Definable functions in the simply typed lambda-calculus

0.13 MB·English

by  Mateusz Zakrzewski

Definable Functions in the Simply Typed
λ-Calculus
Mateusz Zakrzewski
Instituteof Informatics,
7 Warsaw University,
0 Banacha 2, 02-097 Warsaw, Poland.
0 [email protected]
2

n
a Abstract. It is a common knowledge that the integer functions defin-
J
able in the simply typed λ-calculus are exactly the extended polynomi-
4
als. This is indeed the case when one interprets integers over the type

(p → p) → p → p where p is a base type and/or equality is taken as
]
O β-conversion. It is commonly believed that the same holds for
βη-equality and for integers represented overany fixed typeof theform
L
(τ →τ)→τ →τ.
.
s Inthispaperweshowthat thisopinion isnotquitetrue.Weprovethat
c
theclass of functions strictly definablein thesimply typedλ-calculusis
[
considerablylargerthantheextendedpolynomials.Next,weinvestigate

1 which functionsbelong to that class.
v
2
2 1 Introduction
0
1
We assume that all types are constructed from only one base type, which we
0
7 denote by o. Natural numbers are represented as Church numerals: 0=λfx.x,
0 1=λfx.fx, 2=λfx.f(fx), 3=λfx.f(f(fx)), ...Everytype that can be assigned
/ to all Church numerals is of the form (τ →τ)→τ →τ, for some τ.
s
c We say that a function f over a free algebra A, of arity k, is definable
v: (or representable) in the untyped λ-calculus, if there exists a term E, such that
Xi Eρ(a1)ρ(a2)...ρ(ak)is equal(up to β- or βη-conversion)to ρ(f(a1,a2,...,ak))
for all a ,a ,...,a ∈A. By ρ(a) we mean the term which represents a (as de-
1 2 k
r fined in [4]) e.g. if A = N and a ∈ A then ρ(A) is a Church numeral. In typed
a
λ-calculi,thereareadditionalrequirementsthatmustbemetbyadefinablefunc-
tion. We say that a function is non-uniformly definable, if Eρ(a )ρ(a )...ρ(a )
1 2 k
can be typed for all a ,a ,...,a . A function is skewly definable, if there are
1 2 k
types τ ,τ ,...,τ , such that Eρ(a )ρ(a )...ρ(a ) can be typed for all a ,a ,
1 2 k 1 2 k 1 2
...,a , with type τ assigned to ρ(a ) (i = 1...k). If τ are all the same and
k i i i
equal to the type of Eρ(a )ρ(a )...ρ(a ), we say that E represents f strictly
1 2 k
(or f is strictly definable).
In the simply typed λ-calculus, functions over natural numbers that are
strictlydefinablewithτ beingabasetype(oratypevariable)havebeencharac-
terizedby Schwichtenberg(see [1]) and areknownas the extendedpolynomials.
The resultof Schwichtenberghas been generalizedto arbitraryfree algebras[5].
Leivanthas shownthat, insimple types,everynon-uniformly definable function
(over any algebra) is skewly definable. The exact characterizationof skewly de-
finable functions is not known. It has been shown (see [4]) that when equality
is taken as β-conversion, every function that is strictly definable is strictly de-
finable with τ being a base type. It was also claimed in [4] that the same holds
whenequalityistakenasβη-conversion.Surprisingly,asweshowbelow,itisnot
trueevenforfunctions overnaturalnumbers.Inotherwords,thereexiststrictly
definable functions that are not extended polynomials.
This paper aims to investigate which functions are strictly definable over
natural numbers, with arbitrary τ and βη-conversion. We define F as the class
ofstrictlydefinablefunctionsandGasaclassthatcontainsextendedpolynomials
and two additional functions (or, more precisely, two function schemas) and is
closed under composition. We prove that for every function f ∈ G, there exists
a type τ such that f is definable with numerals of type (τ → τ) → τ → τ. It
follows that G is a subset of F. The proof yields an effective procedure that can
be used to find τ and a term which represents f.
We conjecture that G exactly characterizes strictly definable functions, i.e.
