Zero-Divisors and Their Graph Languages

We introduce the use of formal languages in place of zero-divisor graphs used to study theoretic properties of commutative rings. We show that a regular language called a graph language can be constructed from the set of zero-divisors of a commutative ring. We then prove that graph languages are equivalent to their associated graphs. We go on to define several properties of graph languages

Equational Reasoning about Programs with General Recursion and Call-by-value Semantics

Dependently typed programming languages provide a mechanism for integrating verification and programming by encoding invariants as types. Traditionally, dependently typed languages have been based on constructive type theories, where the connection between proofs and programs is based on the Curry-Howard correspondence. This connection comes at a price, however, as it is necessary for the languages to be normalizing to preserve logical soundness. Trellys is a call-by-value dependently typed programming language currently in development that is designed to integrate a type theory with unsound programming features, such as general recursion, Type:Type, and others. In this paper we outline one core language design for Trellys, and demonstrate the use of the key language constructs to facilitate sound reasoning about potentially unsound programs

Irrelevance, Heterogeneous Equality, and Call-by-value Dependent Type Systems

We present a full-spectrum dependently typed core language which includes both nontermination and computational irrelevance (a.k.a. erasure), a combination which has not been studied before. The two features interact: to protect type safety we must be careful to only erase terminating expressions. Our language design is strongly influenced by the choice of CBV evaluation, and by our novel treatment of propositional equality which has a heterogeneous, completely erased elimination form