Computability theory

Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability. In these areas, recursion theory overlaps with proof theory and effective descriptive set theory (Dell XPS M1210 Battery) .

The basic questions addressed by recursion theory are "What does it mean for a function from the natural numbers to themselves to be computable?" and "Can noncomputable functions be classified into a hierarchy based on their level of noncomputability?". The answers to these questions have led to a rich theory that is still being actively researched (Dell Studio XPS 1340 Battery) .

The field is also closely related to computer science. Recursion theorists in mathematical logic often study the theory of relative computability, reducibility notions and degree structures described in this article. This contrasts with the theory of subrecursive hierarchies, formal methods and formal languages that is common in the study of computability theory in computer science (Dell Studio XPS 1640 Battery) .

There is considerable overlap in knowledge and methods between these two research communities, however, and no firm line can be drawn between them.

Computable and uncomputable sets

Recursion theory originated with work of Kurt Gödel, Alonzo Church, Alan Turing, Stephen Kleene and Emil Post in the 1930s (Dell Vostro 1710 Battery) .

The fundamental results the researchers obtained established Turing computability as the correct formalization of the informal idea of effective calculation. These results led Stephen Kleene (1952) to coin the two names "Church's thesis" (Kleene 1952:300) and "Turing's Thesis" (p. 376). Nowadays these are often considered as a single hypothesis, the Church–Turing thesis, which states that any function that is computable by an algorithm is a computable function (Sony VGP-BPS13 battery) .

Although initially skeptical, by 1946 Gödel argued in favor of this thesis.

"Tarski has stressed in his lecture (and I think justly) the great importance of the concept of general recursiveness (or Turing's computability). It seems to me that this importance is largely due to the fact that with this concept one has for the first time succeeded in giving an absolute notion to an interesting epistemological notion, i.e., one not depending on the formalism chosen (Sony VGP-BPL9 battery) ."

With a definition of effective calculation came the first proofs that there are problems in mathematics that cannot be effectively decided. Church (1936a, 1936b) and Turing (1936), inspired by techniques used by Gödel (1931) to prove his incompleteness theorems, independently demonstrated that the Entscheidungsproblem is not effectively decidable. This result showed that there is no algorithmic procedure that can correctly decide whether arbitrary mathematical propositions are true or false (Sony VGP-BPL11 battery) .

Many problems of mathematics have been shown to be undecidable after these initial examples were established. In 1947, Markov and Post published independent papers showing that the word problem for semigroups cannot be effectively decided. Extending this result, Pyotr Novikov and William Boone showed independently in the 1950s that the word problem for groups is not effectively solvable (Sony VGP-BPL15 battery) :

there is no effective procedure that, given a word in a finitely presented group, will decide whether the element represented by the word is the identity element of the group. In 1970, Yuri Matiyasevich proved Matiyasevich's theorem, which implies that Hilbert's tenth problem has no effective solution; this problem asked whether there is an effective procedure to decide whether aDiophantine equation over the integers has a solution in the integers (Dell Inspiron E1505 battery) .

The list of undecidable problems gives additional examples of problems with no computable solution.

The study of which mathematical constructions can be effectively performed is sometimes called recursive mathematics; the Handbook of Recursive Mathematics (Ershov et al. 1998) covers many of the known results in this field (Dell Latitude E6400 battery) .

Turing computability

The main form of computability studied in recursion theory was introduced by Turing (1936). A set of natural numbers is said to be a computable set (also called a decidable, recursive, or Turing computable set) if there is a Turing machine that, given a number n, halts with output 1 if n is in the set and halts with output 0 if n is not in the set (HP Pavilion dv6000 Battery) .

A function f from the natural numbers to themselves is a recursive or (Turing) computable function if there is a Turing machine that, on input n, halts and returns output f(n). The use of Turing machines here is not necessary; there are many othermodels of computation that have the same computing power as Turing machines; for example the ?-recursive functions obtained from primitive recursion and the ? operator (Sony VGN-FZ31S battery) .

