05391 Abstracts Collection -- Algebraic and Numerical Algorithms and Computer-assisted Proofs

From 25.09.05 to 30.09.05, the Dagstuhl Seminar 05391 ``Algebraic and Numerical Algorithms and Computer-assisted Proofs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. Links to extended abstracts or full papers are provided, if available

05391 Executive Summary -- Numerical and Algebraic Algorithms and Computer-assisted Proofs

The common goal of self-validating methods and computer algebra methods is to solve mathematical problems with complete rigor and with the aid of computers. The seminar focused on several aspects of such methods for computer-assisted proofs

Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gröbner Bases

International audienc

Automated Generation of User Guidance by Combining Computation and Deduction

Herewith, a fairly old concept is published for the first time and named "Lucas Interpretation". This has been implemented in a prototype, which has been proved useful in educational practice and has gained academic relevance with an emerging generation of educational mathematics assistants (EMA) based on Computer Theorem Proving (CTP). Automated Theorem Proving (ATP), i.e. deduction, is the most reliable technology used to check user input. However ATP is inherently weak in automatically generating solutions for arbitrary problems in applied mathematics. This weakness is crucial for EMAs: when ATP checks user input as incorrect and the learner gets stuck then the system should be able to suggest possible next steps. The key idea of Lucas Interpretation is to compute the steps of a calculation following a program written in a novel CTP-based programming language, i.e. computation provides the next steps. User guidance is generated by combining deduction and computation: the latter is performed by a specific language interpreter, which works like a debugger and hands over control to the learner at breakpoints, i.e. tactics generating the steps of calculation. The interpreter also builds up logical contexts providing ATP with the data required for checking user input, thus combining computation and deduction. The paper describes the concepts underlying Lucas Interpretation so that open questions can adequately be addressed, and prerequisites for further work are provided.Comment: In Proceedings THedu'11, arXiv:1202.453

SC 2 : Satisfiability Checking meets Symbolic Computation (Project Paper)

International audienceSymbolic Computation and Satisfiability Checking are two research areas, both having their individual scientific focus but sharing also common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite their commonalities, the two communities are rather weakly connected. The aim of our newly accepted SC 2 project (H2020-FETOPEN-CSA) is to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap from theory along the way to tools and (industrial) applications. In this paper we report on the aims and on the first activities of this project, and formalise some relevant challenges for the unified SC 2 community

Chaperones convert the energy from ATP into the nonequilibrium stabilization of native proteins.

During and after protein translation, molecular chaperones require ATP hydrolysis to favor the native folding of their substrates and, under stress, to avoid aggregation and revert misfolding. Why do some chaperones need ATP, and what are the consequences of the energy contributed by the ATPase cycle? Here, we used biochemical assays and physical modeling to show that the bacterial chaperones GroEL (Hsp60) and DnaK (Hsp70) both use part of the energy from ATP hydrolysis to restore the native state of their substrates, even under denaturing conditions in which the native state is thermodynamically unstable. Consistently with thermodynamics, upon exhaustion of ATP, the metastable native chaperone products spontaneously revert to their equilibrium non-native states. In the presence of ATPase chaperones, some proteins may thus behave as open ATP-driven, nonequilibrium systems whose fate is only partially determined by equilibrium thermodynamics