Expressiveness and Model of the Polymorphic λ Calculus

Expressiveness and
Model of the
Polymorphic λ Calculus
Shin-Cheng Mu
IIS, Sinica
Friday, April 11, 14

Motivation
• V. Vene in APLAS 2004 [GUV04] gave a new
proof of shortcut deforestation because it was
only proved “informally by theorem-for free.”
• But what’s wrong with theorem for free?
• So I studied polymorphism and realised that
there is a lot I did not know...

Parametricity, or the
“theorems for free.”
• Informally known by functional programmers
as the “theorems for free.”
• A polymorphic function’s type induces a
“theorem”.
• E.g. hd . map f = f . hd, for all hd:: [a]→a,
regradless of its actual deﬁnition.
• The word “theorem” may be mis-leading,
however.
[Wad89]

In this talk
• Review some members of the λ calculus
family:
• Untyped λ calculus: λx . x
• Simply-typed λ calculus: λx:Int . x
• Polymorphic λ calculus: Λα.(λx:α . x),
Δα.α--->τ, f [a] x
• Consider their termination property,
expressiveness, and model.

Untyped λ calculus
• Not always terminating: (λx.(x x))(λx.(x x))
• Recursion: Y f = (λx . f (x x))(λx . f (x x))
• Equivalent to Turing machine (Church &
Kleene).
• Denotational model not trivial: requires Scott
domains.

Simply-typed λ calculus
• Strongly normalisable: all well-typed
expression terminates. Y combinator cannot
be typed.
• Computes total functions only!
• Has a simple model of total functions and
sets.
• Apparently cannot deﬁne some useful
functions. But exactly how expressive?

The computability hierarchy
partial recursive (functions)
total recursive
primitive recursive
elementry recursive

Primitive recursion
• A function that can be implemented using
only for-loops (math world). It always
terminate.
• Zero: 0, succ: (1+), projection: (x1,...,xn)= xi.
• Composition: f o g.
• Primitive recursion:
h(0,x) = f(x)
h(n+1,x) = g (h(n),n,x)

Primitive recursion
• A function that can be implemented using
only for-loops (math world). It always
terminate.
• Zero: 0, succ: (1+), projection: (x1,...,xn)= xi.
• Composition: f o g.
• Primitive recursion:
h(0,x) = f(x)
h(n+1,x) = g (h(n),n,x)
h(0) = c
h(n+1) = g (h(n),n)
Familiar? It’s paramorphism! [Mee90]

Partial and total recursion
• Partial recursive functions: primitive
recursion plus unbounded search:
μf z = least x such that f(x,z) = 0.
• The search may not terminate, thus
introduces the partiality.
• Partial rec. fn. is equiv. to Turing machine.
• Total recursive functions: the subset of parital
rec. fn. that is total. It’s bigger than primitive
recursive functions! (eg. Ackermann’s
function.)

Elementary recursion
• Also called Kalmar elementary functions.　
Functions deﬁnable by 1, +, x, -., and
　SUM f (x,n) = Σn
i=0
f(x,i)
　PRO f (x,n) = Πn
i=0
f(x,i)
• Deﬁnable functions include: mod, div,
isPrime, etc.

Representing natural num.
• Rep. of n over domain τ:
λf:τ--->τ . λx:τ . f (... (f x)) with n occur. of f.
• If we allow different instantiation of τ, we
can deﬁne addition, multiplication, exp(Ii+1-->
Ii---> Ii), predecessor(Ii+3---> Ii)... but not
subtraction.
• Therefore simply-typed calculus is weaker
than elementary functions. Moreover, its
value is bound by elementary fns.

• Use the type Δα.(α--->α)--->(α--->α) to
represent natural numbers.
• addition, multiplication, subtraction,pairs,..
• primrec = Δα.λg:N--->α--->α.λc:α.
λn:N.snd(n[Nxα]
(λz:Nxα.<1+fst z,g (fst z) (snd z)>) <0,c>)
Polymorphism enhances
expressiveness

• Use the type Δα.(α--->α)--->(α--->α) to
represent natural numbers.
• addition, multiplication, subtraction,pairs,..
• primrec = Δα.λg:N--->α--->α.λc:α.
λn:N.snd(n[Nxα]
(λz:Nxα.<1+fst z,g (fst z) (snd z)>) <0,c>)
Polymorphism enhances
expressiveness
primrec = Δα.λg.λc.
λn.snd
(n (λz.<1+fst z,g (fst z) (snd z)>) <0,c>)