The terminology for recursive functions and sets is not completely standardized. The definition in terms of ?-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for sets and functions computable by a Turing machine. The word decidable stems from the German word Entscheidungsproblem which was used in the original papers of Turing and others (Sony VGN-FZ31S battery) .

In contemporary use, the term "computable function" has various definitions: according to Cutland (1980), it is a partial recursive function (which can be undefined for some inputs), while according to Soare (1987) it is a total recursive (equivalently, general recursive) function. This article follows the second of these conventions. Soare (1996) gives additional comments about the terminology (Hp pavilion dv6000 battery) .

Not every set of natural numbers is computable. The halting problem, which is the set of (descriptions of) Turing machines that halt on input 0, is a well known example of a noncomputable set. The existence of many noncomputable sets follows from the facts that there are only countably many Turing machines, and thus only countably many computable sets, but there are uncountably manysets of natural numbers (SONY VGN-FZ38M Battery) .

Although the Halting problem is not computable, it is possible to simulate program execution and produce an infinite list of the programs that do halt. Thus the halting problem is an example of arecursively enumerable set, which is a set that can be enumerated by a Turing machine (other terms for recursively enumerable include computably enumerableand semidecidable) (SONY VGN-FZ31z Battery) .

Equivalently, a set is recursively enumerable if and only if it is the range of some computable function. The recursively enumerable sets, although not decidable in general, have been studied in detail in recursion theory (SONY VGN-FZ31E Battery) .

Areas of research in recursion theory

Beginning with the theory of recursive sets and functions described above, the field of recursion theory has grown to include the study of many closely related topics. These are not independent areas of research: each of these areas draws ideas and results from the others, and most recursion theorists are familiar with the majority of them (SONY VGN-FZ31J Battery) .

Relative computability and the Turing degrees

Main articles: Turing reduction and Turing degree

Recursion theory in mathematical logic has traditionally focused on relative computability, a generalization of Turing computability defined using oracle Turing machines, introduced by Turing (1939). An oracle Turing machine is a hypothetical device which, in addition to performing the actions of a regular Turing machine, is able to ask questions of an oracle, which is a particular set of natural numbers (SONY VGN-FZ31M Battery) .

The oracle machine may only ask questions of the form "Is n in the oracle set?". Each question will be immediately answered correctly, even if the oracle set is not computable. Thus an oracle machine with a noncomputable oracle will be able to compute sets that are not computable without an oracle (SONY VGN-FZ31B Battery) .

Informally, a set of natural numbers A is Turing reducible to a set B if there is an oracle machine that correctly tells whether numbers are in A when run with B as the oracle set (in this case, the setA is also said to be (relatively) computable from Band recursive in B). If a set A is Turing reducible to a set B and B is Turing reducible to A then the sets are said to have the same Turing degree (also called degree of unsolvability) (SONY VGP-BPS13 Battery) .

The Turing degree of a set gives a precise measure of how uncomputable the set is.

The natural examples of sets that are not computable, including many different sets that encode variants of the halting problem, have two properties in common (Dell Inspiron 1320 Battery) :

1. They are recursively enumerable, and
2. Each can be translated into any other via a many-one reduction. That is, given such sets A and B, there is a total computable function f such that A = {x : f(x) ? B}. These sets are said to bemany-one equivalent (or m-equivalent) (Dell Inspiron 1320n Battery) .

Many-one reductions are "stronger" than Turing reductions: if a set A is many-one reducible to a set B, then A is Turing reducible to B, but the converse does not always hold. Although the natural examples of noncomputable sets are all many-one equivalent, it is possible to construct recursively enumerable sets A and B such that A is Turing reducible to B but not many-one reducible to B (Dell Inspiron 1464 Battery) .

