In mathematical logic, G¨odel numbering refers to a function that assigns to each well-formed formula of some formal language a unique natural number called its G¨odel number. This concept was introduced by the logician Kurt G¨odel (1906-1978) for the proof of incompleteness of elementary arithmetic (1931).
Here G¨odel numbering is used to provide an encoding of GOTO programs.
For this, let N∗0 denote the union of all cartesian products Nk0, k ≥ 0. In particular, N00 = {ǫ}, where ǫis the empty string. The Cantor pairing function J2 :N20→N0 is used to define an encoding J :N∗0→N0 as follows:
J(ǫ) = 0, (5.1)
J(x) =J2(0, x) + 1, x∈N0, (5.2)
J(x, y) =J2(J(x), y) + 1, x∈N∗0, y∈N0. (5.3) Note that the second equation is a special case of the third one, since for eachy∈N0,
J(ǫ, y) =J2(J(ǫ), y) + 1 =J2(0, y) + 1 =J(y). (5.4) Example 5.1.We have
J(1,3) =J2(J(1),3) + 1 =J2(J2(0,1) + 1,3) + 1 =J2(2,3) + 1 = 17 + 1 = 18.
♦ Proposition 5.2.The encoding function J is a primitive recursive bijection.
Proof. First, claim thatJ is primitive recursive. Indeed, the functionJ is primitive recursive for strings of length≤1, sinceJ2 is primitive recursive. Assume thatJ is primitive recursive for strings of length
≤k, wherek≥1. For strings of lengthk+1, the functionJ can be written as a composition of primitive recursive functions:
J =ν◦J2(J(π(k+1)1 , . . . , πk(k+1)), πk+1(k+1)). (5.5) By induction hypothesis,Jis primitive recursive for strings of length≤k+1 and thus the claim follows.
Second, let A be the set of all numbers n ∈ N0 such that there is a unique string x ∈ N∗0 with J(x) =n. Claim thatA=N0. Indeed, 0 lies inAsinceJ(ǫ) = 0 andJ(x)>0 for allx6=ǫLetn >0 and assume that the assertion holds for all numbersm < n. Define
u=K2(n−1) and v=L2(n−1). (5.6)
Then J2(u, v) = J2(K2(n−1), L2(n−1)) =n−1. By construction,K2(z)≤z and L2(z)≤z for all z ∈N0. Thusu=K2(n−1)< nand hence u∈A. By induction, there is exactly one stringx∈Nk0 such thatJ(x) =u. ThenJ(x, v) =J2(J(x), v)) + 1 =J2(u, v) + 1 =n.
Assume that J(x, v) = n = J(y, w) for some x,y ∈ N∗0 and v, w ∈ N0. Then by definition, J2(J(x), v) = J2(J(y), w). But the Cantor pairing function is bijective and thus J(x) = J(y) and v=w. SinceJ(x)< nit follows by induction thatx=y. Thusn∈Aand so by the induction axiom,
A=N0. It follows thatJ is bijective. ⊓⊔
The encoding functionJ gives rise to two functionsK, L:N0→N0 defined as
K(n) =K2(n−˙ 1) and L(n) =L2(n−˙ 1), n∈N0. (5.7) Note that the following marginal conditions hold:
K(1) =K(0) =L(0) =L(1) = 0. (5.8)
Proposition 5.3.The functions K and L are primitive recursive, and for each number n≥1, there are unique x∈N∗0 andy∈N0 withJ(x, y) =nsuch that
J(x) =K(n) and y=L(n). (5.9)
Proof. By Proposition 2.27, the functions K2 and L2 are primitive recursive and so K and L are primitive recursive, too.
Let n≥1. By Proposition 5.2, there are unique x∈N∗0 andy ∈ N0 such thatJ(x, y) =n. Thus J2(J(x), y) =n−1. ButJ2(K2(n−1), L2(n−1)) =n−1 and the Cantor pairing functionJ2is bijective.
ThusK2(n−1) =J(x) andL2(n−1) =y, and soK(n) =J(x) andL(n) =y. ⊓⊔ The length of a string can be determined by its encoding.
Proposition 5.4.If xis a string inN∗0 of lengthk, thenkis the smallest number such that Kk(J(x)) = 0.
Proof. In view of the empty string, K0(J(ǫ)) = J(ǫ) = 0. Let x =x1. . . xk be a non-empty string.