G = F, and we gather some evidence for this conjecture proving, for example,
thateveryskewlyrepresentablefiniterangefunctionisstrictlyrepresentableover
(τ →τ) →τ →τ, for some τ. We also prove that all definable functions of the
form: f(m,n ,n ) = if m ∈ A then m else m (for all A ⊆ N) are in G.
1 2 1 2
Finally, we give examples of very simple functions that are not definable.
1.1 Extended Polynomials
Definition 1. The class of extended polynomials is the smallest class of func-
tions over N which contains:
1. the constant functions: 0 and 1,
2. projections,
3. addition,
4. multiplication,
5. the function ifzero(n,m,p)=if n=0 then m else p,
and is closed under composition.
Addition, multiplication and ifzero can be represented by the terms:
λnm.λfx.nf(mfx), λnm.λfx.n(mf)xandλnmp.λfx.n(λy.pfx)(mfx), respec-
tively. For allτ, every k-aryextended polynomialcanbe strictly representedby
a term of type ωk →ω where ω =(τ →τ)→τ →τ.
τ τ τ
1.2 Beyond Extended Polynomials
Consider the following term:
E =λn.λfx.
λa a ...a .
1 2 l
(n(λy.λz z ...z .yz z ...z z )(λz z ...z .z ))
1 2 l 2 3 l 1 1 2 l 1
(x a a ...a )
1 2 l
(fx a a ...a )
1 2 l
(f(fx) a a ...a )
1 2 l
...
((fl−1x) a a ...a ).
1 2 l
If n is assigned a numeral N, the term
(n(λy.λz z ...z .yz z ...z z )(λz z ...z .z ))
1 2 l 2 3 l 1 1 2 l 1
will evaluate (or reduce) to (λz z ...z .z ), where i is equal to (N modl)+1.
1 2 l i
Therefore,E strictlyrepresentsthefunctiong(n)=nmodlandcanbeassigned
the type ω →ω , where τ must be of the form αl →α, for some α.
τ τ
Proposition 2. For every extended polynomial f : N → N, there exists an m,
such that for all n ,n ≥m, if n ≥n then f(n )≥f(n ).
1 2 1 2 1 2
Proof. Simple induction with respect to the structure of a polynomial.
Proposition 2 states that every extended polynomial is non-decreasing for
sufficiently large arguments. Clearly, it is not the case for g. It follows that g is
not an extended polynomial. It is worth noting that for this example to work,
weneedbothβη-conversionandnumeralsoftypeω withτ beingdifferentfrom
τ
the base type.
2 Strictly Definable Functions
In this section, we define a class of functions and prove that all functions which
are in that class are (strictly) definable.
Definition 3. G is the smallestclass of functions which is closedunder compo-
sition and contains:
1. extended polynomials,
2. the function fl(m,n ,n ,...,n )=n , where i=(mmodl)+1,
1 1 2 l i
(for all l ≥2),
3. the function fl(m,n ,n )=if m≤l then n else n , (for all l≥1).
2 1 2 1 2
By F we denote the class of all functions that are strictly definable. Nota-
tion (s)hM ,M ,...,M i means a tuple of length s, represented by the term
1 2 s
λp.pM M ...M . By Π (s)(p), we denote the i-th element of the tuple p of
1 2 s i
length s, represented by p(λx x ...x .x ). The function fl can be represented
1 2 s i 2
by the term:
E =λmn n .λfx.λa a a .ifzero Gm then n fxa a a else n fxa a a ,
1 2 1 2 3 1 1 2 3 2 1 2 3
where:
ifzero x then y else z =x(λt.z)y,
G=λm.Π (l+1)(mFP),
1
F =λp.hΠ (l+1)(p),Π (l+1)(p),...,Π (l+1)(p),succ Π (l+1)(p)i,
2 3 l+1 l+1
P =(l+1)h0,0,...,0i,
succ=λn.λfx.f(nfx).
The term G skewly represents the function g(m) = m−l. Therefore ifzero
Gm then x else y returns the same value as if m≤l then x else y.
