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131-516: In computability theory , the Church–Turing thesis (also known as computability thesis , the Turing–Church thesis , the Church–Turing conjecture , Church's thesis , Church's conjecture , and Turing's thesis ) is a thesis about the nature of computable functions . It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by

262-586: A Turing machine . The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing . Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability : Church, Kleene , and Turing proved that these three formally defined classes of computable functions coincide:

393-431: A " natural law " rather than by "a definition or an axiom". This idea was "sharply" criticized by Church. Thus Post in his 1936 paper was also discounting Kurt Gödel 's suggestion to Church in 1934–1935 that the thesis might be expressed as an axiom or set of axioms. Turing adds another definition, Rosser equates all three : Within just a short time, Turing's 1936–1937 paper "On Computable Numbers, with an Application to

524-435: A (multi-tape) universal Turing machine only suffers a logarithmic slowdown factor in simulating any Turing machine. Computability theory (computation) Basic questions addressed by computability theory include: Although there 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

655-409: A Turing Machine". Turing stated it this way: It was stated ... that "a function is effectively calculable if its values can be found by some purely mechanical process". We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development ... leads to ... an identification of computability with effective calculability. [

786-494: A Turing machine based on the work of Emil Post. Both theses are proven equivalent by use of "Theorem XXX". Thesis I. Every effectively calculable function (effectively decidable predicate) is general recursive . Theorem XXX: The following classes of partial functions are coextensive, i.e. have the same members: (a) the partial recursive functions, (b) the computable functions ... Turing's thesis: Turing's thesis that every function which would naturally be regarded as computable

917-435: A clean formal language , and much of his subsequent work was directed toward that goal. In 1928, David Hilbert and Wilhelm Ackermann posed the question in the form outlined above. In continuation of his "program", Hilbert posed three questions at an international conference in 1928, the third of which became known as "Hilbert's Entscheidungsproblem ". In 1929, Moses Schönfinkel published one paper on special cases of

1048-472: 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

1179-448: A computer in our universe ( physical Church-Turing thesis ) and what can be efficiently computed ( Church–Turing thesis (complexity theory) ). These variations are not due to Church or Turing, but arise from later work in complexity theory and digital physics . The thesis also has implications for the philosophy of mind (see below ). J. B. Rosser  ( 1939 ) addresses the notion of "effective computability" as follows: "Clearly

1310-689: A finite set of Aristotelean logic formulas, it is NLOGSPACE -complete to decide its S a t {\displaystyle {\rm {Sat}}} . It is also NLOGSPACE-complete to decide S a t {\displaystyle {\rm {Sat}}} for a slight extension (Theorem 2.7): ∀ x , ± p ( x ) → ± q ( x ) , ∃ x , ± p ( x ) ∧ ± q ( x ) {\displaystyle \forall x,\pm p(x)\to \pm q(x),\quad \exists x,\pm p(x)\wedge \pm q(x)} Relational logic extends Aristotelean logic by allowing

1441-436: 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

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1572-405: A function is λ-computable if and only if it is Turing computable, and if and only if it is general recursive . This has led mathematicians and computer scientists to believe that the concept of computability is accurately characterized by these three equivalent processes. Other formal attempts to characterize computability have subsequently strengthened this belief (see below ). On the other hand,

1703-534: 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

1834-443: 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

1965-418: A logical fragment is called decidable if there exists a program that can decide, for each Φ {\displaystyle \Phi } finite set of logical formulas in the fragment, whether S a t ( Φ ) {\displaystyle {\rm {{Sat}(\Phi )}}} or not. There is a hierarchy of decidabilities. On the top are the undecidable problems. Below it are

