O-P

a recognizer for the ordinals up to epsilon-0
Major Section:  PROGRAMMING

Using the nonnegative integers and lists we can represent the ordinals up to epsilon-0. The ordinal representation used in ACL2 has changed as of Version_2.8 from that of Nqthm-1992, courtesy of Pete Manoilios and Daron Vroon; additional discussion may be found in ``Ordinal Arithmetic in ACL2'', proceedings of ACL2 Workshop 2003, http://www.cs.utexas.edu/users/moore/acl2/workshop-2003/. Previously, ACL2's notion of ordinal was very similar to the development given in ``New Version of the Consistency Proof for Elementary Number Theory'' in The Collected Papers of Gerhard Gentzen, ed. M.E. Szabo, North-Holland Publishing Company, Amsterdam, 1969, pp 132-213.

The following essay is intended to provide intuition about ordinals. The truth, of course, lies simply in the ACL2 definitions of o-p and o<.

Very intuitively, think of each non-zero natural number as by being denoted by a series of the appropriate number of strokes, i.e.,

0             0
1             |
2             ||
3             |||
4             ||||
...           ...
Then ``omega,'' here written as w, is the ordinal that might be written as
w             |||||...,
i.e., an infinite number of strokes. Addition here is just concatenation. Observe that adding one to the front of w in the picture above produces w again, which gives rise to a standard definition of w: w is the least ordinal such that adding another stroke at the beginning does not change the ordinal.

We denote by w+w or w*2 the ``doubly infinite'' sequence that we might write as follows.

w*2           |||||... |||||...
One way to think of w*2 is that it is obtained by replacing each stroke in 2 (||) by w. Thus, one can imagine w*3, w*4, etc., which leads ultimately to the idea of ``w*w,'' the ordinal obtained by replacing each stroke in w by w. This is also written as ``omega squared'' or w^2, or:
 2
w             |||||... |||||... |||||... |||||... |||||... ...
We can analogously construct w^3 by replacing each stroke in w by w^2 (which, it turns out, is the same as replacing each stroke in w^2 by w). That is, we can construct w^3 as w copies of w^2,
 3              2       2       2       2
w              w  ...  w  ...  w  ...  w ... ...
Then we can construct w^4 as w copies of w^3, w^5 as w copies of w^4, etc., ultimately suggesting w^w. We can then stack omegas, i.e., (w^w)^w etc. Consider the ``limit'' of all of those stacks, which we might display as follows.
       .
      .
     .
    w
   w
  w
 w
w
That is epsilon-0.

Below we begin listing some ordinals up to epsilon-0; the reader can fill in the gaps at his or her leisure. We show in the left column the conventional notation, using w as ``omega,'' and in the right column the ACL2 object representing the corresponding ordinal.

  ordinal            ACL2 representation

0 0 1 1 2 2 3 3 ... ... w '((1 . 1) . 0) w+1 '((1 . 1) . 1) w+2 '((1 . 1) . 2) ... ... w*2 '((1 . 2) . 0) (w*2)+1 '((1 . 2) . 1) ... ... w*3 '((1 . 3) . 0) (w*3)+1 '((1 . 3) . 1) ... ...

2 w '((2 . 1) . 0) ... ...

2 w +w*4+3 '((2 . 1) (1 . 4) . 3) ... ...

3 w '((3 . 1) . 0) ... ...

w w '((((1 . 1) . 0) . 1) . 0) ... ...

w 99 w +w +w4+3 '((((1 . 1) . 0) . 1) (99 . 1) (1 . 4) . 3) ... ...

2 w w '((((2 . 1) . 0) . 1) . 0)

... ...

w w w '((((((1 . 1) . 0) . 1) . 0) . 1) . 0) ... ...

Observe that the sequence of o-ps starts with the natural numbers (which are recognized by natp). This is convenient because it means that if a term, such as a measure expression for justifying a recursive function (see o<) must produce an o-p, it suffices for it to produce a natural number.

The ordinals listed above are listed in ascending order. This is the ordering tested by o<.

The ``epsilon-0 ordinals'' of ACL2 are recognized by the recursively defined function o-p. The base case of the recursion tells us that natural numbers are epsilon-0 ordinals. Otherwise, an epsilon-0 ordinal is a list of cons pairs whose final cdr is a natural number, ((a1 . x1) (a2 . x2) ... (an . xn) . p). This corresponds to the ordinal (w^a1)x1 + (w^a2)x2 + ... + (w^an)xn + p. Each ai is an ordinal in the ACL2 representation that is not equal to 0. The sequence of the ai's is strictly decreasing (as defined by o<). Each xi is a positive integer (as recognized by posp).

Note that infinite ordinals should generally be created using the ordinal constructor, make-ord, rather than cons. The functions o-first-expt, o-first-coeff, and o-rst are ordinals destructors. Finally, the function o-finp and the macro o-infp tell whether an ordinal is finite or infinite, respectively.

The function o< compares two epsilon-0 ordinals, x and y. If both are integers, (o< x y) is just x<y. If one is an integer and the other is a cons, the integer is the smaller. Otherwise, o< recursively compares the o-first-expts of the ordinals to determine which is smaller. If they are the same, the o-first-coeffs of the ordinals are compared. If they are equal, the o-rsts of the ordinals are recursively compared.

Fundamental to ACL2 is the fact that o< is well-founded on epsilon-0 ordinals. That is, there is no ``infinitely descending chain'' of such ordinals. See proof-of-well-foundedness.