Ackermann’s function!
Deﬁne ack
without recursion?
[Rey85]
ack 0 n = n+1
ack (m+1) 0 = ack m 1
ack (m+1) (n+1) = ack m (ack (m+1) n)
• Guess: ack = λm. m aug (1+)
therefore ack (m+1) = aug (ack m)
• ack (m+1) (n+1) = aug (ack m) (n+1) =
ack m (aug (ack m) n) = ack m (ack (m+1) n)
• We want aug f 0 = f 1 and
aug f (n+1) = f (aug f n)
• Solution: aug = λf. λn. (n+1) f 1

Simulating data structures
• Let list a = Δβ.(a--->β--->β) ---> β ---> β
nil = Λβ.λf.λa.a
cons x xs = Λβ.λf.λa. f x (xs f a)
• append xs ys = xs cons ys
• append = λxs:list a. λys:list a .
xs[list a] (λx:a.λzs:list a . cons x zs) ys
[Rey85]

Simulating data structures
• Let list a = Δβ.(a--->β--->β) ---> β ---> β
nil = Λβ.λf.λa.a
cons x xs = Λβ.λf.λa. f x (xs f a)
• append xs ys = xs cons ys
• append = λxs:list a. λys:list a .
xs[list a] (λx:a.λzs:list a . cons x zs) ys
! nil = Λβ.λf:a--->β--->β.λa:a.a
cons x xs = Λβ.λf:a--->β--->β.λa:a. f x (xs f a)
[Rey85]

Where does it stand?
partial recursive (functions)
total recursive
primitive recursive
elementry recursive
Polymorphic λ
Simply-typed λ

Fundamental sequence
• Deﬁne f0(x) = x+1
• fn+1(x)= fx
n(x) = fn(fn...(fn(x))..)... x
applications.
• f1(x)=2x, f2(x)=2x
×x
• fΘ(x)=fΘ[x](x) for limit ordinal Θ.
• Limit ordinals: ω is the size of natural
numbers, ε0 is the limit of ω, ωω
, ωωω
.....

Hierarchy characterised by
rate of growth
• Running time of elementary functions are
bounded by fn
2(x) for some n.
• Time of primitive recursive functions are
bounded by fn(x) for some n< ω.
• Ackermann’s function is essentially fω(x).
• fε0(x) is representable in polymorphic λ
calculus!
• It represents exactly the functions provably
recursive in 2nd-order arithmetic -- a much
larger class than fε0(x).
[FLD83]

Termination
• Still, every function deﬁned in poly-λ
terminates.
• However, the termination cannot be proved
in Peano arithmetic!
• Peano arithmetic: 0, (1+), addition,
induction... covering most techniques we
use in proofs of programs.
• Godel’s incompleteness theorem stated that
there are true theorems not provable in PA.
This was the ﬁrst “interesting” example.
[FLD83]

Model for poly. λ calculus?
• Untyped λ: not terminating, needs domain.
• Simple-typed λ: terminating, set-theoretic.
• λx:Int.x is the id for Int, λx:Char.x for Char.
• Poly. λ: can we avoid using domains?
• First try: a polymorphic function is a
collection of functions indexed by type.
• Λα.λx:α.x is a collection of identity fns.
• However, that would include some ad-hoc
functions.

Parametricity
• Reynolds restricts polymorphic objects to
parametric values:
• Let p: Δα.τ. It is parametric if for all set
assignments s1, s2, and r s1×s2:
〈p s1, p s2〉 [id|α:r]#
τ
• Which later became “theorem for free”.
• Reynolds [Rey83] believed that there is a set-
theoretic model for poly. λ where poly.
objects represent parametric values.
• He then falsiﬁed his conjecture [Rey84].
[Wad89]

No simple model!
• Bool can be represented by B=Δα.α--->α--->α
• Let Ts = (s--->B)--->B for all type s, and Tf =
λh.λg.h(g o f) (for all f: s--->t, Tf: Ts--->Tt).
• Let P=(((s--->B)--->B)--->s)--->s. We can construct a
h: TP--->P s.t. the diagram commutes for all f.
• In fact h is an initial algebra!
• But that would make TP isomorphic to P,
which is a contradiction (they have diff.
cardinalities unless |B|=1).
h
P TP
s Ts
Tgg
f
[Rey84]Polymor
phism is not
set-theoretic.
f. g.

