Easily checked self-reducibility

Abstract

This paper explores two generalizations, within NP, of self-reducibility: kernel constructibility [1,2] and committability [17]. Informally stated, kernel constructible sets have self-reductions that are easy to check (though perhaps hard to compute), and committable sets are those sets for which the potential correctness of a partial proof of set membership can be checked via a query to the same set (that is, via a self-reduction). We study these two notions of self-reducibility on non-dense sets. We show that sparse kernel constructible sets are of low complexity, we extend previous results showing that sparse committable sets are of low complexity, and we provide structural evidence of interest in its own right—namely that if all sparse disjunctively self-reducible sets are in P then FewP ∩ coFewP is not P-bi-immune—that our extension is unlikely to be further extended. We obtain density-based sufficient conditions for kernel-constructibility: sets whose complements are captured by non-dense sets are perforce kernel constructible. Using sparse languages and Kolmogorov complexity theory as tools, we argue that kernel constructibility is orthogonal to standard notions of complexity.

A full version of this paper is available from either author. Acknowledgments: We thank Yenjo Han and Benjamin Alexander for many interesting discussions on the uses of Kolmogorov complexity in complexity theory, from which the Kolmogorov complexity use in this paper grew. We thank Jacobo Torán for allowing us to include here his Proposition 2 and for suggesting Corollary 6. We thank Yenjo Han, Johannes Köbler, Uwe Schöning, and Osamu Watanabe for affording us weekend and remote access to printers and libraries that would have been inaccessible without their help. We are very grateful to Vikraman Arvind, Lance Fortnow, Johannes Köbler, Jacobo Torán, Leen Torenvliet, and anonymous referees for helpful conversations, comments, suggestions, and literature pointers.

Supported in part by the National Science Foundation under research grant CCR-8957604. Work done in part while visiting Universität Ulm.

Supported in part by Ministero della Pubblica Istruzione through “Progetto 40%: Algoritmi, Modelli di Calcolo e Strutture Informative.” Work done in part while visiting the University of Rochester.