Aspects of Programming

Page 1

Introduction to Lambda Calculus
Henk Barendregt
Erik Barendsen
Revised edition
December 1998, March 2000

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Contents
1 Introduction
5
2 Conversion
9
3 The Power of Lambda
17
4 Reduction
23
5 Type Assignment
33
6 Extensions
41
7 Reduction Systems
47
Bibliography
51
3

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Chapter 1
Introduction
Some history
Leibniz had as ideal the following.
(1) Create a ‘universal language’ in which all possible problems can be stated.
(2) Find a decision method to solve all the problems stated in the universal
language.
If one restricts oneself to mathematical problems, point (1) of Leibniz’ ideal
is fulfilled by taking some form of set theory formulated in the language of
first order predicate logic. This was the situation after Frege and Russell (or
Zermelo).
Point (2) of Leibniz’ ideal became an important philosophical question. ‘Can
one solve all problems formulated in the universal language?’ It seems not,
but it is not clear how to prove that. This question became known as the
Entscheidungsproblem.
In 1936 the Entscheidungsproblem was solved in the negative independently
by Alonzo Church and Alan Turing. In order to do so, they needed a formali-
sation of the intuitive notion of ‘decidable’, or what is equivalent ‘computable’.
Church and Turing did this in two different ways by introducing two models of
computation.
(1) Church (1936) invented a formal system called the lambda calculus and
defined the notion of computable function via this system.
(2) Turing (1936/7) invented a class of machines (later to be called Turing
machines) and defined the notion of computable function via these machines.
Also in 1936 Turing proved that both models are equally strong in the sense
that they define the same class of computable functions (see Turing (1937)).
Based on the concept of a Turing machine are the present day Von Neu-
mann computers. Conceptually these are Turing machines with random access
registers. Imperative programming languages such as Fortran, Pascal etcetera
as well as all the assembler languages are based on the way a Turing machine
is instructed: by a sequence of statements.
Functional programming languages, like Miranda, ML etcetera, are based on
the lambda calculus. An early (although somewhat hybrid) example of such a
language is Lisp. Reduction machines are specifically designed for the execution
of these functional languages.
5

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6
Introduction to Lambda Calculus
Reduction and functional programming
A functional program consists of an expression E (representing both the al-
gorithm and the input). This expression E is subject to some rewrite rules.
Reduction consists of replacing a part P of E by another expression P accord-
ing to the given rewrite rules. In schematic notation
E[P] → E[P ],
provided that P → P is according to the rules. This process of reduction
will be repeated until the resulting expression has no more parts that can be
rewritten. This so called normal form E
∗
of the expression E consists of the
output of the given functional program.
An example:
(7 + 4) ∗ (8 + 5 ∗ 3) → 11 ∗ (8 + 5 ∗ 3)
→ 11 ∗ (8 + 15)
→ 11 ∗ 23
→ 253.
In this example the reduction rules consist of the ‘tables’ of addition and of
multiplication on the numerals.
Also symbolic computations can be done by reduction. For example
first of (sort (append (‘dog’, ‘rabbit’) (sort ((‘mouse’, ‘cat’))))) →
→ first of (sort (append (‘dog’, ‘rabbit’) (‘cat’, ‘mouse’)))
→ first of (sort (‘dog’, ‘rabbit’, ‘cat’, ‘mouse’))
→ first of (‘cat’, ‘dog’, ‘mouse’, ‘rabbit’)
→ ‘cat’.
The necessary rewrite rules for append and sort can be programmed easily
in a few lines. Functions like append given by some rewrite rules are called
combinators.
Reduction systems usually satisfy the Church-Rosser property, which states
that the normal form obtained is independent of the order of evaluation of
subterms. Indeed, the first example may be reduced as follows:
(7 + 4) ∗ (8 + 5 ∗ 3) → (7 + 4) ∗ (8 + 15)
→ 11 ∗ (8 + 15)
→ 11 ∗ 23
→ 253,
or even by evaluating several expressions at the same time:
(7 + 4) ∗ (8 + 5 ∗ 3) → 11 ∗ (8 + 15)
→ 11 ∗ 23
→ 253.

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Introduction
7
Application and abstraction
The first basic operation of the λ-calculus is application. The expression
F · A
or
FA
denotes the data F considered as algorithm applied to the data A considered
as input. This can be viewed in two ways: either as the process of computation
FA or as the output of this process. The first view is captured by the notion
of conversion and even better of reduction; the second by the notion of models
(semantics).
The theory is type-free: it is allowed to consider expressions like FF, that
is F applied to itself. This will be useful to simulate recursion.
The other basic operation is abstraction. If M ≡ M[x] is an expression
containing (‘depending on’) x, then λx.M[x] denotes the function x ↦→ M[x].
Application and abstraction work together in the following intuitive formula.
(λx.2 ∗ x + 1)3 = 2 ∗ 3 + 1 (= 7).
That is, (λx.2 ∗ x + 1)3 denotes the function x ↦→ 2 ∗ x + 1 applied to the
argument 3 giving 2∗3+1 which is 7. In general we have (λx.M[x])N = M[N].
This last equation is preferably written as
(λx.M)N = M[x := N],
(β)
where [x := N] denotes substitution of N for x. It is remarkable that although
(β) is the only essential axiom of the λ-calculus, the resulting theory is rather
involved.
Free and bound variables
Abstraction is said to bind the free variable x in M. E.g. we say that λx.yx
has x as bound and y as free variable. Substitution [x := N] is only performed
in the free occurrences of x:
yx(λx.x)[x := N] ≡ yN(λx.x).
In calculus there is a similar variable binding. In ∫
b
a
f(x,y)dx the variable x is
bound and y is free. It does not make sense to substitute 7 for x: ∫
b
a
f(7,y)d7;
but substitution for y makes sense: ∫
b
a
f(x,7)dx.
For reasons of hygiene it will always be assumed that the bound variables
that occur in a certain expression are different from the free ones. This can be
fulfilled by renaming bound variables. E.g. λx.x becomes λy.y. Indeed, these
expressions act the same way:
(λx.x)a = a = (λy.y)a
and in fact they denote the same intended algorithm. Therefore expressions
that differ only in the names of bound variables are identified.

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8
Introduction to Lambda Calculus
Functions of more arguments
Functions of several arguments can be obtained by iteration of application. The
idea is due to Schonfinkel (1924) but is often called currying, after H.B. Curry
who introduced it independently. Intuitively, if f(x,y) depends on two argu-
ments, one can define
F
x
= λy.f(x,y),
F = λx.F
x
.
Then
(Fx)y = F
x
y = f(x,y).
(∗)
This last equation shows that it is convenient to use association to the left for
iterated application:
FM
1
··· M
n
denotes (··((FM
1
)M
2
)··· M
n
).
The equation (∗) then becomes
Fxy = f(x,y).
Dually, iterated abstraction uses association to the right:
λx
1
··· x
n
.f(x
1
,... ,x
n
) denotes λx
1
.(λx
2
.(··· (λx
n
.f(x
1
,... ,x
n
))··)).
Then we have for F defined above
F = λxy.f(x,y)
and (∗) becomes
(λxy.f(x,y))xy = f(x,y).
For n arguments we have
(λx
1
··· x
n
.f(x
1
,··· ,x
n
))x
1
··· x
n
= f(x
1
,... ,x
n
)
by using n times (β). This last equation becomes in convenient vector notation
(λx.f[x])x = f[x];
more generally one has
(λx.f[x])N = f[N].

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Chapter 2
Conversion
In this chapter, the λ-calculus will be introduced formally.
2.1. Definition. The set of λ-terms (notation Λ) is built up from an infinite
set of variables V = {v,v ,v ,...} using application and (function) abstraction.
x ∈ V
⇒ x ∈ Λ,
M,N ∈ Λ ⇒ (MN) ∈ Λ,
M ∈ Λ,x ∈ V
⇒ (λxM) ∈ Λ.
In BN-form this is
variable ::= ‘v’ | variable ‘ ’
λ-term ::= variable | ‘(’ λ-term λ-term ‘)’ | ‘(λ’ variable λ-term ‘)’
2.2. Example. The following are λ-terms.
v ;
(v v);
(λv(v v));
((λv(v v))v );
(((λv(λv (v v)))v )v ).
2.3. Convention. (i) x,y,z,... denote arbitrary variables; M,N,L,... de-
note arbitrary λ-terms. Outermost parentheses are not written.
(ii) M ≡ N denotes that M and N are the same term or can be obtained
from each other by renaming bound variables. E.g.
(λxy)z ≡ (λxy)z;
(λxx)z ≡ (λyy)z;
(λxx)z ≡ z;
(λxx)z ≡ (λxy)z.
(iii) We use the abbreviations
FM
1
··· M
n
≡ (··((FM
1
)M
2
)··· M
n
)
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10
Introduction to Lambda Calculus
and
λx
1
··· x
n
.M ≡ λx
1
(λx
2
(··· (λx
n
(M))··)).
The terms in Example 2.2 now may be written as follows.
y;
yx;
λx.yx;
(λx.yx)z;
(λxy.yx)zw.
Note that λx.yx is (λx(yx)) and not ((λx.y)x).
2.4. Definition. (i) The set of free variables of M, notation FV(M), is de-
fined inductively as follows.
FV(x) = {x};
FV(MN) = FV(M) ∪ FV(N);
FV(λx.M) = FV(M) − {x}.
A variable in M is bound if it is not free. Note that a variable is bound if it
occurs under the scope of a λ.
(ii) M is a closed λ-term (or combinator) if FV(M) = ∅. The set of closed
λ-terms is denoted by Λ
o
.
(iii) The result of substituting N for the free occurences of x in M, notation
M[x := N], is defined as follows.
x[x := N] ≡ N;
y[x := N] ≡ y, if x ≡ y;
(M
1
M
2
)[x := N] ≡ (M
1
[x := N])(M
2
[x := N]);
(λy.M
1
)[x := N] ≡ λy.(M
1
[x := N]).
2.5. Example. Consider the λ-term
λxy.xyz.
Then x and y are bound variables and z is a free variable. The term λxy.xxy
is closed.
2.6. Variable convention. If M
1
,... ,M
n
occur in a certain mathematical
context (e.g. definition, proof), then in these terms all bound variables are
chosen to be different from the free variables.
Note that in the fourth clause of Definition 2.4 (iii) it is not needed to say
‘provided that y ≡ x and y /∈ FV(N)’. By the variable convention this is the
case.
Now we can introduce the λ-calculus as formal theory.

