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Curry-Howard-Lambek correspondence

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[[Category:Theoretical foundations]]
 
[[Category:Theoretical foundations]]
The '''Curry-Howard isomorphism''' is an isomorphism between types (in programming languages) and propositions (in logic). Interestingly, the isomorphism maps programs (functions in Haskell) to (constructive) proofs (and ''vice versa''). (Note there is also a third part to this correspondance, sometimes called the '''Curry-Howard-Lambek''' correspondance, that shows an equivalance to Cartesian closed [[Category theory|categories]])
+
  +
{{Foundations infobox}}
  +
The '''Curry-Howard-Lambek correspondance''' is a three way isomorphism between types (in programming languages), propositions (in logic) and objects of a Cartesian closed [[Category theory|category]]. Interestingly, the isomorphism maps programs (functions in Haskell) to (constructive) proofs in logic (and ''vice versa'').
   
 
__TOC__
 
__TOC__
   
== The Answer ==
+
== Life, the Universe and Everything ==
 
As is well established by now,
 
As is well established by now,
 
<haskell>theAnswer :: Integer
 
<haskell>theAnswer :: Integer
 
theAnswer = 42</haskell>
 
theAnswer = 42</haskell>
The logical interpretation of the program is that the type <hask>Integer</hask> is inhibited (by the value <hask>42</hask>), so the existence of this program ''proves'' the proposition <hask>Integer</hask> (a type without any value is the "bottom" type, a proposition with no proof).
+
The logical interpretation of the program is that the type <hask>Integer</hask> is inhabited (by the value <hask>42</hask>), so the existence of this program ''proves'' the proposition <hask>Integer</hask> (a type without any value is the "bottom" type, a proposition with no proof).
   
 
== Inference ==
 
== Inference ==
A (non-trivial) Haskell function maps a value (of type <hask>a</hask>, say) to another value (of type <hask>b</hask>), therefore, ''given'' a value of type <hask>a</hask> (a proof of <hask>a</hask>), it ''constructs'' a value of type <hask>b</hask> (so the proof is ''transformed'' into a proof of <hask>b</hask>)! So <hask>b</hask> is inhibited if <hask>a</hask> is, and a proof of <hask>a -> b</hask> is established (hence the notation, in case you were wondering).
+
A (non-trivial) Haskell function maps a value (of type <hask>a</hask>, say) to another value (of type <hask>b</hask>), therefore, ''given'' a value of type <hask>a</hask> (a proof of <hask>a</hask>), it ''constructs'' a value of type <hask>b</hask> (so the proof is ''transformed'' into a proof of <hask>b</hask>)! So <hask>b</hask> is inhabited if <hask>a</hask> is, and a proof of <hask>a -> b</hask> is established (hence the notation, in case you were wondering).
<haskell>representation :: Bool -> Integer
+
  +
<haskell>
  +
representation :: Bool -> Integer
 
representation False = 0
 
representation False = 0
representation True = 1</haskell>
+
representation True = 1
says, for example, if <hask>Boolean</hask> is inhibited, so is <hask>Integer</hask> (well, the point here is demonstration, not discovery).
+
</haskell>
  +
  +
says, for example, if <hask>Boolean</hask> is inhabited, so is <hask>Integer</hask> (well, the point here is demonstration, not discovery).
   
 
== Connectives ==
 
== Connectives ==
 
Of course, atomic propositions contribute little towards knowledge, and the Haskell type system incorporates the logical connectives <math>\and</math> and <math>\or</math>, though heavily disguised.
 
Of course, atomic propositions contribute little towards knowledge, and the Haskell type system incorporates the logical connectives <math>\and</math> and <math>\or</math>, though heavily disguised.
Haskell handles <math>\or</math> conjuction in the manner described by Intuitionistic Logic. When a program has type <math>a \or b</math>, the value returned itself indicates which one. The algebraic data types in Haskell has a tag on each alternative, the constructor, to indicate the injections:
+
Haskell handles <math>\or</math> conjuction in the manner described by Intuitionistic Logic. When a program has type <math>A \or B</math>, the value returned itself indicates which one. The algebraic data types in Haskell has a tag on each alternative, the constructor, to indicate the injections:
 
<haskell>data Message a = OK a | Warning a | Error a
 
<haskell>data Message a = OK a | Warning a | Error a
  +
 
p2pShare :: Integer -> Message String
 
p2pShare :: Integer -> Message String
 
p2pShare n | n == 0 = Warning "Share! Freeloading hurts your peers."
 
p2pShare n | n == 0 = Warning "Share! Freeloading hurts your peers."
 
| n < 0 = Error "You cannot possibly share a negative number of files!"
 
| n < 0 = Error "You cannot possibly share a negative number of files!"
| n > 0 = OK ("You are sharing " ++ show n ++ " files."
+
| n > 0 = OK ("You are sharing " ++ show n ++ " files.")
 
