In this paper, we introduce the notions of expansion of ideals in $MV$-algebras, $ (tau,sigma)- $primary, $ (tau,sigma)$-obstinate  and $ (tau,sigma)$-Boolean  in $ MV- $algebras. We investigate the relations of them. For example, we show that every $ (tau,sigma)$-obstinate ideal of an $ MV-$ algebra is $ (tau,sigma)$-primary  and $ (tau,sigma)$-Boolean. In particular, we define an expansion $ ...

We investigate quantifier alternation hierarchies in first-order logic on finite words. Levels in these hierarchies are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language in the levels $\mathcal{B}{\Sigma}_2$ (finite boolean combinations of formulas having only one alternation) and ${\Sigma}_3$ (formulas having on...

This paper introduces a higher-order neural network called the Binary Pi-sigma Network (BPSN), which is a feedforward network with a single \hidden" layer and product units in the output layer. As training proceeds, the BPSN forms an internal representation of the conjunctive normal form expression corresponding to the Boolean function to be learned. This enables the network to have a regular s...

Effectus theory is a relatively new approach to categorical logic that can be seen as an abstract form of generalized probabilistic theories (GPTs). While the scalars GPT are always real unit interval [0,1], in effectus they any effect monoid. Hence, there quite exotic effectuses resulting from more pathological monoids. In this paper we introduce sigma-effectuses, where certain countable sums ...