Kappa calculus

In mathematical logic, category theory, and

computer science, kappa calculus is a

formal system for defining first-order

functions.

Unlike lambda calculus, kappa calculus has no

higher-order functions; its functions are

not first class objects. Kappa-calculus can be

regarded as "a reformulation of the first-order fragment of typed

lambda calculus".

Because its functions are not first-class objects, evaluation of kappa

calculus expressions does not require

closures.

Definition

The definition below has been adapted from the diagrams on pages 205 and 207 of Hasegawa.

= Grammar =

Kappa calculus consists of types and expressions, given by the

grammar below:

: $\tau =
1 \mid
\tau\times\tau \mid
\ldots$

: $e =
x \mid
id_\tau \mid
!_\tau \mid
\operatorname{lift}_\tau(e) \mid
e \circ e \mid
\kappa x:1{\to}\tau . e$

In other words,

1 is a type
If $\tau_1$ and $\tau_2$ are types then $\tau_1\times\tau_2$ is a type.
Every variable is an expression
If {{mvar|τ}} is a type then $id_\tau$ is an expression
If {{mvar|τ}} is a type then $!_\tau$ is an expression
If {{mvar|τ}} is a type and e is an expression then $\operatorname{lift}_\tau(e)$ is an expression
If $e_1$ and $e_2$ are expressions then $e_1\circ e_2$ is an expression
If x is a variable, {{mvar|τ}} is a type, and e is an expression, then $\kappa x{:}1{\to}\tau\;.\;e$ is an expression

The $:1{\to}\tau$ and the subscripts of {{math|id}}, {{math|!}}, and $\operatorname{lift}$ are

sometimes omitted when they can be unambiguously determined from the

context.

Juxtaposition is often used as an abbreviation for a combination of

$\operatorname{lift}$ and composition:

: $e_1 e_2\ \overset\operatorname{def}{=}\ e_1 \circ \operatorname{lift}(e_2)$

= Typing rules =

The presentation here uses sequents ( $\Gamma\vdash e:\tau$ ) rather than hypothetical judgments in order to ease comparison with the simply typed lambda calculus. This requires the additional Var rule, which does not appear in Hasegawa

In kappa calculus an expression has two types: the type of its source and the type of its target. The notation $e:\tau_1{\to}\tau_2$ is used to indicate that expression e has source type ${\tau_1}$ and target type ${\tau_2}$ .

Expressions in kappa calculus are assigned types according to the following rules:

cellpadding="9" style="text-align:center;" \| ${x{:}1{\to}\tau\;\in\;\Gamma}\over{\Gamma \vdash x : 1{\to}\tau }$	(Var)
${}\over{\vdash id_\tau\;:\;\tau\to\tau }$	(Id)
${}\over{\vdash !_\tau\;:\;\tau\to 1 }$	(Bang)
${\Gamma \vdash e_1{:}\tau_1{\to}\tau_2 \;\;\;\;\;\; \Gamma \vdash e_2{:}\tau_2{\to}\tau_3 }\over{\Gamma \vdash e_2\circ e_1 : \tau_1{\to}\tau_3 }$	(Comp)
${\Gamma \vdash e{:}1{\to}\tau_1} \over {\Gamma \vdash \operatorname{lift}_{\tau_2}(e)\;:\;\tau_2\to(\tau_1\times\tau_2) }$ \| (Lift)
${\Gamma,\;x{:}1{\to}\tau_1\;\vdash\;e:\tau_2{\to}\tau_3} \over {\Gamma \vdash \kappa x{:}1{\to}\tau_1\,.\,e\;:\;\tau_1\times\tau_2\to\tau_3 }$ \|(Kappa)

