Kenneth W. Regan

Pseudorandom generators, measure theory, and natural proofs

We prove that if strong pseudorandom number generators exist, then the class of languages that have polynomial-sized circuits (P/poly) is not measurable within exponential time, in terms of the resource-bounded measure theory of Lutz. We prove our result by showing that if P/poly has measure zero in exponential time, then there is a natural proof against P/poly, in the terminology of Razborov and Rudich (1994). We also provide a partial converse of this result.

Parameterized circuit complexity and the W hierarchy

Abstract A parameterized problem 〈 L , k 〉 belongs to W [ t ] if there exists k ′ computed from k such that 〈 L , k 〉 reduces to the weight-k′ satisfiability problem for weft- t circuits. We relate the fundamental question of whether the W [ t ] hierarchy is proper to parameterized problems for constant-depth circuits. We define classes G [ t ] as the analogues of AC 0 depth- t for parameterized problems, and N [ t ] by weight- k ′ existential quantification on G [ t ], by analogy with NP = ∃ · P . We prove that for each t , W [ t ] equals the closure under fixed-parameter reductions of N [ t ]. Then we prove, using Sipsers results on the AC 0 depth- t hierarchy, that both the G [ t ] and the N [ t ] hierarchies are proper. If this separation holds up under parameterized reductions, then the W [ t ] hierarchy is proper. We also investigate the hierarchy H [ t ] defined by alternating quantification over G [ t ]. By trading weft for quantifiers we show that H [ t ] coincides with H [1]. We also consider the complexity of unique solutions, and show a randomized reduction from W [ t ] to Unique W [ t ].

Quasilinear Time Complexity Theory

This paper furthers the study of quasi-linear time complexity initiated by Schnorr [Sch76] and Gurevich and Shelah [GS89]. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilineartime hierarchy. Whereas all previously known versions of the Valiant-Vazirani reduction from NP to parity run in quadratic time, we give a new construction using error-correcting codes that runs in quasilinear time. We show, however, that the important equivalence between search problems and decision problems in polynomial time is unlikely to carry over: if search reduces to decision for SAT in quasi-linear time, then all of NP is contained in quasi-polynomial time. Other connections to work by Stearns and Hunt [SH86, SH90, HS90] on “power indices” of NP languages are made.

Gap-languages and log-time complexity classes

Abstract This paper shows that classical results about complexity classes involving “delayed diagonalization” and “gap languages”, such as Ladners Theorem and Schonings Theorem and independence results of a kind noted by Schoming and Hartmanis, apply at very low levels of complexity, indeed all the way down in Sipsers log-time hierarchy. This paper also investigates refinements of Sipsers classes and notions of log-time reductions, following on from recent work by Cai, Chen, and others.

The Topology of Provability in Complexity Theory

We present a general technique for showing that many properties of recursive languages are not provable. Here “provable” is taken with respect to a given sound, recursively axiomatized formal system J, such as Peano arithmetic. A representative application (Theorems 6.1–6.2) concerns the property of intractability, i.e., non-membership in the class P. It says that there exists a language E such that E is not in P, but the formal assertion ‘E is not in P’ is independent of J. Moreover, given any recursive language A ∉ P, we can construct E such that also E ⩽mP A. Our techniques strengthen similar results in the literature and lead to several other applications pertaining to P-immune sets, oracle separations, and the Berman-Hartmanis conjecture. We explain the phenomenon of unprovability in terms of both recursive properties of the formal systems J under consideration, and topological properties of complexity classes in a natural space which we call R. Provable properties correspond to closed sets of r. The topology provides geometric intuition for recognizing classes which are not closed in R, such as NPP (unless it is empty). We show how independence results follow immediately for these classes. In conclusion we argue that the type of independence result presented here forms an obstacle for day-to-day work in complexity theory, but does not bear directly on the possible independence of the P = NP question from Peano arithmetic or set theory. However, we believe our tools capable of measuring the link between the structure of a given language E ∉ P and the formal strength needed to prove the assertion ‘E ∉ P.’ Research in this direction has already been initiated by D. Joseph (J. Comput. System Sci. 25 (1983), 205–228).

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]

Performance and prediction: bayesian modelling of fallible choice in chess

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Understanding Distributions of Chess Performances

BP[DC]. Among our applications, we simplify the proof by Toda [Tol], [To2] that the polynomial hierarchy PH is contained in BP[⊕P]. We also show that relative to a random oracleR, PHR is properly contained in ⊕PR.