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Conversion
11
2.7. Definition. (i) The principal axiom scheme of the λ-calculus is
(λx.M)N = M[x := N]
(β)
for all M,N ∈ Λ.
(ii) There are also the ‘logical’ axioms and rules.
Equality:
M = M;
M = N
⇒ N = M;
M = N,N = L ⇒ M = L.
Compatibility rules:
M = M
⇒ MZ = M Z;
M = M
⇒ ZM = ZM ;
M = M
⇒ λx.M = λx.M .
(ξ)
(iii) If M = N is provable in the λ-calculus, then we sometimes write λ ⊣
M = N.
As a consequence of the compatibility rules, one can replace (sub)terms by
equal terms in any term context:
M = N ⇒ ··· M ··· = ··· N ··· .
For example, (λy.yy)x = xx, so
λ ⊣ λx.x((λy.yy)x)x = λx.x(xx)x.
2.8. Remark. We have identified terms that differ only in the names of bound
variables. An alternative is to add to the λ-calculus the following axiom scheme
λx.M = λy.M[x := y], provided that y does not occur in M.
(α)
We prefer our version of the theory in which the identifications are made on
syntactic level. These identifications are done in our mind and not on paper.
For implementations of the λ-calculus the machine has to deal with this so
called α-conversion. A good way of doing this is provided by the name-free
notation of de Bruijn, see Barendregt (1984), Appendix C.
2.9. Lemma. λ ⊣ (λx
1
··· x
n
.M)X
1
··· X
n
= M[x
1
:= X
1
]··· [x
n
:= X
n
].
Proof. By the axiom (β) we have
(λx
1
.M)X
1
= M[x
1
:= X
1
].
By induction on n the result follows. □

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12
Introduction to Lambda Calculus
2.10. Example (Standard combinators). Define the combinators
I ≡ λx.x;
K ≡ λxy.x;
K
∗
≡ λxy.y;
S ≡ λxyz.xz(yz).
Then, by Lemma 2.9, we have the following equations.
IM = M;
KMN = M;
K
∗
MN = N;
SMNL = ML(NL).
Now we can solve simple equations.
2.11. Example. ∃G ∀X GX = XXX (there exists G ∈ Λ such that for all
X ∈ Λ one has λ ⊣ GX = XX). Indeed, take G ≡ λx.xxx and we are done.
Recursive equations require a special technique. The following result pro-
vides one way to represent recursion in the λ-calculus.
2.12. Fixedpoint Theorem. (i) ∀F ∃X FX = X. (This means: for all
F ∈ Λ there is an X ∈ Λ such that λ ⊣ FX = X.)
(ii) There is a fixed point combinator
Y ≡ λf.(λx.f(xx))(λx.f(xx))
such that
∀F F(YF) = YF.
Proof. (i) Define W ≡ λx.F(xx) and X ≡ WW. Then
X ≡ WW ≡ (λx.F(xx))W = F(WW) ≡ FX.
(ii) By the proof of (i). □
2.13. Example. (i) ∃G ∀X GX = SGX. Indeed,
∀X GX = SGX ⇐ Gx = SGx
⇐ G = λx.SGx
⇐ G = (λgx.Sgx)G
⇐ G ≡ Y(λgx.Sgx).
Note that one can also take G ≡ YS.
(ii) ∃G ∀X GX = GG: take G ≡ Y(λgx.gg). (Can you solve this without
using Y?)

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Conversion
13
In the lambda calculus one can define numerals and represent numeric func-
tions on them.
2.14. Definition. (i) F
n
(M) with F ∈ Λ and n ∈ N is defined inductively as
follows.
F
0
(M) ≡ M;
F
n+1
(M) ≡ F(F
n
(M)).
(ii) The Church numerals c
0
,c
1
,c
2
,... are defined by
c
n
≡ λfx.f
n
(x).
2.15. Proposition (J.B. Rosser). Define
A
+
≡ λxypq.xp(ypq);
A
∗
≡ λxyz.x(yz);
A
exp
≡ λxy.yx.
Then one has for all n,m ∈ N
(i) A
+
c
n
c
m
= c
n+m
.
(ii) A
∗
c
n
c
m
= c
n∗m
.
(iii) A
exp
c
n
c
m
= c
(n
m
)
, except for m = 0 (Rosser started counting from 1).
In the proof we need the following.
2.16. Lemma. (i) (c
n
x)
m
(y) = x
n∗m
(y).
(ii) (c
n
)
m
(x) = c
(n
m
)
(x), for m > 0.
Proof. (i) Induction on m. If m = 0, then LHS = y = RHS. Assume (i) is
correct for m (Induction Hypothesis: IH). Then
(c
n
x)
m+1
(y) = c
n
x((c
n
x)
m
(y))
= c
n
x(x
n∗m
(y)) by IH,
= x
n
(x
n∗m
(y))
≡ x
n+n∗m
(y)
≡ x
n∗(m+1)
(y).
(ii) Induction on m > 0. If m = 1, then LHS ≡ c
n
x ≡ RHS. If (ii) is correct
for m, then
(c
n
)
m+1
(x) = c
n
((c
n
)
m
(x))
= c
n
(c
(n
m
)
(x)) by IH,
= λy.(c
(n
m
)
(x))
n
(y)
= λy.x
n
m
∗n
(y) by (i),
= c
(n
m+1
)
x.

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14
Introduction to Lambda Calculus
Proof of the proposition. (i) Exercise.
(ii) Exercise. Use Lemma 2.16 (i).
(iii) By Lemma 2.16 (ii) we have for m > 0
A
exp
c
n
c
m
= c
m
c
n
= λx.(c
n
)
m
(x)
= λx.c
(n
m
)
x
= c
(n
m
)
,
since λx.Mx = M if M ≡ λy.M [y] and x /∈ FV(M). Indeed,
λx.Mx ≡ λx.(λy.M [y])x
= λx.M [x]
≡ λy.M [y]
≡ M. □
Exercises
2.1.
(i) Rewrite according to official syntax
M
1
≡ y(λx.xy(λzw.yz)).
(ii) Rewrite according to the simplified syntax
M
2
≡ λv (λv ((((λvv)v )v )((v (λv (v v )))v ))).
2.2.
Prove the following substitution lemma. Let x ≡ y and x /∈ FV(L). Then
M[x := N][y := L] ≡ M[y := L][x := N[y := L]].
2.3.
(i) Prove, using Exercise 2.2,
λ ⊣ M
1
= M
2
⇒ λ ⊣ M
1
[x := N] = M
2
[x := N].
(ii) Show
λ ⊣ M
1
= M
2
&λ ⊣ N
1
= N
2
⇒ λ ⊣ M
1
[x := N
1
] = M
2
[x := N
2
].
2.4.
Prove Proposition 2.15 (i), (ii).
2.5.
Let B ≡ λxyz.x(yz). Simplify M ≡ BXY Z, that is find a ‘simple’ term N such
that λ ⊣ M = N.
2.6.
Simplify the following terms.
(i) M ≡ (λxyz.zyx)aa(λpq.q);
(ii) M ≡ (λyz.zy)((λx.xxx)(λx.xxx))(λw.I);
(iii) M ≡ SKSKSK.
2.7.
Show that
(i) λ ⊣ KI = K
∗
;
(ii) λ ⊣ SKK = I.
2.8.
(i) Write down a closed λ-term F ∈ Λ such that for all M, N ∈ Λ
FMN = M(NM)N.

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Conversion
15
(ii) Construct a λ-term F such that for all M, N, L ∈ Λ
o
FMNL = N(λx.M)(λyz.yLM).
2.9.
Find closed terms F such that
(i) Fx = xI;
(ii) Fxy = xIy.
2.10.
Find closed terms F such that
(i) Fx = F. This term can be called the ‘eater’ and is often denoted by K
∞
;
(ii) Fx = xF;
(iii) FIKK = FK.
2.11.
Show
∀C[ , ] ∃F ∀x Fx = C[F, x]
and take another look at the exercises 2.8, 2.9 and 2.10.
2.12.
Let P, Q ∈ Λ. P and Q are incompatible, notation P ♯ Q, if λ extended with
P = Q as axiom proves every equation between λ-terms, i.e. for all M, N ∈ Λ
one has λ + (P = Q) ⊣ M = N. In this case one says that λ + (P = Q) is
inconsistent.
(i) Prove that for P, Q ∈ Λ
P ♯ Q ⇔ λ + (P = Q) ⊣ true = false,
where true ≡ K, false ≡ K
∗
.
(ii) Show that I ♯ K.
(iii) Find a λ-term F such that FI = x and FK = y.
(iv) Show that K ♯ S.
2.13.
Write down a grammar in BN-form that generates the λ-terms exactly in the
way they are written in Convention 2.3.