In other words,

Var: assuming $x:1{\to}\tau$ lets you conclude that $x:1{\to}\tau$
Id: for any type {{mvar|τ}}, $id_\tau:\tau{\to}\tau$
Bang: for any type {{mvar|τ}}, $!_\tau:\tau{\to}1$
Comp: if the target type of $e_1$ matches the source type of $e_2$ they may be composed to form an expression $e_2\circ e_1$ with the source type of $e_1$ and target type of $e_2$
Lift: if $e:1{\to}\tau_1$ , then $\operatorname{lift}_{\tau_2}(e):\tau_2{\to}(\tau_1\times\tau_2)$
Kappa: if we can conclude that $e:\tau_2\to\tau_3$ under the assumption that $x:1{\to}\tau_1$ , then we may conclude without that assumption that $\kappa x{:}1{\to}\tau_1\,.\,e\;:\;\tau_1\times\tau_2\to\tau_3$

= Equalities =

Kappa calculus obeys the following equalities:

Neutrality: If $f:\tau_1{\to}\tau_2$ then $f{\circ}id_{\tau_1}=f$ and $f=id_{\tau_2}{\circ}f$
Associativity: If $f:\tau_1{\to}\tau_2$ , $g:\tau_2{\to}\tau_3$ , and $h:\tau_3{\to}\tau_4$ , then $(h{\circ}g){\circ}f = h{\circ}(g{\circ}f)$ .
Terminality: If $f{:}\tau{\to}1$ and $g{:}\tau{\to}1$ then $f=g$
Lift-Reduction: $(\kappa x.f)\circ \operatorname{lift}_\tau(c) = f[c/x]$
Kappa-Reduction: $\kappa x. (h\circ \operatorname{lift}_\tau(x)) = h$ if x is not free in h

The last two equalities are reduction rules for the calculus,

rewriting from left to right.

Properties

The type {{val|1}} can be regarded as the unit type. Because of this, any two functions whose argument type is the same and whose result type is {{val|1}} should be equal – since there is only a single value of type {{val|1}} both functions must return that value for every argument (Terminality).

Expressions with type $1{\to}\tau$ can be regarded as "constants" or values of "ground type"; this is because {{val|1}} is the unit type, and so a function from this type is necessarily a constant function. Note that the kappa rule allows abstractions only when the variable being abstracted has the type $1{\to}\tau$ for some {{mvar|τ}}. This is the basic mechanism which ensures that all functions are first-order.

Categorical semantics

Kappa calculus is intended to be the internal language of

contextually complete categories.

Examples

Expressions with multiple arguments have source types which are

"right-imbalanced" binary trees. For example, a function f with three

arguments of types A, B, and C and result type D will have type

: $f : A\times (B\times (C\times 1)) \to D$

If we define left-associative juxtaposition $f\;c$ as an abbreviation

for $(f\circ \operatorname{lift}(c))$ , then – assuming that

$a:1{\to}A$ , $b:1{\to}B$ , and

$c:1{\to}C$ – we can apply this function:

: $f\;a\;b\;c\;:\;1 \to D$

Since the expression $f\;a\;b\;c$ has source type {{val|1}}, it is a "ground value" and may be passed as an argument to another function. If $g:(D\times E){\to}F$ , then

: $g\;(f\;a\;b\;c)\;:\;E \to F$

Much like a curried function of type

$A{\to}(B{\to}(C{\to}D))$ in lambda calculus, partial

application is possible:

: $f\;a\;:\;B\times (C\times 1) \to D$

However no higher types (i.e. $(\tau{\to}\tau){\to}\tau$ ) are involved. Note that because the source type of {{mvar|f a}} is not {{val|1}}, the following expression cannot be well-typed under the assumptions mentioned so far:

: $h\;(f\;a)$

Because successive application is used for multiple

arguments it is not necessary to know the arity of a function in

order to determine its typing; for example, if we know that

$c:1{\to}C$ then the expression

: {{mvar| j c }}

is well-typed as long as {{mvar|j}} has type

: $(C\times\alpha){\to}\beta$ for some {{mvar|α}}

and {{mvar|β}}. This property is important when calculating

the principal type of an expression, something

which can be difficult when attempting to exclude higher-order

functions from typed lambda calculi by restricting the grammar of types.

History

Barendregt originally introduced the term

"functional completeness" in the context of combinatory algebra.

