Misplaced Pages

#885114

130-397: In computability theory , a system of data-manipulation rules (such as a model of computation , a computer's instruction set , a programming language , or a cellular automaton ) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing ). This means that this system

260-589: A common root: This is the case if and only if there do not exist polynomials q 1 , … , q k {\displaystyle q_{1},\ldots ,q_{k}} and indices λ 1 , … , λ k {\displaystyle \lambda _{1},\ldots ,\lambda _{k}} such that This result is known as the Hilbert root theorem , or "Hilberts Nullstellensatz" in German. He also proved that

390-484: A Turing machine, and therefore that if any real-world computer can simulate a Turing machine, it is Turing equivalent to a Turing machine. A universal Turing machine can be used to simulate any Turing machine and by extension the purely computational aspects of any possible real-world computer. To show that something is Turing-complete, it is enough to demonstrate that it can be used to simulate some Turing-complete system. No physical system can have infinite memory, but if

520-510: A city judge, the family moved to Königsberg. David's sister, Elise, was born when he was six. He began his schooling aged eight, two years later than the usual starting age. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium ( Collegium fridericianum , the same school that Immanuel Kant had attended 140 years before); but, after an unhappy period, he transferred to (late 1879) and graduated from (early 1880)

650-476: 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 computably enumerable sets. A deep theorem of Shore and Slaman states that

780-424: A computing device can be simulated by a universal Turing machine . The Church–Turing thesis states that this is a law of mathematics – that a universal Turing machine can, in principle, perform any calculation that any other programmable computer can. This says nothing about the effort needed to write the program , or the time it may take for the machine to perform the calculation, or any abilities

910-409: A control unit. These two elements make this architecture Turing-complete. Even pure functional languages are Turing-complete. Turing completeness in declarative SQL is implemented through recursive common table expressions . Unsurprisingly, procedural extensions to SQL ( PLSQL , etc.) are also Turing-complete. This illustrates one reason why relatively powerful non-Turing-complete languages are rare:

1040-406: A course for mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. He was a cofounder of proof theory and mathematical logic . Hilbert, the first of two children and only son of Otto, a county judge, and Maria Therese Hilbert ( née Erdtmann), the daughter of a merchant, was born in

1170-494: A curve, which is now called Hilbert curve . Approximations to this curve are constructed iteratively according to the replacement rules in the first picture of this section. The curve itself is then the pointwise limit. The text Grundlagen der Geometrie (tr.: Foundations of Geometry ) published by Hilbert in 1899 proposes a formal set, called Hilbert's axioms, substituting for the traditional axioms of Euclid . They avoid weaknesses identified in those of Euclid , whose works at

1300-436: A function is provided by Goodstein's theorem . 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 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

1430-534: A further example of an automorphic property: that of the creative sets, the sets which are many-one equivalent to the halting problem. Besides the lattice of computably 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 c.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

SECTION 10

#1732783202886

1560-443: A hierarchy of computably enumerable sets that are (1,  n  + 1)-recursive but not (1,  n )-recursive. 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

1690-553: A highly influential list consisting of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician. After reworking the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed from

1820-434: A language in which programs are guaranteed to complete and halt cannot compute the computable function produced by Cantor's diagonal argument on all computable functions in that language. A computer with access to an infinite tape of data may be more powerful than a Turing machine: for instance, the tape might contain the solution to the halting problem or some other Turing-undecidable problem. Such an infinite tape of data

1950-417: A method called the priority method ; a proof using this method is called a priority argument . This method is primarily used to construct computably 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

2080-728: A native of Königsberg. News of his death only became known to the wider world several months after he died. The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930. The words were given in response to the Latin maxim: " Ignoramus et ignorabimus " or "We do not know and we shall not know": Wir müssen wissen. Wir werden wissen. We must know. We shall know. The day before Hilbert pronounced these phrases at

2210-405: A paper on a proof for the existence of Friedberg numberings without using the priority method. When Post defined the notion of a simple set as a c.e. set with an infinite complement not containing any infinite c.e. set, he started to study the structure of the computably enumerable sets under inclusion. This lattice became a well-studied structure. Computable sets can be defined in this structure by

