational Axiomatisation of the Fracterm Calculus for Common Meadows

We analyse abstract data types that model numerical structures with a concept of error. Specifically, we focus on arithmetic data types that contain an error flag $\bot$ whose main purpose is to always return a value for division. To rings and fields we add a division operator $x/y$ and study a class of algebras called \textit{common meadows} wherein $x/0 = \bot$. The set of equations true in all common meadows is named the \textit{fracterm calculus of common meadows}. We give a finite equational axiomatisation of the fracterm calculus of common meadows and prove that it is complete and that the fracterm calculus is decidable

Three forms of physical measurement and their computability

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AN ANALOGUE-DIGITAL CHURCH-TURING THESIS

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Classifying the computational power of stochastic physical oracles

Consider a computability and complexity theory in which the classical set-theoretic oracle to a Turing machine is replaced by a physical process, and oracle queries return measurements of physical behaviour. The idea of such physical oracles is relevant to many disparate situations, but research has focussed on physical oracles that were classic deterministic experiments which measure physical quantities. In this paper, we broaden the scope of the theory of physical oracles by tackling non-deterministic systems. We examine examples of three types of non-determinism, namely systems that are: (1) physically nondeterministic, as in quantum phenomena; (2) physically deterministic but whose physical theory is non-deterministic, as in statistical mechanics; and (3) physically deterministic but whose computational theory is non-deterministic caused by error margins. Physical oracles that have probabilistic theories we call stochastic physical oracles. We propose a set SPO of axioms for a basic form of stochastic oracles. We prove that Turing machines equipped with a physical oracle satisfying the axioms SPO compute precisely the non-uniform complexity class BPP//log* in polynomial time. This result of BPP/log* is a computational limit to a great range of classical and non-classical measurement, and of analogue-digital computation in polynomial time under general conditions.info:eu-repo/semantics/publishedVersio

Oracles that measure thresholds: The Turing machine and the broken balance

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A hierarchy for BPP//log* based on counting calls to an oracle

Algorithms whose computations involve making physical measurements can be modelled by Turing machines with oracles that are physical systems and oracle queries that obtain data from observation and measurement. The computational power of many of these physical oracles has been established using non-uniform complexity classes; in particular, for large classes of deterministic physical oracles, with fixed error margins constraining the exchange of data between algorithm and oracle, the computational power has been shown to be the non-uniform class BPP//log⋆ . In this paper, we consider non-deterministic oracles that can be modelled by random walks on the line. We show how to classify computations within BPP//log⋆ by making an infinite non-collapsing hierarchy between BPP//log⋆ and BPP . The hierarchy rests on the theorem that the number of calls to the physical oracle correlates with the size of the responses to queries.info:eu-repo/semantics/publishedVersio

Naive Fracterm Calculus

An outline is provided of a new perspective on elementary arithmetic, based on addition, multiplication, subtraction and division, which is informal and unique and may be considered naive when contrasted with a plurality of algebraic and logical, axiomatic formalisations of elementary arithmetic