Kappa calculus arose out of efforts by Lambek

name="Lambek"/> to formulate an appropriate analogue of functional

completeness for arbitrary categories (see Hermida and Jacobs,

name=HermidaJacobs/> section 1). Hasegawa later developed kappa

calculus into a usable (though simple) programming language including

arithmetic over natural numbers and primitive recursion.

name="Hasegawa"/> Connections to arrows

were later investigated by Power, Thielecke, and others.

Variants

It is possible to explore versions of kappa calculus with

substructural types such as linear, affine,

and ordered types. These extensions require eliminating or

restricting the $!_\tau$ expression. In such circumstances

the {{math|×}} type operator is not a true cartesian product,

and is generally written {{math|⊗}} to make this clear.

References

{{cite journal

| author-last = Lambek

| author-first = Joachim

| author-link = Joachim Lambek

| date = August 1, 1973

| title = Functional completeness of cartesian categories

| journal = Annals of Mathematical Logic

| publication-date = March 1974

| volume = 6

| issue = 3–4

| pages = 259–292

| doi = 10.1016/0003-4843(74)90003-5

| issn = 0003-4843

| doi-access =

}}

{{cite web |author=Adam |date=August 31, 2010 |title=What are κappa-categories? |website=MathOverflow |url=https://mathoverflow.net/q/37180}}

{{cite journal

| author1-last = Hermida

| author1-first = Claudio

| author2-last = Jacobs

| author2-first = Bart

| date = December 1995

| title = Fibrations with indeterminates: contextual and functional completeness for polymorphic lambda calculi

| journal = Mathematical Structures in Computer Science

| volume = 5

| issue = 4

| pages = 501–531

| doi = 10.1017/S0960129500001213

| s2cid = 3428512

| issn = 1469-8072| url = https://ir.cwi.nl/pub/1375

}}

{{cite book

| date = October 1, 1984

| editor-last = Barendregt

| editor-first = Hendrik Pieter

| title = The Lambda Calculus: Its Syntax and Semantics

| url = https://www.elsevier.com/books/the-lambda-calculus/barendregt/978-0-444-87508-2

| series = Studies in Logic and the Foundations of Mathematics

| edition = Revised

| location = Amsterdam, North Holland

| publisher = Elsevier Science

| volume = 103

| isbn = 978-0-444-87508-2}}

{{cite book

| author-last = Hasegawa

| author-first = Masahito

| title = Category Theory and Computer Science

| chapter = Decomposing typed lambda calculus into a couple of categorical programming languages

| year = 1995

| editor1-last = Pitt

| editor1-first = David

| editor2-last = Rydeheard

| editor2-first = David E.

| editor3-last = Johnstone

| editor3-first = Peter

| editor3-link = Peter Johnstone (mathematician)

| series = Lecture Notes in Computer Science

| publisher = Springer-Verlag Berlin Heidelberg

| volume = 953

| pages = 200–219

| doi = 10.1007/3-540-60164-3_28

| isbn = 978-3-540-60164-7

| issn = 0302-9743

| citeseerx = 10.1.1.53.715

}}

{{cite web |author=Adam |date=August 31, 2010 |title=What are κappa-categories? |website=MathOverflow |url=https://mathoverflow.net/q/37180}}

{{cite book

| author1-last = Power

| author1-first = John

| author2-last = Thielecke

| author2-first = Hayo

| title = Automata, Languages and Programming

| chapter = Closed Freyd- and κ-categories

| year = 1999

| editor1-last = Wiedermann

| editor1-first = Jiří

| editor2-last = van Emde Boas

| editor2-first = Peter

| editor2-link = Peter van Emde Boas

| editor3-last = Nielsen

| editor3-first = Mogens

| series = Lecture Notes in Computer Science

| publisher = Springer-Verlag Berlin Heidelberg

| volume = 1644

| pages = 625–634

| doi = 10.1007/3-540-48523-6_59

| isbn = 978-3-540-66224-2

| issn = 0302-9743| citeseerx = 10.1.1.42.2151

}}

Category:Logical calculi