We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r,’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite p...

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives equivalence between the validity of weighted Hardy–Sobolev inequality and quasiadditivity capacity with respect to Whitney covers $\Omega$. Important ingredients proof include use discrete convolution as test function Maz'ya type characterization inequalities.

We aim to generalize Büchi’s fundamental theorem on the coincidence of recognizable and MSO-definable languages to a weighted timed setting. For this, we investigate subclasses of weighted timed automata and show how we can extend existing timed MSO logics with weights. Here, we focus on the class of weighted event-recording automata and define a weighted extension of the full logic MSOer(Σ) in...

This paper, gives an answer for the Question 1.1 posed by Hitzler (Generalized metrics and topology in logic programming semantics, 2001) by means of "Topological aspects of d-metric space with d-neighborhood system". We have investigated the topological aspects of a d-neighborhood system obtained from dislocated metric space (simply d-metric space) which has got useful applications in the sema...

It is highly desirable for a computational model to have a logic characterization like in the seminal work of Büchi that connects MSO with finite automata. For example, weighted automata are the quantitative extension of finite automata for computing functions over words and they can be naturally characterized by a subfragment of weighted logic introduced by Droste and Gastin. Recently, cost re...

We aim to generalize Büchi’s fundamental theorem on the coincidence of recognizable and MSO-definable languages to a weighted timed setting. For this, we investigate weighted timed automata and show how we can extend Wilke’s relative distance logic with weights taken from an arbitrary semiring. We show that every formula in our logic can effectively be transformed into a weighted timed automato...