Applicative theories for logarithmic complexity classes

We present applicative theories of words corresponding to weak, and especially logarithmic, complexity classes. The theories for the logarithmic hierarchy and alternating logarithmic time formalise function algebras with concatenation recursion as main principle. We present two theories for logarithmic space where the first formalises a new two-sorted algebra which is very similar to Cook and Bellantoni's famous two-sorted algebra B for polynomial time [4]. The second theory describes logarithmic space by formalising concatenation- and sharply bounded recursion. All theories contain the predicates WW representing words, and VV representing temporary inaccessible words. They are inspired by Cantini's theories [6] formalising B.

A novel colourimetric technique to assess chewing function using two-coloured specimens: Validation and application

OBJECTIVES Chewing efficiency may be evaluated using cohesive specimen, especially in elderly or dysphagic patients. The aim of this study was to evaluate three two-coloured chewing gums for a colour-mixing ability test and to validate a new purpose built software (ViewGum©). METHODS Dentate participants (dentate-group) and edentulous patients with mandibular two-implant overdentures (IOD-group) were recruited. First, the dentate-group chewed three different types of two-coloured gum (gum1-gum3) for 5, 10, 20, 30 and 50 chewing cycles. Subsequently the number of chewing cycles with the highest intra- and inter-rater agreement was determined visually by applying a scale (SA) and opto-electronically (ViewGum©, Bland-Altman analysis). The ViewGum© software determines semi-automatically the variance of hue (VOH); inadequate mixing presents with larger VOH than complete mixing. Secondly, the dentate-group and the IOD-group were compared. RESULTS The dentate-group comprised 20 participants (10 female, 30.3±6.7 years); the IOD-group 15 participants (10 female, 74.6±8.3 years). Intra-rater and inter-rater agreement (SA) was very high at 20 chewing cycles (95.00-98.75%). Gums 1-3 showed different colour-mixing characteristics as a function of chewing cycles, gum1 showed a logarithmic association; gum2 and gum3 demonstrated more linear behaviours. However, the number of chewing cycles could be predicted in all specimens from VOH (all p<0.0001, mixed linear regression models). Both analyses proved discriminative to the dental state. CONCLUSION ViewGum© proved to be a reliable and discriminative tool to opto-electronically assess chewing efficiency, given an elastic specimen is chewed for 20 cycles and could be recommended for the evaluation of chewing efficiency in a clinical and research setting. CLINICAL SIGNIFICANCE Chewing is a complex function of the oro-facial structures and the central nervous system. The application of the proposed assessments of the chewing function in geriatrics or special care dentistry could help visualising oro-functional or dental comorbidities in dysphagic patients or those suffering from protein-energy malnutrition.