Here is how E can be typed:
⊢E :ω3 →ω ,
τ τ
τ =(αl+1 →α)→α,
α=ωτ′.
The type τ is the type of an (l+1)-tuple of numerals (hence α = ωτ′, for
some τ′). To prove that all functions which are in G are definable, we will need
more flexibility as to what types can be assigned to the terms which represent
them. For example, if (for any s≥l+1) the function g is represented like this:
G=λn.Π (s)(nFP),
s−l
F =λp.(s)hΠ (s)(p),Π (s)(p),...,Π (s)(p),succ Π (s)(p)i,
2 3 s s
P =(s)h0,0,...,0i.
then E can be assigned the type ω3 → ω with τ being the type of a tuple of
τ τ
length s. A flexible representationof fl is as follows:
1
E =λm.λn n ...n .λfx.λa.Π (s)(mFP),
1 2 l l
where:
F =λp.(s)hΠ (s)(p),Π (s)(p),...,Π(s)(p),Π (s)(p),Π (s)(p)
2 3 l 1 l+1
Π (s)(p),...,Π (s)(p)i,
l+2 s
P =(s)hM ,M ,...,M ,M ,X,X,...,Xi,
2 3 l 1
X =xa,
∀ M =n fxa.
1≤i≤l i i
The term E, when applied to arguments, evaluates similarly to the term
which represents the remainder modulo l (as given in the introduction). The
main difference is that we can now use tuples of arbitrary length s≥l. We can
assign the type ωl+1 →ω (with τ =(αs →α)→α) to E.
τ τ
It is easy to see that G can be equivalently defined as follows:
Definition 4. G is the smallestclass of functions which is closedunder compo-
sition and contains:
1. the constant functions: 0 and 1,
2. projections,
3. addition,
4. multiplication,
5. the function ifzero(n,m,p)=if n=0 then m else p,
6. the function fl(m,n ,n ,...,n ) = n , where i = (mmodl)+1, (for all
1 1 2 l i
l ≥2),
7. the function fl(m,n ,n )=if m≤l then n else n , (for all l≥1).
2 1 2 1 2
Definition 5. G is the smallest class of functions such that for all k:
1. if ∀nf(n)=0 then f ∈G,
2. if ∀nf(n)=1 then f ∈G,
3. ∀ if ∀nf(n)=n then f ∈G,
1≤i≤k i
4. if g ,g ∈G and f(n)=g (n)+g (n), then f ∈G,
1 2 1 2
5. if g ,g ∈G and f(n)=g (n)·g (n), then f ∈G,
1 2 1 2
6. if g,h ,h ∈ G and f(n) = if g(n) = 0 then h (n) else h (n), then
1 2 1 2
f ∈G,
7. ∀ if g ∈G,∀ h ∈G and f(n)=h (n),wherej =(g(n)mod
2≤l≤k−1 1≤i≤l i j
l)+1, then f ∈G,
8. ∀ if g,h ,h ∈G and f(n)=if g(n)≤l then h (n) else h (n), then
l≥1 1 2 1 2
f ∈G,
where n=n ,n ,...,n .
1 2 k
Lemma 6. Definition 5 is yet another equivalent definition of G:
Proof. Let G be the class specified by Definition 4 and let G be the class
1 2
specified by Definition 5. A simple induction on Definition 4 is enough to prove
that G ⊆G . It is easy to see that G ⊆G .
1 2 2 1
Theorem 7. For all k-ary functions f ∈ G and for almost all s, there exists
an E such that:
1. ∀n ,n ,...,n Eρ(n )ρ(n )...ρ(n )= ρ(f(n ,n ,...,n )),
1 2 k 1 2 k βη 1 2 k
2. ⊢E :ω k →ω ,
τ(s) τ(s)
where τ(s) = (αs → α) → α with α = ω (i.e. τ(s) is the type of an s-tuple of
o
numerals).
Proof. InductiononDefinition5.Herewethoroughlydiscussonecase(theother
cases are similar). Suppose f(n) = g (n)+ g (n), where g ,g ∈ G. By the
1 2 1 2
induction hypothesis:
For almost all s, there exists a G such that:
1
1. ∀n ,n ,...,n G ρ(n )ρ(n )...ρ(n )= ρ(g (n ,n ,...,n )),
1 2 k 1 1 2 k βη 1 1 2 k
2. ⊢G :ω k →ω .
