Physicist reckons two-button calculator can do all elementary math
Paper says a single binary operator could replace a lot of scientific heavy lifting
UPDATED Every now and then, a researcher comes up with something that sounds either wrong or unoriginal to outsiders – yet carries just enough of a chance of being correct, novel, and consequential to demand a closer look.
On this occasion, the honor goes to Andrzej Odrzywołek, a postdoctoral researcher at the Institute of Theoretical Physics, Jagiellonian University, Kraków.
In a recently updated, yet-to-be-peer-reviewed paper, Odrzywołek says he has, in essence, developed a two-button calculator that can compute the standard repertoire of a scientific calculator familiar to a high school math and science student. You might have to push the buttons a number of times, but the point is the underlying simplicity.
A single two-input gate already suffices for all Boolean logic in digital hardware; Odrzywołek's claim is that continuous mathematics may have an analogous primitive. It can generate elementary functions from a single operator that would otherwise require multiple distinct operations. These include trigonometric functions such as sine, cosine, and tangent; algebraic functions; and arithmetic operations such as addition, subtraction, multiplication, and division. The two-input gate also produces constants including π, e (Euler's number, 2.71828...), and i (the square root of minus one).
The proposed operator is eml(x, y) = exp(x) - ln(y). Eml is the exponential-minus-log function, exp is the exponential function, and ln is the natural logarithm (or the logarithm to the base e).
"A calculator with just two buttons, EML and the digit 1, can compute everything a full scientific calculator does. This is not a mere mathematical trick. Because one repeatable element suffices, mathematical expressions become uniform circuits, much like electronics built from identical transistors, opening new ways to encoding, evaluating, and discovering formulas across scientific computing," the paper says.
It even has a diagram showing how the functions cascade from the proposed operator.
Not everyone agrees, as a lively discussion on Hacker News demonstrates. A couple of points first, though: the paper is about elementary functions in continuous mathematics, not discrete computation.
The author also points out there is no agreed list of elementary functions, and creates a list of the 36 most commonly used to get around the problem. The task, then, is to show whether every primitive on the list can be expressed as a finite composition of these two ingredients: eml(x, y) and a terminal symbol (e.g. the constant 1).
Odrzywołek also says that direct symbolic verification of the kind used in a formal proof is "intractable."
"The methods here are designed for speed and exhaustiveness, not for proof-level rigor. They use floating-point numerical evaluation and heuristic filtering," the paper's Supplementary Information says. The numerical evaluation is the first of three steps, followed by verification and application.
Whether the approach and the conclusion stand the test of time is unanswered. The author has yet to respond to our question about whether he is submitting the paper for peer review, but anyone with the requisite chops can kick the tires here [PDF]. ®
Updated to add on April 17th:
Odrzywołek told The Register he's planning to submit the paper to a specialized journal. Watch this space.