It can be shown that every recursively enumerable set is many-one reducible to the halting problem, and thus the halting problem is the most complicated recursively enumerable set with respect to many-one reducibility and with respect to Turing reducibility. Post (1944) asked whether every recursively enumerable set is either computable or Turing equivalent to the halting problem, that is, whether there is no recursively enumerable set with a Turing degree intermediate between those two (Dell Inspiron 1564 Battery) .

As intermediate results, Post defined natural types of recursively enumerable sets like the simple, hypersimple and hyperhypersimple sets. Post showed that these sets are strictly between the computable sets and the halting problem with respect to many-one reducibility. Post also showed that some of them are strictly intermediate under other reducibility notions stronger than Turing reducibility (Dell Inspiron 1764 Battery) .

But Post left open the main problem of the existence of recursively enumerable sets of intermediate Turing degree; this problem became known as Post's problem. After ten years, Kleene and Post showed in 1954 that there are intermediate Turing degrees between those of the computable sets and the halting problem, but they failed to show that any of these degrees contains a recursively enumerable set (Dell Studio 1450 Battery) .

Very soon after this, Friedberg and Muchnik independently solved Post's problem by establishing the existence of recursively enumerable sets of intermediate degree. This groundbreaking result opened a wide study of the Turing degrees of the recursively enumerable sets which turned out to possess a very complicated and non-trivial structure (Dell Studio 1457 Battery) .

There are uncountably many sets that are not recursively enumerable, and the investigation of the Turing degrees of all sets is as central in recursion theory as the investigation of the recursively enumerable Turing degrees. Many degrees with special properties were constructed: hyperimmune-free degrees where every function computable relative to that degree is majorized by a (unrelativized) computable function (Dell Latitude D610 Battery) ;

high degrees relative to which one can compute a function f which dominates every computable function g in the sense that there is a constant c depending on gsuch that g(x) <> for all x > c; random degrees containing algorithmically random sets; 1-genericdegrees of 1-generic sets; and the degrees below the halting problem of limit-recursive sets (Toshiba NB100 Battery) .

The study of arbitrary (not necessarily recursively enumerable) Turing degrees involves the study of the Turing jump. Given a set A, the Turing jump of A is a set of natural numbers encoding a solution to the halting problem for oracle Turing machines running with oracle A. The Turing jump of any set is always of higher Turing degree than the original set, and a theorem of Friedburg shows that any set that computes the Halting problem can be obtained as the Turing jump of another set (Toshiba Satellite M65 battery) .

Post's theorem establishes a close relationship between the Turing jump operation and the arithmetical hierarchy, which is a classification of certain subsets of the natural numbers based on their definability in arithmetic.

Much recent research on Turing degrees has focused on the overall structure of the set of Turing degrees and the set of Turing degrees containing recursively enumerable sets. A deep theorem of Shore and Slaman (1999) states that the function mapping a degree x to the degree of its Turing jump is definable in the partial order of the Turing degrees. A recent survey by Ambos-Spies and Fejer (2006) gives an overview of this research and its historical progression (Toshiba Satellite M60 battery) .

Other reducibilities

Main article: Reduction (recursion theory)

An ongoing area of research in recursion theory studies reducibility relations other than Turing reducibility. Post (1944) introduced several strong reducibilities, so named because they imply truth-table reducibility. A Turing machine implementing a strong reducibility will compute a total function regardless of which oracle it is presented with (Dell Latitude D830 Battery) .

Weak reducibilities are those where a reduction process may not terminate for all oracles; Turing reducibility is one example.

