James S. Royer

The isomorphism conjecture fails relative to a random oracle

Berman and Hartmanis [1977] conjectured that there is a polynomial-time computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [1985] gave a structural definition of a class of NP-complete sets-the k-creative sets-and defined a class of sets (the K f k s) that are necessarily k-creative. They went on to conjecture that certain of these K f t s are not isomorphic to the standard NP-complete sets. Clearly, the Berman-Hartmanis and Joseph-Young conjectures cannot both be correct. We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K f k is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scrambling functions, we show that much more powerful one-way functions-the annihilating functions-exist relative to a random oracle. Random oracles are the first examples of oracles relative to which the isomorphism conjecture fails with respect to higher classes such as EXP and NEXP.

A connotational theory of program structure

Motivations, background, and basic definitions.- Effective numberings, completions, and control structures.- Some special effective numberings.- Characterizations of acceptability.- Independence of control structures.- General programming properties of effective numberings of subrecursive classes.

The ismorphism conjecture fails relative to a random oracle

Berman and Hartmanis [BH77] conjectured that there is a polynomial-time computable isomorphism between any two languages m-complete (“Karp” complete) for NP. Joseph and Young [JY85] discovered a structurally defined class of NP-complete sets and conjectured that certain of these sets (the <italic>K<supscrpt>k</supscrpt><subscrpt>ƒ</subscrpt></italic>s) are not isomorphic to the standard NP-complete sets for some one-way functions ƒ. These two conjectures cannot both be correct. We introduce a new family of strong one-way functions, the <italic>scrambling</italic> functions. If ƒ is a scrambling function, then <italic>K<supscrpt>k</supscrpt><subscrpt>fnof;</subscrpt></italic> is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. As evidence for the existence of scrambling functions, we show that much more powerful one-way functions--the <italic>annihilating</italic> functions--exist relative to a random oracle.

The Structure of Complete Degrees

The notion of NP-completeness has cut across many fields and has provided a means of identifying deep and unexpected commonalities. Problems from areas as diverse as combinatorics, logic, and operations research turn out to be NP-complete and thus computationally equivalent in the sense discussed in the next paragraph. PSPACE-completeness, NEXP-completeness, and completeness for other complexity classes have likewise been used to show commonalities in a variety of other problems. This paper surveys investigations into how strong these commonalities are.

On 1-truth-table-hard languages

Abstract The polynomial-time 1-tt-hard sets for EXP and RE are polynomial-time many—one-hard

Inductive inference of approximations

In this paper we investigate inductive inference identification criteria which permit infinitely many errors in explanations, but which require that the “density” of these errors be no more than a certain, prespectified amount. We introduce three hierarchies of such criteria, each of which has the same order type as the real unit interval. These three hierarchies are progressively more strict in the way they measure density of errors of explanations. The strictest of the three turns out to have all of its members, save one, incomparable to the identification criterion which permits finitely many errors in explanations.

On characterizations of the basic feasible functionals, Part I

We introduce a typed programming formalism, type-2 inflationary tiered loop programs or ITLP2, that characterizes the type-2 basic feasible functionals. ITLP2 is based on Bellantoni and Cooks (1992) and Leivants (1995) type-theoretic characterization of polynomial-time, and turns out to be closely related to Kapron and Cooks (1991; 1996) machine-based characterization of the type-2 basic feasible functionals.

A hierarchy based on output multiplicity

Abstract The class NP k V consists of those partial, multivalued functions that can be computed by a nondeterministic, polynomial time-bounded transducer that has at most k distinct values on any input. We define the output-multiplicity hierarchy to consist of the collection of classes NP k V for all positive integers k ≥ 1. In this paper we investigate the strictness of the output-multiplicity hierarchy and establish three main results pertaining to this: 1. 1. If for any k > 1, the class NP k V collapses into the class NP( k − 1)V, then the polynomial hierarchy collapses to Σ 2 P . 2. 2. If the converse of the above result is true, then any proof of this converse cannot relativize. We exhibit an oracle relative to which the polynomial hierarchy collapses to P NP , but the output-multiplicity hierarchy is strict. 3. 3. Relative to a random oracle, the output-multiplicity hierarchy is strict. This result is in contrast to the still open problem of the strictness of the polynomial hierarchy relative to a random oracle. In introducing the technique for the third result we prove a related result of interest: relative to a random oracle UP ≠ NP.

On closure properties of bounded two-sided error complexity classes

AbstractWe show that if a complexity classC is closed downward under polynomial-time majority truth-table reductions (≤mttp), then practically every other “polynomial” closure property it enjoys is inherited by the corresponding bounded two-sided error class BP[C]. For instance, the Arthur-Merlin game class AM [B1] enjoys practically every closure property of NP. Our main lemma shows that, for any relativizable classD which meets two fairly transparent technical conditions, we haveCBP[C]