</haskell>
 
</haskell>
 
So any one of <hask>OK String</hask>, <hask>Warning String</hask> or <hask>Error String</hask> proves the proposition <hask>Message String</hask>, leaving out any two constructors would not invalidate the program. At the same time, a proof of <hask>Message String</hask> can be pattern matched against the constructors to see which one it proves.
 
So any one of <hask>OK String</hask>, <hask>Warning String</hask> or <hask>Error String</hask> proves the proposition <hask>Message String</hask>, leaving out any two constructors would not invalidate the program. At the same time, a proof of <hask>Message String</hask> can be pattern matched against the constructors to see which one it proves.
On the other hand, to prove <hask>String</hask> is inhibited from the proposition <hask>Message String</hask>, it has to be proven that you can prove <hask>String</hask> from any of the alternatives...
+
On the other hand, to prove <hask>String</hask> is inhabited from the proposition <hask>Message String</hask>, it has to be proven that you can prove <hask>String</hask> from any of the alternatives...
 
<haskell>
 
<haskell>
 
show :: Message String -> String
 
show :: Message String -> String
Line 35: Line 36:
 
</haskell>
 
</haskell>
 
The <math>\and</math> conjuction is handled via an isomorphism in Closed Cartesian Categories in general (Haskell types belong to this category): <math>\mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y)</math>.
 
The <math>\and</math> conjuction is handled via an isomorphism in Closed Cartesian Categories in general (Haskell types belong to this category): <math>\mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y)</math>.
  +
That is, instead of a function from <math>X \times Y</math> to <math>Z</math>, we can have a function that takes an argument of type <math>X</math> and returns another function of type <math>Y \rightarrow Z</math>, that is, a function that takes <math>Y</math> to give (finally) a result of type <math>Z</math>: this technique is (known as currying) logically means <math>A \and B \rightarrow C \iff A \rightarrow (B \rightarrow C)</math>.
  +
  +
''(insert quasi-funny example here)''
  +
  +
So in Haskell, currying takes care of the <math>\and</math> connective. Logically, a proof of <math>A \and B</math> is a pair <math>(a, b)</math> of proofs of the propositions. In Haskell, to have the final <math>C</math> value, values of both <math>A</math> and <math>B</math> have to be supplied (in turn) to the (curried) function.
   
 
== Theorems for free! ==
 
== Theorems for free! ==
 
Things get interesting when polymorphism comes in. The composition operator in Haskell proves a very simple theorem.
 
Things get interesting when polymorphism comes in. The composition operator in Haskell proves a very simple theorem.
<haskell>(.) :: (a -> b) -> (b -> c) -> (a -> c)
+
(.) f g x = f (g x)</haskell>
+
<haskell>
  +
(.) :: (a -> b) -> (b -> c) -> (a -> c)
  +
(.) f g x = f (g x)
  +
</haskell>
  +
 
The type is, actually, <hask>forall a b c. (a -> b) -> (b -> c) -> (a -> c)</hask>, to be a bit verbose, which says, logically speaking, for all propositions <hask>a, b</hask> and <hask>c</hask>, if from <hask>a</hask>, <hask>b</hask> can be proven, and if from <hask>b</hask>, <hask>c</hask> can be proven, then from <hask>a</hask>, <hask>c</hask> can be proven (the program says how to go about proving: just compose the given proofs!)
 