2340-489: 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 computably enumerable classes of functions are learnable while the class REC of all computable functions is not learnable. Many related models have been considered and also the learning of classes of computably enumerable sets from positive data is a topic studied from Gold's pioneering paper in 1967 onwards. Computability theory includes

2470-444: A proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) was "the object". Not all were convinced. While Kronecker would die soon afterwards, his constructivist philosophy would continue with the young Brouwer and his developing intuitionist "school", much to Hilbert's torment in his later years. Indeed, Hilbert would lose his "gifted pupil" Weyl to intuitionism—"Hilbert

2600-471: A property. Another important question is the existence of automorphisms in computability-theoretic structures. One of these structures is that one of computably 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

2730-637: A scientist after 1925, and certainly not a Hilbert." Hilbert was elected to the American Philosophical Society in 1932. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933. Those forced out included Hermann Weyl (who had taken Hilbert's chair when he retired in 1930), Emmy Noether and Edmund Landau . One who had to leave Germany, Paul Bernays , had collaborated with Hilbert in mathematical logic, and co-authored with him

SECTION 20

#1732783202886

2860-410: A series of annual conferences. David Hilbert David Hilbert ( / ˈ h ɪ l b ər t / ; German: [ˈdaːvɪt ˈhɪlbɐt] ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory ,

2990-434: 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. This is the computability-theoretic branch of learning theory. It is based on E. Mark Gold 's model of learning in

3120-418: 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 computably enumerable sets A and B such that A is Turing reducible to B but not many-one reducible to B . It can be shown that every computably enumerable set is many-one reducible to the halting problem, and thus

3250-445: A set is c.e. if and only if it is the range of some computable function. The c.e. sets, although not decidable in general, have been studied in detail in computability theory. Beginning with the theory of computable sets and functions described above, the field of computability 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

3380-399: 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 set A is also said to be ( relatively ) computable from B and 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

3510-551: A short time. Others have been discussed throughout the 20th century, with a few now taken to be unsuitably open-ended to come to closure. Some continue to remain challenges. The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the Bulletin of the American Mathematical Society . In an account that had become standard by the mid-century, Hilbert's problem set

3640-464: A single hypothesis, the Church–Turing thesis , which states that any function that is computable by an algorithm is a computable function . 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

3770-404: A solid and complete logical foundation. He believed that in principle this could be done by showing that: He seems to have had both technical and philosophical reasons for formulating this proposal. It affirmed his dislike of what had become known as the ignorabimus , still an active issue in his time in German thought, and traced back in that formulation to Emil du Bois-Reymond . This program

3900-528: A strong reducibility will compute a total function regardless of which oracle it is presented with. Weak reducibilities are those where a reduction process may not terminate for all oracles; Turing reducibility is one example. The strong reducibilities include: Further reducibilities (positive, disjunctive, conjunctive, linear and their weak and bounded versions) are discussed in the article Reduction (computability theory) . The major research on strong reducibilities has been to compare their theories, both for

4030-455: A subset of the natural numbers) is random or not by invoking a notion of randomness for finite objects. 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. This branch of computability theory analyzed the following question: For fixed m and n with 0 <  m  <  n , for which functions A

Turing completeness - Misplaced Pages Continue

4160-526: A universal computer, because the instructions that compute them do not allow for an infinite loop. In the early 20th century, David Hilbert led a program to axiomatize all of mathematics with precise axioms and precise logical rules of deduction that could be performed by a machine. Soon it became clear that a small set of deduction rules are enough to produce the consequences of any set of axioms. These rules were proved by Kurt Gödel in 1930 to be enough to produce every theorem. The actual notion of computation

4290-413: A very complicated and non-trivial structure. There are uncountably many sets that are not computably enumerable, and the investigation of the Turing degrees of all sets is as central in computability theory as the investigation of the computably enumerable Turing degrees. Many degrees with special properties were constructed: hyperimmune-free degrees where every function computable relative to that degree

