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Algebraic systems of equations define functions using recursion
where parameter passing is permitted. This generalizes the
notion of a rational system of equations where parameter passing is
prohibited. It has been known for some time that algebraic systems
in Greibach Normal Form have unique solutions. This paper presents a categorical approach to algebraic systems of
equations which generalizes the traditional approach in two ways i)
we define algebraic equations for locally finitely presentable
...
We describe a sequent calculus μLJ with primitives for inductive and coinductive datatypes and equip it with reduction rules allowing a sound translation of Gödel’s system T. We introduce the notion of a μ-closed category, relying on a uniform interpretation of open μLJ formulas as strong functors. We show that any μ-closed category is a sound model for μLJ. We then turn to the construction of a concrete μ-closed category based on Hyland-Ong game semantics. The model relies on three main ingredients:...
For an arbitrary category, we consider the least class of functors containing the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by -terms. We call the category -bicomplete if every -term defines a functor. We provide concrete examples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved...
For an arbitrary category, we consider the least class of functors
containing the projections and closed under finite products, finite
coproducts, parameterized initial algebras and parameterized final
coalgebras, i.e. the class of functors that are definable by
μ-terms. We call the category μ-bicomplete if every μ-term
defines a functor. We provide concrete examples of such categories and
explicitly characterize this class of functors for the category of
sets and functions. This goal is achieved...
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