On a conjecture of Bergstra and Tucker

AbstractBergstra and Tucker (1983) conjectured that a semicomputable (abstract) data type has a finite hidden enrichment specification under its initial algebra semantics. In a previous paper (1987) we tried to solve the entire conjecture and we found a weak solution. Here, following the line and the proof techniques of the previous paper, we examine a nontrivial case in which the conjecture has a positive answer

Training deep neural networks with low precision multiplications

Multipliers are the most space and power-hungry arithmetic operators of the digital implementation of deep neural networks. We train a set of state-of-the-art neural networks (Maxout networks) on three benchmark datasets: MNIST, CIFAR-10 and SVHN. They are trained with three distinct formats: floating point, fixed point and dynamic fixed point. For each of those datasets and for each of those formats, we assess the impact of the precision of the multiplications on the final error after training. We find that very low precision is sufficient not just for running trained networks but also for training them. For example, it is possible to train Maxout networks with 10 bits multiplications.Comment: 10 pages, 5 figures, Accepted as a workshop contribution at ICLR 201

Fracpairs and fractions over a reduced commutative ring

In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce "fracpairs" as pairs subject to laws consistent with the use of the pair as a fraction, but do not exclude denominators to be zero. We investigate fracpairs over a reduced commutative ring (a commutative ring that has no nonzero nilpotent elements) and provide these with natural definitions for addition, multiplication, and additive and multiplicative inverse. We find that modulo a simple congruence these fracpairs constitute a "common meadow", which is a commutative monoid both for addition and multiplication, extended with a weak additive inverse, a multiplicative inverse except for zero, and an additional element "a" that is the image of the multiplicative inverse on zero and that propagates through all operations. Considering "a" as an error-value supports the intuition. The equivalence classes of fracpairs thus obtained are called common cancellation fractions (cc-fractions), and cc-fractions over the integers constitute a homomorphic pre-image of the common meadow Qa, the field Q of rational numbers expanded with an a-totalized inverse. Moreover, the initial common meadow is isomorphic to the initial algebra of cc-fractions over the integer numbers. Next, we define canonical term algebras for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and provide some negative results concerning their associated term rewriting properties. Then we consider reduced commutative rings in which the sum of two squares plus one cannot be a zero divisor: by extending the equivalence relation on fracpairs we obtain an initial algebra that is isomorphic to Qa. Finally, we express negative conjectures concerning alternative specifications for these (concrete) datatypes.Comment: 25 pages, 8 table

Education, division & derivative: Putting a Sky above a Field or a Meadow Comments on the field, meadow, dynamic quotient and derivative, as seen from research in mathematics education (elementary, highschool & matricola)

Abstract The sky is a tentative suggestion for extension group theory, that has supervariables with flexible domains, that would allow the formulation of dynamic rational functions, such that division by zero is prevented by manipulation of the domain. This would allow an algebraic approach to the derivative. Meadows are no alternative to such an approach. The world needs Academic Schools in which teaching is merged with empirical research on didactics. Summary (1) A sky is a tentative suggestion for group theory to use a field or meadow as foundation, and then include supervariables and expressions, such that a supervariable has a flexible domain. These variables are symbols and not just placeholders for numbers. The sky has functions with domains and ranges that use such supervariables. The dynamic quotient is defined with supervariables. Its outcome and range require both the algebra of expressions and the manipulation of the domain for the denominator. The dynamic quotient allows an algebraic definition of the derivative. For school mathematics, this allows an algebraic middle ground between Mathematical Analysis (with limits) and Calculus (mainly technique). Though the dynamic quotient is embedded in a sky in the algebra of groups in research mathematics, the algebra that is required in practice in school mathematics is the common algebra of expressions, though with the manipulation of domains. (2) It has been reported in the literature that advantages of a meadow over a field are: (i) it has a function name for the inverse rather than the mere statement of existence, (ii) there is less testing on zero values, with faster computer calculation, (iii) there is an outcome 1 / 0 = â ("additional", e.g. Undefined or Indeterminate). However, the dynamic quotient still has its own exception switch. Since it is not useful to have all derivatives equal to â, this outcome of division in a meadow does not present an alternative to the standard derivative or the algebraic approach via the dynamic quotient. (3) The analysis supports the alert in the AMS Notices by H. Wu (2011:372) of "(...) the mathematics community to the urgent need of active participation in the education enterprise." For improvement in education in elementary school, highschool and matricola, and its research, we would rather see more attention for empirics and computer algebra than that sky. Like Academic Hospitals have research for cure and not dissection, the world needs Academic Schools in which teaching is merged with empirical research on didactics