Time-space trade-offs for branching programs

AbstractBranching program depth and the logarithm of branching program complexity are lower bounds on time and space requirements for any reasonable model of sequential computation. In order to gain more insight to the complexity of branching programs and to the problems of time-space trade-offs one considers, on one hand, width-restricted and, on the other hand, depth-restricted branching programs. We present these computation models and the trade-off results already proved. We prove a new result of this type by presenting an effectively defined Boolean function whose complexity in depth-restricted one-time-only branching programs is exponential while its complexity even in width-2 branching programs is polynomial

Switching functions whose monotone complexity is nearly quadratic

AbstractA sequence of monotone switching functions hn:{0,1}n→ {0,1}n is constructed, such that the monotone complexity of hn grows faster than Ω(n2 log−2n). Previously the best lower bounds of this nature were several Ω(n32 bounds due to Pratt, Paterson, Mehlhorn and Galil and Savage

The complexity of the parity function in unbounded fan-in, unbounded depth circuits

AbstractAlmost everything is known on the complexity of the parity function in fan-in 2 circuits over various bases. Also the minimal depth of polynomial-size, unbounded fan-in {∧, ∨, ⌝\dl;} circuits for the parity function has been studied. Here the complexity without any depth restriction is considered. For the basis {∧, ∨, ⌝\dl;} almost optimal bounds, and for the basis of NOR gates and the basis of all threshold functions optimal bounds on the number of gates are obtained. For the basis {∧, ∨, ⌝\dl;} the minimal number of wires is determined. For threshold circuits an exponential gap between synchronous and asynchronous circuits is proved. The results not only answer open questions in complexity theory but also have implications for the real-life circuit design

Homozygous NMNAT2 mutation in sisters with polyneuropathy and erythromelalgia

none14siWe identified a homozygous missense mutation in the gene encoding NAD synthesizing enzyme NMNAT2 in two siblings with childhood onset polyneuropathy with erythromelalgia. No additional homozygotes for this rare allele, which leads to amino acid substitution T94M, were present among the unaffected relatives tested or in the 60,000 exomes of the ExAC database. For axons to survive, axonal NMNAT2 activity has to be maintained above a threshold level but the T94M mutation confers a partial loss of function both in the ability of NMNAT2 to support axon survival and in its enzymatic properties. Electrophysiological tests and histological analysis of sural nerve biopsies in the patients were consistent with loss of distal sensory and motor axons. Thus, it is likely that NMNAT2 mutation causes this pain and axon loss phenotype making this the first disorder associated with mutation of a key regulator of Wallerian-like axon degeneration in humans. This supports indications from numerous animal studies that the Wallerian degeneration pathway is important in human disease and raises important questions about which other human phenotypes could be linked to this gene.openPeter Huppke; Eike Wegener; Jonathan Gilley; Carlo Angeletti; Ingo Kurth; Joost P.H. Drenth; Christine Stadelmann; Alonso Barrantes-Freer; Wolfgang Brück; Holger Thiele; Peter Nürnberg; Jutta Gärtner; Giuseppe Orsomando; Michael P. ColemanHuppke, Peter; Wegener, Eike; Gilley, Jonathan; Angeletti, Carlo; Kurth, Ingo; Drenth, Joost P. H.; Stadelmann, Christine; Barrantes-Freer, Alonso; Brück, Wolfgang; Thiele, Holger; Nürnberg, Peter; Gärtner, Jutta; Orsomando, Giuseppe; Coleman, Michael P