4420-409: Is a one-one numbering of all partial-computable functions; it is necessarily not an admissible numbering. Later research dealt also with numberings of other classes like classes of computably enumerable sets. Goncharov discovered for example a class of computably enumerable sets for which the numberings fall into exactly two classes with respect to computable isomorphisms. Post's problem was solved with

4550-407: Is able to ask questions of an oracle , which is a particular set of natural numbers. 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 a Turing machine without an oracle cannot. Informally,

4680-503: Is able to recognize or decode other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all programming languages today are Turing-complete. A related concept is that of Turing equivalence  – two computers P and Q are called equivalent if P can simulate Q and Q can simulate P. The Church–Turing thesis conjectures that any function whose values can be computed by an algorithm can be computed by

4810-426: Is an automorphism of the computably enumerable sets under the structure just mentioned, then every maximal set is mapped to another maximal set. In 1974, Soare 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

4940-419: Is an effective procedure to decide whether a Diophantine equation over the integers has a solution in the integers. The main form of computability studied in the field was introduced by Turing in 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

5070-550: Is by characterizing which computable functions the system can prove to be total . For example, in primitive recursive arithmetic any computable function that is provably total is actually primitive recursive , while Peano arithmetic proves that functions like the Ackermann function , which are not primitive recursive, are total. Not every total computable function is provably total in Peano arithmetic, however; an example of such

5200-452: Is called a Turing oracle . Even a Turing oracle with random data is not computable ( with probability 1 ), since there are only countably many computations but uncountably many oracles. So a computer with a random Turing oracle can compute things that a Turing machine cannot. All known laws of physics have consequences that are computable by a series of approximations on a digital computer. A hypothesis called digital physics states that this

5330-423: Is closed under Turing reducibility. 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-computable although some of its members are total computable functions. Admissible numberings are those into which all others can be translated. A Friedberg numbering (named after its discoverer)

Turing completeness - Misplaced Pages Continue

5460-478: Is computable if it is ( m ,  n )-recursive for some m ,  n with 2 m  >  n . 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 2 m  <  n  + 1. There are uncountably many of these sets and also some computably enumerable but noncomputable sets of this type. Later, Degtev established

5590-475: Is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of subrecursive hierarchies , formal methods , and formal languages . The study of which mathematical constructions can be effectively performed is sometimes called recursive mathematics . Computability theory originated in

5720-463: Is either a finite variant of the given maximal set or is co-finite. 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 computable 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; in 1991, Harrington and Soare found eventually such

5850-418: Is impossible to determine whether this characteristic will hold. This impossibility poses problems when analyzing real-world computer programs. For example, one cannot write a tool that entirely protects programmers from writing infinite loops or protects users from supplying input that would cause infinite loops. One can instead limit a program to executing only for a fixed period of time ( timeout ) or limit

5980-418: Is in the set and halts with output 0 if n is not in the set. A function f from natural numbers to natural numbers is a (Turing) computable , or recursive 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 other models of computation that have the same computing power as Turing machines; for example

6110-404: Is it possible to compute for any different n inputs x 1 ,  x 2 , ...,  x n a tuple of n numbers y 1 , y 2 , ..., y n such that at least m of the equations A ( x k ) = y k are true. Such sets are known as ( m ,  n )-recursive sets. The first major result in this branch of computability theory is Trakhtenbrot's result that a set

6240-511: 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." With a definition of effective calculation came the first proofs that there are problems in mathematics that cannot be effectively decided . In 1936, Church and Turing were inspired by techniques used by Gödel to prove his incompleteness theorems - in 1931, Gödel independently demonstrated that

6370-527: Is majorized by a (unrelativized) computable function; 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 g such that g(x) < f(x) for all x > c ; random degrees containing algorithmically random sets ; 1-generic degrees of 1-generic sets; and the degrees below the halting problem of limit-computable sets. The study of arbitrary (not necessarily computably enumerable) Turing degrees involves

6500-440: Is no accident because the universe itself is computable on a universal Turing machine. This would imply that no computer more powerful than a universal Turing machine can be built physically. The computational systems (algebras, calculi) that are discussed as Turing-complete systems are those intended for studying theoretical computer science . They are intended to be as simple as possible, so that it would be easier to understand