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Page 17

Chapter 3
The Power of Lambda
We have seen that the function plus, times and exponentiation on N can be
represented in the λ-calculus using Church’s numerals. We will now show that
all computable (recursive) functions can be represented in the λ-calculus. In
order to do this we will use first a different system of numerals.
Truth values and a conditional can be represented in the λ-calculus.
3.1. Definition. (i) true ≡ K, false ≡ K
∗
.
(ii) If B is considered as a Boolean, i.e. a term that is either true or false,
then
if B then P else Q
can be represented by
BPQ.
3.2. Definition (Pairing). For M,N ∈ Λ write
[M,N] ≡ λz.if z then M else N
(≡ λz.zMN).
Then
[M,N]true = M,
[M,N]false = N,
and hence [M,N] can serve as an ordered pair.
We can use this pairing construction for an alternative representation of
natural numbers due to Barendregt (1976).
3.3. Definition. For each n ∈ N, the numeral n is defined inductively as
follows.
0
≡ I,
n + 1
≡ [false, n ].
17

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18
Introduction to Lambda Calculus
3.4. Lemma (Successor, predecessor, test for zero). There exist combinators S
+
,
P
−
, and Zero such that
S
+
n
=
n + 1 ,
P
−
n + 1
=
n ,
Zero 0
= true,
Zero n + 1
= false.
Proof. Take
S
+
≡ λx.[false,x],
P
−
≡ λx.xfalse,
Zero ≡ λx.xtrue. □
3.5. Definition (Lambda definability). (i) A numeric function is a map
ϕ : N
p
→ N
for some p. In this case ϕ is called p-ary.
(ii) A numeric p-ary function ϕ is called λ-definable if for some combinator
F
F n
1
··· n
p
= ϕ(n
1
,... ,n
p
)
(∗)
for all n
1
,... ,n
p
∈ N. If (∗) holds, then ϕ is said to be λ-defined by F.
3.6. Definition. The initial functions are the numeric functions U
n
i
, S
+
, Z
defined by
U
n
i
(x
1
,... ,x
n
) = x
i
, (1 ≤ i ≤ n);
S
+
(n) = n + 1;
Z(n) = 0.
Let P(n) be a numeric relation. As usual
µm[P(m)]
denotes the least number m such that P(m) holds if there is such a number;
otherwise it is undefined.
3.7. Definition. Let A be a class of numeric functions.
(i) A is closed under composition if for all ϕ defined by
ϕ(n) = χ(ψ
1
(n),... ,ψ
m
(n))
with χ,ψ
1
,... ,ψ
m
∈ A, one has ϕ ∈ A.
(ii) A is closed under primitive recursion if for all ϕ defined by
ϕ(0,n) = χ(n),
ϕ(k + 1,n) = ψ(ϕ(k,n),k,n)
with χ,ψ ∈ A, one has ϕ ∈ A.

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Page 19

The Power of Lambda
19
(iii) A is closed under minimalization if for all ϕ defined by
ϕ(n) = µm[χ(n,m) = 0]
with χ ∈ A such that
∀n ∃m χ(n,m) = 0,
one has ϕ ∈ A.
3.8. Definition. The class R of recursive functions is the smallest class of
numeric functions that contains all initial functions and is closed under compo-
sition, primitive recursion and minimalization.
So R is an inductively defined class. The proof that all recursive functions
are λ-definable is in fact by a corresponding induction argument. The result is
originally due to Kleene (1936).
3.9. Lemma. The initial functions are λ-definable.
Proof. Take as defining terms
U
n
i
≡ λx
1
··· x
n
.x
i
,
S
+
≡ λx.[false,x] (see Lemma 3.4)
Z ≡ λx. 0 . □
3.10. Lemma. The λ-definable functions are closed under composition.
Proof. Let χ,ψ
1
,... ,ψ
m
be λ-defined by G,H
1
,... ,H
m
respectively. Then
ϕ(n) = χ(ψ
1
(n),... ,ψ
m
(n))
is λ-defined by
F ≡ λx.G(H
1
x)··· (H
m
x). □
As to primitive recursion, let us first consider an example. The addition
function can be specified as follows.
Add(0,y) = y,
Add(x + 1,y) = 1 + Add(x,y) = S
+
(Add(x,y)).
An intuitive way to compute Add(m,n) us the following.
Test whether m = 0.
If yes: give output n;
if no: compute Add(m − 1,n) and give its successor as output.
Therefore we want a term Add such that
Addxy = if Zerox then y else S
+
(Add(P
−
x)y).
This equation can be solved using the fixedpoint combinator: take
Add ≡ Y(λaxy.if Zero x then y else S
+
(a(P
−
x)y)).
The general case is treated as follows.

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Page 20

20
Introduction to Lambda Calculus
3.11. Lemma. The λ-definable functions are closed under primitive recursion.
Proof. Let ϕ be defined by
ϕ(0,n) = χ(n),
ϕ(k + 1,n) = ψ(ϕ(k,n),k,n),
where χ,ψ are λ-defined by G,H respectively. Now we want a term F such
that
Fxy = if Zero x then Gy else H(F(P
−
x)y)(P
−
x)y
≡ D(F,x,y), say.
It is sufficient to find an F such that
F = λxy.D(F,x,y)
= (λfxy.D(f,x,y))F.
Now such an F can be found by the Fixedpoint Theorem and we are done. □
3.12. Lemma. The λ-definable functions are closed under minimalization.
Proof. Let ϕ be defined by
ϕ(n) = µm[χ(n,m) = 0],
where χ is λ-defined by G. Again by the Fixedpoint Theorem there is a term
H such that
Hxy = if Zero(Gxy) then y else Hx(S
+
y)
= (λhxy.E(h,x,y))Hxy, say.
Set F ≡ λx.Hx 0 . Then F λ-defines ϕ:
F n
= H n 0
=
0
if G n 0 = 0
= H n 1
else
=
1
if G n 1 = 0
= H n 2
else
=
2
if . ..
= ... □
3.13. Theorem. All recursive functions are λ-definable.
Proof. By the lemmas 3.9–3.12. □
The converse also holds. So for numeric functions we have ϕ is recursive iff
ϕ is λ-definable. Moreover also for partial functions a notion of λ-definability
exists. If ψ is a partial numeric function, then we have
ψ is partial recursive ⇔ ψ is λ-definable.

---

Page 21

The Power of Lambda
21
3.14. Theorem. With respect to the Church numerals c
n
all recursive func-
tions can be λ-defined.
Proof. Define
S
+
c
≡ λxyz.y(xyz),
P
−
c
≡ λxyz.x(λpq.q(py))(Kz)I
1
,
Zero
c
≡ λx.x(Kfalse)true.
Then these terms represent the successor, predecessor and test for zero. Then
as before all recursive functions can be λ-defined. □
An alternative proof uses ‘translators’ between the numerals n and c
n
.
3.15. Proposition. There exist terms T, T
−1
such that for all n
Tc
n
=
n ;
T
−1
n
= c
n
.
Proof. Construct T, T
−1
such that
T ≡ λx.xS
+
0 .
T
−1
= λx.if Zerox then c
0
else S
+
c
(T
−1
(P
−
x)). □
3.16. Corollary (Second proof of Theorem 3.14). Let ϕ be a recursive func-
tion (of arity 2 say). Let F represent ϕ with respect to the numerals n .
Define
F
c
≡ λxy.T
−1
(F(Tx)(Ty)).
Then F
c
represents ϕ with respect to the Church numerals. □
The representation of pairs in the lambda calculus can also be used to solve
multiple fixedpoint equations.
3.17. Multiple Fixedpoint Theorem. Let F
1
,... ,F
n
be λ-terms. Then we
can find X
1
,... ,X
n
such that
X
1
= F
1
X
1
··· X
n
,
.
.
.
X
n
= F
n
X
1
··· X
n
.
Observe that for n = 1 this is the ordinary Fixedpoint Theorem (2.12).
Proof. We treat the case n = 2. So we want
X
1
= F
1
X
1
X
2
,
X
2
= F
2
X
1
X
2
.
1
Term found by J. Velmans.

---

Page 22

22
Introduction to Lambda Calculus
The trick is to construct X
1
and X
2
simultaneously, as a pair. By the ordinary
Fixedpoint Theorem we can find an X such that
X = [F
1
(Xtrue)(Xfalse),F
2
(Xtrue)(Xfalse)].
Now define X
1
≡ Xtrue, X
2
≡ Xfalse. Then the result follows. This can be
generalized to arbitrary n. □
3.18. Example. There exist G,H ∈ Λ such that
Gxy = Hy(Kx),
Hx = G(xx)(S(H(xx))).
Indeed, we can replace the above equations by
G = λxy.Hy(Kx),
H = λx.G(xx)(S(H(xx))),
and apply the Multiple Fixedpoint Theorem with F
1
≡ λghxy.hy(Kx) and
F
2
≡ λghx.g(xx)(S(h(xx))).
Exercises
3.1.
(i) Find a λ-term Mult such that for all m, n ∈ N
Mult n m = n · m .
(ii) Find a λ-term Fac such that for all n ∈ N
Fac n = n! .
3.2.
The simple Ackermann function ϕ is defined as follows.
ϕ(0, n) = n + 1,
ϕ(m + 1, 0) = ϕ(m, 1),
ϕ(m + 1, n + 1) = ϕ(m, ϕ(m + 1, n)).
Find a λ-term F that λ-defines ϕ.
3.3.
Construct λ-terms M
0
, M
1
, . . . such that for all n one has
M
0
= x,
M
n+1
= M
n+2
M
n
.
3.4.
Verify that P
−
c
(see the first proof of Theorem 3.14) indeed λ-defines the pre-
decessor function with respect to the Church numerals.