1 τ(s) τ(s)
For almost all s, there exists a G such that:
2
1. ∀n ,n ,...,n G ρ(n )ρ(n )...ρ(n )= ρ(g (n ,n ,...,n )),
1 2 k 2 1 2 k βη 2 1 2 k
2. ⊢G :ω k →ω .
2 τ(s) τ(s)
Itfollowsthatforalmostalls,thefunctionf canberepresentedbytheterm
E =λn.λfx.(G n)f(G nfx),whichissimilartothetermλn n .λfx.n f(n fx)
1 2 1 2 1 2
(the one used to represent addition), but there are two differences:
1. it takes k arguments of type ω instead of two,
τ(s)
2. n is replaced with G n (i=1,2) in the part to the right of“λfx”.
i i
The terms needed in other cases can be obtained similarly. There is always
only a finite number of tuple lengths for which E cannot be constructed. ⊓⊔
Corollary 8. G ⊆F.
3 Limitations of Definability
InthissectionweshowsomepropertiesoftheclassG tonarrowthegapbetween
thefunctionsthatareknowntobedefinableandthefunctionsthatareknownto
be undefinable. We make extensive use of the following theorem (provedin [2]).
Theorem 9. Forarbitrary M of type α, there existsa finitemodel M suchthat
for every term N of type α:
M = N if and only if M|=M =N.
βη
Lemma 10. Let M be a finite model, and let α = ω , for some τ. For every
τ
c ∈ M the function h(m,n ,n ) = if (Jρ(m)αKM = c) then n else n is
ωτ 1 2 1 2
definable.
Proof. LetlbethesmallestnumbersuchthatJρ(l+t)αKM =Jρ(l)αKM,forsome
t≥1. Let t be the smallest such t. If c=Jρ(s)αKM, where s<l, then
min
h(m,n ,n )=
1 2
if (m=s) then n else n =
1 2
if m≤s then (if m≤s−1 then n else n ) else n .
2 1 2
The function h is a composition of functions which belong to G. Therefore h is
in G. If c=Jρ(s)αKM, where s≥l, then
h(m,n ,n )=
1 2
if m≡s (mod t ) then n else n .
min 1 2
Again, h can be expressed as a composition of functions which belong to G.
If c6=Jρ(s)αKM for all s∈N then h(m,n ,n )=n . Therefore h∈G.
1 2 2
Theorem 11. If f is of the form f(m,n ,n ) = if m ∈ A then n else n ,
1 2 1 2
for some A⊆N, then f is not definable or f ∈G.
Proof. Suppose f is definable. Therefore, g(m)= f(m,1,0) is also definable. It
is clear that the following holds:
f(m,n ,n )=ifzero g(m) then n else n .
1 2 2 1
Let G of type α → α be a term which represents g. The following is the result
of applying Theorem 9 to ρ(0)α:
1. N = ρ(0)α if and only if M|=N =ρ(0) for all N of type α,
βη
2. M is finite.
Let B = {Jρ(m)KM | m ∈ N, JGKM · Jρ(m)KM = Jρ(0)αKM}. Then B =
{b ,b ,...,b }, for some k ≥0.
1 2 k
The function f can be expressed as follows:
f(m,n ,n )=
1 2
if (Jρ(m)αKM =b ) then n else
1 2
if (Jρ(m)αKM =b ) then n else
2 2
...
if (Jρ(m)αKM =b ) then n else n .
k 2 1
ItfollowsfromLemma10thatf isacompositionoffunctionswhichbelongtoG.
Therefore f ∈G. ⊓⊔
The next theorem is a generalizationof Theorem 11.
Theorem 12. Every skewly representable finite range function is strictly repre-
sentable.
