nLab exception monad

Contents

Context

Computation

Categorical algebra

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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

semantics

Contents

Idea

In computer science, an exception is a nominal failure of a program which however is handled in some manner, so that the failing computation still terminates in a controlled way.

In the simplest case this may mean that the program outputs an error message instead of continuing with the intended computation.

In terms of monads in computer science, the effect of (possibly) throwing an exception is modeled by the monad:

  1. which turns a given output data type into the coproduct type

    DDMsg D \;\mapsto\; D \sqcup Msg

    with the intended type of handling/messages (typically Msg=Msg = string);

  2. whose bind-operation on any prog:D 1D 2Msgprog \colon D_1 \to D_2 \sqcup Msg is given by the codiagonal map :MsgMsgMsg\nabla \;\colon\; Msg \sqcup Msg \;\to\; Msg as

    | | | | |prog[]D 2MsgprogId MsgD 3StringMsgid D 3D 3Msg, \phantom{\mathclap{\vert^{\vert^{\vert^{\vert^{\vert}}}}}} prog\big[-\big] \;\coloneqq\; D_2 \sqcup Msg \xrightarrow{\; prog \sqcup Id_{Msg} \;} D_3 \sqcup String \sqcup Msg \xrightarrow{\; id_{D_3} \sqcup \nabla \;} D_3 \sqcup Msg \,,

    meaning that whatever exception message has already been thrown gets carried further along.

If here Msg=*Msg = \ast is the unit type, then the exception monad is also known as the maybe monad, modelling the effect that a program may fail, without however transmitting any further information about the failure.

References

Exception handling as a monad in computer science:

Lecture notes:

See also:

Last revised on November 3, 2022 at 11:48:18. See the history of this page for a list of all contributions to it.