The strong reducibilities include:

• One-one reducibility: A is one-one reducible (or 1-reducible) to B if there is a total computable injective function f such that each n is in A if and only if f(n) is in B (Dell Studio 1735 Battery) .
• Many-one reducibility: This is essentially one-one reducibility without the constraint that f be injective. A is many-one reducible (or m-reducible) to B if there is a total computable function f such that each n is in A if and only if f(n) is in B.
• Truth-table reducibility: A is truth-table reducible to B if A is Turing reducible to B via an oracle Turing machine that computes a total function regardless of the oracle it is given (Dell Latitude D620 Battery) .
• Because of compactness of Cantor space, this is equivalent to saying that the reduction presents a single list of questions (depending only on the input) to the oracle simultaneously, and then having seen their answers is able to produce an output without asking additional questions regardless of the oracle's answer to the initial queries. Many variants of truth-table reducibility have also been studied (Dell Inspiron Mini 10 Battery) .

Further reducibilities (positive, disjunctive, conjunctive, linear and their weak and bounded versions) are discussed in the article Reduction (recursion theory).

The major research on strong reducibilities has been to compare their theories, both for the class of all recursively enumerable sets as well as for the class of all subsets of the natural numbers (Sony VGN-FW11S Battery) .

Furthermore, the relations between the reducibilities has been studied. For example, it is known that every Turing degree is either a truth-table degree or is the union of infinitely many truth-table degrees.

Reducibilities weaker than Turing reducibility (that is, reducibilities that are implied by Turing reducibility) have also been studied. The most well known are arithmetical reducibility and hyperarithmetical reducibility (Sony VGN-FW11M Battery) .

These reducibilities are closely connected to definability over the standard model of arithmetic.

Rice's theorem and the arithmetical hierarchy

Rice showed that for every nontrivial class C (which contains some but not all r.e. sets) the index set E = {e: the eth r.e. set We is in C} has the property that either the halting problem or its complement is many-one reducible to E, that is, can be mapped using a many-one reduction to E (see Rice's theorem for more detail) (Dell Studio 1555 battery) .

But, many of these index sets are even more complicated than the halting problem. These type of sets can be classified using the arithmetical hierarchy. For example, the index set FIN of class of all finite sets is on the level ?2, the index set REC of the class of all recursive sets is on the level ?3, the index set COFIN of all cofinite sets is also on the level ?3 and the index set COMP of the class of all Turing-complete sets ?4. These hierarchy levels are defined inductively, ?n+1 contains just all sets which are recursively enumerable relative to ?n; ?1 contains the recursively enumerable sets. The index sets given here are even complete for their levels, that is, all the sets in these levels can be many-one reduced to the given index sets (Dell Latitude E5400 Battery) .

Reverse mathematics

Main article: Reverse mathematics

The program of reverse mathematics asks which set-existence axioms are necessary to prove particular theorems of mathematics in subsystems of second-order arithmetic. This study was initiated by Harvey Friedman and was studied in detail by Stephen Simpson and others; Simpson (1999) gives a detailed discussion of the program (Dell Latitude E4200 Battery) .

The set-existence axioms in question correspond informally to axioms saying that the powerset of the natural numbers is closed under various reducibility notions. The weakest such axiom studied in reverse mathematics is recursive comprehension, which states that the powerset of the naturals is closed under Turing reducibility (Dell Vostro A840 Battery) .

Numberings

A numbering is an enumeration of functions; it has two parameters, e and x and outputs the value of the e-th function in the numbering on the input x. Numberings can be partial-recursive although some of its members are total recursive, that is, computable functions. Acceptable or Gödel numberings are those into which all others can be translated. A Friedberg numbering (named after its discoverer) is a one-one numbering of all partial-recursive functions; it is necessarily not an acceptable numbering (Dell Inspiron 300M Battery) .

Later research dealt also with numberings of other classes like classes of recursively enumerable sets. Goncharov discovered for example a class of recursively enumerable sets for which the numberings fall into exactly two classes with respect to recursive isomorphisms.

The priority method

For further explanation, see the section Post's problem and the priority method in the article Turing degree.

Post's problem was solved with a method called the priority method; a proof using this method is called a priority argument(Dell Studio 1737 battery) .