---

Page 23

Chapter 4
Reduction
There is a certain asymmetry in the basic scheme (β). The statement
(λx.x
2
+ 1)3 = 10
can be interpreted as ‘10 is the result of computing (λx.x
2
+ 1)3’, but not vice
versa. This computational aspect will be expressed by writing
(λx.x
2
+ 1)3 →→ 10
which reads ‘(λx.x
2
+ 1)3 reduces to 10’.
Apart from this conceptual aspect, reduction is also useful for an analysis
of convertibility. The Church-Rosser theorem says that if two terms are con-
vertible, then there is a term to which they both reduce. In many cases the
inconvertibility of two terms can be proved by showing that they do not reduce
to a common term.
4.1. Definition. (i) A binary relation R on Λ is called compatible (with the
operations) if
M R N ⇒ (ZM) R (ZN),
(MZ) R (NZ) and
(λx.M) R (λx.N).
(ii) A congruence relation on Λ is a compatible equivalence relation.
(iii) A reduction relation on Λ is a compatible, reflexive and transitive rela-
tion.
4.2. Definition. The binary relations →
β
, →→
β
and =
β
on Λ are defined in-
ductively as follows.
(i) 1. (λx.M)N →
β
M[x := N];
2. M →
β
N ⇒ ZM →
β
ZN, MZ →
β
NZ and λx.M →
β
λx.N.
(ii) 1. M →→
β
M;
2. M →
β
N ⇒ M →→
β
N;
3. M →→
β
N,N →→
β
L ⇒ M →→
β
L.
23

---

Page 24

24
Introduction to Lambda Calculus
(iii) 1. M →→
β
N ⇒ M =
β
N;
2. M =
β
N ⇒ N =
β
M;
3. M =
β
N,N =
β
L ⇒ M =
β
L.
These relations are pronounced as follows.
M →→
β
N : Mβ-reduces to N;
M →
β
N : Mβ-reduces to Nin one step;
M =
β
N : M is β-convertible to N.
By definition →
β
is compatible, →→
β
is a reduction relation and =
β
is a con-
gruence relation.
4.3. Example. (i) Define
ω ≡ λx.xx,
Ω ≡ ωω.
Then Ω →
β
Ω.
(ii) KIΩ →→
β
I.
Intuitively, M =
β
N if M is connected to N via →
β
-arrows (disregarding
the directions of these). In a picture this looks as follows.
M
•
d
d‚
© 
  d
d‚
•
•
•
•
N
d
d‚ © 
  d
d‚
© 
 
d
d‚ © 
 
•
•
•
•
d
d‚ © 
 
•
4.4. Example. KIΩ =
β
II. This is demonstrated by the following reductions.
KIΩ
d
d‚
(λy.I)Ω
II
d
d‚
© 
 
I
4.5. Proposition. M =
β
N ⇔ λ ⊣ M = N.
Proof. By an easy induction. □
4.6. Definition. (i) A β-redex is a term of the form (λx.M)N. In this case
M[x := N] is its contractum.
(ii) A λ-term M is a β-normal form (β-nf) if it does not have a β-redex as
subexpression.
(iii) A term M has a β-normal form if M =
β
N and N is a β-nf, for some
N.

---

Page 25

Reduction
25
4.7. Example. (λx.xx)y is not a β-nf, but has as β-nf the term yy.
An immediate property of nf’s is the following.
4.8. Lemma. Let M be a β-nf. Then
M →→
β
N ⇒ N ≡ M.
Proof. This is true if →→
β
is replaced by →
β
. Then the result follows by
transitivity. □
4.9. Church-Rosser Theorem. If M →→
β
N
1
, M →→
β
N
2
, then for some N
3
one has N
1
→→
β
N
3
and N
2
→→
β
N
3
; in diagram
M
©
©
 
 
 
 
 
 
d
d
d
d
d
d
‚
‚
N
1
N
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
‚
‚
©
©
.
.
.
.
.
.
.
.
.
.
.
.
.
.
N
3
The proof is postponed until 4.19.
4.10. Corollary. If M =
β
N, then there is an L such that M →→
β
L and
N →→
β
L.
An intuitive proof of this fact proceeds by a tiling procedure: given an arrow
path showing M =
β
N, apply the Church-Rosser property repeatedly in order
to find a common reduct. For the example given above this looks as follows.
M
•
d
d‚
© 
  d
d‚
•
•
•
•
N
d
d‚ © 
  d
d‚
© 
 
d
d‚ © 
 
•
•
•
•
.
.
.
.
.
.
.
.
.
.
.
.
.
.
‚
‚
d
d‚ © 
 
©
©
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
•
©
©
.
.
.
.
.
.
•
.
.
.
.
.
.
.
.
.
.
.
.
.
.
‚
‚
•
This is made precise below.

---

Page 26

26
Introduction to Lambda Calculus
Proof. Induction on the generation of =
β
.
Case 1. M =
β
N because M →→
β
N. Take L ≡ N.
Case 2. M =
β
N because N =
β
M. By the IH there is a common β-reduct
L
1
of N, M. Take L ≡ L
1
.
Case 3. M =
β
N because M =
β
N , N =
β
N. Then
M
(IH)
N
(IH)
N
d
d
d
d
‚
‚
©
©
 
 
 
 
d
d
d
d
‚
‚
©
©
 
 
 
 
L
1
(CR)
L
2
.
.
.
.
.
.
.
.
.
.
‚
‚
©
©
.
.
.
.
.
.
.
.
.
.
L
□
4.11. Corollary. (i) If M has N as β-nf, then M →→
β
N.
(ii) A λ-term has at most one β-nf.
Proof. (i) Suppose M =
β
N with N in β-nf. By Corollary 4.10 M →→
β
L
and N →→
β
L for some L. But then N ≡ L, by Lemma 4.8, so M →→
β
N.
(ii) Suppose M has β-nf’s N
1
, N
2
. Then N
1
=
β
N
2
(=
β
M). By Corollary
4.10 N
1
→→
β
L, N
2
→→
β
L for some L. But then N
1
≡ L ≡ N
2
by Lemma
4.8. □
4.12. Some consequences. (i) The λ-calculus is consistent, i.e. λ ⊣ true =
false. Otherwise true =
β
false by Proposition 4.5, which is impossible by
Corollary 4.11 since true and false are distinct β-nf’s. This is a syntactic
consistency proof.
(ii) Ω has no β-nf. Otherwise Ω →→
β
N with N in β-nf. But Ω only reduces
to itself and is not in β-nf.
(iii) In order to find the β-nf of a term M (if it exists), the various subex-
pressions of M may be reduced in different orders. By Corollary 4.11 (ii) the
β-nf is unique.
The proof of the Church-Rosser theorem occupies 4.13–4.19. The idea of
the proof is as follows. In order to prove Theorem 4.9, it is sufficient to show
the Strip Lemma:
M
© 
 
β
 
 
d
d
d
d
β
d
d
d
d
‚
‚
N
1
.
.
.
.
.
.
.
.
.
.
.
.
β
.
.
.
.
.
.
.
.
.
.
.
.
‚
‚
N
2
©
©
.
.
.
.
.
β
.
.
.
.
.
N
3
In order to prove this lemma, let M →
β
N
1
be a one step reduction resulting
from changing a redex R in M in its contractum R in N
1
. If one makes a

---

Page 27

Reduction
27
bookkeeping of what happens with R during the reduction M →→
β
N
2
, then by
reducing all ‘residuals’ of R in N
2
the term N
3
can be found. In order to do the
necessary bookkeeping an extended set Λ ⊇ Λ and reduction β is introduced.
The underlining serves as a ‘tracing isotope’.
4.13. Definition (Underlining). (i) Λ is the set of terms defined inductively
as follows.
x ∈ V
⇒ x ∈ Λ,
M,N ∈ Λ ⇒ (MN) ∈ Λ,
M ∈ Λ,x ∈ V
⇒ (λx.M) ∈ Λ,
M,N ∈ Λ,x ∈ V
⇒ ((λx.M)N) ∈ Λ.
(ii) The underlined reduction relations →
β
(one step) and →→
β
are defined
starting with the contraction rules
(λx.M)N →
β
M[x := N],
(λx.M)N →
β
M[x := N].
Then →
β
is extended in order to become a compatible relation (also with respect
to λ-abstraction). Moreover, →→
β
is the transitive reflexive closure of →
β
.
(iii) If M ∈ Λ, then |M| ∈ Λ is obtained from M by leaving out all underlin-
ings. E.g. |(λx.x)((λx.x)(λx.x))| ≡ I(II).
4.14. Definition. The map ϕ : Λ → Λ is defined inductively as follows.
ϕ(x) ≡ x,
ϕ(MN) ≡ ϕ(M)ϕ(N),
ϕ(λx.M) ≡ λx.ϕ(M),
ϕ((λx.M)N) ≡ ϕ(M)[x := ϕ(N)].
In other words, ϕ contracts all redexes that are underlined, from the inside to
the outside.
Notation. If |M| ≡ N or ϕ(M) ≡ N, then this will be denoted by
M
| |
E
N or M
ϕ
E
N.
4.15. Lemma.
M ···············
β
···············
EE
N
| |
c
c
| |
M
β
EE
N
M ,N ∈ Λ,
M,N ∈ Λ.

