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The rational numbers as an abstract data type
, 2007
"... We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of ..."
Abstract

Cited by 54 (38 self)
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We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0−1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system?
Can Newtonian systems, bounded in space, time, mass and energy compute all functions?
"... In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and anal ..."
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Cited by 15 (4 self)
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In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and analysis of simple examples that are models of simple subtheories of physical theories. Then we illustrate the methodology, by presenting a simple example for Newtonian Kinematics, and a critique that leads to a substantial extension of the methodology. The example proves that for any set A of natural numbers there exists a 3dimensional Newtonian kinematic system MA, with an infinite family of particles Pn whose total mass is bounded, and whose observable behaviour can decide whether or not n ∈ A for all n ∈ N in constant time. In particular, the example implies that simple Newtonian kinematic systems that are bounded in space, time, mass and energy can compute all possible sets and functions on discrete data. The system is a form of marble run and is a model of a small fragment of Newtonian Kinematics. Next, we use the example to extend the methodology. The marble run shows that a formal theory for computation by physical systems needs strong conditions on the notion of experimental procedure and, specifically, on methods for the construction of equipment. We propose to extend the methodology by defining languages to express experimental procedures and the construction of equipment. We conjecture that the functions computed by experimental computation in Newtonian Kinematics are “equivalent” to those computed by algorithms, i.e. the partial computable functions.
University of Wales Swansea,
"... In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy, especially on the issue of the existence of hypercomputers. First, we present some principles for making a theoretical analysis of computation by physical systems. We focus on the rol ..."
Abstract
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In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy, especially on the issue of the existence of hypercomputers. First, we present some principles for making a theoretical analysis of computation by physical systems. We focus on the role of simple examples that are models of simple subtheories of physical theories. Secondly, we present such a simple example for Newtonian Kinematics and explore its meaning. We prove that for any set A of natural numbers there exists a 3dimensional Newtonian kinematic system MA, with an infinite family of particles Pn whose total mass is bounded, and whose observable behaviour can decide whether or not n ∈ A for all n ∈ N in constant time. In particular, the theorem implies that simple Newtonian mechanical systems that are bounded in space, time, mass and energy can compute all possible functions on discrete data. The system is a form of marble run and is a model of a small fragment of Newtonian Kinematics. Then, we use the example to reflect on the nature of computation by physical systems. The proof of the theorem shows how any information (coded by some A) can be embedded in the static structure of simple kinematic equipment and retrieved by a simple experimental procedure. This suggests that the idea of equipment is problematic when computing under Newtonian mechanics. The example shows that a formal theory for experimental computation needs restrictive conditions on the notion of experimental procedure and the construction of equipment. We conjecture that using such a restricted mechanics the functions computed by experimental computation are “equivalent ” to those computed by algorithms, i.e. the partial computable functions.