Multisorted

Multisorted, also called many-sorted, describes systems that explicitly distinguish between multiple domains of elements, or sorts, rather than operating over a single carrier set. The use of multiple sorts helps model heterogeneous data and keep operations well typed.

In universal algebra, multisorted algebras generalize single-sorted algebras by associating to each sort s a carrier set A_s and by providing operations whose input and output sorts are specified. For example, a two-sorted algebra might have sorts Int and Bool with a predicate isZero: Int -> Bool and a function compare: Int × Int -> Int, among others. This framework allows operations to act on appropriate combinations of sorts.

In logic, many-sorted first-order logic extends standard first-order logic by including multiple domains. Variables are declared with sorts; quantifiers range over the domain of their sort; interpretations assign to each sort a nonempty domain, and symbols have declared input and output sorts. This approach clarifies typing constraints and can simplify reasoning about heterogeneous structures.

Applications of multisorted formalisms appear across formal specification, type systems, databases and knowledge representation, and areas such as term rewriting and constraint solving. They provide a natural way to model data and operations that inherently involve different types or domains, while preserving rigorous semantic and syntactic discipline.

Advantages include clearer typing, modular modeling, and the ability to represent diverse data within a unified framework. Challenges involve managing multiple domains, cross-sort operations, and the associated complexity in tasks such as unification, model checking, or automated reasoning.