Delta-Decision Procedures for Exists-Forall Problems over the Reals

Solving nonlinear SMT problems over real numbers has wide applications in robotics and AI. While significant progress is made in solving quantifier-free SMT formulas in the domain, quantified formulas have been much less investigated. We propose the first delta-complete algorithm for solving satisfiability of nonlinear SMT over real numbers with universal quantification and a wide range of nonlinear functions. Our methods combine ideas from counterexample-guided synthesis, interval constraint propagation, and local optimization. In particular, we show how special care is required in handling the interleaving of numerical and symbolic reasoning to ensure delta-completeness. In experiments, we show that the proposed algorithms can handle many new problems beyond the reach of existing SMT solvers

The suitability of sham treatments for use as placebo controls in trials of spinal manipulative therapy: A pilot study

Despite the augmented use and dependence on manual therapy (MT), there are still calls from both within and outside the MT professions to provide robust evidence that spinal manipulative therapy (SMT) induces therapeutic effects beyond placebo. To facilitate this, placebo or 'sham' treatments, the development of which is notoriously difficult, must be used in rigorously controlled trials. The aim of this study was to investigate the suitability of different shams as controls in SMT trials.A repeated measures, single-blind, randomised trial was conducted on 10 asymptomatic subjects. Pain pressure thresholds (PPTs) were measured at 2 sites, local and systemic, before and after the application of either high-velocity low-amplitude thrust (HVLAT), sham functional technique (SFT), sham ultrasound (SUS) or no intervention control (NIC) to the cervico-thoracic (CT) junctional area. Treatment credibility was then assessed using a 4-point Likert Scale in response to 4 statements.Results demonstrated no significant change in PPT following any of the interventions, irrespective of site tested. The effect sizes for all interventions were considered small (d = <0.2). There were significant differences in Likert Scale responses for each statement (P < 0.001), with SUS eliciting significantly different responses as compared to SFT and NIC but not, predominantly, with HVLAT.SUS is implicated as being the most effective sham, having high fidelity to subjects' perceptions of a 'real' treatment, whilst being therapeutically inert i.e. results in no significant change in clinical status. © 2012 Elsevier Ltd