🤖 AI Summary
Traditional coequational methods suffer from limitations in expressiveness and usability. This work proposes “equational path constraints” as an algebraic alternative: by assigning a pair of values to each path in a coalgebra and enforcing their equality, it algebraically characterizes finite-behavior properties, thereby enabling an axiomatic definition of covarieties and the construction of final coalgebras. The approach establishes a connection with coequations in the setting of monads and provides an upper bound on the number of colors required. Combining category theory, coalgebraic techniques, and Adámek–Barr final sequence constructions, the method is successfully validated across multiple case studies—including automata commutativity, differential equations, bi-infinite streams, and modal frame conditions—demonstrating its effectiveness and broad applicability.
📝 Abstract
Axiomatizing covarieties of coalgebras for an endofunctor is less intuitive than axiomatizing varieties of algebras via equations (Dahlqvist and Schmid, 2022). Existing techniques come from coalgebraic modal logic, pattern avoidance specifications, and hidden algebra. We introduce equational path constraints, a well-behaved and relatively easy to describe class of finitary behavioural properties that provide an algebra-flavoured alternative to coequations. The basic idea is to assign a pair of values to each path through a coalgebra and posit that the two values coincide. We show that equational path constraints define covarieties and construct final coalgebras relative to equational path constraints in some concrete cases. We connect equational path constraints to coequations when values computed from paths live in a monad, and we compute an upper bound on the number of colours needed to express the coequation. One of our constructions is reminiscent of the initial/terminal sequences of (Adámek, 1974) and (Barr, 1993). Motivating examples include commutativity conditions in automata theory, differential equations, bi-infinite streams, and frame conditions.