This theory defines a semantics for lambda-terms in the sense that it defines the
meaning of a
term` M `as the
(equivalence) class of all terms lambda-convertible to ` M`.

It will be useful to introduce some preliminary notions.

These concepts of binder and scope are similar to the concepts of declaration and scope in programming langauges like for instance C, Pascal, or ML.

For example, consider the term

+++ (\x. ... (\x. --- ) *** ) ^^^The parts ... and *** are in the scope of the outermost \x. The part --- is in the scope of the innermost \x. The parts +++ and ^^^ are out of the scope of both \x.

An occurrence of x is* free *in M
if it is not in the scope of any binder for x.

For example, consider the term

(\x. x y (\x.\y. y z x ) x ) y xWe have 4 occurrences of x, of which only the last is free. The first and the third are bound by the outermost \x and the second is bound by the innermost \x. As for the other variables, we have 3 occurrences of y, of which the first and the last are free, and we have only one occurrence of z, free.

The free variables of M are all the variables which occur free in M. Formally:

FV(x) = {x}

FV(\x.M) = FV(M) - {x}

FV(M N) = FV(M) union FV(N)

x[N/x] = N

y[N/x] = y

(\y.M)[N/x] = \y.(M[N/x])

(\x.M)[N/x] = \x.M

(M P)[N/x] = (M[N/x])(P[N/x])

The condition "N does not contain free variables which are bound in M" is meant to avoid "variable capture". For example, given M = \y.zx, and N = y, if we simply replaced x by N in M, we would obtain the term \y.zy, which is intuitively incorrect since y, which was free in N, would become bound in M[N/x]. (Or in other words, M was constant on y and would become not constant.)

\x.M =provided that y is neither free nor bound in M._{alpha}\y.M[y/x]

(\x.M) N =provided that N does not contain free variables which is bound in M (in a subterm containing x), so to avoid variable capture._{beta}M[N/x]

Example: (\x. x+1) 5 =_{beta} 5+1 = 6

Variable assumption: To avoid the "variable capture" problem, and more in general, confusion with names, we will assume that all variables that occur free have names different from those of the bound variables, and that all bound variables have different names as well. Note that this assumption is not restrictive since we can always alpha-rename the bound variables.

For example, when we have a term like

x (\x.x(\x.x)x)we will write, instead

x (\y.y(\z.z)y)

Another solution would be to define the notion of substitution so that it renames the variables automatically when needed. For this approach see for instance Hindley and Seldin, Definition 1.1.

- M = N => M P = N P (Congruence 1)
- M = N => P M = P N (Congruence 2)
- M = N => \x.M = \x.N (Xi rule)

As a definition of computable function, we will use the system of "Recursive Functions" defined by Gödel in 1931.

- Zero: Z(n) = 0
- Successor: S(n) = n+1
- Projection:
U
^{p}_{i}(n_{1},...,n_{p}) = n_{i}

(f o (gExample: the function f(n) = n + 2 can be defined as the composition of the successor function with itself, namely f = S o S._{1},...,g_{k})) (n_{1},...,n_{h}) = f(g_{1}(n_{1},...,n_{h}),..., g_{k}(n_{1},...,n_{h}))

f(0,nExamples:_{1},...,n_{k}) = g(n_{1},...,n_{k})

f(n+1,n_{1},...,n_{k}) = h(n, f(n,n_{1},...,n_{k}), n_{1},...,n_{k})

- The function plus(n,m) = n + m can be defined by primitive recursion from g = U
^{1}_{1}and h = S o U^{3}_{2}. - The function times(n,m) = n * m can be defined by primitive recursion from g = Z
and h = plus o (U
^{3}_{3}, U^{3}_{2}).

f(nwhere mu is the minimalization operator: mu n. P(n) returns the least natural number n such that P(n) holds. If such n does not exists, then the result is undefined._{1},...,n_{k}) = mu n. (g(n,n_{1},...,n_{k}) = 0)

Note that f could be computed by general iteration, i.e. using a while loop of the form

n := 0; while (g(n,n_{1},...,n_{k}) <> 0) do n := n+1;

Example: Consider the function
log_{2} n = k, where k is the number, if it exists, such that
2^{k} = n (the result is undefined otherwise).
Then log_{2} can be defined by minimalization from the function
g(k,n) = | 2^{k} - n |, where | m | represents the absolute value of m.

**Definition** The set of *Recursive Functions* is the
smallest set which contains the initial functions and is closed w.r.t.
composition, primitive recursion, and minimalization.

If we exclude minimalization from this construction, we get the set of
*Primitive Recursive Fuctions*, which are all total.
Minimalization is the only operator which introduces non-termination, i.e.
partiality.