Later models
• Using topos [Pit87]Poly. is set-theoretic, constructively.
• Frame [BM84][MM85][Wad89].
• Functorial approach [BFS90].
• Operational aspects [Pit00].
• Used to prove short-cut fusion [Joh03].

What’s the use of
parametricity?
• It’s still a key concept! Recall representation
of lists: llist a = Δβ.(a--->β--->β) ---> β ---> β.
• In general, the least fixed-point of functor F
(or the inital F-algebra) can be represented by
Δβ.(F β--->β) ---> β... iff parametricity holds!
• Types defined as least-fixed-points are called
inductive.
• nil = Λβ.λf.λa.a, cons x xs = Λβ.λf.λa. f x
(xs f a), x = cons 1 (cons 2 (cons 3 nil).
[Has94]

Inductive and coinductive
datatypes
• The greatest ﬁxed-point of functor F can be
represented by ∃x.(x → F x, x), iff
parametricity holds. It’s called coinductive.
• from n = (λm.Right (m,m+1), n) :: glist a
• When least and greatest ﬁxed-points
coincide (i.e. exists a force:: glist a →llist a),
we can do hylomorphism.
[Has94]
[How96]

Pointed types
• In a model where parametricity and
extensionality holds, the following are
equivalent:
• Inductive and coinductive types coincide;
• Exists a ﬁxed-point operator for values;
• Exists a ﬁxed-point operator for types.
[Has94] ?

Conclusion
Termination Expressiveness Model
Untyped NO = Turing
machine
domain
Simply typed YES < elementry fn. set &
functio
n
Polymorphic
YES but not
provable in
PA
> Ackermann
domain or
non-trivial
sets

What I learnt from this history
study...
• Adding polymorphism to a language strongly
enhances its power.
• The concept of fold, etc., ﬁnds its root in very
fundamental research.
• Category theory has played an important
role in early stage of computing science.
• Parametricity is an important assumption
leading to many useful properties.

References
• [FLD83] S.Fortune, D. Leivant, M. O’Donnell, The expressiveness of
simple and second-order type structures. In Journal of the ACM, 30(1), pp
151-185.
• [Rey83] J.C. Reynolds. Types, abstractions and parametric polymorphism.
In Information Processing 83, pp 513-523.
• [BM84]K.B. Bruce, A.R. Meyer, The semantics of second-order
polymorphic lambda calculus. In Semantics of Data Types, LNCS 173.
• [Rey84] J.C. Reynolds, Polymorphism is not set theoretic. In Semantics of
data Types, LNCS 173, pp 145-156.
• [MM85] J.C. Mitchell, A.R. Meyer, Second-order logical relations. In
Logics of Programs, LNCS 193.
• [Rey85] J.C. Reynolds, Three approaches to type structure. In
Mathematical Foundations of Software Development, LNCS 185.

References
• [Pit87] A.M. Pitts, Polymorphism is set theoretic, constructively. In
Category Theory and Computer Science, LNCS 283, pp 12-39.
• [Wad89] P. Wadler, Theorems for free! In Int. Sym. on Functional
Programming Languages and Computer Architecture, ‘89.
• [BFS90] E.S. Bainbridge, P.J. Freyd, A. Scedrov, P.J. Scott, Functorial
polymorphism. In Theoretical computer science v.70, pp 36-54.
• [Mee90] L. Meertens, Paramorphisms. In Formal Aspects of Computing.
• [Has94] R. Hasegawa. Categorical data types in parametric
polymorphism. Math. Structures in Computer Science, March 1994.
• [How96] B.T. Howard. Inductive, coinductive, and pointed types. ICFP 96.
• [Pit00] A.M. Pitts, Parametric polymorphism and operational equivalence.
In Math. Struct. in Comp. Science v.10, pp 1-39.
• [Joh03] P. Johann, Short cut fusion is correct. In J. Functional Programming
v.13, pp 797-814.
• [GUV04]N. Ghani, T. Uustalu, V, Vene, Build, augment and destory. In
APLAS 2004, LNCS 3002, pp 327-347.

Expressiveness and Model of the Polymorphic λ Calculus

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