A simultaneous generalization of independence and disjointness in boolean algebras

We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these algebras, which we call n-independent. The properties of these classes (n-free and omega-free boolean algebras) are investigated. These include connections to hypergraph theory and cardinal invariants on these algebras. Related cardinal functions, $n$Ind, which is the supremum of the cardinalities of n-independent subsets; i_n, the minimum size of a maximal n-independent subset; and i_omega, the minimum size of an omega-independent subset, are introduced and investigated. The values of i_n and i_omega on P(omega)/fin are shown to be independent of ZFC.Comment: Sumbitted to Orde

Type Directed Specification Refinement

Specification languages serve a fundamentally different purpose than general-purpose programming languages, and their type systems reflect these needs. Specification type systems must record and track more information for us to reason about a system adequately, and this added expressiveness may lead to an undecidable typing analysis. System level design begins with a high-level specification that is continually refined and expanded with implementation details, constraints, and typing information, down to a concrete specification. During this refinement process, the system is underspecified, and many static analyses aren't applicable until the system is fully specified. However, partial specifications contain valuable information that can inform the refinement process--we can locally inspect parts of the specification from a typing perspective to look for inferrable information or inconsistencies early on to aid the refinement process. This work defines a typing analysis that gathers constraints and typing information to inform the specification refinement process. It explores localized techniques such as local type inference and tracking of values as a means of influencing the specification refinement process