Halting Problem

We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language.

We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'.

We then adopt what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective.

We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways.

We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N.

We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences.

The Turing Machine (TM) presents itself as the very landmark and initial design of digital automata present in all modern general-purpose digital computers and whose design on computable numbers implies deeply ontological as well as epistemological foundations for today's computers. These lines of work attempt to briefly analyze the fundamental epistemological problem that rose in the late 19th and early 20th century whereby "machine cognition" emerges. The epistemological roots addressed in the TM and notably in its "Halting Problem" uncovers the tension between determinism and uncertainty, regarded here as the primal and inherent features of machine cognition.

At the core of engineering is human problem-solving. Creating a cognitive model of the task of problem-solving is helpful for planning and organizing engineering tasks. One possibility rarely considered in modeling cognitive processes is the use of Turing Oracles. Copeland (1998) put forth the possibility that the mind could be viewed as an oracle machine, but he never applied that idea practically. Oracles enable the modeling of processes in the mind which are not computationally based. Using oracles resolves many of the surprising results of computational problem-solving which arise as a result of the Tractable Cognition Thesis and similar mechanistic models of the mind. However, as research into the use of Turing Oracles in problem-solving is new, there are many methodological issues.

Calculating the complexity of software projects is important to software engineering as it helps in estimating the likely locations of bugs as well as the number of resources required to modify certain program areas. Cyclomatic complexity is one of the pri- mary estimators of software complexity which operates by counted branch points in software code. However, cyclomatic complexity assumes that all branch points are equally complex. Some types of branch points require more creativity and foresight to understand and program correctly than others. Specifically, when knowledge of the behavior of a loop or recursion requires solving a problem similar to the halting problem, that loop has intrinsically more complexity than other types of loops or con- ditions. Halting-problem-like problems can be detected by looking for loops whose termination conditions are not intrinsically bound in the looping construct. These types of loops are counted to find the program complexity. This metric is orthogonal to cyclomatic complexity (which remains useful) rather than as a substitute for it.

We show in this article that uncomputability is also a relative property of subrecursive classes built on a recursive relative incompressible function, which acts as a higher-order "yardstick" of irreducible information for the respective sub-recursive class. We define the concept of a Turing submachine, and a recursive relative version for the Busy Beaver function and for the halting probability (or Chaitin's constant) Ω; respectively the Busy Beaver Plus (BBP) function and a time-bounded halting probability. Therefore, we prove that the computable BBP function defined on any Turing submachine is neither computable nor compressible by any program running on this submachine. In addition, we build a Turing submachine that can use lower approximations to its own time-bounded halting probability to calculate the values of its Busy Beaver Plus function, in the " same " manner that universal Turing machines use approximations to Ω to calculate Busy Beaver values. Thus, the algorithmic information carried by the BBP function is relatively incompressible (and uncomputable) at the same time that it still is occasionally reached by submachines. We point that this phenomenon enriches the research on the relativization and simulation of uncomputability and irreducible information.