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Applied Proof Theory / Proof Mining
Algorithmic Model Theory

Faculty: Klaus Keimel; Ulrich Kohlenbach; Martin Otto; Thomas Streicher; Martin Ziegler;

Christian Herrmann; Thomas Ihringer

Retired: Peter Burmeister; Rudolf Wille

The research group primarily represents the subject area of Mathematical Logic viewed as an applied foundational discipline between mathematics and computer science. Research activities focus on the application of proof theoretic, recursion theoretic, category theoretic, algebraic and model theoretic methods from mathematical logic to mathematics and computer science.

Beside classical mathematical logic (with proof theory, recursion theory and model theory) this involves constructive type theory, categorical logic, universal algebra, domain theory, lattice theory, finite model theory, and algorithmic issues.

Within mathematics, a primary field of applications in the proof- and recursion-theoretic setting (Kohlenbach) is the extraction of new information from proofs in algebra, analysis, functional analysis, hyperbolic geometry and numerical mathematics (proof mining). This involves qualitative aspects (e.g., independence of existence assertions from certain parameters) as well as quantitative aspects of computability and complexity of solutions (extraction of algorithms and bounds from proofs, exact real arithmetic, "computational mathematics": Kohlenbach, Streicher, sZiegler). Model theoretic investigations (Herrmann, Otto) make intra-mathematical links with algebra and discrete mathematics (Ihringer).

Concerning Logic in Computer Science and the mathematical foundations of computer science, major activities revolve around issues of semantics. On the one hand this involves the mathematical foundation of the semantics and the logic of programming languages (Keimel, Streicher); on the other hand, logics and formal systems are investigated in the sense of model theoretic semantics, w.r.t. expressiveness and definability, with an emphasis on computational aspects (algorithmic model theory, finite model theory: Otto). We investigate complexity issues from the point of view of functional programs (implicit computational complexity: Kohlenbach), in the descriptive (Otto) and the resource-oriented and structural sense (running time, memory, information theory and topology: Ziegler). Besides specific application domains in computer science, as, e.g., verification, data bases and knowledge representation, there is work on foundational issues in the areas of computability and complexity, as well as type theory and category theory.

Overall, the unit forms an internationally well connected cluster of expertise, with a characteristic emphasis on the connections that mathematical logic has to offer, both w.r.t. to other areas within mathematics and w.r.t. to the Logic in Computer Science spectrum.

Research group on Formal Concept Analysis. Based on lattice and order theoretic foundations, this group in the former AG1 (General Algebra and Discrete Mathematics) focuses on graphical logic systems for concept analysis in knowledge acquisition and processing applications (Burmeister, Wille). This research continues to be pursued in close co-operation with the ErnstSchröderZentrum für Begriffliche Wissensverarbeitung.

National and International Cooperations and Projects:
CCA (Computability and Complexity in Analysis)
MAP (Mathematics, Algorithms and Proofs)
Algorithmische Modelltheorie (Aachen/Berlin/Freiburg/Mainz/Marburg)
EPSRC project: Algorithmic Model Theory for Specific Domains (Otto)
ErnstSchröderZentrum, NaviCon

Martin Otto
Last modified: Fri May 21 09:40:48 CEST 2004