nLab
modal type theory

Context

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 rulecompositionsubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive 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 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

homotopy levels

semantics

Modalities, Closure and Reflection

Contents

Idea

Modal type theory is a flavor of type theory with type formation rules for modalities, hence type theory which on propositions reduces to modal logic.

Following (Moggi91, Benton-Bierman-de Paiva 95, Kobayashi 97) modal type theory is specifically understood as being a type theory equipped with a monad on its type system, representing the intended modality. Since monads in computer science embody a notion of computation, some authors also speak also of computational type theory here (Benton-Bierman-de Paiva 95, Fairtlough-Mendler 02).

According to (Benton-Bierman-de Paiva 95, p. 1-2) this matches well with the default interpretation of (S4) modal logic as being about the modality TT of “possibility”:

The starting point for Moggi’s work is an explicit semantic distinction between computations and values. If AA is an object which interprets the values of a particular type, then T(A)T(A) is the object which models computation of that type AA. [...][...] For a wide variety of notions of computation, the unary operator T()T(-) turns out to have the categorical structure of a strong monad on an underlying cartesian closed category of values. [...][...] On a purely intuitive level and particularly if one thinks about non-termination, there is certainly something appealing about the idea that a computation of type AA represents the possibility of a value of type AA.

When the underlying type theory is homotopy type theory these modalities are a “higher” generalization of traditional modalities, with “higher” in the sense of higher category theory: they have categorical semantics in (∞,1)-categories given by (∞,1)-monads. See (Shulman 12) for remarks on such higher modalities.

Properties

Relation to monads

At least in many cases, modalities in type theory are identified with monads or comonads on the underlying type universe, or on the subuniverse of propositions.

See for instance (Benton-Bierman-de Paiva, section 3.2), (Kobayashi), (Gabbay-Nanevski, section 8), (Gaubault-Larrecq, Goubault, section 5.1), (Park-Harper, section 2.6), (Shulman) as examples of the first, and (Moggi, def. 4.7), (Awodey-Birkedal, section 4.2) as examples of the second.

Examples

Geometric modality – Grothendieck topology

As a special case of the modalities-are-monads relation, a Grothendieck topology on the site underlying a presheaf topos is equivalently encoded in the sheafification monad PSh(C)Sh(C)PSh(C)PSh(C) \to Sh(C) \hookrightarrow PSh(C) induced by the sheaf topos geometric embedding. More generally, any geometric subtopos is equivalently represented by a left-exact idempotent monad.

When restricted to act on (-1)-truncated objects (i.e. subterminal objects or more generally monomorphisms), this becomes a universal closure operator. When internalized as an operation on the subobject classifier, this becomes the corresponding Lawvere-Tierney operator. This modal perspective on sheafification was maybe first made explicit by Bill Lawvere:

A Grothendieck ‘topology’ appears most naturally as a modal operator of the nature ‘it is locally the case that’ (Lawvere).

More discussion along these lines is in (Goldblatt), where this kind of modality is called a geometric modality.

For higher toposes, it is no longer true in general that a subtopos is determined by its behavior on the (1)(-1)-truncated objects, but we can still regard the entire sheafification monad as a higher modality in the internal homotopy type theory.

Closure modality

The canonical monad on a local topos gives rise to a closure modality on propositions in its internal language, as described in (Awodey-Birkedal).

Cohesive and differential modality

By adding to homotopy type theory three (higher) modalities that encode discrete types and codiscrete types and hence implicitly a non-(co)-discrete notion of cohesion one obtained cohesive homotopy type theory. Adding furthermore modalities for infinitesimal (co)discreteness yields differential homotopy type theory.

References

The clear identification of modal operators on types with monads is due to

  • Eugenio Moggi, Notions of computation and monads. Information and Computation, 93(1), 1991. (pdf)

This was observed (independently) to systematically yield constructive modal logic in (see also at computational type theory)

  • P.N. Benton , G.M. Bierman , Valeria de Paiva, Computational Types from a Logical Perspective I (1995) (web)

and

  • M. Fairlough, Michael Mendler, Propositional lax logic, Information and computation 137(1):1-33 (1997)

and

  • Satoshi Kobayashi, Monad as modality, Theoretical Computer Science, Volume 175, Issue 1, 30 (1997), Pages 29–74

The modal systems “CL” and “PLL” in (Benton-Bierman-Paiva) and (Fairlough-Mendler), respectively, turn out to be equivalent to the geometric modality of Goldblatt. The system CS4 in (Kobayashi) yields a constructive version of S4 modal logic.

Explicit mentioning of type theory equipped with such a monad as modal type theory or computational type theory is in

  • Matt Fairtlough, Michael Mendler, On the Logical Content of Computational Type Theory: A Solution to Curry’s Problem, Types for Proofs and Programs Lecture Notes in Computer Science Volume 2277, 2002, pp 63-78

Discussion of modal operators explicitly in dependent type theory (and with a brief mentioning of the relation to monads) is in

A survey of the field of modal type theory is in the collections

  • M. Fairtlough, M. Mendler, Eugenio Moggi (eds.), Modalities in Type Theory, Mathematical Structures in Computer Science, Vol. 11, No. 4, (2001)

and

  • Valeria de Paiva, Rajeev Goré, Michael Mendler, Modalities in constructive logics and type theories, Special issue of the Journal of Logic and Computation, editorial pp. 439-446, Vol. 14, No. 4, Oxford University Press, (2004) (pdf)

and

  • Valeria de Paiva, Brigitte Pientka (eds.) Intuitionistic Modal Logic and Applications (IMLA 2008), . Inf. Comput. 209(12): 1435-1436 (2011) (web)

The historically first modal type theory, the interpretation of sheafification as a modality in the internal language of a Grothendieck topos goes back to the notion of Lawvere-Tierney operator

  • Bill Lawvere, Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)

reviewed in

  • Robert Goldblatt, Grothendieck topology as geometric modality, Mathematical Logic Quarterly, Volume 27, Issue 31-35, pages 495–529, (1981)

Modal type theory with an eye towards homotopy type theory is discussed in

Monadic modal type theory with idempotent monads/monadic reflection is discussed in

Formalization of modalities in homotopy type theory is discussed also around def. 1.11 of

See also

  • Frank Pfenning, Towards modal type theory (2000) (pdf)

  • Frank Pfenning, Intensionality, Extensionality, and Proof Irrelevance in Modal Type Theory, Pages 221–230 of: Symposium on Logic in Computer Science (2001) (web)

  • Giuseppe Primiero, A multi-modal dependent type theory (pdf)

  • Murdoch Gabbay, Aleksandar Nanevski, Denotation of contextual modal type theory (CMTT): syntax and metaprogramming (pdf)

A modality in the internal language of a local topos is discussed in section 4.2 of

  • Jean Goubault-Larrecq, Éric Goubault, On the geometry of intuitionistic S4 proofs, Homology, homotopy and applications vol 5(2) (2003)

A list of related references is also kept at

Revised on January 13, 2014 08:39:13 by Urs Schreiber (89.204.155.62)