2096-417: 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

2227-405: 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

2358-489: 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

2489-471: 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

2620-1463: A relational predicate. For example, "Everybody loves somebody" can be written as ∀ x , b o d y ( x ) , ∃ y , b o d y ( y ) ∧ l o v e ( x , y ) {\textstyle \forall x,{\rm {{body}(x),\exists y,{\rm {{body}(y)\wedge {\rm {{love}(x,y)}}}}}}} . Generally, we have 8 kinds of sentences: ∀ x , p ( x ) → ( ∀ y , q ( x ) → ± r ( x , y ) ) , ∀ x , p ( x ) → ( ∃ y , q ( x ) ∧ ± r ( x , y ) ) ∃ x , p ( x ) ∧ ( ∀ y , q ( x ) → ± r ( x , y ) ) , ∃ x , p ( x ) ∧ ( ∃ y , q ( x ) ∧ ± r ( x , y ) ) {\displaystyle {\begin{aligned}\forall x,p(x)\to (\forall y,q(x)\to \pm r(x,y)),&\quad \forall x,p(x)\to (\exists y,q(x)\wedge \pm r(x,y))\\\exists x,p(x)\wedge (\forall y,q(x)\to \pm r(x,y)),&\quad \exists x,p(x)\wedge (\exists y,q(x)\wedge \pm r(x,y))\end{aligned}}} It

2751-407: A result". In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turing-machine or equivalent mechanical device". Turing's "definitions" given in a footnote in his 1938 Ph.D. thesis Systems of Logic Based on Ordinals , supervised by Church, are virtually the same: We shall use

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2882-485: A rigorous, formal proof. To establish that a function is computable by Turing machine, it is usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude "by the Church–Turing thesis" that the function is Turing computable (equivalently, partial recursive). Dirk van Dalen gives the following example for the sake of illustrating this informal use of

3013-477: A series of annual conferences. Entscheidungsproblem In mathematics and computer science , the Entscheidungsproblem ( German for 'decision problem'; pronounced [ɛntˈʃaɪ̯dʊŋspʁoˌbleːm] ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it

3144-464: A series of constraints on the behavior of a computor —"a human computing agent who proceeds mechanically". These constraints reduce to: The matter remains in active discussion within the academic community. The thesis can be viewed as nothing but an ordinary mathematical definition. Comments by Gödel on the subject suggest this view, e.g. "the correct definition of mechanical computability was established beyond any doubt by Turing". The case for viewing

3275-434: 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

3406-418: 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

3537-445: 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

3668-509: A set of axioms which would embody the generally accepted properties of this notion, and to do something on that basis. But Gödel offered no further guidance. Eventually, he would suggest his recursion, modified by Herbrand's suggestion, that Gödel had detailed in his 1934 lectures in Princeton NJ (Kleene and Rosser transcribed the notes). But he did not think that the two ideas could be satisfactorily identified "except heuristically". Next, it

3799-399: 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

3930-528: 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

4061-455: 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

Church–Turing thesis - Misplaced Pages Continue

4192-413: 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

4323-412: Is NEXPTIME -complete (Theorem 3.18). With x , y , z {\displaystyle x,y,z} , it is RE -complete to decide its S a t {\displaystyle {\rm {Sat}}} , and co-RE-complete to decide F i n S a t {\displaystyle {\rm {FinSat}}} (Theorem 3.15), thus undecidable. The monadic predicate calculus

4454-464: Is NLOGSPACE -complete to decide its S a t {\displaystyle {\rm {Sat}}} (Theorem 2.15). Relational logic can be extended to 32 kinds of sentences by allowing ± p , ± q {\displaystyle \pm p,\pm q} , but this extension is EXPTIME -complete (Theorem 2.24). The first-order logic fragment where the only variable names are x , y {\displaystyle x,y}

4585-409: 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

4716-407: 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,

4847-426: 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

4978-419: 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

5109-463: Is argued, any machine must satisfy". His most-important fourth, "the principle of causality" is based on the "finite velocity of propagation of effects and signals; contemporary physics rejects the possibility of instantaneous action at a distance". From these principles and some additional constraints—(1a) a lower bound on the linear dimensions of any of the parts, (1b) an upper bound on speed of propagation (the velocity of light), (2) discrete progress of

5240-541: 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

5371-423: 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)

Church–Turing thesis - Misplaced Pages Continue

5502-478: 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

5633-521: Is computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX. Kleene, finally, uses for the first time the term the "Church-Turing thesis" in a section in which he helps to give clarifications to concepts in Alan Turing's paper "The Word Problem in Semi-Groups with Cancellation", as demanded in a critique from William Boone. An attempt to better understand

5764-401: Is correct. In fact, Gödel (1936) proposed something stronger than this; he observed that there was something "absolute" about the concept of "reckonable in S 1 ": It may also be shown that a function which is computable ['reckonable'] in one of the systems S i , or even in a system of transfinite type, is already computable [reckonable] in S 1 . Thus the concept 'computable' ['reckonable']

