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Logical Approaches to Barriers in Computing and Complexity

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Logical Approaches to Barriers in Computing and Complexity Arnold Beckmann Christine Gaßner Benedikt L¨owe (Eds.) Logical Approaches to Barriers in Computing and Complexity International Workshop, Greifswald, Germany, February 2010 Organized by the Alfried Krupp Wissenschaftskolleg Greifswald under the auspices of the Deutsche Vereinigung f¨urMathematische Logik und f¨ur Grundlagen der Exakten Wissenschaften (DVMLG), the Polskie Towarzystwo Logiki i Filozofii Nauki (PTLiFN), the Association for Computability in Europe (ACiE) and the European Association for Computer Science Logic (EACSL) Abstract Booklet Funded by the Alfried Krupp von Bohlen und Halbach-Stiftung, Essen and the Deutsche Forschungsgemeinschaft, Bonn Preface This booklet contains the extended abstracts of talks presented at the workshop on Logical Approaches to Barriers in Computing and Complexity, held on February 17{20, 2010, in Greifswald, Germany. The workshop was initiated by the Deutsche Vereinigung f¨urMathematische Logik und f¨urGrundlagen der Exakten Wissenschaften (DVMLG), the Polskie Towarzystwo Logiki i Filozofii Nauki (PTLiFN), the Association for Computability in Europe (ACiE) and the European Association for Computer Science Logic (EACSL) and organized under the auspices of these four learn`edsocieties. The workshop was funded by the Alfried Krupp von Bohlen und Halbach-Stiftung, Essen and the Deutsche Forschungsgemeinschaft, Bonn, and it was run by and held at the Alfried Krupp Wissenschaftskolleg in Greifswald, Germany. Scientific Scope. Computability theory and complexity theory have their origins in logic, with connec- tions to foundations of mathematics. The fundamental goal in these areas is to understand the limits of computability (that is analysing which problems can be solved on nowadays and future computers in principle) and effective computability (that is understanding the class of problems which can be solved quickly and with restricted resources.) Logic provides a multifarious toolbox of techniques to analyse questions like this, some of which promise to provide a deep insight in the structure of limit of compu- tation. The workshop had a focus on the following related aspects: logical descriptions of complexity (e.g., descriptive complexity, bounded arithmetic), complexity classes of abstract, algebraic and infinite struc- tures, barriers in proving complexity results, and Kolmogorov complexity and randomness. Descriptive complexity and bounded arithmetic are two complementary approaches to describe computational com- plexity in logical terms. The former is focused on decision problems, while the latter is more concerned with search problems. Both environments render questions about complexity classes in a natural way, leading to important open problems in their areas (e.g. finding logics to capture certain complexity classes, or the separation problem for bounded arithmetic.) Another path to gain more understanding of complexity classes is to study them in more general settings, e.g. on general algebraic structures or for computational models with infinite resources. Questions arising in this area are how complexity classes are rendered in them, and what their relationship is. It is well known that any proof of P 6=NP will have to overcome two barriers: relativization and natural proofs. To this has recently been added a third barrier, called algebraic relativization or algebrization. The idea is that, when we relativize some com- plexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a finite field or ring. The workshop intended to address current questions in all these areas, and initiate communication and research between them. In particular, the aim was to inform researcher in these related but separated areas about current approaches and ideas, to initiate new approaches to current questions. Some of these aspects were particularly timely: recently, research in these areas became more intense. Part of this is the new conference series CiE (run by the Association for Computability in Europe) whose range of interests includes those of our workshop, creating an important focus on the emerging topics of the field. This workshop was intended as a research-oriented follow-up to the CiE conferences, allowing researchers ample time for discussions and joint work. Programme. The Programme Committee had been designed by the four organising scientific organ- isations and consisted of thirteen members: Zofia Adamowicz (Warschau), Franz Baader (Dresden), Arnold Beckmann (chair, Swansea), Sam Buss (La Jolla, CA), Manfred Droste (Leipzig), Christine Gaßner (Greifswald), Peter Koepke (Bonn), Benedikt L¨owe (Amsterdam), Janos Makowsky (Haifa), Elvira Mayordomo (Zaragoza), Damian