Generalizations of Rice's Theorem, Applicable to Executable and Non-Executable Formalisms

13 pages•Published: June 22, 2012

Abstract

We formulate and prove two Rice-like theorems
that characterize limitations on nameability of properties
within a given naming scheme for partial functions.
Such a naming scheme can, but need not be, an executable formalism.
A programming language is an example of an executable naming scheme,
where the program text names the partial function it implements.
Halting is an example of a property
that is not nameable in that naming scheme.

The proofs reveal requirements on the naming scheme
to make the characterization work.
Universal programming languages satisfy these requirements,
but also other formalisms can satisfy them.
We present some non-universal programming languages
and a non-executable specification language
satisfying these requirements.
Our theorems have
Turing's well-known Halting Theorem and Rice's Theorem as special cases,
by applying them to a universal programming language or Turing Machines as naming scheme.
Thus, our proofs separate the nature of the naming scheme
(which can, but need not, coincide with computability) from the diagonal argument.
This sheds further light on how far reaching and simple the `diagonal' argument is in itself.

Keyphrases: execution based computation formalisms, generalized rice theorem, halting theorem, specification based computation formalisms

In: Andrei Voronkov (editor). Turing-100. The Alan Turing Centenary, vol 10, pages 168-180.

Links:	https://easychair.org/publications/paper/QNGJ
	https://doi.org/10.29007/jnl6