the language recognized by it, Rice's theorem does not apply. Using Rogers' characterization of acceptable programming systems, Rice's theorem may essentially be generalized from Turing machines to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the behavior of computer programs. As an example, consider the following variant of the halting problem. Let P be the following property of partial functions F of one argument: P(F) means that F is defined for the argument '1'. It is obviously non-trivial, since there are partial functions that are defined at 1, and others that are undefined at 1. The 1-halting problem is the problem of deciding of any algorithm whether it defines a function with this property, i.e., whether the algorithm halts on input 1. By Rice's theorem, the 1-halting problem is undecidable. Similarly the question of whether a Turing machine T terminates on an initially empty tape (rather than with an initial word w given as second argument in addition to a description of T, as in the full halting problem) is still undecidable. Formal statement Let denote the natural numbers, and let denote the class of unary (partial) computable functions. Let be an admissible numbering of the computable functions. Denote by the th (partial) computable function. We identify each property that a computable function may have with the subset of consisting of the functions with that property. Thus, given a set , a computable function has property if and only if . For each property there is an associated membership decision problem of determining, given , whether . Rice's theorem states that the decision problem is decidable (also called recursive or computable) if and only if or . Examples According to Rice's theorem, if there is at least one partial computable function in a particular class C of partial computable functions and another partial computable function not in C then the problem of deciding whether a particular program computes a function in C is undecidable. For example, Rice's theorem shows that each of the following sets of partial computable functions is undecidable (that is, the set is not recursive, or not computable): The class of partial computable functions that return 0 for every input, and its complement. The class of partial computable functions that return 0 for at least one input, and its complement. The class of partial computable functions that are constant,