Subset of lambda calculus
In mathematical logic, category theory, andcomputer science, kappa calculus is aformal system for defining first-orderfunctions.
Unlike lambda calculus, kappa calculus has nohigher-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".[1]
Because its functions are not first-class objects, evaluation of kappa
calculus expressions does not requireclosures.
Definition
The definition below has been adapted from the diagrams on pages 205 and 207 of Hasegawa.[1]
Grammar
Kappa calculus consists of types and expressions, given by the
grammar below:
In other words,
- 1 is a type
- If and are types then is a type.
- Every variable is an expression
- If τ is a type then is an expression
- If τ is a type then is an expression
- If τ is a type and e is an expression then is an expression
- If and are expressions then is an expression
- If x is a variable, τ is a type, and e is an expression, then is an expression
The and the subscripts of id, !, and are
sometimes omitted when they can be unambiguously determined from the
context.
Juxtaposition is often used as an abbreviation for a combination of and composition:
Typing rules
The presentation here uses sequents () 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[1]
In kappa calculus an expression has two types: the type of its source and the type of its target. The notation is used to indicate that expression e has source type and target type .
Expressions in kappa calculus are assigned types according to the following rules:
In other words,
- Var: assuming lets you conclude that
- Id: for any type τ,
- Bang: for any type τ,
- Comp: if the target type of matches the source type of they may be composed to form an expression with the source type of and target type of
- Lift: if , then
- Kappa: if we can conclude that under the assumption that , then we may conclude without that assumption that
Equalities
Kappa calculus obeys the following equalities:
- Neutrality: If then and
- Associativity: If , , and , then .
- Terminality: If and then
- Lift-Reduction:
- Kappa-Reduction: 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 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 1 should be equal – since there is only a single value of type 1 both functions must return that value for every argument (Terminality).
Expressions with type can be regarded as "constants" or values of "ground type"; this is because 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 for some 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 ofcontextually 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
If we define left-associative juxtaposition as an abbreviation
for , then – assuming that, , and – we can apply this function:
Since the expression has source type 1, it is a "ground value" and may be passed as an argument to another function. If , then
Much like a curried function of type in lambda calculus, partial
application is possible:
However no higher types (i.e. ) are involved. Note that because the source type of f a is not 1, the following expression cannot be well-typed under the assumptions mentioned so far:
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 then the expression
- j c
is well-typed as long as j has type
- for some α
and 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[2] the term
"functional completeness" in the context of combinatory algebra.
Kappa calculus arose out of efforts by Lambek[3] to formulate an appropriate analogue of functional
completeness for arbitrary categories (see Hermida and Jacobs,[4] section 1). Hasegawa later developed kappa
calculus into a usable (though simple) programming language including
arithmetic over natural numbers and primitive recursion.[1] Connections to arrowswere later investigated[5] by Power, Thielecke, and others.
Variants
It is possible to explore versions of kappa calculus withsubstructural types such as linear, affine,
and ordered types. These extensions require eliminating or
restricting the expression. In such circumstances
the × type operator is not a true cartesian product,
and is generally written ⊗ to make this clear.
References
- ^ a b c d Hasegawa, Masahito (1995). "Decomposing typed lambda calculus into a couple of categorical programming languages". In Pitt, David; Rydeheard, David E.; Johnstone, Peter (eds.). Category Theory and Computer Science. Lecture Notes in Computer Science. Vol. 953. Springer-Verlag Berlin Heidelberg. pp. 200–219. CiteSeerX 10.1.1.53.715. doi:10.1007/3-540-60164-3_28. ISBN 978-3-540-60164-7. ISSN 0302-9743.
- Adam (August 31, 2010). "What are κappa-categories?". MathOverflow.
- ^ Barendregt, Hendrik Pieter, ed. (October 1, 1984). The Lambda Calculus: Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics. Vol. 103 (Revised ed.). Amsterdam, North Holland: Elsevier Science. ISBN 978-0-444-87508-2.
- ^ Lambek, Joachim (August 1, 1973). "Functional completeness of cartesian categories". Annals of Mathematical Logic. 6 (3–4) (published March 1974): 259–292. doi:10.1016/0003-4843(74)90003-5. ISSN 0003-4843.
- Adam (August 31, 2010). "What are κappa-categories?". MathOverflow.
- ^ Hermida, Claudio; Jacobs, Bart (December 1995). "Fibrations with indeterminates: contextual and functional completeness for polymorphic lambda calculi". Mathematical Structures in Computer Science. 5 (4): 501–531. doi:10.1017/S0960129500001213. ISSN 1469-8072. S2CID 3428512.
- ^ Power, John; Thielecke, Hayo (1999). "Closed Freyd- and κ-categories". In Wiedermann, Jiří; van Emde Boas, Peter; Nielsen, Mogens (eds.). Automata, Languages and Programming. Lecture Notes in Computer Science. Vol. 1644. Springer-Verlag Berlin Heidelberg. pp. 625–634. CiteSeerX 10.1.1.42.2151. doi:10.1007/3-540-48523-6_59. ISBN 978-3-540-66224-2. ISSN 0302-9743.