We associate with every commutative continuous semiring S and alphabet S a category whose objects are all sets and a morphism X → Y is determined by a function from X into the semiring of formal series St(Y ∪+ ∗).y of finite words over Y ∪+, an X × Y -matrix over St(Y ∪+∗).y, and a function from X into the continuous St(Y ∪+∗).y-semimodule St(Y ∪+)ωy of series of ω- words over Y ∪+. When S is also an ω-semiring (equipped with an infinite product operation), then we define a fixed point operation over our category and show that it satisfies all identities of iteration categories. We then use this fixed point operation to give semantics to recursion schemes defining series of finite and infinite words. In the particular case when the semiring is the Boolean semiring, we obtain the context-free languages of finite and !-words.
ASJC Scopus subject areas
- Theoretical Computer Science
- Computational Theory and Mathematics