The type is, actually, <hask>forall a b c. (a -> b) -> (b -> c) -> (a -> c)</hask>, to be a bit verbose, which says, logically speaking, for all propositions <hask>a, b</hask> and <hask>c</hask>, if from <hask>a</hask>, <hask>b</hask> can be proven, and if from <hask>b</hask>, <hask>c</hask> can be proven, then from <hask>a</hask>, <hask>c</hask> can be proven (the program says how to go about proving: just compose the given proofs!)
  +
  +
== Negation ==
  +
Of course, there's not much you can do with just truth. <hask>forall b. a -> b</hask> says that given <hask>a</hask>, we can infer anything. Therefore we will take <hask>forall b. a -> b</hask> as meaning <hask>not a</hask>. Given this, we can prove several more of the axioms of logic.
  +
  +
<haskell>
  +
type Not x = (forall a. x -> a)
  +
  +
doubleNegation :: x -> Not (Not x)
  +
doubleNegation k pr = pr k
  +
  +
contraPositive :: (a -> b) -> (Not b -> Not a)
  +
contraPositive fun denyb showa = denyb (fun showa)
  +
  +
deMorganI :: (Not a, Not b) -> Not (Either a b)
  +
deMorganI (na, _) (Left a) = na a
  +
deMorganI (_, nb) (Right b) = nb b
  +
  +
deMorganII :: Either (Not a) (Not b) -> Not (a,b)
  +
deMorganII (Left na) (a, _) = na a
  +
deMorganII (Right nb) (_, b) = nb b
  +
</haskell>
  +
  +
== Type classes ==
  +
A type class in Haskell is a proposition ''about'' a [[type]].
  +
  +
<haskell>
  +
class Eq a where
  +
(==) :: a -> a -> Bool
  +
(/=) :: a -> a -> Bool
  +
</haskell>
  +
  +
means, logically, there is a type <hask>a</hask> for which the type <hask>a -> a -> Bool</hask> is inhabited, or, from <hask>a</hask> it can be proved that <hask>a -> a -> Bool</hask> (the class promises two different proofs for this, having names <hask>==</hask> and <hask>/=</hask>).
  +
This proposition is of existential nature (not to be confused with [[existential type]]). A proof for this proposition (that there is a type that conforms to the specification) is (obviously) a set of proofs of the advertised proposition (an implementation), by an <hask>instance</hask>
  +
declaration:
  +
  +
<haskell>
  +
instance Eq Bool where
  +
True == True = True
  +
False == False = True
  +
_ == _ = False
  +
  +
(/=) a b = not (a == b)
  +
</haskell>
  +
  +
A not-so-efficient sort implementation would be:
  +
  +
<haskell>
  +
sort [] = []
  +
sort (x : xs) = sort lower ++ [x] ++ sort higher
  +
where (lower,higher) = partition (< x) xs
  +
</haskell>
  +
  +
Haskell infers its type to be <hask>forall a. (Ord a) => [a] -> [a]</hask>. It means, if a type <hask>a</hask> satisfies the proposition about propositions <hask>Ord</hask> (that is, has an ordering defined, as is necessary for comparison), then <hask>sort</hask> is a proof of <hask>[a] -> [a]</hask>. For this to work, somewhere, it should be proved (that is, the comparison functions defined) that <hask>Ord a</hask> is true.
  +
  +
== Multi-parameter type classes ==
  +
Haskell makes frequent use of multiparameter type classes. Type classes constitute a Prolog-like logic language, and multiparameter type classes define a relation between types.
  +
=== [[Functional dependencies]] ===
  +
These type level functions are set-theoretic. That is, <hask> class TypeClass a b | a -> b</hask> defines a relation between types <hask>a</hask> and <hask>b</hask>, and requires that there would not be different instances of <hask>TypeClass a b</hask> and <hask>TypeClass a c</hask> for different <hask>b</hask> and <hask>c</hask>, so that, essentially, <hask>b</hask> can be inferred as soon as <hask>a</hask> is known. This is precisely functions as relations as prescribed by set theory.
  +
  +
== Indexed types ==
  +
''(please someone complete this, should be quite interesting, I have no idea what it should look like logically)''

Latest revision as of 21:35, 21 February 2010


Haskell theoretical foundations

General:
Mathematics - Category theory
Research - Curry/Howard/Lambek

Lambda calculus:
Alpha conversion - Beta reduction
Eta conversion - Lambda abstraction

Other:
Recursion - Combinatory logic
Chaitin's construction - Turing machine
Relational algebra

The Curry-Howard-Lambek correspondance is a three way isomorphism between types (in programming languages), propositions (in logic) and objects of a Cartesian closed category. Interestingly, the isomorphism maps programs (functions in Haskell) to (constructive) proofs in logic (and vice versa).

Contents


[edit] 1 Life, the Universe and Everything

As is well established by now,

theAnswer :: Integer
theAnswer = 42
The logical interpretation of the program is that the type
Integer
is inhabited (by the value
42
), so the existence of this program proves the proposition
Integer
(a type without any value is the "bottom" type, a proposition with no proof).

[edit] 2 Inference

A (non-trivial) Haskell function maps a value (of type
a
, say) to another value (of type
b
), therefore, given a value of type
a
(a proof of
a
), it constructs a value of type
b
(so the proof is transformed into a proof of
b
)! So
b
is inhabited if
a
is, and a proof of
a -> b
is established (hence the notation, in case you were wondering).
representation :: Bool -> Integer
representation False = 0
representation True = 1
says, for example, if
Boolean
is inhabited, so is
Integer
(well, the point here is demonstration, not discovery).

