3 research outputs found
Mathematical Proof Between Generations
A proof is one of the most important concepts of mathematics. However, there
is a striking difference between how a proof is defined in theory and how it is
used in practice. This puts the unique status of mathematics as exact science
into peril. Now may be the time to reconcile theory and practice, i.e.
precision and intuition, through the advent of computer proof assistants. For
the most time this has been a topic for experts in specialized communities.
However, mathematical proofs have become increasingly sophisticated, stretching
the boundaries of what is humanly comprehensible, so that leading
mathematicians have asked for formal verification of their proofs. At the same
time, major theorems in mathematics have recently been computer-verified by
people from outside of these communities, even by beginning students. This
article investigates the gap between the different definitions of a proof and
possibilities to build bridges. It is written as a polemic or a collage by
different members of the communities in mathematics and computer science at
different stages of their careers, challenging well-known preconceptions and
exploring new perspectives.Comment: 17 pages, 1 figur
Hilbert's Tenth Problem in Coq (Extended Version)
We formalise the undecidability of solvability of Diophantine equations, i.e.
polynomial equations over natural numbers, in Coq's constructive type theory.
To do so, we give the first full mechanisation of the
Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively
enumerable problem -- in our case by a Minsky machine -- is Diophantine. We
obtain an elegant and comprehensible proof by using a synthetic approach to
computability and by introducing Conway's FRACTRAN language as intermediate
layer. Additionally, we prove the reverse direction and show that every
Diophantine relation is recognisable by -recursive functions and give a
certified compiler from -recursive functions to Minsky machines.Comment: submitted to LMC
The DPRM Theorem in Isabelle (Short Paper)
Hilbert\u27s 10th problem asks for an algorithm to tell whether or not a given diophantine equation has a solution over the integers. The non-existence of such an algorithm was shown in 1970 by Yuri Matiyasevich. The key step is known as the DPRM theorem: every recursively enumerable set of natural numbers is Diophantine. We present the formalization of Matiyasevich\u27s proof of the DPRM theorem in Isabelle. To represent recursively enumerable sets in equations, we implement and arithmetize register machines. Using several number-theoretic lemmas, we prove that exponentiation has a diophantine representation. Further, we contribute a small library of number-theoretic implementations of binary digit-wise relations. Finally, we discuss and contribute an is_diophantine predicate. We expect the complete formalization of the DPRM theorem in the near future; at present it is complete except for a minor gap in the arithmetization proofs of register machines and extending the is_diophantine predicate by two binary digit-wise relations