5895-463: 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

6026-416: Is general recursive [Kleene's italics] Since a precise mathematical definition of the term effectively calculable (effectively decidable) has been wanting, we can take this thesis ... as a definition of it ... ...the thesis has the character of an hypothesis—a point emphasized by Post and by Church. If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about

6157-417: Is in a certain definite sense 'absolute', while practically all other familiar metamathematical concepts (e.g. provable, definable, etc.) depend quite essentially on the system to which they are defined ... Proofs in computability theory often invoke the Church–Turing thesis in an informal way to establish the computability of functions while avoiding the (often very long) details which would be involved in

6288-418: 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

6419-404: 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

6550-527: 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

6681-410: Is no way to extend the notion of computable function." All these contributions involve proofs that the models are computationally equivalent to the Turing machine; such models are said to be Turing complete . Because all these different attempts at formalizing the concept of "effective calculability/computability" have yielded equivalent results, it is now generally assumed that the Church–Turing thesis

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6812-465: Is now known as the Church–Turing thesis . The origin of the Entscheidungsproblem goes back to Gottfried Leibniz , who in the seventeenth century, after having constructed a successful mechanical calculating machine , dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements. He realized that the first step would have to be

6943-428: Is now known as the counter machine model. In the late 1960s and early 1970s researchers expanded the counter machine model into the register machine , a close cousin to the modern notion of the computer . Other models include combinatory logic and Markov algorithms . Gurevich adds the pointer machine model of Kolmogorov and Uspensky (1953, 1958): "... they just wanted to ... convince themselves that there

7074-610: Is of considerable interest for program verification and circuit verification. Pure Boolean logical formulas are usually decided using SAT-solving techniques based on the DPLL algorithm . For more general decision problems of first-order theories, conjunctive formulas over linear real or rational arithmetic can be decided using the simplex algorithm , formulas in linear integer arithmetic ( Presburger arithmetic ) can be decided using Cooper's algorithm or William Pugh 's Omega test . Formulas with negations, conjunctions and disjunctions combine

7205-405: Is provable using the rules of logic . In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notion of " effectively calculable " is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus ). This assumption

7336-473: 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,

7467-575: Is the class of first-order formulas with quantifier prefix ∃ ⋯ ∃ ∀ ⋯ ∀ {\displaystyle \exists \cdots \exists \forall \cdots \forall } , equality symbols, and no function symbols . For example, Turing's 1936 paper (p. 263) observed that since the halting problem for each Turing machine is equivalent to a first-order logical formula of form ∀ ∃ ∀ ∃ 6 {\displaystyle \forall \exists \forall \exists ^{6}} ,

7598-464: Is the footnote quoted above.] One of the important problems for logicians in the 1930s was the Entscheidungsproblem of David Hilbert and Wilhelm Ackermann , which asked whether there was a mechanical procedure for separating mathematical truths from mathematical falsehoods. This quest required that the notion of "algorithm" or "effective calculability" be pinned down, at least well enough for

7729-631: Is the fragment where each formula contains only 1-ary predicates and no function symbols. Its S a t {\displaystyle {\rm {Sat}}} is NEXPTIME-complete (Theorem 3.22). Any first-order formula has a prenex normal form. For each possible quantifier prefix to the prenex normal form, we have a fragment of first-order logic. For example, the Bernays–Schönfinkel class , [ ∃ ∗ ∀ ∗ ] = {\displaystyle [\exists ^{*}\forall ^{*}]_{=}} ,

7860-416: Is universally valid, i.e., valid in every structure . Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of first-order logic , a statement is universally valid if and only if it can be deduced using logical rules and axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement

7991-403: Is unsolvable: there is no algorithm that can determine whether a well formed formula has a beta normal form . Many years later in a letter to Davis (c. 1965), Gödel said that "he was, at the time of these [1934] lectures, not at all convinced that his concept of recursion comprised all possible recursions". By 1963–1964 Gödel would disavow Herbrand–Gödel recursion and the λ-calculus in favor of

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8122-540: 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

