nLab LF



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
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/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


Deduction and Induction

Constructivism, Realizability, Computability




LF (Pfenning 91, Harper-Honsell-Plotkin 93) is a certain logical framework based on dependent intuitionistic type theory. Variants include the Edinburgh logical framework, abbreviated “Elf”, and linear LF, abbreviated LLF (Pfenning 96), the latter capturing also dependent linear type theory.


The logical framework LF originates around

  • Frank Pfenning, Logic Programming in the LF Logical Framework (1991) (web)

  • Robert Harper, F. Honsell, G. Plotkin, A framework for defining logics, Journal for the association for computing machinery 40(1):143-184 (1993)

For Elf see

Extension to a framework for dependent linear type theory called LLF is discussed in

  • Iliano Cervesato, Frank Pfenning, A Linear Logical Framework, 1996, (web)

  • Kevin Watkins, Iliano Cervesato, Frank Pfenning, David Walker, A concurrent logical framework I: Judgments and properties, CMU technical report CMU-CS-02-101, revised May 2003 (web)

category: software

Last revised on May 2, 2014 at 04:19:19. See the history of this page for a list of all contributions to it.