23 research outputs found
Informatics: Science or TĂ©chne?
Informatics is generally understood as a “new technology” and is therewith discussed according to technological aspects such as speed, data retrieval, information control and so on. Its widespread use from home appliances to enterprises and universities is not the result of a clear-cut analysis of its inner possibilities but is rather dependent on all sorts of ideological promises of unlimited progress. We will discuss the theoretical definition of informatics proposed in 1936 by Alan Turing in order to show that it should be taken as final and complete. This definition has no relation to the technology because Turing defines computers as doing the work of solving problems with numbers. This formal definition implies nonetheless a relation to the non-formalized elements around informatics, which we shall discuss through the Greek notion of téchne
Using Isabelle/HOL to verify first-order relativity theory
Logicians at the Rényi Mathematical Institute in Budapest have spent several years developing versions of relativity theory (special, general, and other variants) based wholly on first-order logic, and have argued in favour of the physical decidability, via exploitation of cosmological phenomena, of formally unsolvable questions such as the Halting Problem and the consistency of set theory. As part of a joint project, researchers at Sheffield have recently started generating rigorous machine-verified versions of the Hungarian proofs, so as to demonstrate the soundness of their work. In this paper, we explain the background to the project and demonstrate a first-order proof in Isabelle/HOL of the theorem “no inertial observer can travel faster than light”. This approach to physical theories and physical computability has several pay-offs, because the precision with which physical theories need to be formalised within automated proof systems forces us to recognise subtly hidden assumptions
Constructive Many-One Reduction from the Halting Problem to Semi-Unification
Semi-unification is the combination of first-order unification and
first-order matching. The undecidability of semi-unification has been proven by
Kfoury, Tiuryn, and Urzyczyn in the 1990s by Turing reduction from Turing
machine immortality (existence of a diverging configuration). The particular
Turing reduction is intricate, uses non-computational principles, and involves
various intermediate models of computation. The present work gives a
constructive many-one reduction from the Turing machine halting problem to
semi-unification. This establishes RE-completeness of semi-unification under
many-one reductions. Computability of the reduction function, constructivity of
the argument, and correctness of the argument is witnessed by an axiom-free
mechanization in the Coq proof assistant. Arguably, this serves as
comprehensive, precise, and surveyable evidence for the result at hand. The
mechanization is incorporated into the existing, well-maintained Coq library of
undecidability proofs. Notably, a variant of Hooper's argument for the
undecidability of Turing machine immortality is part of the mechanization.Comment: CSL 2022 - LMCS special issu
Asymptotic distribution of integers with certain prime factorizations
Let be the sequence of prime numbers and let
be a positive integer. We give a strong asymptotic formula for the distribution
of the set of integers having prime factorizations of the form
p_{m^{k_1}}p_{m^{k_{2}}...p_{m^{k_{n}}} with .
Such integers originate in various combinatorial counting problems; when ,
they arise as Matula numbers of certain rooted trees.Comment: 11 page