Paolo Mancosu‘s and my paper on Heinrich Behmann’s 1921 lecture on the decision problem is out in the new issue of the Bulletin of Symbolic Logic. (Hey you are a member of the Association for Symbolic Logic, right? Comes with subscriptions to the Bulletin, the Review, and the Journal of Symbolic Logic!)
This is the abstract:
Heinrich Behmann (1891–1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved—independently of Löwenheim and Skolem’s earlier work—the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included.
If you can’t wait for your copy of the Bulletin in the mail, there’s a free preprint on arXiv.
First I just thought, “How weird! Applying many-valued logic to slime moulds.” But then I read it and not only is this a bona-fide application of p-adic logic to the behavior of slime moulds, no, the slime moulds are used as computers in this application! And my own work is used! So, yay to p-adic logic and slime mould computers!
Andrew Schumann, p-Adic valued logical calculi in simulations of the slime mould behaviour. J. Applied Non-Classical Logics 2015, forthcoming.
In this paper we consider possibilities for applying p-adic valued logic BL to the task of designing an unconventional computer based on the medium of slime mould (order Physarales, class Myxomecetes, subclass Myxogastromycetidae), the giant amoebozoa that looks for attractants and reaches them by means of propagating complex networks. If it is assumed that at any time step t of propagation the slime mould can discover and reach not more than \(p-1\) attractants, then this behaviour can be coded in terms of p-adic numbers. As a result, this behaviour implements some p-adic valued arithmetic circuits and can verify p-adic valued logical propositions.
[Image credit: Andrew Adamatzky, Physarum Machines (World Scientific, 2010), courtesy of Andrew Adamatzky]