This method is primarily used to construct recursively enumerable sets with particular properties. To use this method, the desired properties of the set to be constructed are broken up into an infinite list of goals, known as requirements, so that satisfying all the requirements will cause the set constructed to have the desired properties. Each requirement is assigned to a natural number representing the priority of the requirement; so 0 is assigned to the most important priority, 1 to the second most important, and so on (Dell Inspiron E1505 battery) .

The set is then constructed in stages, each stage attempting to satisfy one of more of the requirements by either adding numbers to the set or banning numbers from the set so that the final set will satisfy the requirement. It may happen that satisfying one requirement will cause another to become unsatisfied; the priority order is used to decide what to do in such an event (Dell Latitude E6400 battery) .

Priority arguments have been employed to solve many problems in recursion theory, and have been classified into a hierarchy based on their complexity (Soare 1987). Because complex priority arguments can be technical and difficult to follow, it has traditionally been considered desirable to prove results without priority arguments, or to see if results proved with priority arguments can also be proved without them (Dell RM791 battery) .

For example, Kummer published a paper on a proof for the existence of Friedberg numberings without using the priority method.

The lattice of recursively enumerable sets

When Post defined the notion of a simple set as an r.e. set with an infinite complement not containing any infinite r.e. set, he started to study the structure of the recursively enumerable sets under inclusion. This lattice became a well-studied structure (Dell XPS M1530 battery) .

Recursive sets can be defined in this structure by the basic result that a set is recursive if and only if the set and its complement are both recursively enumerable. Infinite r.e. sets have always infinite recursive subsets; but on the other hand, simple sets exist but do not have a coinfinite recursive superset. Post (1944) introduced already hypersimple and hyperhypersimple sets; later maximal sets were constructed which are r.e. sets such that every r.e. superset is either a finite variant of the given maximal set or is co-finite (Dell XPS M2010 battery) .

Post's original motivation in the study of this lattice was to find a structural notion such that every set which satisfies this property is neither in the Turing degree of the recursive sets nor in the Turing degree of the halting problem. Post did not find such a property and the solution to his problem applied priority methods instead; Harrington and Soare (1991) found eventually such a property (Dell Vostro 1000 battery) .

Automorphism problems

Another important question is the existence of automorphisms in recursion-theoretic structures. One of these structures is that one of recursively enumerable sets under inclusion modulo finite difference; in this structure, A is below B if and only if the set difference B ? A is finite. Maximal sets (as defined in the previous paragraph) have the property that they cannot be automorphic to non-maximal sets, that is, if there is an automorphism of the recursive enumerable sets under the structure just mentioned, then every maximal set is mapped to another maximal set (Acer Aspire One battery) .

Soare (1974) showed that also the converse holds, that is, every two maximal sets are automorphic. So the maximal sets form an orbit, that is, every automorphism preserves maximality and any two maximal sets are transformed into each other by some automorphism. Harrington gave a further example of an automorphic property: that of the creative sets, the sets which are many-one equivalent to the halting problem (Sony VGP-BPS13A/B Battery) .

Besides the lattice of recursively enumerable sets, automorphisms are also studied for the structure of the Turing degrees of all sets as well as for the structure of the Turing degrees of r.e. sets. In both cases, Cooper claims to have constructed nontrivial automorphisms which map some degrees to other degrees; this construction has, however, not been verified and some colleagues believe that the construction contains errors and that the question of whether there is a nontrivial automorphism of the Turing degrees is still one of the main unsolved questions in this area (Slaman and Woodin 1986, Ambos-Spies and Fejer 2006) (Dell Precision M70 Battery) .

Kolmogorov complexity

Main article: Kolmogorov complexity

The field of Kolmogorov complexity and algorithmic randomness was developed during the 1960s and 1970s by Chaitin, Kolmogorov, Levin, Martin-Löf and Solomonoff (the names are given here in alphabetical order; much of the research was independent, and the unity of the concept of randomness was not understood at the time) (Toshiba Satellite L305 Battery)

The main idea is to consider a universal Turing machine U and to measure the complexity of a number (or string) x as the length of the shortest input p such that U(p) outputs x. This approach revolutionized earlier ways to determine when an infinite sequence (equivalently, characteristic function of a subset of the natural numbers) is random or not by invoking a notion of randomness for finite objects (Toshiba Satellite T4900 Battery) .

