Squeeze theorem

Abstract: Introduction Few of those who have seen a modern high-speed digital computer digest and transform a mass of data in less time than it takes to follow the process in the mind can suppress a certain amount of speculation concerning the future of such machines. Under the assumption that the computer is operating at the mere threshhold of its capacity in performing the tasks we have thus far delegated to it, a long-range program directed at the problem of "intelligent" behavior and learning in machines has been established at the IBM Research Center in New York (Gelernter and Rochester, 1958). In particular the technique of heuristic programming is under detailed investigation as a means to the end of applying large-scale digital computers to the solution of a difficult class of problems currently considered to be beyond their capabilities; namely those problems that seem to require the agent of human intelligence and ingenuity for their solution. It is difficult to characterize such problems further, except, perhaps, to remark rather vaguely that they generally involve complex decision processes in a potentially infinite and uncontrollable environment. If, however, we should restrict the universe of problems to those that amount to the discovery of a proof for a theorem in some well-defined formal system, then the distinguishing characteristics of those problems of special interest to us are brought clearly into focus. We should like our machine to be able to prove many of the theorems presented to it in a formal system that is manifestly undecidable. Further, as the machine 134 gains "experience" in proving theorems, we should expect it to be able to solve problems that were earlier beyond its capabilities. The requirement that a machine should deal with undecidable systems places a fundamental restriction on its modus operandi. Finding a suitable algorithm, the obvious technique for the solution of problems on a digital computer, is no longer acceptable for the simple reason that no such algorithm exists. An exhaustive search for the initial axioms and theorems of the proof, combined with exhaustive development of the proof sequence by systematically applying the rules of transformation until the required proof has been produced, has been shown to be much too time-consuming for so simple a logic as propositional calculus (Newell, Shaw and Simon, 1957 a). It is a fortiori out of the question for any of the more interesting logics. A remaining alternative is …