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FEALORA Report Summary

Project ID: 339691
Funded under: FP7-IDEAS-ERC
Country: Czech Republic

Mid-Term Report Summary - FEALORA (Feasibility, logic and randomness in computational complexity)

In this project we study certain aspects of computational complexity, which is a subject that plays an important role in theoretical computer science. The aim is to fully understand how difficult is to solve particular computational problems and classify them according to their complexity. To show that a computational problem is hard, one must prove that there is no efficient algorithm that solves the problem, which is typically very difficult. Although many results have been obtained that show impossibility of efficient algorithms in specific situations, the most important problems remain notoriously open. The prevailing feeling is that it will be necessary to develop completely new mathematical methods to solve these problems.

Project FEALORA focuses on the study of logical aspects of complexity, as we believe that logic will play important role in the solution of the big open problems of computational complexity theory. Specifically, we study the thesis that logical complexity of a mathematical statement, represented as the necessity of using strong axioms, or long proofs, may be connected with the computational complexity of a computational problem associated with the statement. In order to state this thesis formally, some conjectures were proposed. During the first period of the project we have gained better understanding of the thesis by a systematic study of these conjectures. We found new plausible conjectures, proved more relations between them and classified them.

Furthermore, project FEALORA is also aimed at developing new methods of proving lower bounds on computational complexity and proof complexity on the one hand, and proving limitations of the known methods on the other. We have been studying “border line” problems, the open problems about proving complexity lower bounds which seems solvable with the currently available theoretical means. Specifically, we proved several new lower bounds on the complexity of certain problems and using new concepts we found reductions that possibly may be used to prove more lower bounds.

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Czech Republic
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