nLab
Horn 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 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
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, (idemponent) 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

(0,1)(0,1)-Category theory

Contents

Idea

A Horn theory is a theory in which every axiom has a certain special form.

Definition

Let Σ\Sigma be a signature. A term of Σ\Sigma is an expression built out of variables and function symbols. (For example, xy 1zx y^{-1} z is a term in the language of groups.) An atomic formula relative to Σ\Sigma is a formula that consists of a relation symbol of Σ\Sigma (including equality) applied to terms.

A Horn clause is a logical expression of the form

ϕ 1ϕ nϕ\phi_1 \wedge \ldots \wedge \phi_n \vdash \phi

where each ϕ i\phi_i and ϕ\phi is atomic. A (universal) Horn theory is a theory in which every axiom is a Horn clause.

Revised on August 25, 2012 21:44:45 by Urs Schreiber (82.113.106.150)