6630-502: Is not Mathematics. This is Theology. Klein , on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein, Hilbert extended his method in a second article, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to the Annalen . After having read the manuscript, Klein wrote to him, saying: Without doubt this

SECTION 50

#1732783202886

6760-561: Is still recognizable in the most popular philosophy of mathematics , where it is usually called formalism . For example, the Bourbaki group adopted a watered-down and selective version of it as adequate to the requirements of their twin projects of (a) writing encyclopedic foundational works, and (b) supporting the axiomatic method as a research tool. This approach has been successful and influential in relation with Hilbert's work in algebra and functional analysis, but has failed to engage in

6890-478: 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. 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,

7020-429: Is the halting problem : create an algorithm that takes as input a program in some Turing-complete language and some data to be fed to that program, and determines whether the program, operating on the input, will eventually stop or will continue forever. It is trivial to create an algorithm that can do this for some inputs, but impossible to do this in general. For any characteristic of the program's eventual output, it

7150-563: Is the most important work on general algebra that the Annalen has ever published. Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say: I have convinced myself that even theology has its merits. For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. Although Kronecker had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea"—in other words (to quote Reid): "Through

7280-416: Is the set of regular languages , which are generated by regular expressions and which are recognized by finite automata . A more powerful but still not Turing-complete extension of finite automata is the category of pushdown automata and context-free grammars , which are commonly used to generate parse trees in an initial stage of program compiling . Further examples include some of the early versions of

7410-416: Is their defined relationships that are discussed. Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points ( line segments ), and congruence of angles . The axioms unify both the plane geometry and solid geometry of Euclid in a single system. Hilbert put forth

7540-608: 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. Many problems in 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

7670-514: The Berlin Group whose leading founders had studied under Hilbert in Göttingen ( Kurt Grelling , Hans Reichenbach and Walter Dubislav ). Around 1925, Hilbert developed pernicious anemia , a then-untreatable vitamin deficiency whose primary symptom is exhaustion; his assistant Eugene Wigner described him as subject to "enormous fatigue" and how he "seemed quite old," and that even after eventually being diagnosed and treated, he "was hardly

7800-548: The Blum–Shub–Smale machine model have formalized computation on the reals. 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 . Gödel's proofs show that

7930-531: The Cantor's theorem , there are uncountably many sets of natural numbers. 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 a computably enumerable (c.e.) set , which is a set that can be enumerated by a Turing machine (other terms for computably enumerable include recursively enumerable and semidecidable ). Equivalently,

SECTION 60

#1732783202886

8060-615: The Province of Prussia , Kingdom of Prussia , either in Königsberg (according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk ) near Königsberg where his father worked at the time of his birth. His paternal grandfather was David Hilbert, a judge and Geheimrat . His mother Maria had an interest in philosophy, astronomy and prime numbers , while his father Otto taught him Prussian virtues . After his father became

8190-414: The arithmetical hierarchy by permitting quantification over sets of natural numbers in addition to quantification over individual numbers. These areas are linked to the theories of well-orderings and trees; for example the set of all indices of computable (nonbinary) trees without infinite branches is complete for level Π 1 1 {\displaystyle \Pi _{1}^{1}} of

8320-415: The calculus of variations , commutative algebra , algebraic number theory , the foundations of geometry , spectral theory of operators and its application to integral equations , mathematical physics , and the foundations of mathematics (particularly proof theory ). He adopted and defended Georg Cantor 's set theory and transfinite numbers . In 1900, he presented a collection of problems that set

8450-580: The law of excluded middle in an infinite extension. Hilbert sent his results to the Mathematische Annalen . Gordan, the house expert on the theory of invariants for the Mathematische Annalen , could not appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was: Das ist nicht Mathematik. Das ist Theologie. This

8580-587: 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. But Post left open the main problem of the existence of computably 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

8710-603: The spherical harmonic functions" ). Hilbert remained at the University of Königsberg as a Privatdozent ( senior lecturer ) from 1886 to 1895. In 1895, as a result of intervention on his behalf by Felix Klein , he obtained the position of Professor of Mathematics at the University of Göttingen . During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world. He remained there for

