What's Up with Downward Collapse: Using the Easy-Hard Technique to Link Boolean and Polynomial Hierarchy Collapses

During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what beforehand were thought to be) lower complexity levels. For example, for each $k\geq 2$ it is now known that if \psigkone=\psigktwo then \ph=\sigmak. This article surveys the history, the results, and the technique---the so-called easy-hard method---of these nine papers.Comment: 37 pages. an extended abstract appeared in SIGACT News, 29, 10-22, 199

On the Structure of Learnability Beyond P/Poly

Motivated by the goal of showing stronger structural results about the complexity of learning, we study the learnability of strong concept classes beyond P/poly, such as PSPACE/poly and EXP/poly. We show the following: 1) (Unconditional Lower Bounds for Learning) Building on [Adam R. Klivans et al., 2013], we prove unconditionally that BPE/poly cannot be weakly learned in polynomial time over the uniform distribution, even with membership and equivalence queries. 2) (Robustness of Learning) For the concept classes EXP/poly and PSPACE/poly, we show unconditionally that worst-case and average-case learning are equivalent, that PAC-learnability and learnability over the uniform distribution are equivalent, and that membership queries do not help in either case. 3) (Reducing Succinct Search to Decision for Learning) For the decision problems R_{Kt} and R_{KS} capturing the complexity of learning EXP/poly and PSPACE/poly respectively, we show a succinct search to decision reduction: for each of these problems, the problem is in BPP iff there is a probabilistic polynomial-time algorithm computing circuits encoding proofs for positive instances of the problem. This is shown via a more general result giving succinct search to decision results for PSPACE, EXP and NEXP, which might be of independent interest. 4) (Implausibility of Oblivious Strongly Black-Box Reductions showing NP-hardness of learning NP/poly) We define a natural notion of hardness of learning with respect to oblivious strongly black-box reductions. We show that learning PSPACE/poly is PSPACE-hard with respect to oblivious strongly black-box reductions. On the other hand, if learning NP/poly is NP-hard with respect to oblivious strongly black-box reductions, the Polynomial Hierarchy collapses

Size-Change Termination, Monotonicity Constraints and Ranking Functions

Size-Change Termination (SCT) is a method of proving program termination based on the impossibility of infinite descent. To this end we may use a program abstraction in which transitions are described by monotonicity constraints over (abstract) variables. When only constraints of the form x>y' and x>=y' are allowed, we have size-change graphs. Both theory and practice are now more evolved in this restricted framework then in the general framework of monotonicity constraints. This paper shows that it is possible to extend and adapt some theory from the domain of size-change graphs to the general case, thus complementing previous work on monotonicity constraints. In particular, we present precise decision procedures for termination; and we provide a procedure to construct explicit global ranking functions from monotonicity constraints in singly-exponential time, which is better than what has been published so far even for size-change graphs.Comment: revised version of September 2

The boom-bust cycle in Finland and Sweden 1984-1995 in an international perspective

This paper compares the boom-bust cycle in Finland and Sweden 1984-1995 with the average boom-bust pattern in industrialized countries as calculated from an international sample for the period 1970-2002. Two clear conclusions emerge. First, the Finnish-Swedish experience is much more volatile than the average boom-bust pattern. This holds for virtually every time series examined. Second, the bust and the recovery in the two Nordic countries differ markedly more from the international pattern than the boom phase does. The bust is considerably deeper and the recovery comes earlier and is more rapid. We explain the highly volatile character of the Finnish and Swedish boom-bust episode by the design of economic policies in the 1980s and 1990s. The boom-bust cycle in Finland and Sweden 1984-1995 was driven by financial liberalization and a hard currency policy, causing large pro-cyclical swings in the real rate of interest transmitted via the financial sector into the real sector and then into the public finances. JEL Classification: E32, E62, E6