We investigate the phenomenon that every monad is a linear state monad. We do this by studying a fully-complete state-passing translation from an impure call-by-value language to a new linear type theory: the enriched call-by-value calculus. The results are not specific to store, but can be applied to any computational effect expressible using algebraic operations, even to effects that are not ...

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical log...

The category theoretic structures of monads and comonads can be used as an abstraction mechanism for simplifying both language semantics and programs. Monads have been used to structure impure computations, whilst comonads have been used to structure context-dependent computations. Interestingly, the class of computations structured by monads and the class of computations structured by comonads...

Labelled Markov processes are continuous-state fully probabilistic labelled transition systems. They can be seen as co-algebras of a suitable monad on the category of measurable space. The theory as developed so far included a treatment of bisimulation, logical characterization of bisimulation, weak bisimulation, metrics, universal domains for LMPs and approximations. Much of the theory involve...

Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of Galois functors over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of a...

The paper investigates non-deterministic, probabilistic and quantum walks, from the perspective of coalgebras and monads. Nondeterministic and probabilistic walks are coalgebras of a monad (powerset and distribution), in an obvious manner. It is shown that also quantum walks are coalgebras of a new monad, involving additional control structure. This new monad is also used to describe Turing mac...

Birkhoff’s completeness theorem of equational logic asserts the coincidence of the model-theoretic and proof-theoretic consequence relations. Goguen and Meseguer, giving a sound and adequate system of inference rules for finitary many-sorted equational deduction, generalized the completeness theorem of Birkhoff to the completeness of finitary many-sorted equational logic and provided simultaneo...

CLF (Concurrent LF) [CPWW02a] is a logical framework for specifying and implementing deductive and concurrent systems from areas, such as programming language theory, security protocol analysis, process algebras, and logics. Celf is an implementation of the CLF type theory that extends the LF type theory by linear types to support representation of state and a monad to support representation of...

In this paper, we look at two categorical accounts of computational effects (strong monad as a model of the monadic metalanguage, adjunction as a model of call-bypush-value with stacks), and we adapt them to incorporate global exceptions. In each case, we extend the calculus with a construct, due to Benton and Kennedy, that fuses exception handling with sequencing. This immediately gives us an ...