ON $λ$-DEFINABILITY OF ARITHMETICAL FUNCTIONS WITH INDETERMINATE VALUES OF ARGUMENTS - Nigiyan, S.A. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2016)
In this paper the arithmetical functions with indeterminate values of arguments are regarded. It is known that every $λ$-definable arithmetical function with indeterminate values of arguments is monotonic and computable. The $λ$-definability of every computable, monotonic, 1-ary arithmetical function with indeterminate values of arguments is proved. For computable, monotonic, $k$-ary, $k ≥ 2$, arithmetical functions with indeterminate values of arguments, the so-called diagonal property is defined. It is proved that every computable, monotonic, $k$-ary, $k ≥ 2$, arithmetical function with indeterminate values of arguments, which has the diagonal property, is not $λ$-definable. It is proved that for any $k ≥ 2$; the problem of $λ$-definability for computable, monotonic, $k$-ary arithmetical functions with indeterminate values of arguments is algorithmic unsolvable. It is also proved that the problem of diagonal property of such functions is algorithmic unsolvable, too.
Keywords: arithmetical functions, indeterminate value of argument, monotonicity, computability, strong computability, $λ$-definability, algorithmic unsolvability