8840-412: The μ-recursive functions obtained from primitive recursion and the μ operator. The terminology for computable functions and sets is not completely standardized. The definition in terms of μ-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

8970-416: The 1830s until the 1940s, mechanical calculating machines such as adders and multipliers were built and improved, but they could not perform a conditional branch and therefore were not Turing-complete. In the late 19th century, Leopold Kronecker formulated notions of computability, defining primitive recursive functions . These functions can be calculated by rote computation, but they are not enough to make

9100-614: The 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel —in a round table discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem. Gödel's incompleteness theorems show that even elementary axiomatic systems such as Peano arithmetic are either self-contradicting or contain logical propositions that are impossible to prove or disprove within that system. Hilbert's first work on invariant functions led him to

9230-427: The 1930s, with the work of Kurt Gödel , Alonzo Church , Rózsa Péter , Alan Turing , Stephen Kleene , and Emil Post . The fundamental results the researchers obtained established Turing computability as the correct formalization of the informal idea of effective calculation. In 1952, these results led Kleene to coin the two names "Church's thesis" and "Turing's thesis". Nowadays these are often considered as

9360-456: The 1950s that the word problem for groups is not effectively solvable: 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 (using results of Julia Robinson ) Matiyasevich's theorem , which implies that Hilbert's tenth problem has no effective solution; this problem asked whether there

9490-416: The German word Entscheidungsproblem which was used in the original papers of Turing and others. In contemporary use, the term "computable function" has various definitions: according to Nigel J. Cutland , it is a partial recursive function (which can be undefined for some inputs), while according to Robert I. Soare it is a total recursive (equivalently, general recursive) function. This article follows

9620-412: The abstraction of a universal computer is defined as a device with a Turing-complete instruction set, infinite memory, and infinite available time. In computability theory , several closely related terms are used to describe the computational power of a computational system (such as an abstract machine or programming language ): Turing completeness is significant in that every real-world design for

9750-646: 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 . 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 , artificial neural networks and continuous-time control theory , modelled by differential equations and continuous dynamical systems . For example, models of computation such as

9880-423: The basic result that a set is computable if and only if the set and its complement are both computably enumerable. Infinite c.e. sets have always infinite computable subsets; but on the other hand, simple sets exist but do not always have a coinfinite computable superset. Post introduced already hypersimple and hyperhypersimple sets; later maximal sets were constructed which are c.e. sets such that every c.e. superset

10010-530: The changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. New editions followed the 7th, but the main text was essentially not revised. Hilbert's approach signaled the shift to the modern axiomatic method . In this, Hilbert

10140-609: The class of all computably enumerable sets as well as for the class of all subsets of the natural numbers. 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 . These reducibilities are closely connected to definability over

10270-427: The computable sets and the halting problem, but they failed to show that any of these degrees contains a computably enumerable set. Very soon after this, Friedberg and Muchnik independently solved Post's problem by establishing the existence of computably enumerable sets of intermediate degree. This groundbreaking result opened a wide study of the Turing degrees of the computably enumerable sets which turned out to possess

10400-405: The computably 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. 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

10530-405: 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. 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

10660-484: The correspondence between vanishing ideals and their vanishing sets is bijective between affine varieties and radical ideals in C [ x 1 , … , x n ] {\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]} . In 1890, Giuseppe Peano had published an article in the Mathematische Annalen describing the historically first space-filling curve . In response, Hilbert designed his own construction of such

10790-460: The demonstration in 1888 of his famous finiteness theorem . Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. To solve what had become known in some circles as Gordan's Problem , Hilbert realized that it

10920-532: The departure of the Jews." Hilbert replied, "Suffered? It doesn't exist any longer, does it?" By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld , a theoretical physicist and also

11050-417: 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. 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

11180-476: The function mapping a degree x to the degree of its Turing jump is definable in the partial order of the Turing degrees. A survey by Ambos-Spies and Fejer gives an overview of this research and its historical progression. An ongoing area of research in computability theory studies reducibility relations other than Turing reducibility. Post introduced several strong reducibilities , so named because they imply truth-table reducibility. A Turing machine implementing

