Notions of Computation Determine Monads

Abstract

We model notions of computation using algebraic operations and equations. We show that these generate several of the monads of primary interest that have been used to model computational effects, with the striking omission of the continuations monad. We focus on semantics for global and local state, showing that taking operations and equations as primitive yields a mathematical relationship that reflects their computational relationship.

Combining Computational Effects: Commutativity and Sum

“…Modulo a few side conditions, the answer is yes; we make fundamental use of the correspondence between strong monads and a generalised notion of Lawvere theory in order to provide it (23]. That correspondence is computationally natural and is already implicit in our previous work on computational effects (20,21,22]. We are unaware of any direct justification for the existence of T ® T'.…”

“…Here, we are following an algebraic programme that shifts focus away from monads to the study of natural operations that yield the required effects (see (20,21] for other recent work along these lines), with the monads then corresponding to natural theories for these operations (22]. For instance, rather than emphasise the side-effects monad (S x -) 8 , we emphasise the operations lookup and update associated with side-effects, and the equations that relate them (22].…”

“…For instance, rather than emphasise the side-effects monad (S x -) 8 , we emphasise the operations lookup and update associated with side-effects, and the equations that relate them (22]. In the case where S = vLoc, lookup can be considered as a Loc-indexed family of V-ary operations, and update as a Loc x V-indexed family; the idea is that lookup,(x) proceeds with xv if the contents of lis v and update{l,v)(Y) proceeds with y, having updated l with v. Again, rather than emphasise the powerdomain P, we emphasise the operation of nondeterministic choice V with its equations for associativity, symmetry, and idempotence (7,19].…”

Towards a Coalgebraic Chomsky Hierarchy

“…Analogously, we show that certain (nondeterministic) transformations of the pushdown store form the underlying effect of pushdown automata, etc. By instantiating the operational analysis of computational effects from [24] to our setting we arrive at syntactic fixpoint expressions representing T-automata and prove a Kleenestyle theorem for them, thus generalizing previous work of the third author [34]. A crucial ingredient of our framework is the generalized powerset construction [35], which serves as a coalgebraic counterpart of classical Rabin-Scott determinization algorithm [26].…”

“…These not only formally generalize various computation models and tools, but also identify core principles and reasoning patterns behind them. An example of this kind is given by the notion of computational monad [22], which made an impact both on the theory of programming (as an organization tool for denotational semantics [8,24]) and on the practice (e.g. being implemented as a programming language feature of Haskell [1] and F# [36]).…”