Complex Algebras of Arithmetic

Abstract

An arithmetic circuit is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper we investigate the algebraic structure of complex algebras of natural numbers and make some observations regarding the complexity of various theories of such algebras.

From Contact Relations to Modal Operators, and Back

“…(see Fig 7) Remark 1. From the above proposition we can see that C P is a counterpart of a possible contact relation from [8]. However, the right-to-left implication from Proposition 31 is not always true about the m operator (in the sense Let us make a notion of the fineness of a partition precise, and let us observe that the finer the partition, the fewer regions are in contact.…”

“…Given a topological space X, let RC(X) be the complete Boolean algebra of all regular closed subsets of X, i.e., the sets A ∈ P(X) for which A = Cl Int A. 8 The Boolean operations are given by the following identities:…”

“…In the final section, we examine a proper subclass of C4 c that will serve as a geometrical interpretation of both the contact relation that satisfies (C4 c ) and the modal operator defined via the contact. The inspiration for this kind of interpretation comes from [8,17,24]. It is routine to verify that C P is a contact relation which satisfies (C4 c ).…”