Biannual Report - Fachbereich Mathematik - Technische Universität ...

The groupoid model introduced in [1] was the first model validating V. Voevodsky’s Univalence
Axiom. In my Habilitation Thesis from 1993 I constructed realizability models
faithfully reflecting most of the intensionality phenomena of Intensional Type Theory
(ITT). My aim is now to consider groupoid models inside realizability models in order
to combine the advantages of both models.
Contact: T. Streicher
References
[1] M. Hofmann and T. Streicher. The groupoid interpretation of type theory. In Twenty-five years
of constructive type theory, pages 83–111. Oxford University Press, 1998.
Project: Problems Complete for the Blum-Shub-Smale Model
The Blum-Shub-Smale (BSS) Model is a generalization of the classical Turing machine
from bits to rings and other structures. It extends the classical theory of computing and
complexity and translates open questions such as P-versus-NP-versus-EXP to settings where
the algebraic structure of the (usually continuous) underlying space both requires and
allows for new proof methods. In particular, the class of decision problems NP-complete
over the reals and the complex numbers is a promising field of research with currently
only roughly 5 examples known complete for it – compared to 500 in the discrete case.
We identify and establish new natural problems complete for BSS complexity classes.
Partner: C. Herrmann, TU Darmstadt; P. Scheiblechner, Hochschule Luzern
Contact: M. Ziegler, C. Herrmann
References
[1] T. Gärtner and M. Ziegler. Real analytic machines and degrees. Logical Methods in Computer
Science, 7:1–20, 2011.
[2] C. Herrmann and M. Ziegler. Computational complexity of quantum satisfiability. In Proceedings
of the 26th IEEE Symposium on Logic in Computer Science (LiCS), pages 175–184. IEEE
Computer Society, 2011.
[3] J. Sokoli. On the complexity of satisfiability over vector product terms. Master’s thesis, TU
Darmstadt, Feb 2013.
Project: Computability and Complexity in Numerics
Recursive Analysis, as initiated by Alan Turing in his 1937 publication, combines the theory
of (discrete) computation with numerical analysis in the sense of rational approximation to
real numbers with arbitrarily prescribable absolute error. We delineate the border between
computability and incomputability, and explore the inner structure of the former in both
sound logical frameworks of descriptive and of computational complexity theory. That
is, we classify practical problems and standard numerical tasks over the reals in terms of
quantitative and qualtitative hierarchies such as Borel’s, fragments of second order logic,
and computational complexity P ⊆ NP ⊆ #P ⊆ CH ⊆ PSPACE ⊆ EXP. Upper bounds here
refer to algorithms in the strict sense of formally proven correct on a fully specified set
of inputs with guaranteed worst-case running time; uniform lower bounds are derived by
adversary arguments adapted from Information-Based Complexity (IBC).
56 1 Research