Kolmogorov complexity became not only a subject of independent study but is also applied to other subjects as a tool for obtaining proofs. There are still many open problems in this area. For that reason, a recent research conference in this area was held in January 2007[2] and a list of open problems[3] is maintained by Joseph Miller and Andre Nies (Toshiba PA3399U-2BRS battery) .

Frequency computation

This branch of recursion theory analyzed the following question: For fixed m and n with 0 < m < n, for which functions A is it possible to compute for any different n inputs x1, x2, ..., xn a tuple of nnumbers y1,y2,...,yn such that at least m of the equations A(xk) = yk are true. Such sets are known as (m, n)-recursive sets. The first major result in this branch of Recursion Theory is Trakhtenbrot's result that a set is computable if it is (m, n)-recursive for some m, n with 2m > n (Toshiba Satellite A200 Battery) .

On the other hand, Jockusch's semirecursive sets (which were already known informally before Jockusch introduced them 1968) are examples of a set which is (m, n)-recursive if and only if 2m < n + 1. There are uncountably many of these sets and also some recursively enumerable but noncomputable sets of this type. Later, Degtev established a hierarchy of recursively enumerable sets that are (1, n + 1)-recursive but not (1, n)-recursive (Toshiba Satellite 1200 Battery) .

After a long phase of research by Russian scientists, this subject became repopularized in the west by Beigel's thesis on bounded queries, which linked frequency computation to the above mentioned bounded reducibilities and other related notions. One of the major results was Kummer's Cardinality Theory which states that a set A is computable if and only if there is an n such that some algorithm enumerates for each tuple of n different numbers up to n many possible choices of the cardinality of this set of n numbers intersected with A; these choices must contain the true cardinality but leave out at least one false one (Toshiba Satellite M300 Battery) .

Inductive inference

This is the recursion-theoretic branch of learning theory. It is based on Gold's model of learning in the limit from 1967 and has developed since then more and more models of learning. The general scenario is the following: Given a class S of computable functions, is there a learner (that is, recursive functional) which outputs for any input of the form (f(0),f(1),...,f(n)) a hypothesis (Dell Latitude XT2 Tablet PC Battery) .

A learner M learns a function f if almost all hypotheses are the same index e of f with respect to a previously agreed on acceptable numbering of all computable functions; M learns S if M learns every f in S. Basic results are that all recursively enumerable classes of functions are learnable while the class REC of all computable functions is not learnable (Toshiba Portege 335CT Battery) .

Many related models have been considered and also the learning of classes of recursively enumerable sets from positive data is a topic studied from Gold's pioneering paper in 1967 onwards.

Generalizations of Turing computability

Recursion theory includes the study of generalized notions of this field such as arithmetic reducibility, hyperarithmetical reducibility and ?-recursion theory, as described by Sacks (1990) (Dell Vostro A90 Battery) .

These generalized notions include reducibilities that cannot be executed by Turing machines but are nevertheless natural generalizations of Turing reducibility. These studies include approaches to investigate the analytical hierarchy which differs from the arithmetical hierarchy by permitting quantification over sets of natural numbers in addition to quantification over individual numbers (Toshiba Satellite P15 Battery) .

These areas are linked to the theories of well-orderings and trees; for example the set of all indices of recursive (nonbinary) trees without infinite branches is complete for level of the analytical hierarchy. Both Turing reducibility and hyperarithmetical reducibility are important in the field of effective descriptive set theory. The even more general notion of degrees of constructibility is studied in set theory (Toshiba Satellite Pro M10 Battery) .