8253-401: The Entscheidungsproblem was then given by Alonzo Church in 1935–36 ( Church's theorem ) and independently shortly thereafter by Alan Turing in 1936 ( Turing's proof ). Church proved that there is no computable function which decides, for two given λ-calculus expressions, whether they are equivalent or not. He relied heavily on earlier work by Stephen Kleene . Turing reduced the question of

8384-548: 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

8515-527: 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,

8646-414: 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

8777-587: 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

8908-412: 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

9039-456: 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

9170-459: The Church–Turing thesis states that the above three formally-defined classes of computable functions coincide with the informal notion of an effectively calculable function. Although the thesis has near-universal acceptance, it cannot be formally proven, as the concept of effective calculability is only informally defined. Since its inception, variations on the original thesis have arisen, including statements about what can physically be realized by

9301-499: The Church–Turing thesis: Example: Each infinite recursively enumerable (RE) set contains an infinite recursive set . Proof: Let A be infinite RE. We list the elements of A effectively, n 0 , n 1 , n 2 , n 3 , ... From this list we extract an increasing sublist: put m 0  = n 0 , after finitely many steps we find an n k such that n k > m 0 , put m 1  = n k . We repeat this procedure to find m 2 > m 1 , etc. this yields an effective listing of

9432-660: The Entscheidungsproblem" appeared. In it he stated another notion of "effective computability" with the introduction of his a-machines (now known as the Turing machine abstract computational model). And in a proof-sketch added as an "Appendix" to his 1936–1937 paper, Turing showed that the classes of functions defined by λ-calculus and Turing machines coincided. Church was quick to recognise how compelling Turing's analysis was. In his review of Turing's paper he made clear that Turing's notion made "the identification with effectiveness in

9563-459: The Entscheidungsproblem. Such more general decision problems are of practical interest. Some first-order theories are algorithmically decidable ; examples of this include Presburger arithmetic , real closed fields , and static type systems of many programming languages . On the other hand, the first-order theory of the natural numbers with addition and multiplication expressed by Peano's axioms cannot be decided with an algorithm. By default,

9694-465: 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

9825-486: The Turing machine as the definition of "algorithm" or "mechanical procedure" or "formal system". A hypothesis leading to a natural law? : In late 1936 Alan Turing 's paper (also proving that the Entscheidungsproblem is unsolvable) was delivered orally, but had not yet appeared in print. On the other hand, Emil Post 's 1936 paper had appeared and was certified independent of Turing's work. Post strongly disagreed with Church's "identification" of effective computability with

9956-468: The above example completely rigorous, one would have to carefully construct a Turing machine, or λ-function, or carefully invoke recursion axioms, or at best, cleverly invoke various theorems of computability theory. But because the computability theorist believes that Turing computability correctly captures what can be computed effectively, and because an effective procedure is spelled out in English for deciding

10087-646: 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

10218-496: The application of the mathematical theory developed from the definition. For the acceptance of the hypothesis, there are, as we have suggested, quite compelling grounds. The Church–Turing Thesis : Stephen Kleene, in Introduction To Metamathematics , finally goes on to formally name "Church's Thesis" and "Turing's Thesis", using his theory of recursive realizability. Kleene having switched from presenting his work in

10349-423: 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

10480-439: The citations in the section are from Pratt-Hartmann (2023). The classical Entscheidungsproblem asks that, given a first-order formula, whether it is true in all models. The finitary problem asks whether it is true in all finite models. Trakhtenbrot's theorem shows that this is also undecidable. Some notations: S a t ( Φ ) {\displaystyle {\rm {{Sat}(\Phi )}}} means

10611-609: 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

10742-427: 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

10873-405: 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

11004-433: 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 the 1930s, with the work of Kurt Gödel , Alonzo Church , Rózsa Péter , Alan Turing , Stephen Kleene , and Emil Post . The fundamental results

11135-405: 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

11266-815: The decidable problems. Furthermore, the decidable problems can be divided into a complexity hierarchy. Aristotelian logic considers 4 kinds of sentences: "All p are q", "All p are not q", "Some p is q", "Some p is not q". We can formalize these kinds of sentences as a fragment of first-order logic: ∀ x , p ( x ) → ± q ( x ) , ∃ x , p ( x ) ∧ ± q ( x ) {\displaystyle \forall x,p(x)\to \pm q(x),\quad \exists x,p(x)\wedge \pm q(x)} where p , q {\displaystyle p,q} are atomic predicates, and + q := q , − q := ¬ q {\displaystyle +q:=q,\;-q:=\neg q} . Given

