Question

After discovering the Boost preprocessor’s capabilities I found myself wondering: Is the C99 preprocessor Turing complete?

If not, what does it lack to not qualify?

Answer 1

Here is an example of abusing the preprocessor to implement a Turing machine. Note that an external build script is needed to feed the preprocessor’s output back into its input, so the preprocessor in and of itself isn’t Turing complete. Still, it’s an interesting project.

From the description of the afore-linked project:

the preprocessor is not Turing complete, at least not if
the program is preprocessed only once. This is true even if
the program is allowed to include itself. (The reason being
that for a given program, the preprocessor has only a finite
number of states, plus a stack consisting of the places which
the file has been included from. This is only a push-down automaton.)

The answer by Paul Fultz II is quite impressive and certainly closer than I thought the preprocessor could ever get, but it’s not a true Turing machine. The C preprocessor has certain limits that prevent it from executing an arbitrary program like a Turing machine could, even if you had infinite memory and time. Section 5.2.4.1 of the C spec gives the following minimum limits for a C compiler:

• 63 nesting levels of parenthesized expressions within a full expression
• 63 significant initial characters in an internal identifier or a macro name
• 4095 macro identifiers simultaneously defined in one preprocessing translation unit
• 4095 characters in a logical source line

The counter mechanism below requires a macro definition per value, so the macro definition limit will limit how many times you can loop (EVAL(REPEAT(4100, M, ~)) would yield undefined behavior). This essentially puts a cap on the complexity of the program that you can execute. The nesting and complexity of the multi-level expansions may hit one of the other limits as well.

This is fundamentally different than the “infinite memory” limitation. In this case, the spec specifically says that a standards-conforming C compiler is only required to conform to these limits, even if it has infinite time, memory, etc. Any input file exceeding these limits can be processed in an unpredictable or undefined manner (or outright rejected). Some implementations may have higher limits, or no limits at all, but that’s considered “implementation-specific” and not part of the standard. It may be possible to use Paul Fultz II’s method to implement something like a Turing machine on some specific compiler implementation that has no finite limits, but in a general sense of “can this be done on any arbitrary, standards-conforming C99 pre-processor”, the answer is no. Since the limit here is built into the language itself and not simply a side-effect of our inability to construct an infinite computer, I say that breaks Turing completeness.