Hollow Man has a direct-to-video prequel.
NOTOC. In mathematical logic and type theory , the λ-cube is a framework for exploring the axes of refinement in Coquand 's calculus of constructions , starting from the simply typed lambda calculus -LRB- written as λ → in the cube diagram to the right -RRB- as the vertex of a cube placed at the origin , and the calculus of constructions -LRB- higher-order dependently typed polymorphic lambda calculus ; written as λPω in the diagram -RRB- as its diametrically opposite vertex .. mathematical logic. mathematical logic. type theory. type theory. Coquand. Thierry Coquand. calculus of constructions. calculus of constructions. simply typed lambda calculus. simply typed lambda calculus. Each axis of the cube represents a new form of abstraction  :. Values depending on types , or polymorphism .. polymorphism. Polymorphism ( computer science ). System F , aka second order lambda calculus -LRB- written as λ2 in the diagram -RRB- , is obtained by imposing only this property .. System F. System F. Types depending on types , or type operators .. type operators. type constructor. Simply typed lambda calculus with type operators -LRB- λω in the diagram -RRB- is obtained by imposing only this property .. type operators. type constructor. Simply typed lambda calculus with type operators. Simply typed lambda calculus with type operators. Combined with System F it yields System Fω -LRB- written as λω without the underline in the diagram -RRB- .. System F. System F. Types depending on values , or dependent types .. dependent types. Dependent type theory. Imposing only this property yields λΠ -LRB- written as λP in the diagram -RRB- , a type system closely related to LF .. Dependent type#First order dependent type theory. LF ( logical framework ). All eight calculi include the most basic form of abstraction , values depending on values , ordinary functions as in the simply typed lambda calculus .. simply typed lambda calculus. simply typed lambda calculus. The richest calculus in the cube , with all three abstractions , is the calculus of constructions .. calculus of constructions. calculus of constructions. All eight calculi are strongly normalizing .. strongly normalizing. Normalization property ( lambda-calculus ). Subtyping however is not represented in the cube , even though systems like , known as higher-order bounded quantification , which combines subtyping and polymorphism are of practical interest , and can be further generalized to bounded type operators .. polymorphism. Polymorphism ( computer science ). type operators. type constructor. Subtyping. Subtyping. higher-order bounded quantification. higher-order bounded quantification. Further extensions to allow the definition of purely functional objects ; these systems were generally developed after the lambda cube paper was published .. purely functional objects. purely functional objects. The idea of the cube is due to the mathematician Henk Barendregt -LRB- 1991 -RRB- .. Henk Barendregt. Henk Barendregt. The framework of pure type systems generalizes the lambda cube in the sense that all corners of the cube , as well as many other systems can be represented as instances of this general framework .. This framework predates the lambda cube by a couple of years .. In his 1991 paper , Barendregt also defines the corners of the cube in this framework .. In his Habilitation a diriger les recherches Lambda-Prolog de A a Z. .. ou presque , Olivier Ridoux gives a cut-out template of the Lambda cube -LRB- p. 70 -RRB- and also a dual representation of the cube as an octahedron , where the 8 vertices are replaced by faces , as well as a dual representation as a dodecahedron , where the 12 edges are replaced by faces .