---

Page 28

28
Introduction to Lambda Calculus
Proof. First suppose M →
β
N. Then N is obtained by contracting a redex
in M and N can be obtained by contracting the corresponding redex in M .
The general statement follows by transitivity. □
4.16. Lemma. (i) Let M,N ∈ Λ. Then
ϕ(M[x := N]) ≡ ϕ(M)[x := ϕ(N)].
(ii)
M
β
EE
N
ϕ
c
c
ϕ
ϕ(M) ·············
β
·············
EE
ϕ(N)
M,N ∈ Λ.
Proof. (i) By induction on the structure of M, using the Substitution Lemma
(see Exercise 2.2) in case M ≡ (λy.P)Q. The condition of that lemma may be
assumed to hold by our convention about free variables.
(ii) By induction on the generation of →→
β
, using (i). □
4.17. Lemma.
M
© 
 
 
| |
 
 
 
d
d
d
ϕ
d
d
d‚
N ················
β
················
EE
L
M ∈ Λ,
N,L ∈ Λ.
Proof. By induction on the structure of M. □
4.18. Strip lemma.
M
© 
 
β
 
 
d
d
d
d
β
d
d
d
d
‚
‚
N
1
.
.
.
.
.
.
.
.
.
.
.
.
.
β
.
.
.
.
.
.
.
.
.
.
.
.
.
‚
‚
N
2
©
©
.
.
.
.
.
β
.
.
.
.
.
N
3
M,N
1
,N
2
,N
3
∈ Λ.
Proof. Let N
1
be the result of contracting the redex occurrence R ≡ (λx.P)Q
in M. Let M ∈ Λ be obtained from M by replacing R by R ≡ (λx.P)Q. Then

---

Page 29

Reduction
29
|M | ≡ M and ϕ(M ) ≡ N
1
. By the lemmas 4.15, 4.16 and 4.17 we can erect
the diagram
M
© 
 
β
 
 
r
r
r
r
r
r
r
r
β
r
r
r
r
r
r
r
r
j
j
sd
d
| |
d
d
N
1
'
ϕ
M
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
β
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
j
j
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
j
j
β
N
2
©
©
.
.
.
.
.
β .
.
.
.
.
sd
d
| |
d
d
N
3
'
ϕ
N
2
which proves the Strip Lemma. □
4.19. Proof of the Church-Rosser Theorem. If M →→
β
N
1
, then M ≡
M
1
→
β
M
2
→
β
··· →
β
M
n
≡ N
1
. Hence the CR property follows from the
Strip Lemma and a simple diagram chase:
M
© 
 
d
d
d
d
d
d
d
d
d
d
d
d
‚
‚
M
1
© 
 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
‚
‚
•
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
‚
‚
•
© 
 
N
1
N
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
‚
‚
©
©
.
.
.
.
.
.
•
©
©
.
.
.
.
.
.
•
.
.
.
•
©
©
.
.
.
.
.
.
•
□
4.20. Definition. For M ∈ Λ the reduction graph of M, notation G
β
(M), is
the directed multigraph with vertices {N | M →→
β
N} and directed by →
β
.

---

Page 30

30
Introduction to Lambda Calculus
4.21. Example. G
β
(I(Ix)) is
x
I x
( )
I
xI
sometimes simply drawn as
It can happen that a term M has a nf, but at the same time an infinite
reduction path. Let Ω ≡ (λx.xx)(λx.xx). Then Ω → Ω → ··· so KIΩ →
KIΩ → ···, and KIΩ →→ I. Therefore a so called strategy is necessary in
order to find the normal form. We state the following theorem; for a proof see
Barendregt (1984), Theorem 13.2.2.
4.22. Normalization Theorem. If M has a normal form, then iterated con-
traction of the leftmost redex leads to that normal form.
In other words: the leftmost reduction strategy is normalizing. This fact
can be used to find the normal form of a term, or to prove that a certain term
has no normal form.
4.23. Example. KΩI has an infinite leftmost reduction path, viz.
KΩI →
β
(λy.Ω)I →
β
Ω →
β
Ω →
β
··· ,
and hence does not have a normal form.
The functional language (pure) Lisp uses an eager or applicative evaluation
strategy, i.e. whenever an expression of the form FA has to be evaluated, A is
reduced to normal form first, before ‘calling’ F. In the λ-calculus this strat-
egy is not normalizing as is shown by the two reduction paths for KIΩ above.
There is, however, a variant of the lambda calculus, called the λI-calculus, in
which the eager evaluation strategy is normalizing. In this λI-calculus terms
like K, ‘throwing away’ Ω in the reduction KIΩ →→ I do not exist. The ‘ordi-
nary’ λ-calculus is sometimes referred to as λK-calculus; see Barendregt (1984),
Chapter 9.
Remember the fixedpoint combinator Y. For each F ∈ Λ one has YF =
β
F(YF), but neither YF →→
β
F(YF) nor F(YF) →→
β
YF. In order to solve

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Reduction
31
reduction equations one can work with A.M. Turing’s fixedpoint combinator,
which has a different reduction behaviour.
4.24. Definition. Turing’s fixedpoint combinator Θ is defined by setting
A ≡ λxy.y(xxy),
Θ ≡ AA.
4.25. Proposition. For all F ∈ Λ one has
ΘF →→
β
F(ΘF).
Proof.
ΘF
≡ AAF
→
β
(λy.y(AAy))F
→
β
F(AAF)
≡ F(ΘF). □
4.26. Example. ∃G ∀X GX →→ X(XG). Indeed,
∀X GX →→ X(XG) ⇐ G →→ λx.x(xG)
⇐ G →→ (λgx.x(xg))G
⇐ G ≡ Θ(λgx.x(xg)).
Also the Multiple Fixedpoint Theorem has a ‘reducing’ variant.
4.27. Theorem. Let F
1
,... ,F
n
be λ-terms. Then we can find X
1
,... ,X
n
such
that
X
1
→→ F
1
X
1
··· X
n
,
.
.
.
X
n
→→ F
n
X
1
··· X
n
.
Proof. As for the equational Multiple Fixedpoint Theorem 3.17, but now
using Θ. □
Exercises
4.1.
Show ∀M ∃N [N in β-nf and NI →→
β
M].
4.2.
Construct four terms M with G
β
(M) respectively as follows.

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32
Introduction to Lambda Calculus
4.3.
Show that there is no F ∈ Λ such that for all M, N ∈ Λ
F(MN) = M.
4.4.*
Let M ≡ AAx with A ≡ λaxz.z(aax). Show that G
β
(M) contains as subgraphs
an n-dimensional cube for every n ∈ N.
4.5.
(A. Visser)
(i) Show that there is only one redex R such that G
β
(R) is as follows.
(ii) Show that there is no M ∈ Λ with G
β
(M) is
[Hint. Consider the relative positions of redexes.]
4.6.*
(C. Bohm) Examine G
β
(M) with M equal to
(i) HIH, H ≡ λxy.x(λz.yzy)x.
(ii) LLI, L ≡ λxy.x(yy)x.
(iii) QIQ, Q ≡ λxy.xyIxy.
4.7.*
(J.W. Klop) Extend the λ-calculus with two constants δ, ε. The reduction
rules are extended to include δMM → ε. Show that the resulting system is
not Church-Rosser.
[Hint. Define terms C, D such that
Cx →→ δx(Cx)
D →→ CD
Then D →→ ε and D →→ Cε in the extended reduction system, but there is no
common reduct.]
4.8.
Show that the term M ≡ AAx with A ≡ λaxz.z(aax) does not have a normal
form.
4.9.
(i) Show λ ⊣ WWW = ω
3
ω
3
, with W ≡ λxy.xyy and ω
3
≡ λx.xxx.
(ii) Show λ ⊣ B
x
= B
y
with B
z
≡ A
z
A
z
and A
z
≡ λp.ppz.
4.10.
Draw G
β
(M) for M equal to:
(i) WWW, W ≡ λxy.xyy.
(ii) ωω, ω ≡ λx.xx.
(iii) ω
3
ω
3
, ω
3
≡ λx.xxx.
(iv) (λx.Ixx)(λx.Ixx).
(v) (λx.I(xx))(λx.I(xx)).
(vi) II(III).
4.11.
The length of a term is its number of symbols times 0.5 cm. Write down a
λ-term of length < 30 cm with normal form > 10
10
10
light year.
[Hint. Use Proposition 2.15 (ii). The speed of light is c = 3 × 10
10
cm/s.]

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Chapter 5
Type Assignment
The lambda calculus as treated so far is usually referred to as a type-free theory.
This is so, because every expression (considered as a function) may be applied to
every other expression (considered as an argument). For example, the identity
function I ≡ λx.x may be applied to any argument x to give as result that same
x. In particular I may be applied to itself.
There are also typed versions of the lambda calculus. These are introduced
essentially in Curry (1934) (for the so called Combinatory Logic, a variant of
the lambda calculus) and in Church (1940). Types are usually objects of a
syntactic nature and may be assigned to lambda terms. If M is such a term
and a type A is assigned to M, then we say ‘M has type A’ or ‘M in A’; the
denotation used for this is M : A. For example in some typed systems one has
I : (A→A), that is, the identity I may get as type A→A. This means that if
x being an argument of I is of type A, then also the value Ix is of type A. In
general, A→B is the type of functions from A to B.
Although the analogy is not perfect, the type assigned to a term may be
compared to the dimension of a physical entity. These dimensions prevent us
from wrong operations like adding 3 volt to 2 amp`ere. In a similar way types
assigned to lambda terms provide a partial specification of the algorithms that
are represented and are useful for showing partial correctness.
Types may also be used to improve the efficiency of compilation of terms
representing functional algorithms. If for example it is known (by looking at
types) that a subexpression of a term (representing a funtional program) is
purely arithmetical, then fast evaluation is possible. This is because the ex-
pression then can be executed by the alu of the machine and not in the slower
way in which symbolic expressions are evaluated in general.
The two original papers of Curry and Church introducing typed versions of
the lambda calculus give rise to two different families of systems. In the typed
lambda calculi `a la Curry terms are those of the type-free theory. Each term
has a set of possible types. This set may be empty, be a singleton or consist
of several (possibly infinitely many) elements. In the systems `a la Church the
terms are annotated versions of the type-free terms. Each term has (up to an
equivalence relation) a unique type that is usually derivable from the way the
term is annotated.
The Curry and Church approaches to typed lambda calculus correspond to
33