Niwinski (Warschau), Wolfgang Thomas (Aachen), and Martin Ziegler (Darmstadt). The DVMLG was represented by its vice president L¨owe and members of executive board Beckmann und Makowsky, the PTLiFN by the member of executive board Niwinski, CiE by the members of the executive board Beckmann, L¨owe and Mayordomo, and the EACSL by its president Makowsky, vice president Niwinski and member of executive board L¨owe. The Programme Committee has chosen six scientists as keynote speakers: Alessandra Carbone who spoke on Logical structures, cyclic graphs and genus of proofs, Lance Fortnow on Hardness of Instance Compression and Its Applications, Erich Gr¨adelon Fixed point logics, games, and polynomial-time com- plexity, Pascal Koiran on Shallow circuits with high-powered inputs, Leszek Ko lodziejczyk on The scheme IV of induction for sharply bounded arithmetic formulas, and Antonina Kolokolova on Axiomatic approach to barriers in complexity. The programme was enriched through an Offentliche¨ Abendveranstaltung (pub- lic evening lecture) given by Janos Makowsky on Spektrum Problem von H. Scholz und G. Asser: Ein halbes Jahrhundert danach. The workshop received 45 contributed submissions which were reviewed by the programme committee. The committee decided to accept 34 talks whose abstracts are printed in this booklet. The Programme Committee also decided to add to the workshop a Special Session on the topic Complexity in Arbitrary Structures which was organised by Christine Gaßner and Martin Ziegler, see the separate introduction to the Special Session on page V. Organization and Acknowledgements. The workshop was organized by Christine Gaßner (University of Greifswald). We are delighted to acknowledge and thank the Stiftung Alfried Krupp Kolleg Greifswald and the Deutsche Forschungsgemeinschaft for their essential financial support. The high scientic quality of the workshop was possible through the conscientious work of the Programme Committee, and the Special Session organizers. We thank Andrej Voronkov for his EasyChair system which facilitated the work of the Programme Committee and the editors considerably. Swansea, Greifswald and Amsterdam, January 2010 Arnold Beckmann Christine Gaßner Benedikt L¨owe Preface V Special Session on Complexity in Arbitrary Structures The classical notion of computability is closely related to the classical theory of recursive functions. This theory deals with functions on natural numbers, on words over a finite alphabet, or on some other countable sets. For functions on real numbers and other uncountable universes, several non-equivalent definitions of computability have been introduced. Here, real closed and algebraically closed fields are the starting point for the development of new theories of computing. Analytical or algebraic properties of the ring over the real numbers are used for defining notions of computability for real functions. One of the best known models is the Type-2 Turing machine where real numbers are approximated by rational numbers and the classical Turing machine is extended to infinite converging computations. Other settings consider real numbers as entities and one assumes that it is possible to execute exact arithmetic instructions and comparisons. Algebraic properties of rings and fields yield many interesting results in models of this kind, such as algebraic circuits and the real Turing machine (i.e. the BSS machine over the reals). More general models cover computation over arbitrary fields or other structures. Most complexity classes well-known from classical (i.e. discrete) computational complexity theory have an analogue in the theories to these computation models and play partially an analogous role in the corresponding theories. A lot of motivation to investigate these derived classes primary comes from the classical theory as well as from practical requirements. One hope is that we gain a better understanding of the difficulty of the P-versus-NP question. This famous problem is open for Turing machines as well as for many algebraic models of computation, but has been answered (in fact both in the positive and in the negative) for several structures. Further extending the list of structures with, provably, P = NP or P 6= NP will provide new insights in the original millennium prize problem. Another benefit is the development and study of fast algorithms for processing real numbers. In the classical theory as well as in the theory of computation over arbitrary structures, most proof techniques are of a logical nature. For more restricted models, topological and algebraic properties are of additional relevance. Practical real number computation can be separated into a variety of subfields with distinct objectives and paradigms; e.g. exact vs. approximate input, fixed-precision (floating point) vs. unbounded
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