how type information is encoded in ACL2

To help you experiment with type-sets we briefly note the following utility functions.

(type-set-quote x) will return the type-set of the object x. For example, (type-set-quote "test") is 2048 and (type-set-quote '(a b c)) is 512.

(type-set 'term nil nil nil nil (ens state) (w state) nil nil nil) will return the type-set of term. For example,

(type-set '(integerp x) nil nil nil nil (ens state) (w state) nil nil nil)
will return (mv 192 nil). 192, otherwise known as *ts-boolean*, is the type-set containing t and nil. The second result may be ignored in these experiments. Term must be in the translated, internal form shown by :trans. See trans and see term.

(type-set-implied-by-term 'x nil 'term (ens state)(w state) nil) will return the type-set deduced for the variable symbol x assuming the translated term, term, true. The second result may be ignored in these experiments. For example,

(type-set-implied-by-term 'v nil '(integerp v)
                          (ens state) (w state) nil)
returns 11.

(convert-type-set-to-term 'x ts (ens state) (w state) nil) will return a term whose truth is equivalent to the assertion that the term x has type-set ts. The second result may be ignored in these experiments. For example

(convert-type-set-to-term 'v 523 (ens state) (w state) nil)
returns a term expressing the claim that v is either an integer or a non-nil true-list. 523 is the logical-or of 11 (which denotes the integers) with 512 (which denotes the non-nil true-lists).

The ``actual primitive types'' of ACL2 are listed in *actual-primitive-types*, whose elements are shown below. Each actual primitive type denotes a set -- sometimes finite and sometimes not -- of ACL2 objects and these sets are pairwise disjoint. For example, *ts-zero* denotes the set containing 0 while *ts-positive-integer* denotes the set containing all of the positive integers.

*TS-ZERO*                  ;;; {0}
*TS-POSITIVE-INTEGER*      ;;; positive integers
*TS-POSITIVE-RATIO*        ;;; positive non-integer rationals
*TS-NEGATIVE-INTEGER*      ;;; negative integers
*TS-NEGATIVE-RATIO*        ;;; negative non-integer rationals
*TS-COMPLEX-RATIONAL*      ;;; complex rationals
*TS-NIL*                   ;;; {nil}
*TS-T*                     ;;; {t}
*TS-NON-T-NON-NIL-SYMBOL*  ;;; symbols other than nil, t
*TS-PROPER-CONS*           ;;; null-terminated non-empty lists
*TS-IMPROPER-CONS*         ;;; conses that are not proper
*TS-STRING*                ;;; strings
*TS-CHARACTER*             ;;; characters

The actual primitive types were chosen by us to make theorem proving convenient. Thus, for example, the actual primitive type *ts-nil* contains just nil so that we can encode the hypothesis ``x is nil'' by saying ``x has type *ts-nil*'' and the hypothesis ``x is non-nil'' by saying ``x has type complement of *ts-nil*.'' We similarly devote a primitive type to t, *ts-t*, and to a third type, *ts-non-t-non-nil-symbol*, to contain all the other ACL2 symbols.

Let *ts-other* denote the set of all Common Lisp objects other than those in the actual primitive types. Thus, *ts-other* includes such things as floating point numbers and CLTL array objects. The actual primitive types together with *ts-other* constitute what we call *universe*. Note that *universe* is a finite set containing one more object than there are actual primitive types; that is, here we are using *universe* to mean the finite set of primitive types, not the infinite set of all objects in all of those primitive types. *Universe* is a partitioning of the set of all Common Lisp objects: every object belongs to exactly one of the sets in *universe*.

Abstractly, a ``type-set'' is a subset of *universe*. To say that a term, x, ``has type-set ts'' means that under all possible assignments to the variables in x, the value of x is a member of some member of ts. Thus, (cons x y) has type-set {*ts-proper-cons* *ts-improper-cons*}. A term can have more than one type-set. For example, (cons x y) also has the type-set {*ts-proper-cons* *ts-improper-cons* *ts-nil*}. Extraneous types can be added to a type-set without invalidating the claim that a term ``has'' that type-set. Generally we are interested in the smallest type-set a term has, but because the entire theorem-proving problem for ACL2 can be encoded as a type-set question, namely, ``Does p have type-set complement of *ts-nil*?,'' finding the smallest type-set for a term is an undecidable problem. When we speak informally of ``the'' type-set we generally mean ``the type-set found by our heuristics'' or ``the type-set assumed in the current context.''

Note that if a type-set, ts, does not contain *ts-other* as an element then it is just a subset of the actual primitive types. If it does contain *ts-other* it can be obtained by subtracting from *universe* the complement of ts. Thus, every type-set can be written as a (possibly complemented) subset of the actual primitive types.

By assigning a unique bit position to each actual primitive type we can encode every subset, s, of the actual primitive types by the nonnegative integer whose ith bit is on precisely if s contains the ith actual primitive type. The type-sets written as the complement of s are encoded as the twos-complement of the encoding of s. Those type-sets are thus negative integers. The bit positions assigned to the actual primitive types are enumerated from 0 in the same order as the types are listed in *actual-primitive-types*. At the concrete level, a type-set is an integer between *min-type-set* and *max-type-set*, inclusive.

For example, *ts-nil* has bit position 6. The type-set containing just *ts-nil* is thus represented by 64. If a term has type-set 64 then the term is always equal to nil. The type-set containing everything but *ts-nil* is the twos-complement of 64, which is -65. If a term has type-set -65, it is never equal to nil. By ``always'' and ``never'' we mean under all, or under no, assignments to the variables, respectively.

Here is a more complicated example. Let s be the type-set containing all of the symbols and the natural numbers. The relevant actual primitive types, their bit positions and their encodings are:

actual primitive type       bit    value

*ts-zero* 0 1 *ts-positive-integer* 1 2 *ts-nil* 6 64 *ts-t* 7 128 *ts-non-t-non-nil-symbol* 8 256

Thus, the type-set s is represented by (+ 1 2 64 128 256) = 451. The complement of s, i.e., the set of all objects other than the natural numbers and the symbols, is -452.