ThenJ(x) =n≥1. By Proposition 5.3,J(x1. . . xk−1) =K(n). By induction,J(x1. . . xi) =Kk−i(n) for each 0 ≤i ≤k. In particular, for each 0≤ i < k, Kk−i(n) is non-zero, since x1. . . xi is not the empty string. Moreover,Kk(n) =J(ǫ) = 0. The result follows. ⊓⊔
Note that the converse of the above assertion is also valid. As an example, take a numberx∈N0, a string of length one. Then by (5.4),J(ǫ, x) =J(x) =J2(0, x) + 1 =n≥1 and so by (5.9), K(n) =J(ǫ) = 0 andL(n) =x.
In view of Proposition 5.4, define the mapping f :N∗0×N0→N0 as
f : (x, k)7→Kk(J(x)). (5.10)
Minimalization of this function yield thelength functionlg :N∗0→N0 given by lg(x) =µf(x) =
kifksmallest withf(x, k) = 0 andf(x, i)>0 for 0≤i≤k,
↑ otherwise. (5.11)
Proposition 5.5.The length functionlg is primitive recursive.
Proof. The length function is partial recursive, since it is obtained by minimalization of a primitive recursive function. But each string x has finite length and so the repeated application of K to J(x) yields 0 after a finite number of steps. Hence, the length function is total. Moreover, the length of a stringx=x1. . . xnis not larger than its decimal value which is less than 10nand so the minimalization is actually bounded. Hence, the length function is primitive recursive. ⊓⊔ Proposition 5.6.Let n≥1. The inverse value J−1(n)is given by
(Kk−1(n), L◦Kk−2(n), . . . , L◦K(n), L(n)), (5.12)
The primitive recursive bijection J allows to encode the standard GOTO programs, SGOTO pro-grams for short. These are GOTO propro-gramsP =s0;s1;. . .;sq that have a canonical labelling in the sense thatλ(sl) =l, 0≤l≤q. It is clear that for each GOTO program there is a (semantically) equiv-alent SGOTO program. SGOTO programs are used in the following since they will permit a slightly simpler G¨odel numbering than arbitrary GOTO programs. In the following, let PSGOTO denote the class of SGOTO programs.
The numberI(sl) is called theG¨odel numberof the instructionsl, 0≤l≤q. The functionIis primitive recursive, since it is defined by cases and the functions involved are primitive recursive.
Note that the function I allows to identify the lth instruction of an SGOTO program given its G¨odel numbere. Indeed, the residue ofemodulo 3 provides the type of instruction and the quotient of emodulo 3 gives the encoding of the parameters of the instruction. More concretely, write
e= 3n+t, (5.14)
wheren=÷(e,3) andt= mod(e,3). Then the instruction can be decoded by using Proposition 5.6 as follows: Proposition 5.7.The functionΓ :PSGOTO→N0 is bijective and primitive recursive.
Proof. The mappingΓ is bijective sinceJ is bijective and the instructions encoded byI are uniquely determined as shown above. Moreover, the functionΓ is a composition of primitive recursive functions
and thus is primitive recursive. ⊓⊔
The SGOTO programP with G¨odel numbereis denoted byPe. This G¨odel numbering provides a list of all SGOTO programs
P0, P1, P2, . . . . (5.17)
Conversely, each numberecan be assigned the SGOTO program P such thatΓ(P) =e. For this, the length of the string encoded bye is first determined by the minimalization given in Proposition 5.4.
Suppose the string has lengthn+ 1, wheren≥0. Then the task is to find x∈Nn0 and y ∈N0 such that J(x, y) =n. But by (5.9),K(n) =J(x) and L(n) =y and so the preimage ofn underJ can be repeatedly determined. Finally, when the string is given, the preimage (instruction) of each number is established as described in (5.15).
For each number eand each numbern≥0, denote the n-ary partial recursive function computing the SGOTO program with G¨odel numbereby
φ(n)e =kPekn,1. (5.18)
If f is ann-ary partial recursive function, each number e ∈N0 with the propertyf =φ(n)e is called anindex of f. The index of a partial recursive functionf provides the G¨odel number of an SGOTO program computing it. The list of all SGOTO program in (5.17) yiels a list of alln-ary partial recursive functions:
φ(n)0 , φ(n)1 , φ(n)2 , . . . . (5.19) Note that the list contains repetitions, since eachn-ary partial recursive function has infinitely many indices.