state monad



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Modalities, Closure and Reflection



The state monad is a monad (in computer science) used to implement computational side-effects in functional programming.

A functional program with input of type XX, output of type YY and mutable state type WW is a function (morphism) of type X×WY×WX \times W \longrightarrow Y \times W.

Under the (Cartesian product \dashv internal hom)-adjunction this is equivalently given by its adjunct, which is a function of type

X[W,W×Y]. X \longrightarrow [W, W \times Y ] \,.

Here the operation [W,W×()][W, W\times (-)] is the monad on the type system which is induced by the above adjunction; and this latter function is naturally regarded as a morphism in the Kleisli category of this monad.

In the context of monads in computer science, this monad [W,W×()]:HH[W, W\times (-)] \colon \mathbf{H} \to \mathbf{H} is called the state monad for mutable states of type WW.


Realization in dependent type theory

In a locally Cartesian closed category/dependent type theory H\mathbf{H}, then to every type WW is associated its base change adjoint triple

H /W WW * WH. \mathbf{H}_{/W} \stackrel{\overset{\sum_W}{\longrightarrow}}{\stackrel{\overset{W^\ast}{\longleftarrow}}{\underset{\prod_W}{\longrightarrow}}} \mathbf{H} \,.

In terms of this the state monad is the composite

State= WW * WW * State = \prod_W W^\ast \sum_W W^\ast

of context extension followed by dependent sum, followed by context extension, followed by dependent product.

Here WW *=[W,]\prod_W W^\ast = [W,-] is called the function monad or reader monad and WW *=W×()\sum_W W^\ast = W \times (-) is the writer comonad.

Last revised on March 10, 2019 at 23:32:06. See the history of this page for a list of all contributions to it.