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34
Introduction to Lambda Calculus
two paradigms in programming. In the first of these a program may be written
without typing at all. Then a compiler should check whether a type can be
assigned to the program. This will be the case if the program is correct. A
well-known example of such a language is ML, see Milner (1984). The style of
typing is called implicit typing. The other paradigm in programming is called
explicit typing and corresponds to the Church version of typed lambda calculi.
Here a program should be written together with its type. For these languages
type-checking is usually easier, since no types have to be constructed. Examples
of such languages are Algol 68 and Pascal. Some authors designate the Curry
systems as ‘lambda calculi with type assignment’ and the Church systems as
‘systems of typed lambda calculus’.
Within each of the two paradigms there are several versions of typed lambda
calculus. In many important systems, especially those `a la Church, it is the case
that terms that do have a type always possess a normal form. By the unsolv-
ability of the halting problem this implies that not all computable functions can
be represented by a typed term, see Barendregt (1990), Theorem 4.2.15. This
is not so bad as it sounds, because in order to find such computable functions
that cannot be represented, one has to stand on one’s head. For example in
λ2, the second order typed lambda calculus, only those partial recursive func-
tions cannot be represented that happen to be total, but not provably so in
mathematical analysis (second order arithmetic).
Considering terms and types as programs and their specifications is not the
only possibility. A type A can also be viewed as a proposition and a term M in A
as a proof of this proposition. This so called propositions-as-types interpretation
is independently due to de Bruijn (1970) and Howard (1980) (both papers
were conceived in 1968). Hints in this direction were given in Curry and Feys
(1958) and in Lauchli (1970). Several systems of proof checking are based
on this interpretation of propositions-as-types and of proofs-as-terms. See e.g.
de Bruijn (1980) for a survey of the so called automath proof checking system.
Normalization of terms corresponds in the formulas-as-types interpretation to
normalisation of proofs in the sense of Prawitz (1965). Normal proofs often
give useful proof theoretic information, see e.g. Schwichtenberg (1977).
In this section a typed lambda calculus will be introduced in the style of
Curry. For more information, see Barendregt (1992).
The system λ→-Curry
Originally the implicit typing paradigm was introduced in Curry (1934) for the
theory of combinators. In Curry and Feys (1958) and Curry et al. (1972) the
theory was modified in a natural way to the lambda calculus assigning elements
of a given set T of types to type free lambda terms. For this reason these calculi
`a la Curry are sometimes called systems of type assignment. If the type σ ∈ T
is assigned to the term M ∈ Λ one writes ⊣ M : σ, sometimes with a subscript
under ⊣ to denote the particular system. Usually a set of assumptions Γ is
needed to derive a type assignment and one writes Γ ⊣ M : σ (pronounce this
as ‘Γ yields M in σ’). A particular Curry type assignment system depends on
two parameters, the set T and the rules of type assignment. As an example we

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Type Assignment
35
now introduce the system λ→-Curry.
5.1. Definition. The set of types of λ→, notation Type(λ→), is inductively
defined as follows. We write T = Type(λ→). Let V = {α,α ,...} be a set of
type variables. It will be convenient to allow type constants for basic types such
as Nat, Bool. Let B be such a collection. Then
α ∈ V ⇒ α ∈ T,
B ∈ B ⇒ B ∈ T,
σ,τ ∈ T ⇒ (σ→τ) ∈ T (function space types).
For such definitions it is convenient to use the following abstract syntax to
form T.
T = V | B | T→T
with
V = α | V
(type variables).
Notation. (i) If σ
1
,... ,σ
n
∈ T then
σ
1
→σ
2
→··· →σ
n
stands for
(σ
1
→(σ
2
→··· →(σ
n−1
→σ
n
)··));
that is, we use association to the right.
(ii) α,β,γ,... denote arbitrary type variables.
5.2. Definition. (i) A statement is of the form M : σ with M ∈ Λ and σ ∈ T.
This statement is pronounced as ‘M in σ’. The type σ is the predicate and the
term M is the subject of the statement.
(ii) A basis is a set of statements with only distinct (term) variables as
subjects.
5.3. Definition. Type derivations in the system λ→ are built up from as-
sumptions x:σ, using the following inference rules.
M : σ→τ
N : σ
MN : τ
x : σ
·
·
·
M : τ
λx.M : σ→τ
5.4. Definition. (i) A statement M : σ is derivable from a basis Γ, notation
Γ ⊣ M : σ
(or Γ ⊣
λ→
M : σ if we wish to stress the typing system) if there is a derivation
of M : σ in which all non-cancelled assumptions are in Γ.
(ii) We use ⊣ M : σ as shorthand for ∅ ⊣ M : σ.

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36
Introduction to Lambda Calculus
5.5. Example. (i) Let σ ∈ T. Then ⊣ λfx.f(fx) : (σ→σ)→σ→σ, which is
shown by the following derivation.
f : σ→σ
(2)
f : σ→σ
(2)
x : σ
(1)
fx : σ
f(fx) : σ
(1)
λx.f(fx) : σ→σ
(2)
λfx.f(fx) : (σ→σ)→σ→σ
The indices (1) and (2) are bookkeeping devices that indicate at which appli-
cation of a rule a particular assumption is being cancelled.
(ii) One has ⊣ K : σ→τ→σ for any σ,τ ∈ T, which is demonstrated as
follows.
x : σ
(1)
λy.x : τ→σ
(1)
λxy.x : σ→τ→σ
(iii) Similarly one can show for all σ ∈ T
⊣ I : σ→σ.
(iv) An example with a non-empty basis is the statement
y:σ ⊣ Iy : σ.
Properties of λ→
Several properties of type assignment in λ→ are valid. The first one analyses
how much of a basis is necessary in order to derive a type assignment.
5.6. Definition. Let Γ = {x
1
:σ
1
,... ,x
n
:σ
n
} be a basis.
(i) Write dom(Γ) = {x
1
,... ,x
n
} and σ
i
= Γ(x
i
). That is, Γ is considered
as a partial function.
(ii) Let V
0
be a set of variables. Then Γ ↾ V
0
= {x:σ | x ∈ V
0
&σ = Γ(x)}.
(iii) For σ,τ ∈ T substitution of τ for α in σ is denoted by σ[α := τ].
5.7. Basis Lemma. Let Γ be a basis.
(i) If Γ ⊇ Γ is another basis, then
Γ ⊣ M : σ ⇒ Γ ⊣ M : σ.
(ii) Γ ⊣ M : σ ⇒ FV(M) ⊆ dom(Γ).
(iii) Γ ⊣ M : σ ⇒ Γ ↾ FV(M) ⊣ M : σ.
Proof. (i) By induction on the derivation of M : σ. Since such proofs will
occur frequently we will spell it out in this simple situation in order to be shorter
later on.

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Type Assignment
37
Case 1. M : σ is x:σ and is element of Γ. Then also x:σ ∈ Γ and hence
Γ ⊣ M : σ.
Case 2. M : σ is (M
1
M
2
) : σ and follows directly from M
1
: (τ→σ) and
M
2
: τ for some τ. By the IH one has Γ ⊣ M
1
: (τ→σ) and Γ ⊣ M
2
: τ. Hence
Γ ⊣ (M
1
M
2
) : σ.
Case 3. M : σ is (λx.M
1
) : (σ
1
→σ
2
) and follows directly from Γ,x : σ
1
⊣
M
1
: σ
2
. By the variable convention it may be assumed that the bound variable
x does not occur in dom(Γ ). Then Γ ,x:σ
1
is also a basis which extends Γ,x:σ
1
.
Therefore by the IH one has Γ ,x:σ
1
⊣ M
1
: σ
2
and so Γ ⊣ (λx.M
1
) : (σ
1
→σ
2
).
(ii) By induction on the derivation of M : σ. We only treat the case that
M : σ is (λx.M
1
) : (σ
1
→σ
2
) and follows directly from Γ,x:σ
1
⊣ M
1
: σ
2
. Let y ∈
FV(λx.M
1
), then y ∈ FV(M
1
) and y ≡ x. By the IH one has y ∈ dom(Γ,x:σ
1
)
and therefore y ∈ dom(Γ).
(iii) By induction on the derivation of M : σ. We only treat the case that
M : σ is (M
1
M
2
) : σ and follows directly from M
1
: (τ→σ) and M
2
: τ for some
τ. By the IH one has Γ ↾ FV(M
1
) ⊣ M
1
: (τ→σ) and Γ ↾ FV(M
2
) ⊣ M
2
: τ. By
(i) it follows that Γ ↾ FV(M
1
M
2
) ⊣ M
1
: (τ→σ)and Γ ↾ FV(M
1
M
2
) ⊣ M
2
: τ
and hence Γ ↾ FV(M
1
M
2
) ⊣ (M
1
M
2
) : σ. □
The second property analyses how terms of a certain form get typed. It is
useful among other things to show that certain terms have no types.
5.8. Generation Lemma. (i) Γ ⊣ x : σ ⇒ (x:σ) ∈ Γ.
(ii) Γ ⊣ MN : τ ⇒ ∃σ [Γ ⊣ M : (σ→τ) &Γ ⊣ N : σ].
(iii) Γ ⊣ λx.M : ρ ⇒ ∃σ,τ [Γ,x:σ ⊣ M : τ &ρ ≡ (σ→τ)].
Proof. By induction on the structure of derivations. □
5.9. Proposition (Typability of subterms). Let M be a subterm of M. Then
Γ ⊣ M : σ ⇒ Γ ⊣ M : σ
for some Γ and σ .
The moral is: if M has a type, i.e. Γ ⊣ M : σ for some Γ and σ, then every
subterm has a type as well.
Proof. By induction on the generation of M. □
5.10. Substitution Lemma.
(i) Γ ⊣ M : σ ⇒ Γ[α := τ] ⊣ M : σ[α := τ].
(ii) Suppose Γ,x:σ ⊣ M : τ and Γ ⊣ N : σ. Then Γ ⊣ M[x := N] : τ.
Proof. (i) By induction on the derivation of M : σ.
(ii) By induction on the derivation showing Γ,x:σ ⊣ M : τ. □
The following result states that the set of M ∈ Λ having a certain type in
λ→ is closed under reduction.
5.11. Subject Reduction Theorem. Suppose M →→
β
M . Then
Γ ⊣ M : σ ⇒ Γ ⊣ M : σ.

