We formulate an infinitary proof system for transitive closure logic, which is the logic obtained by augmenting first-order logics with a transitive closure operator. Our system is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first order logic with inductive definitions, our infinitary system is c...

Many problems in data mining can be viewed as a special case of the problem of enumerating the closed elements of an independence system with respect to some specific closure operator. Motivated by real-world applications, e.g., in track mining, we consider a generalization of this problem to strongly accessible set systems and arbitrary closure operators. For this more general problem setting,...

We prove some unconditional cases of the Existential Closedness problem for modular $j$-function. For this, we show that any finitely generated field can find a "convenient" set generators. This is done by showing in equipped with functions replicating algebraic behaviour $j$-function and its derivatives, one define natural closure operator three equivalent different ways.

Abstract In topological modal logic, it is well known that the Cantor derivative more expressive than closure, and ‘elsewhere’, or ‘difference’, operator ‘somewhere’ operator. 2014, Kudinov Shehtman asked whether combination of closure elsewhere becomes strictly when adding derivative. this paper we give an affirmative answer: in fact, alone can define properties spaces not expressible with els...

This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental ...

This paper is devoted to the study of nondeterministic closure automata, that is, nondeterministic finite automata (nfas) equipped with a strict closure operator on the set of states and continuous transition structure. We prove that for each regular language L there is a unique minimal nondeterministic closure automaton whose underlying nfa accepts L. Here minimality means no proper sub or quo...

the main goal of this paper is to introduce and study a new class of function via the notions of $e$-$theta$-open sets and $e$-$theta$-closure operator which are defined by özkoç and aslım cite{10} called weakly $er$-open functions and $e$-$theta$-open functions. moreover, we investigate not only some of their basic properties but also their relationships with other types of already existing to...

In belief revision, the result of a contraction should be a logically closed set (closure) such that the input is not inferred (success). Traditionally, these two postulates depend on the same consequence operator. The present work investigates the consequences of using two different consequence operators for each of these desiderata: the classic Cn to compute success and a second weaker operat...

1.1 What is a Group? A group is a set of objects with an associated operation which satisfies certain properties. For a group G which is a set of elements {T, T ′, T ′′, ...} with operator ◦, we require: 1. Closure. ∀T, T ′ ∈ G : T ◦ T ′ ∈ G. This property of closure should be familiar to anyone with a linear algebra background, as a vector space also fulfills this property, with addition as th...