Continuous computability theory

Computability theory for digital computation is well developed. Computability theory is less well developed for analog computation that occurs in analog computers, analog signal processing, analog electronics, neural networks and continuous-time control theory, modelled by differential equations and continuous dynamical systems (Toshiba Portege 3110 Battery) .

[edit]Relationships between definability, proof and computability

There are close relationships between the Turing degree of a set of natural numbers and the difficulty (in terms of the arithmetical hierarchy) of defining that set using a first-order formula. One such relationship is made precise by Post's theorem. A weaker relationship was demonstrated by Kurt Gödel in the proofs of his completeness theorem and incompleteness theorems (Toshiba Portege R600 Battery) .

Gödel's proofs show that the set of logical consequences of an effective first-order theory is a recursively enumerable set, and that if the theory is strong enough this set will be uncomputable. Similarly, Tarski's indefinability theorem can be interpreted both in terms of definability and in terms of computability (Toshiba Satellite 1900 Battery) .

Recursion theory is also linked to second order arithmetic, a formal theory of natural numbers and sets of natural numbers. The fact that certain sets are computable or relatively computable often implies that these sets can be defined in weak subsystems of second order arithmetic. The program of reverse mathematics uses these subsystems to measure the noncomputability inherent in well known mathematical theorems. Simpson (1999) discusses many aspects of second-order arithmetic and reverse mathematics (Toshiba Portege R200 Battery) .

The field of proof theory includes the study of second-order arithmetic and Peano arithmetic, as well as formal theories of the natural numbers weaker than Peano arithmetic. One method of classifying the strength of these weak systems is by characterizing which computable functions the system can prove to be total (see Fairtlough and Wainer (1998)). For example, in primitive recursive arithmeticany computable function that is provably total is actually primitive recursive, while Peano arithmetic proves that functions like the Ackerman function, which are not primitive recursive, are total (SONY VAIO VGN-FZ21m Battery) .

Not every total computable function is provably total in Peano arithmetic, however; an example of such a function is provided by Goodstein's theorem.

Name of the subject

The field of mathematical logic dealing with computability and its generalizations has been called "recursion theory" since its early days. Robert I. Soare, a prominent researcher in the field, has proposed (Soare 1996) that the field should be called "computability theory" instead. He argues that Turing's terminology using the word "computable" is more natural and more widely understood than the terminology using the word "recursive" introduced by Kleene (SONY VAIO VGN-FZ18m Battery) .

Many contemporary researchers have begun to use this alternate terminology. These researchers also use terminology such aspartial computable function and computably enumerable (c.e.) set instead of partial recursive function and recursively enumerable (r.e.) set. Not all researchers have been convinced, however, as explained by Fortnow and Simpson. Some commentators argue that both the names recursion theory and computability theory fail to convey the fact that most of the objects studied in recursion theory are not computable (Dell Vostro A90 Battery) .

Rogers (1967) has suggested that a key property of recursion theory is that its results and structures should be invariant under computable bijections on the natural numbers (this suggestion draws on the ideas of the Erlangen program in geometry). The idea is that a computable bijection merely renames numbers in a set, rather than indicating any structure in the set, much as a rotation of the Euclidean plane does not change any geometric aspect of lines drawn on it (Dell Vostro A860 Battery) .

Since any two infinite computable sets are linked by a computable bijection, this proposal identifies all the infinite computable sets (the finite computable sets are viewed as trivial). According to Rogers, the sets of interest in recursion theory are the noncomputable sets, partitioned into equivalence classes by computable bijections of the natural numbers (Dell Vostro 2510 Battery) .

Professional organizations

The main professional organization for recursion theory is the Association for Symbolic Logic, which holds several research conferences each year. The interdisciplinary research AssociationComputability in Europe (CiE) also organizes a series of annual conferences. CiE 2012 will be the Turing Centenary Conference, held in Cambridge as part of the Alan Turing Year (Dell Vostro 1510 Battery) .

(Dell Studio 1457 Battery)