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38
Introduction to Lambda Calculus
Proof. Induction on the generation of →→
β
using the Generation Lemma 5.8
and the Substitution Lemma 5.10. We treat the prime case, namely that M ≡
(λx.P)Q and M ≡ P[x := Q]. Well, if
Γ ⊣ (λx.P)Q : σ
then it follows by the Generation Lemma that for some τ one has
Γ ⊣ (λx.P) : (τ→σ) and Γ ⊣ Q : τ.
Hence once more by the Generation Lemma
Γ,x:τ ⊣ P : σ and Γ ⊣ Q : τ
and therefore by the Substitution Lemma
Γ ⊣ P[x := Q] : σ. □
Terms having a type are not closed under expansion. For example,
⊣ I : (σ→σ), but ⊣ KI (λx.xx) : (σ→σ).
See Exercise 5.1. One even has the following stronger failure of subject expan-
sion, as is observed in van Bakel (1992).
5.12. Observation. There are M,M ∈ Λ and σ,σ ∈ T such that M →→
β
M
and
⊣ M : σ,
⊣ M : σ ,
but
⊣ M : σ.
Proof. Take M ≡ λxy.y,M ≡ SK, σ ≡ α→(β→β) and σ ≡ (β→α)→(β→β);
do Exercise 5.1. □
All typable terms have a normal form. In fact, the so-called strong nor-
malization property holds: if M is a typable term, then all reductions starting
from M are finite.
Decidability of type assignment
For the system of type assignment several questions may be asked. Note that
for Γ = {x
1
:σ
1
,... ,x
n
:σ
n
} one has
Γ ⊣ M : σ ⇔ ⊣ (λx
1
:σ
1
··· λx
n
:σ
n
.M) : (σ
1
→··· →σ
n
→σ),
therefore in the following one has taken Γ = ∅. Typical questions are
(1) Given M and σ, does one have ⊣ M : σ?
(2) Given M, does there exist a σ such that ⊣ M : σ?
(3) Given σ, does there exist an M such that ⊣ M : σ?

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Type Assignment
39
These three problems are called type checking, typability and inhabitation re-
spectively and are denoted by M : σ?, M : ? and ? : σ.
Type checking and typability are decidable. This can be shown using the
following result, independently due to Curry (1969), Hindley (1969), and Milner
(1978).
5.13. Theorem. (i) It is decidable whether a term is typable in λ→.
(ii) If a term M is typable in λ→, then M has a principal type scheme, i.e.
a type σ such that every possible type for M is a substitution instance of σ.
Moreover σ is computable from M.
5.14. Corollary. Type checking for λ→ is decidable.
Proof. In order to check M : τ it suffices to verify that M is typable and that
τ is an instance of the principal type of M. □
For example, a principal type scheme of K is α→β→α.
Polymorphism
Note that in λ→ one has
⊣ I : σ→σ
for all σ ∈ T.
In the polymorphic lambda calculus this quantification can be internalized by
stating
⊣ I : ∀α.α→α.
The resulting system is the polymorphic of second-order lambda calculus due
to Girard (1972) and Reynolds (1974).
5.15. Definition. The set of types of λ2 (notation T = Type(λ2)) is specified
by the syntax
T = V | B | T→T | ∀V.T.
5.16. Definition. The rules of type assignment are those of λ→, plus
M : ∀α.σ
M : σ[α := τ]
M : σ
M : ∀α.σ
In the latter rule, the type variable α may not occur free in any assumption on
which the premiss M : σ depends.
5.17. Example. (i) ⊣ I : ∀α.α→α.
(ii) Define Nat ≡ ∀α.(α→α)→α→α. Then for the Church numerals c
n
≡
λfx.f
n
(x) we have ⊣ c
n
: Nat.
The following is due to Girard (1972).
5.18. Theorem. (i) The Subject Reduction property holds for λ2.
(ii) λ2 is strongly normalizing.
Typability in λ2 is not decidable; see Wells (1994).

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40
Introduction to Lambda Calculus
Exercises
5.1.
(i) Give a derivation of
⊣ SK : (α→β)→(α→α).
(ii) Give a derivation of
⊣ KI : β→(α→α).
(iii) Show that ⊣ SK : (α→β→β).
(iv) Find a common β-reduct of SK and KI. What is the most general type for
this term?
5.2.
Show that λx.xx and KI(λx.xx) have no type in λ→.
5.3.
Find the most general types (if they exist) for the following terms.
(i) λxy.xyy.
(ii) SII.
(iii) λxy.y(λz.z(yx)).
5.4.
Find terms M, N ∈ Λ such that the following hold in λ→.
(i) ⊣ M : (α→β)→(β→γ)→(α→γ).
(ii) ⊣ N : (((α→β)→β)→β)→(α→β).
5.5.
Find types in λ2 for the terms in the exercises 5.2 and 5.3.

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Page 41

Chapter 6
Extensions
In Chapter 3 we have seen that all computable functions can be expressed
in the lambda calculus. For reasons of efficiency, reliability and convenience
this language will be extended. The set of λ-terms Λ will be extended with
constants. Some of the constants are selected to represent primitive data (such
as numbers) and operations on these (such as addition). Some new reduction
rules (the so called δ-rules) are introduced to express the operational semantics
of these operations. Even if these constants and operations can be implemented
in the lambda calculus, it is worthwhile to have primitive symbols for them.
The reason is that in an implementation of the lambda calculus addition of the
Church numerals runs less efficient than the usual implementation in hardware
of addition of binary represented numbers. Having numerals and addition as
primitives therefore creates the possibility to interprete these efficiently.
From now on we allow constants in λ-terms. Let C be a set of constants.
6.1. Definition. The set of lambda terms with constants, notation Λ(C), is
defined inductively as follows.
C ∈ C ⇒ C ∈ Λ(C),
x ∈ V
⇒ x ∈ Λ(C),
M,N ∈ Λ(C) ⇒ (MN) ∈ Λ(C),
M ∈ Λ(C),x ∈ V
⇒ (λx.M) ∈ Λ(C).
This definition given as an abstract syntax is as follows.
Λ(C) ::= C | V | Λ(C)Λ(C) | λV Λ(C).
6.2. Definition (δ-reduction). Let X ⊆ Λ(C) be a set of closed normal forms.
Usually we take X ⊆ C. Let f : X
k
→Λ be an ‘externally defined’ function. In
order to represent f, a so-called δ-rule may be added to the λ-calculus. This is
done as follows.
(1) A special constant in C is selected and is given some name, say δ (= δ
f
).
(2) The following contraction rules are added to those of the λ-calculus:
δM
1
··· M
k
→ f(M
1
,... ,M
k
),
for M
1
,... ,M
k
∈ X.
41

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42
Introduction to Lambda Calculus
Note that for a given function f this is not one contraction rule but in fact
a rule schema. The resulting extension of the λ-calculus is called λδ. The
corresponding notion of (one step) reduction is denoted by (→
βδ
) →→
βδ
.
So δ-reduction is not an absolute notion, but depends on the choice of f.
6.3. Theorem (G. Mitschke). Let f be a function on closed normal forms.
Then the resulting notion of reduction →→
βδ
satisfies the Church-Rosser prop-
erty.
Proof. Follows from Theorem 15.3.3 in Barendregt (1984). □
The notion of normal form generalises to βδ-normal form. So does the
concept of leftmost reduction. The βδ-normalforms can be found by a leftmost
reduction (notation →→
lβδ
).
6.4. Theorem. If M →→
βδ
N and N is in βδ-nf, then M →→
lβδ
N.
Proof. Analogous to the proof of the theorem for β-normal forms (4.22). □
6.5. Example. One of the first versions of a δ-rule is in Church (1941). Here
X is the set of all closed normal forms and for M,N ∈ X we have
δ
C
MN → true,
if M ≡ N;
δ
C
MN → false,
if M ≡ N.
Another possible set of δ-rules is for the Booleans.
6.6. Example. The following constants are selected in C.
true,false,not,and,ite (for if then else).
The following δ-rules are introduced.
not true → false;
not false → true;
and true true → true;
and true false → false;
and false true → false;
and false false → false;
ite true → true (≡ λxy.x);
ite false → false (≡ λxy.x).
It follows that
ite true xy →→ x,
ite false xy →→ y.