11310-476: The halting problem is the most complicated computably enumerable set with respect to many-one reducibility and with respect to Turing reducibility. In 1944, Post asked whether every computably enumerable set is either computable or Turing equivalent to the halting problem, that is, whether there is no computably enumerable set with a Turing degree intermediate between those two. As intermediate results, Post defined natural types of computably enumerable sets like

11440-550: The halting problem. These type of sets can be classified using the arithmetical hierarchy . For example, the index set FIN of the class of all finite sets is on the level Σ 2 , the index set REC of the class of all recursive sets is on the level Σ 3 , the index set COFIN of all cofinite sets is also on the level Σ 3 and the index set COMP of the class of all Turing-complete sets Σ 4 . These hierarchy levels are defined inductively, Σ n +1 contains just all sets which are computably enumerable relative to Σ n ; Σ 1 contains

11570-592: The important book Grundlagen der Mathematik (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert– Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyl's successor was Helmut Hasse . About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust . Rust asked whether "the Mathematical Institute really suffered so much because of

11700-599: The later "foundationalist" Russell–Whitehead or "encyclopedist" Nicolas Bourbaki , and from his contemporary Giuseppe Peano . The mathematical community as a whole could engage in problems of which he had identified as crucial aspects of important areas of mathematics. The problem set was launched as a talk, "The Problems of Mathematics", presented during the course of the Second International Congress of Mathematicians held in Paris. The introduction of

11830-424: 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, computable functional) which outputs for any input of the form ( f (0),  f (1), ...,  f ( n )) a hypothesis. A learner M learns a function f if almost all hypotheses are the same index e of f with respect to

11960-413: The limitation of finite memory is ignored, most programming languages are otherwise Turing-complete. In colloquial usage, the terms "Turing-complete" and "Turing-equivalent" are used to mean that any real-world general-purpose computer or computer language can approximately simulate the computational aspects of any other real-world general-purpose computer or computer language. In real life, this leads to

12090-811: The limits of computation. Here are a few: Most programming languages (their abstract models, maybe with some particular constructs that assume finite memory omitted), conventional and unconventional, are Turing-complete. This includes: Some rewrite systems are Turing-complete. Turing completeness is an abstract statement of ability, rather than a prescription of specific language features used to implement that ability. The features used to achieve Turing completeness can be quite different; Fortran systems would use loop constructs or possibly even goto statements to achieve repetition; Haskell and Prolog, lacking looping almost entirely, would use recursion . Most programming languages are describing computations on von Neumann architectures , which have memory (RAM and register) and

12220-404: The machine may possess that have nothing to do with computation. Charles Babbage 's analytical engine (1830s) would have been the first Turing-complete machine if it had been built at the time it was designed. Babbage appreciated that the machine was capable of great feats of calculation, including primitive logical reasoning, but he did not appreciate that no other machine could do better. From

12350-803: The more powerful the language is initially, the more complex are the tasks to which it is applied and the sooner its lack of completeness becomes perceived as a drawback, encouraging its extension until it is Turing-complete. The untyped lambda calculus is Turing-complete, but many typed lambda calculi, including System F , are not. The value of typed systems is based in their ability to represent most typical computer programs while detecting more errors. Rule 110 and Conway's Game of Life , both cellular automata , are Turing-complete. Some software and video games are Turing-complete by accident, i.e. not by design. Software: Games: Social media: Computational languages: Biology: Many computational languages exist that are not Turing-complete. One such example

12480-575: The more science-oriented Wilhelm Gymnasium . Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg , the "Albertina". In early 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but had gone to Berlin for three semesters), returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius (i.e., an associate professor). An intense and fruitful scientific exchange among

12610-482: The names recursion theory and computability theory fail to convey the fact that most of the objects studied in computability theory are not computable. In 1967, Rogers has suggested that a key property of computability 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

12740-401: The non-computability inherent in well known mathematical theorems. In 1999, Simpson discussed many aspects of second-order arithmetic and reverse mathematics. 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