Proof. Let f be a k-ary skewly representablefinite rangefunction, and let A be
the range of f. The function f is equal to the function f′, which is defined as
follows:
f′(m)=
if (f(m)=a ) then a else
1 1
if (f(m)=a ) then a else
2 2
...
if (f(m)=a ) then a else a ,
p−1 p−1 p
where
A={a ,a ,...,a },
1 2 p
m=m ,m ,...,m .
1 2 k
The function f is a composition of functions of the form:
h (m,n ,n )=if f(m)=a then n else n .
i 1 2 i 1 2
We show that h ∈ G, for arbitrary 1 ≤ i ≤ p. Let E of type α → α →
i 1 2
··· →α → β be a term which skewly represents f. The following is the result
k
of applying Theorem 9 to ρ(a )β:
i
1. N = ρ(a )β if and only if M|=N =ρ(a ) for all N of type β,
βη i i
2. M is finite.
Let B ={hJρ(m )α1KM,Jρ(m )α2KM,...,Jρ(m )αkKMi|m ,m ,...,m ∈N,
1 2 k 1 2 k
JEKM·Jρ(m )α1KM·Jρ(m )α2KM·····Jρ(m )αkKM =Jρ(a )βKM}.
1 2 k i
The set B consists of the arguments at which the value of f is equal to a , in
i
model M. The function h can be expressed as follows:
i
h (m,n ,n )=if Jρ(m)KM ∈B then n else n ,
i 1 2 1 2
where
m=m ,m ,...,m ,
1 2 k
Jρ(m)KM =hJρ(m )α1KM,Jρ(m )α2KM,...,Jρ(m )αkKMi.
1 2 k
ThesetBisfinite.ItfollowsfromLemma10thath isacompositionoffunctions
i
which belong to G. Therefore h ∈G. ⊓⊔
i
Proposition 13. The predecessor function is not strictly definable.
Proof. Suppose pred(n) = n − 1 is strictly definable. Let E be a term that
represents pred. Therefore, subtraction can be skewly represented by the term
λnm.mEn.Ontheotherhand,weknowthatsubtractionisnotskewlydefinable
(see [3]).
Proposition 14. Division by two is not strictly definable.
Proof. Suppose f(n) = n div 2 is strictly definable. Let E be a term that
strictly represents f. Let g be the function which is skewly represented by the
term G = λnm.mEn of type α → β → α, where β = ω . The following is the
α
result of applying Theorem 9 to ρ(0)α:
1. N = ρ(0)α if and only if M|=N =ρ(0) for all N of type α,
βη
2. M is finite.
Let l be the smallest number such that Jρ(l+t)αKM = Jρ(l)αKM, for some
t ≥ 1. Let t be the smallest such t. Similarly, let l′ be the smallest number
min
suchthatJρ(l′+t′)βKM =Jρ(l′)βKM,forsomet′ ≥1,andlett′ bethesmallest
min
such t′.
On one hand, we have:
JGKM·Jρ(l)αKM·Jρ(l′)βKM =JGKM·Jρ(l)αKM·Jρ(l′+l·t′ )βKM =
min
Jρ(g(l,l′+l·t′ ))αKM =Jρ(0)αKM.
min
On the other hand, we have:
JGKM·Jρ(l)αKM·Jρ(l′)βKM =JGKM·Jρ(l+2l′ ·t )αKM·Jρ(l′)βKM =
min
Jρ(g(l+2l′ ·t ,l′))αKM 6=Jρ(0)αKM.
min
4 Conclusion
We have given examples of functions that are strictly definable and are not ex-
tended polynomials. It is clear that there are many considerably different kinds
of strict definability, depending on the type of numerals. There is also an unex-
pecteddifferencebetweenβ-andβη-conversionregardingλ-definability.Acorol-
laryofProposition13isthatfunctionswhicharestrictlydefinablewitharbitrary
type of numerals form a proper subset of the set of skewly definable functions.
Theorem 12 states that it is not the case for functions with finite range.
We conjecture that G exactly characterizes strictly definable functions (or
G =F). We have shownthat G ⊆F. The functions: fl and fl are compositions
1 2
of functions of the form if m ∈ A then n else n , for all l. It follows from
1 2
Theorem 11 that every strictly definable function of that form is already in G.
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