Non-computable Functions Research Articles

Et and an infinitary church’s thesis

Martin Davis, in Computability & Unsolvability (1958), said of Church's thesis, "For how can we ever exclude the possibility of our being presented, some day (perhaps by some extraterrestrial visitors), with a (perhaps extremely complex) device or 'oracle' that 'computes' a noncomputable function?" The recent television revival of the Stephen Spielberg film ET presented a script sadly deficient with respect to Davis's question and to mathematics generally. To be fair, this film was made for American juveniles, for whom any mathematical references would have been mysterious. Still, one can imagine a scenario for an actual ET visit. In this scenario, the ET landing occurs at Princeton, NJ, and the colloquy takes place in English, between ET and a mathematician from the Institute for Advanced Study (who agreed reluctantly to be the designated Presidential Representative only after assurance that he would not first have to be approved by any Senate committee) .. . ET: You have some questions for me?

Strong time bounds: Non-computable bounds and a hierarchy theorem

A RAM program is said to run within a “strong” time bound T if on every sequence of n inputs it terminates within T( n) instruction executions. There are some programs whose execution time in this sense is a non-computable function of n. It is shown that such programs are essential in the sense that some functions can be computed within a non-computable time bound but not within any computable time bound. Nevertheless, strong time bounds are subject to a powerful hierarchy theorem. The condition such as being time constructable which normally applies to the “lower” function in such theorems is replaced by a condition of being the minimum strong time bound for some program.

Pictures of functions and their acceptability by automata

Pictures of functions of one variable in the natural number theory are introduced as patterns over {0,1} in the first quadrant. It is proved that the set of pictures of all computable functions of one variable is obtained as the bottom planes of three-dimensional arrays accepted by some nondeterministic automaton on ω3-tapes. To our surprise, it is also shown that a set of pictures of functions containing a noncomputable function is deterministically obtained in a similar way. Further, it is proved that the set of initial segments of pictures of all computable functions and/or the noncomputable function is accepted by a deterministic Turing rectangular array acceptor.

The Determination of the Value of Rado's Noncomputable Function | sum(k) for Four-State Turing Machines

The Determination of the Value of Rado's Noncomputable Function | sum(k) for Four-State Turing Machines

New Results for Rado's Sigma Function for Binary Turing Machines

A computer program was written and executed to search for better lower bounds to Rado's noncomputable sigma and shift functions for binary Turing machines. Former results in this search (called by Rado the Busy Beaver logical game) are reviewed and new bounds found by this program are presented.

High Speed Binary Parallel Adder

A study of the output of a heuristic computer program reveals two four-state binary Turing machines which yield the highest known score for four states in Rado's co-called Busy Beaver logical game. There is evidence which supports the conjecture that this score of 13 is the particular value of ?(4), where ?is a noncomputable integer function associated with this game. It is also conjectured that S(4) = 106, where S is another noncomputable function, the maximum shift number, of interest in Rado's study. Complete solution of the problem for four states has been reduced to a relatively small set of machines.

The Conjectured Highest Scoring Machines for Rado's Σ(k) for the Value k = 4

A study of the output of a heuristic computer program reveals two four-state binary Turing machines which yield the highest known score for four states in Rado's co-called Busy Beaver logical game. There is evidence which supports the conjecture that this score of 13 is the particular value of Σ(4), where Σ is a noncomputable integer function associated with this game. It is also conjectured that S(4) = 106, where S is another noncomputable function, the maximum shift number, of interest in Rado's study. Complete solution of the problem for four states has been reduced to a relatively small set of machines.

On Non-Computable Functions

The construction of non-computable functions used in this paper is based on the principle that a finite, non-empty set of non-negative integers has a largest element. Also, this principle is used only for sets which are exceptionally well-defined by current standards. No enumeration of computable functions is used, and in this sense the diagonal process is not employed. Thus, it appears that an apparently self-evident principle, of constant use in every area of mathematics, yields non-constructive entities.