12870-763: The notion of computation is essentially unique. In 1941 Konrad Zuse completed the Z3 computer. Zuse was not familiar with Turing's work on computability at the time. In particular, the Z3 lacked dedicated facilities for a conditional jump, thereby precluding it from being Turing complete. However, in 1998, it was shown by Rojas that the Z3 is capable of simulating conditional jumps, and therefore Turing complete in theory. To do this, its tape program would have to be long enough to execute every possible path through both sides of every branch. The first computer capable of conditional branching in practice, and therefore Turing complete in practice,

13000-420: The others, and most computability theorists are familiar with the majority of them. Computability theory in mathematical logic has traditionally focused on relative computability , a generalization of Turing computability defined using oracle Turing machines , introduced by Turing in 1939. An oracle Turing machine is a hypothetical device which, in addition to performing the actions of a regular Turing machine,

13130-515: The pixel shader languages embedded in Direct3D and OpenGL extensions. In total functional programming languages, such as Charity and Epigram , all functions are total and must terminate. Charity uses a type system and control constructs based on category theory , whereas Epigram uses dependent types . The LOOP language is designed so that it computes only the functions that are primitive recursive . All of these compute proper subsets of

13260-434: The power of flow-control instructions (for example, providing only loops that iterate over the items of an existing array). However, another theorem shows that there are problems solvable by Turing-complete languages that cannot be solved by any language with only finite looping abilities (i.e., languages that guarantee that every program will eventually finish to a halt). So any such language is not Turing-complete. For example,

13390-412: The practical concepts of computing virtualization and emulation . Real computers constructed so far can be functionally analyzed like a single-tape Turing machine (which uses a "tape" for memory); thus the associated mathematics can apply by abstracting their operation far enough. However, real computers have limited physical resources, so they are only linear bounded automaton complete. In contrast,

13520-501: The problems at the Congress, which were published in the acts of the Congress. In a subsequent publication, he extended the panorama, and arrived at the formulation of the now-canonical 23 Problems of Hilbert. See also Hilbert's twenty-fourth problem . The full text is important, since the exegesis of the questions still can be a matter of inevitable debate, whenever it is asked how many have been solved. Some of these were solved within

13650-613: 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. Priority arguments have been employed to solve many problems in computability theory, and have been classified into a hierarchy based on their complexity. 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. For example, Kummer published

13780-420: The research was independent, and the unity of the concept of randomness was not understood at the time). 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

13910-708: The rest of his life. Among Hilbert's students were Hermann Weyl , chess champion Emanuel Lasker , Ernst Zermelo , and Carl Gustav Hempel . John von Neumann was his assistant. At the University of Göttingen , Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church . Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911), and Wilhelm Ackermann (1925). Between 1902 and 1939 Hilbert

14040-413: The same Turing degree (also called degree of unsolvability ). 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: Many-one reductions are "stronger" than Turing reductions: if a set A is many-one reducible to

14170-474: The second of these conventions. In 1996, Soare gave additional comments about the terminology. 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 according to

14300-657: The set of logical consequences of an effective first-order theory is a computably 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. Computability 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

14430-521: The sets of interest in computability theory are the noncomputable sets, partitioned into equivalence classes by computable bijections of the natural numbers. The main professional organization for computability theory is the Association for Symbolic Logic , which holds several research conferences each year. The interdisciplinary research Association Computability in Europe ( CiE ) also organizes

14560-471: The speech that Hilbert gave said: Who among us would not be happy to lift the veil behind which is hidden the future; to gaze at the coming developments of our science and at the secrets of its development in the centuries to come? What will be the ends toward which the spirit of future generations of mathematicians will tend? What methods, what new facts will the new century reveal in the vast and rich field of mathematical thought? He presented fewer than half

14690-453: The standard model of arithmetic. Rice showed that for every nontrivial class C (which contains some but not all c.e. sets) the index set E = { e : the e th c.e. set W e 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). But, many of these index sets are even more complicated than

14820-427: The study of generalized notions of this field such as arithmetic reducibility , hyperarithmetical reducibility and α-recursion theory , as described by Sacks in 1990. 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

14950-546: 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. Post's theorem establishes

