Borut Robič

The Foundations of Computability Theory

Introduction.- The Foundational Crisis of Mathematics.- Formalism.- Hilberts Attempt at Recovery.- The Quest for a Formalization.- The Turing Machine.- The First Basic Results.- Incomputable Problems.- Methods of Proving the Incomputability.- Computation with External Help.- Degrees of Unsolvability.- The Turing Hierarchy of Unsolvability.- The Class D of Degrees of Unsolvability.- C.E. Degrees and the Priority Method.- The Arithmetical Hierarchy.- Further Reading.- App. A, Mathematical Background.- References.- Index.

The Foundational Crisis of Mathematics

The need for a formal definition of the concept of algorithm was made clear during the first few decades of the twentieth century as a result of events taking place in mathematics. At the beginning of the century, Cantor’s naive set theory was born. This theory was very promising because it offered a common foundation to all the fields of mathematics. However, it treated infinity incautiously and boldly. This called for a response, which soon came in the form of logical paradoxes. Because Cantor’s set theory was unable to eliminate them—or at least bring them under control—formal logic was engaged. As a result, three schools of mathematical thought—intuitionism, logicism, and formalism—contributed important ideas and tools that enabled an exact and concise mathematical expression and brought rigor to mathematical research.

The Quest for a Formalization

The difficulties that arose at the beginning of the twentieth century shook the foundations of mathematics and led to several fundamental questions: “What is an algorithm? What is computation? What does it mean when we say that a function or problem is computable?” Because of Hilbert’s Program, intuitive answers to these questions no longer sufficed. As a result, a search for appropriate definitions of these fundamental concepts followed. In the 1930s it was discovered—miraculously, as Godel put it—that all these notions can be formalized, i.e, mathematically defined; indeed, they were formalized in several completely different yet equivalent ways. After this, they finally became amenable to mathematical analysis and could be rigorously treated and used. This opened the door to the seminal results of the 1930s that marked the beginning of Computability Theory.

The Turing Machine

The Turing machine convincingly formalized the concepts of “algorithm,” “computation,” and “computable.” It convinced researchers by its simplicity, generality, mechanical operation, and resemblance to human activity when solving computational problems, and by Turing’s reasoning and analysis of “computable” functions and his argumentation that partial “computable” functions are exactly Turing computable functions. Turing considered several variants that are generalizations of the basic model of his machine. But he also proved that they add nothing to the computational power of the basic model. This strengthened the belief in the Turing machine as an appropriate model of computation. Turing machines can be encoded and consequently enumerated. This enabled the construction of the universal Turing machine that is capable of computing anything that can be computed by any other Turing machine. This seminal discovery laid the theoretical grounds for several all important practical consequences, the general-purpose computer and the operating system being the most notable. The Turing machine is a versatile model of computation: it can be used to compute values of a function, or to generate elements of a set, or to decide about the membership of an object in a set. The last led to the notions of decidable and semi-decidable sets that would later prove to be very important in solving general computational problems.

The Arithmetical Hierarchy

In this chapter we will introduce a different view of sets of natural numbers. Sometimes such a set can be defined by a property of its members, where the property is expressed by a formula of Formal Arithmetic. Sets defined by formulas of the same complexity constitute an arithmetical class. Different complexities of formulas give rise to different arithmetical classes. There is also an ordering between these classes, so they form the so-called Arithmetical hierarchy. We will show that the Arithmetical hierarchy is closely connected with the Jump hierarchy.

The First Basic Results

In the previous chapters we have defined the basic notions and concepts of a theory that we are interested in, Computability Theory. In particular, we have rigorously defined its basic notions, i.e., the notions of algorithm, computation, and computable function. We have also defined some new notions, such as the decidability and semi-decidability of a set, that will play key roles in the next chapter (where we will further develop Computability Theory). As a side product of the previous chapters we have also discovered some surprising facts, such as the existence of the universal Turing machine. It is now time to start using this apparatus and deduce the first theorems of Computability Theory. In this chapter we will first prove several simple but useful theorems about decidable and semi-decidable sets and their relationship. Then we will deduce the so-called Padding Lemma and, based on it, introduce the extremely important concept of the index set. This will enable us to deduce two influential theorems, the Parametrization Theorem and the Recursion Theorem. We will not be excessively formal in our deductions; instead, we will equip them with meaning and motivation wherever appropriate.

Degrees of Unsolvability

In Part II, we proved that besides computable problems there are also incomputable ones. So, given a computational problem, it makes sense to talk about its degree of unsolvability. Of course, at this point we only know of two such degrees: one is shared by all computable problems, and the other is shared by all incomputable ones. (This will change, however, in the next chapter.) Nevertheless, the main aim of this chapter is to formalize the intuitive notion of the degree of unsolvability. Building on the concept of the oracle Turing machine, we will first define the concept of the Turing reduction, the most general reduction between computational problems. We will then proceed in a natural way to the definition of Turing degree—the formal counterpart of the intuitive notion of the degree of unsolvability.

Hilbert’s Attempt at Recovery

Hilbert’s Program was a promising formalistic attempt to recover mathematics. It would use formal axiomatic systems to put mathematics on a sound footing and eliminate all the paradoxes. Unfortunately, the program was severely shaken by Godel’s astonishing and far-reaching discoveries about the general properties of formal axiomatic systems and their theories. Thus Hilbert’s attempt fell short of formalists’ expectations. Nevertheless, although shattered, the program left open an important question about the existence of a certain algorithm—a question that was to lead to the birth of Computability Theory.

The Turing Hierarchy of Unsolvability

At this point we only know of two degrees of unsolvability: the T-degree shared by all the decidable decision problems, and the T-degree shared by all undecidable decision problems that are T-equivalent to the Halting Problem. In this chapter we will prove that, surprisingly, for every undecidable decision problem there exists a more difficult decision problem. This will in effect mean that there is an infinite hierarchy of degrees of unsolvability and that there is no most difficult decision problem.

C.E. Degrees and the Priority Method

Among the Turing degrees, the so-called computably enumerable (c.e.) degrees are all important. This is because they stem from c.e. sets, the sets that often spring up in practice. In this chapter we will present the basic facts of c.e. degrees. We will then describe Post’s Problem, a problem about c.e. degrees that was posed by Emil Post in 1944. After a series of attempts by Post and others, the problem was finally solved in 1956 by Muchnik and Friedberg. They simultaneously and independently devised a method, called the Priority Method, and applied it to solve the problem. We will describe Post’s Problem and the Priority Method.