Gudmund Skovbjerg Frandsen

An arithmetic model of computation equivalent to threshold circuits

We define a new structured and general model of computation: circuits using arbitrary fan-in arithmetic gates over the characteristic-two finite fields (F2n). These circuits have only one input and one output. We show how they correspond naturally to boolean computations with n inputs and n outputs. We show that if circuit sizes are polynomially related, then the arithmetic circuit depth and the threshold circuit depth to compute a given function differ by at most a constant factor. We use threshold functions with arbitrary weights; however, we show that when compared to the usual threshold model, the depth measure of this generalised model differs only by at most a constant factor (at polynomial size). The fan-in of our arithmetic model is also unbounded in the most generous sense: circuit size is measured as the number of Σ and Π-gates; there is no bound on the number of “wires”. We show that these results are provable for any reasonable correspondence between strings of n-bits and elements of F2n. And we find two such distinct characterizations. Thus, we show that arbitrary fan-in arithmetic computations over F2n constitute a precise abstraction of Boolean threshold computations with the pleasant property that various algebraic laws have been recovered.

A singular choice for multiple choice

How should multiple choice tests be scored and graded, in particular when students are allowed to check several boxes to convey partial knowledge? Many strategies may seem reasonable, but we demonstrate that five self-evident axioms are sufficient to determine completely the correct strategy. We also discuss how to measure robustness of the obtained grades. Our results have practical advantages and also suggest criteria for designing multiple choice questions.

Some results on uniform arithmetic circuit complexity

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fieldsF2n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial-size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fieldsF2n may be identified with ann-input andn-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME), then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

Dynamic matrix rank

We consider maintaining information about the rank of a matrix under changes of the entries. For n×n matrices, we show an upper bound of O(n1.575) arithmetic operations and a lower bound of Ω(n) arithmetic operations per change. The upper bound is valid when changing up to O(n0.575) entries in a single column of the matrix. Both bounds appear to be the first non-trivial bounds for the problem. The upper bound is valid for arbitrary fields, whereas the lower bound is valid for algebraically closed fields. The upper bound uses fast rectangular matrix multiplication, and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic computation of rational functions.

What is an Efficient Implementation of the \lambda-calculus?

We propose to measure the efficiency of any implementation of the λ-calculus as a function of a new parameter v, that is itself a function of any λ-expression.

The computational complexity of some problems of linear algebra

In this paper we consider the computational complexity of some problems dealing with matrix rank. Let E, S be subsets of a commutative ring R. Let x1, x2, ..., x t be variables. Given a matrix M= M(x1, x2, ..., x t ) with entries chosen from E ∪ {x1, x2, ..., x t }, we want to determine

Binary GCD Like Algorithms for Some Complex Quadratic Rings

\max rank_S (M) = \mathop {max}\limits_{(a_1 ,a_2 ,...a_t ) \in S^t } rank M(a_1 ,a_2 ,...a_t )

Reviewing bounds on the circuit size of the hardest functions

and