A Lower Bound of 2n2^n Conditional Branches for Boolean Satisfiability on Post Machines

We establish a lower bound of $2^n$ conditional branches for deciding the satisfiability of the conjunction of any two Boolean formulas from a set called a full representation of Boolean functions of $n$ variables - a set containing a Boolean formula to represent each Boolean function of $n$ variables. The contradiction proof first assumes that there exists a Post machine (Post's Formulation 1) that correctly decides the satisfiability of the conjunction of any two Boolean formulas from such a set by following an execution path that includes fewer than $2^n$ conditional branches. By using multiple runs of this Post machine, with one run for each Boolean function of $n$ variables, the proof derives a contradiction by showing that this Post machine is unable to correctly decide the satisfiability of the conjunction of at least one pair of Boolean formulas from a full representation of $n$-variable Boolean functions if the machine executes fewer than $2^n$ conditional branches. This lower bound of $2^n$ conditional branches holds for any full representation of Boolean functions of $n$ variables, even if a full representation consists solely of minimized Boolean formulas derived by a Boolean minimization method. We discuss why the lower bound fails to hold for satisfiability of certain restricted formulas, such as 2CNF satisfiability, XOR-SAT, and HORN-SAT. We also relate the lower bound to 3CNF satisfiability. The lower bound does not depend on sequentiality of access to the boxes in the symbol space and will hold even if a machine is capable of non-sequential access.Comment: This article draws heavily from arXiv:1406.597

Model Verification and the Likelihood Principle

The likelihood principle (LP) is typically understood as a constraint on any measure of evidence arising from a statistical experiment. It is not sufficiently often noted, however, that the LP assumes that the probability model giving rise to a particular concrete data set must be statistically adequate—it must “fit” the data sufficiently. In practice, though, scientists must make modeling assumptions whose adequacy can nevertheless then be verified using statistical tests. My present concern is to consider whether the LP applies to these techniques of model verification. If one does view model verification as part of the inferential procedures that the LP intends to constrain, then there are certain crucial tests of model verification that no known method satisfying the LP can perform. But if one does not, the degree to which these assumptions have been verified is bracketed from the evidential evaluation under the LP. Although I conclude from this that the LP cannot be a universal constraint on any measure of evidence, proponents of the LP may hold out for a restricted version thereof, either as a kind of “ideal” or as defining one among many different forms of evidence

A novel apparatus/protocol designed for optogenetic manipulation and recording of individual neurons during a motivation and working memory task in the rodent

Innovative molecular tools allow neuroscientists to study neural circuitry associated with specific behaviors. Consequently, behavioral methods must be developed to interface with these new molecular tools in order for neuroscientists to identify the causal elements underlying behavior and decision-making processes. Here we present an apparatus and protocol for a novel Go/No-Go behavioral paradigm to study the brain attention and motivation/reward circuitry in awake, head-restrained rodents. This experimental setup allows: (1) Painless and stable restraint of the head and body; (2) Rapid acquisition to simple or complex operant tasks; (3) Repeated electrophysiological single and multiple unit recordings during ongoing behavior; (4) Pharmacological and viral manipulation of various brain regions via targeted guide cannula, and; (5) Optogenetic cell-type specific activation and silencing with simultaneous electrophysiological recording. In addition to the experimental advantages, the head-restraint system is relatively inexpensive and training parameters can be easily modulated to the specifications of the experimenter. The system runs on custom LabView software. In summary, our novel apparatus and protocol allows researchers to study and manipulate components of behavior, such as motivation, impulsivity, and reward-related working memory during an ongoing operant behavioral task without interference from non task-related behaviors. For more information on the custom apparatus, software or to collaborate please visit www.neuro-cloud.net/nature-precedings/dolzani