15080-435: The subject of algebra , a field is called algebraically closed if and only if every polynomial over it has a root in it. Under this condition, Hilbert gave a criterion for when a collection of polynomials ( p λ ) λ ∈ Λ {\displaystyle (p_{\lambda })_{\lambda \in \Lambda }} of n {\displaystyle n} variables has

15210-469: The terminology using the word "recursive" introduced by Kleene. Many contemporary researchers have begun to use this alternate terminology. These researchers also use terminology such as partial 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

15340-426: The three began, and Minkowski and Hilbert especially would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann , titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special binary forms , in particular

15470-594: The time were still used textbook-fashion. It is difficult to specify the axioms used by Hilbert without referring to the publication history of the Grundlagen since Hilbert changed and modified them several times. The original monograph was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated

15600-492: The total computable functions, since the full set of total computable functions is not computably enumerable . Also, since all functions in these languages are total, algorithms for recursively enumerable sets cannot be written in these languages, in contrast with Turing machines. Although (untyped) lambda calculus is Turing-complete, simply typed lambda calculus is not. Computability theory Basic questions addressed by computability theory include: Although there

15730-581: Was admitted into a psychiatric clinic, Hilbert said, "From now on, I must consider myself as not having a son." His attitude toward Franz brought Käthe considerable sorrow. Hilbert considered the mathematician Hermann Minkowski to be his "best and truest friend". Hilbert was baptized and raised a Calvinist in the Prussian Evangelical Church . He later left the Church and became an agnostic . He also argued that mathematical truth

15860-465: Was also a kind of manifesto that opened the way for the development of the formalist school, one of three major schools of mathematics of the 20th century. According to the formalist, mathematics is manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought. In 1920, Hilbert proposed a research project in metamathematics that became known as Hilbert's program. He wanted mathematics to be formulated on

15990-461: Was anticipated by Moritz Pasch 's work from 1882. Axioms are not taken as self-evident truths. Geometry may treat things , about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point , line , plane , and others, could be substituted, as Hilbert is reported to have said to Schoenflies and Kötter , by tables, chairs, glasses of beer and other such objects. It

16120-589: Was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker". Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert responded: Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists. In

16250-657: Was editor of the Mathematische Annalen , the leading mathematical journal of the time. He was elected an International Member of the United States National Academy of Sciences in 1907. In 1892, Hilbert married Käthe Jerosch (1864–1945), who was the daughter of a Königsberg merchant, "an outspoken young lady with an independence of mind that matched [Hilbert's]." While at Königsberg, they had their one child, Franz Hilbert (1893–1969). Franz suffered throughout his life from mental illness, and after he

16380-516: Was independent of the existence of God or other a priori assumptions. When Galileo Galilei was criticized for failing to stand up for his convictions on the Heliocentric theory , Hilbert objected: "But [Galileo] was not an idiot. Only an idiot could believe that scientific truth needs martyrdom; that may be necessary in religion, but scientific results prove themselves in due time." Like Albert Einstein , Hilbert had closest contacts with

16510-602: Was isolated soon after, starting with Gödel's incompleteness theorem . This theorem showed that axiom systems were limited when reasoning about the computation that deduces their theorems. Church and Turing independently demonstrated that Hilbert's Entscheidungsproblem (decision problem) was unsolvable, thus identifying the computational core of the incompleteness theorem. This work, along with Gödel's work on general recursive functions , established that there are sets of simple instructions, which, when put together, are able to produce any computation. The work of Gödel showed that

16640-432: Was necessary to take a completely different path. As a result, he demonstrated Hilbert's basis theorem , showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not a constructive proof —it did not display "an object"—but rather, it was an existence proof and relied on use of

16770-411: Was studied in detail by Stephen Simpson and others; in 1999, Simpson gave a detailed discussion of the program. 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

16900-527: Was the ENIAC in 1946. Zuse's Z4 computer was operational in 1945, but it did not support conditional branching until 1950. Computability theory uses models of computation to analyze problems and determine whether they are computable and under what circumstances. The first result of computability theory is that there exist problems for which it is impossible to predict what a (Turing-complete) system will do over an arbitrarily long time. The classic example

#885114