Presenting Distributive Laws

Abstract

Abstract. Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages.

Combining Bialgebraic Semantics and Equations

“…Second, in the current work we only consider free monads. One can possibly incorporate equations which already hold, by using the theory of [5]. Finally, it is worthwhile to consider categorical generalisations to allow, e.g., to study structural congruences for calculi with names.…”

“…Their Example 32 discusses the definition of the parallel operator as in (2) above, but a general theory for structural congruence is missing. In [5] it is shown how to obtain a distributive law for a monad that is the quotient of another one by imposing extra equations, under the condition that the distributive law respects the equations. However, this condition requires that the equations already hold semantically, which is fundamentally different from the present paper where we define behaviour by imposing equations on an operational specification.…”

Structural congruence for bialgebraic semantics

“…Second, in the current work we only consider free monads. One may incorporate equations which already hold, for instance by using the theory of [6].…”

“…The left square and lower left triangle commute by definition of τ * (Equation (6) in Section 2.6), and the lower right triangle by a unit law of the monad. Thus (F µ ∅ × µ ∅ ) • τ * T ∅ is an algebra homomorphism extending id, and since β is by definition an algebra homomorphism extending id and homomorphic extensions are unique, we have β = (F µ ∅ × µ ∅ ) • τ * T ∅ .…”

“…In [6], it is shown how to obtain a distributive law for a monad that is the quotient of another one by imposing extra equations, under the condition that the distributive law respects the equations. However, this condition requires that the equations already hold semantically, which is fundamentally different from the present paper where we define behaviour by imposing equations on an operational specification.…”