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Extensions
43
Now we introduce as δ-rules some operations on the set of integers
Z = {... ,−2,−1,0,1,2,...}.
6.7. Example. For each n ∈ Z a constant in C is selected and given the name
n. (We will express this as follows: for each n ∈ Z a constant n ∈ C is chosen.)
Moreover the following constants in C are selected:
plus,minus,times,divide,equal,error.
Then we introduce the following δ-rules (schemes). For m,n ∈ Z
plus nm → n + m;
minus nm → n − m;
times nm → n ∗ m;
divide nm → n ÷ m, if m = 0;
divide n0 → error;
equal nn → true;
equal nm → false, if n = m.
We may add rules like
plus nerror → error.
Similar δ-rules can be introduced for the set of reals.
Again another set of δ-rules is concerned with characters.
6.8. Example. Let Σ be some linearly ordered alphabet. For each symbol
s ∈ Σ we choose a constant ‘s’ ∈ C. Moreover we choose two constants δ
≤
and
δ
=
in C and formulate the following δ-rules.
δ
≤
‘s
1
’‘s
2
’ → true,
if s
1
precedes s
2
in the ordering of Σ;
δ
≤
‘s
1
’‘s
2
’ → false,
otherwise.
δ
=
‘s
1
’‘s
2
’ → true,
if s
1
= s
2
;
δ
=
‘s
1
’‘s
2
’ → false,
otherwise.
It is also possible to represent ‘multiple valued’ functions F by putting as
δ-rule
δn → m, provided that F(n) = m.
Of course the resulting λδ-calculus does not satisfy the Church-Rosser theorem
and can be used to deal with non-deterministic computations. We will not
pursue this possibility, however.
We can extend the type assignment system λ→ to deal with constants by
adding typing axioms of the form
C : σ.

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44
Introduction to Lambda Calculus
For the system with integers this would result in the following. Let Z,B ∈ B
be basic type constants (with intended interpretation Z and booleans, respec-
tively). Then one adds the following typing axioms to λ→.
true : B, false : B,
not : B→B, and : B→B→B,
n : Z, error : Z,
plus : Z→Z→Z, minus : Z→Z→Z, times : Z→Z→Z, divide : Z→Z→Z,
equal : Z→Z→B.
6.9. Example. ⊣ λxy.times x(plus xy) : Z→Z→Z, as is shown by the follow-
ing derivation.
times : Z→Z→Z
x : Z
(2)
times x : Z→Z
plus : Z→Z→Z
x : Z
(2)
plus x : Z→Z
y : Z
(1)
plus xy : Z
times x(plus xy) : Z
(1)
λy.times x(plus xy) : Z→Z
(2)
λxy.times x(plus xy) : Z→Z→Z
The Strong Normalization property for (plain) λ→ implies that not all re-
cursive functions are definable in the system. The same holds for the above
λδ-calculus with integers. The following system of type assignment is such
that all computable functions are representable by a typed term. Indeed, the
system also assigns types to non-normalizing terms by introducing a primitive
fixedpoint combinator Y having type (σ→σ)→σ for every σ.
6.10. Definition. (i) The λY δ-calculus is an extension of the λδ-calculus in
which there is a constant Y with reduction rule
Y f → f(Y f).
(ii) Type assignment to λY δ-terms is defined by adding the axioms
Y : (σ→σ)→σ
for each σ ∈ T. The resulting system is denoted by λY δ→.
Because of the presence of Y , not all terms have a normal form. Without
proof we state the following.
6.11. Theorem. (i) The λY δ-calculus satisfies the Church-Rosser property.
(ii) If a term in the λY δ-calculus has a normal form, then it can be found
using leftmost reduction.
(iii) The Subject Reduction property holds for λY δ→.

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Extensions
45
6.12. Theorem. All computable functions can be represented in the λY δ-
calculus by a term typable in λY δ→.
Proof. The construction uses the primitive numerals n. If we take S
+
Y δ
≡
λx.plus x1, P
−
Y δ
≡ λx.minus x1, and Zero
Y δ
≡ λx.equal x0, then the proof
of Theorem 3.13 can be imitated using Y instead of the fixedpoint combinator
Y. The types for the functions defined using Y are natural. □
One could also add Y to the system λ2 using the (single) axiom
Y : ∀α.(α→α)→α.
Exercises
6.1.
Let k
n
be defined by k
0
≡ I and k
n+1
≡ K(k
n
). Show that on the k
n
the
recursive functions can be represented by terms in the λδ
C
-calculus.
6.2.
Write down a λδ-term F in the system of Example 6.7 such that
Fn →→ n! + n.
6.3.
Write down a λδ-term F in the system of Example 6.8 such that for s
1
, s
2
, t
1
, t
2
∈
Σ we have
F[‘s
1
’, ‘t
1
’][‘s
2
’, ‘t
2
’] →→ true,
if (s
1
, t
1
) precedes (s
2
, t
2
) in the
lexicographical ordering of Σ × Σ;
→→ false,
otherwise.
6.4.
Give suitable typing axioms (in λ→ and λ2) for the constants in Example 6.6.

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Page 47

Chapter 7
Reduction Systems
In this chapter we consider some alternative models of computation based on
rewriting. The objects in these models are terms built up from constants with
arity in N and variables, using application.
7.1. Definition. Let C be a set of constants. The set of terms over C (notation
T = T (C)) is defined as follows.
x ∈ V
⇒ x ∈ T ,
C ∈ C,t
1
... ,t
k
∈ T
⇒ C(t
1
,... ,t
n
) ∈ T ,
where n = arity(C).
Recursive programming schemes
The simplest reduction systems are recursive programming schemes (RPS).
The general form of an RPS has as language the terms T (C). On these a
reduction relation is defined as follows.
C
1
(x
1
,... ,x
n
1
) → t
1
,
.
.
.
C
k
(x
1
,... ,x
n
k
) → t
k
,
where n
i
= arity(C
i
). Here we have
(1) The C’s are all different constants.
(2) The free variables in t
i
are among the x
1
,... ,x
n
i
.
(3) In the t’s there may be arbitrary C’s.
For example, the system
C(x,y) → D(C(x,x),y),
D(x,y) → C(x,D(x,y))
is an RPS.
The λ-calculus is powerful enough to ‘implement’ all these RPS’s. We can
find λ-terms with the specified reduction behaviour.
47

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Page 48

48
Introduction to Lambda Calculus
7.2. Theorem. Each RPS can be represented in λ-calculus. For example (see
above), there are terms C and D such that
Cxy →→
β
D(Cxx)y,
Dxy →→
β
Cx(Dxy).
Proof. By the reducing variant 4.27 of the Multiple Fixedpoint Theorem. □
Without proof we mention the following.
7.3. Theorem. Every RPS satisfies the Church-Rosser theorem.
Term rewrite systems
More general than the RPS’s are the so called term rewrite systems (TRS’s),
which use pattern matching in function definitions. A typical example is
A(0,y) → y,
A(S(x),y) → S(A(x,y)).
Then, for example, A(S(0),S(S(0))) →→ S(S(S(0))).
The difference with RPS’s is that in a TRS the arguments of a rewrite rule
may have some structure. A constant in a TRS that does not have a contraction
rule (i.e. no rewrite rule starts with that constant) is called a constructor. The
other constants are called functions.
Not all TRS’s satisfy the Church-Rosser property. Consider the system
A(x) → B,
A(B) → C.
Then A(B) reduces both to B and to C. It is said that the two rules overlap.
The following rule overlaps with itself:
D(D(x)) → E.
Then D(D(D(D))) reduces to E and to D(E).
See Klop (1992) for a survey and references on TRS’s.
Combinatory logic (CL) is a reduction system related to λ-calculus. Terms
in CL consist of (applications of) constants I, K, S and variables, without
arity restrictions. The contraction rules are
Ix → x,
Kxy → x,
Sxyz → xz(yz).
(Note that KI is a nf.) Then KII → I, and SII(SII) has no normal
form. This CL can be represented as a TRS by considering I,K,S as (0-ary)
constructors, together with a function Ap with arity 2, as follows.
Ap(I,x) → x,
Ap(Ap(K,x),y) → x,
Ap(Ap(Ap(S,x),y),z) → Ap(Ap(x,z),Ap(y,z)).

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Page 49

Reduction Systems
49
The CL-term SII(SII) is translated into Ω ≡ Ap(ω,ω) where ω ≡ Ap(Ap(S,I),I).
The Normalization Theorem does not extend to TRS’s. Consider the above
TRS-version of CL, together with the rules
or(x,true) → true,
or(true,x) → true,
or(false,false) → false.
The expression
or(A,B)
can, in general, not be normalized by contracting always the leftmost redex. In
fact A and B have to be evaluated in parallel. Consider e.g. the terms
or(Ω,Ap(I,true))
and
or(Ap(I,true),Ω).
Therefore this system is called non-sequential.
Combinatory reduction systems
Even more general than TRS’s are the combinatory reduction systems (CRS)
introduced in Klop (1980). These are TRS’s together with arbitrary variable
binding operations. We have in fact
CRS
 
 
 
 
d
d
d
d
λ
TRS
d
d
d
d
 
 
 
 
RPS
Exercises
7.1.
(Toyama et al. (1989a), see also (1989b)) A TRS is called strongly normalizing
(SN) if there is no term that has an infinite reduction path. So (the TRS version
of) CL(S, K) is not SN, but CL(I, K) (with the obvious reduction rules) is.
Define the following two TRS’s.
R
1
:
F(4, 5, 6, x) → F(x, x, x, x),
F(x, y, z, w) → 7,

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50
Introduction to Lambda Calculus
1
2
3
c
d
d
d
d‚
© 
 
 
 
d
d
d
d‚
© 
 
 
 
c
4
5
6
d
d
d
d‚ c© 
 
 
 
7
R
2
:
G(x, x, y) → x,
G(x, y, x) → x,
G(y, x, x) → x.
Show that both R
1
and R
2
are SN, but the union R
1
∪ R
2
is not.

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Page 51

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