In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be embedded. The importance of Weihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the Weihrauch reductions between theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. As crucial corner points of this classification scheme the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations are studied. It is proved that parallelized LLPO is equivalent to Weak Kőnig's Lemma and hence to the Hahn—Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof, based on a computational version of Kleene's ternary logic, that the class of weakly computable operations is closed under composition. Moreover, weakly computable operations on computable metric spaces are characterized as operations that admit upper semi-computable compact-valued selectors and it is proved that any single-valued weakly computable operation is already computable in the ordinary sense.
The Journal of Symbolic Logic (JSL) was founded in 1936 and it has become the leading research journal in the field. It is issued quarterly. Volume 71, being published during 2006, will consist of approximately 1300 pages. The Journal is distributed with The Bulletin of Symbolic Logic. The Journal and The Bulletin are the official organs of the Association for Symbolic Logic, an international organization for supporting research in symbolic logic and furthering the exchange of ideas among mathematicians, philosophers, computer scientists, linguists, and others interested in this field. The main purpose of The Journal is to publish original scholarly work in symbolic logic. The Journal intends to represent the entire field of symbolic logic, which has become very broad, including its connections with mathematics and philosophy as well as newer aspects related to computer science and linguistics. The Journal invites the submission of research papers and expository articles in all areas of symbolic logic. These may have technical, philosophical, or historical emphases. In order to be considered for publication, papers should be prepared following the JSL Guidelines and should be submitted to one of the JSL Editors. The Journal currently has no backlog and the expected time from submission to publication is about one year.
The Association for Symbolic Logic is an international organization supporting research and critical studies in logic. Its primary function is to provide an effective forum for the presentation, publication, and critical discussion of scholarly work in this area of inquiry. Among its many activities, the Association organizes and sponsors meetings and summer schools throughout the world, and publishes books and journals. Logic is an ancient discipline that has undergone striking modern developments through the introduction of rigorous formal methods, stimulated largely by foundational problems in mathematics. "Symbolic logic" is a term intended to encompass the entire field of logical inquiry, undertaken in this modern spirit. The Association was founded in 1936, at a time when great advances in logic were beginning to be made. Its first members were mainly mathematicians and philosophers who perceived a common ground and sought to strengthen it. Recent research in other areas such as computer science, linguistics, and cognitive science has also been inspired by logic, and the current membership and activities of the Association reflects such expanding interests.