[edit] 3 Connectives

Of course, atomic propositions contribute little towards knowledge, and the Haskell type system incorporates the logical connectives \and and \or, though heavily disguised. Haskell handles \or conjuction in the manner described by Intuitionistic Logic. When a program has type A \or B, the value returned itself indicates which one. The algebraic data types in Haskell has a tag on each alternative, the constructor, to indicate the injections:

data Message a = OK a | Warning a | Error a
 
p2pShare :: Integer -> Message String
p2pShare n | n == 0 = Warning "Share! Freeloading hurts your peers."
           | n < 0 = Error "You cannot possibly share a negative number of files!"
           | n > 0 = OK ("You are sharing " ++ show n ++ " files.")
So any one of
OK String
,
Warning String
or
Error String
proves the proposition
Message String
, leaving out any two constructors would not invalidate the program. At the same time, a proof of
Message String
can be pattern matched against the constructors to see which one it proves. On the other hand, to prove
String
is inhabited from the proposition
Message String
, it has to be proven that you can prove
String
from any of the alternatives...
show :: Message String -> String
show (OK s) = s
show (Warning s) = "Warning: " ++ s
show (Error s) = "ERROR! " ++ s

The \and conjuction is handled via an isomorphism in Closed Cartesian Categories in general (Haskell types belong to this category): \mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y). That is, instead of a function from X \times Y to Z, we can have a function that takes an argument of type X and returns another function of type Y \rightarrow Z, that is, a function that takes Y to give (finally) a result of type Z: this technique is (known as currying) logically means A \and B \rightarrow C \iff A \rightarrow (B \rightarrow C).

(insert quasi-funny example here)

So in Haskell, currying takes care of the \and connective. Logically, a proof of A \and B is a pair (a,b) of proofs of the propositions. In Haskell, to have the final C value, values of both A and B have to be supplied (in turn) to the (curried) function.

[edit] 4 Theorems for free!

Things get interesting when polymorphism comes in. The composition operator in Haskell proves a very simple theorem.

(.) :: (a -> b) -> (b -> c) -> (a -> c)
(.) f g x = f (g x)
The type is, actually,
forall a b c. (a -> b) -> (b -> c) -> (a -> c)
, to be a bit verbose, which says, logically speaking, for all propositions
a, b
and
c
, if from
a
,
b
can be proven, and if from
b
,
c
can be proven, then from
a
,
c
can be proven (the program says how to go about proving: just compose the given proofs!)

[edit] 5 Negation

Of course, there's not much you can do with just truth.
forall b. a -> b
says that given
a
, we can infer anything. Therefore we will take
forall b. a -> b
as meaning
not a
. Given this, we can prove several more of the axioms of logic.
type Not x = (forall a. x -> a)
 
doubleNegation :: x -> Not (Not x)
doubleNegation k pr = pr k
 
contraPositive :: (a -> b) -> (Not b -> Not a)
contraPositive fun denyb showa = denyb (fun showa)
 
deMorganI :: (Not a, Not b) -> Not (Either a b)
deMorganI (na, _) (Left a) = na a
deMorganI (_, nb) (Right b) = nb b
 
deMorganII :: Either (Not a) (Not b) -> Not (a,b)
deMorganII (Left na) (a, _) = na a
deMorganII (Right nb) (_, b) = nb b

[edit] 6 Type classes

A type class in Haskell is a proposition about a type.

class Eq a where
    (==) :: a -> a -> Bool
    (/=) :: a -> a -> Bool
means, logically, there is a type
a
for which the type
a -> a -> Bool
is inhabited, or, from
a
it can be proved that
a -> a -> Bool
(the class promises two different proofs for this, having names
==
and
/=
). This proposition is of existential nature (not to be confused with existential type). A proof for this proposition (that there is a type that conforms to the specification) is (obviously) a set of proofs of the advertised proposition (an implementation), by an
instance

declaration:

instance Eq Bool where
    True  == True  = True
    False == False = True
    _     == _     = False
 
(/=) a b = not (a == b)

A not-so-efficient sort implementation would be:

sort [] = []
sort (x : xs) = sort lower ++ [x] ++ sort higher
                     where (lower,higher) = partition (< x) xs
Haskell infers its type to be
forall a. (Ord a) => [a] -> [a]
. It means, if a type
a
satisfies the proposition about propositions
Ord
(that is, has an ordering defined, as is necessary for comparison), then
sort
is a proof of
[a] -> [a]
. For this to work, somewhere, it should be proved (that is, the comparison functions defined) that
Ord a
is true.

[edit] 7 Multi-parameter type classes

Haskell makes frequent use of multiparameter type classes. Type classes constitute a Prolog-like logic language, and multiparameter type classes define a relation between types.

[edit] 7.1 Functional dependencies

These type level functions are set-theoretic. That is,
 class TypeClass a b | a -> b
defines a relation between types
a
and
b
, and requires that there would not be different instances of
TypeClass a b
and
TypeClass a c
for different
b
and
c
, so that, essentially,
b
can be inferred as soon as
a
is known. This is precisely functions as relations as prescribed by set theory.

[edit] 8 Indexed types

(please someone complete this, should be quite interesting, I have no idea what it should look like logically)