11397-541: The decision problem, that was prepared by Paul Bernays . As late as 1930, Hilbert believed that there would be no such thing as an unsolvable problem. Before the question could be answered, the notion of "algorithm" had to be formally defined. This was done by Alonzo Church in 1935 with the concept of "effective calculability" based on his λ-calculus , and by Alan Turing the next year with his concept of Turing machines . Turing immediately recognized that these are equivalent models of computation . A negative answer to

11528-417: 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

11659-486: The difficulties of satisfiability testing with that of decision of conjunctions; they are generally decided nowadays using SMT-solving techniques, which combine SAT-solving with decision procedures for conjunctions and propagation techniques. Real polynomial arithmetic, also known as the theory of real closed fields , is decidable; this is the Tarski–Seidenberg theorem , which has been implemented in computers by using

11790-439: The existence of CC and RC (Church's and Rosser's proofs) presupposes a precise definition of 'effective'. 'Effective method' is here used in the rather special sense of a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps". Thus the adverb-adjective "effective" is used in a sense of "1a: producing a decided, decisive, or desired effect", and "capable of producing

11921-414: The existence of an 'algorithm' or 'general method' able to solve the Entscheidungsproblem to the question of the existence of a 'general method' which decides whether any given Turing machine halts or not (the halting problem ). If 'algorithm' is understood as meaning a method that can be represented as a Turing machine, and with the answer to the latter question negative (in general), the question about

12052-415: The existence of an algorithm for the Entscheidungsproblem also must be negative (in general). In his 1936 paper, Turing says: "Corresponding to each computing machine 'it' we construct a formula 'Un(it)' and we show that, if there is a general method for determining whether 'Un(it)' is provable, then there is a general method for determining whether 'it' ever prints 0". The work of both Church and Turing

12183-425: The expression "computable function" to mean a function calculable by a machine, and let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions. The thesis can be stated as: Every effectively calculable function is a computable function . Church also stated that "No computational procedure will be considered as an algorithm unless it can be represented as

12314-476: 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

12445-637: 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." 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

12576-476: 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

12707-550: 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

12838-424: 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

12969-547: The list the functions " reckonable in the system S 1 " of Kurt Gödel 1936, and Emil Post 's (1943, 1946) " canonical [also called normal ] systems ". In the 1950s Hao Wang and Martin Davis greatly simplified the one-tape Turing-machine model (see Post–Turing machine ). Marvin Minsky expanded the model to two or more tapes and greatly simplified the tapes into "up-down counters", which Melzak and Lambek further evolved into what

13100-436: The machine, and (3) deterministic behavior—he produces a theorem that "What can be calculated by a device satisfying principles I–IV is computable." In the late 1990s Wilfried Sieg analyzed Turing's and Gandy's notions of "effective calculability" with the intent of "sharpening the informal notion, formulating its general features axiomatically, and investigating the axiomatic framework". In his 1997 and 2002 work Sieg presents

13231-477: 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

13362-401: 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

13493-450: The notion of "effective computability" led Robin Gandy (Turing's student and friend) in 1980 to analyze machine computation (as opposed to human-computation acted out by a Turing machine). Gandy's curiosity about, and analysis of, cellular automata (including Conway's game of life ), parallelism, and crystalline automata, led him to propose four "principles (or constraints) ... which it

13624-467: The ordinary (not explicitly defined) sense evident immediately". In a few years (1939) Turing would propose, like Church and Kleene before him, that his formal definition of mechanical computing agent was the correct one. Thus, by 1939, both Church (1934) and Turing (1939) had individually proposed that their "formal systems" should be definitions of "effective calculability"; neither framed their statements as theses . Rosser (1939) formally identified

13755-420: 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,

13886-551: The problem S a t ( ∀ ∃ ∀ ∃ 6 ) {\displaystyle {\rm {{Sat}(\forall \exists \forall \exists ^{6})}}} is undecidable. The precise boundaries are known, sharply: Börger et al. (2001) describes the level of computational complexity for every possible fragment with every possible combination of quantifier prefix, functional arity, predicate arity, and equality/no-equality. Having practical decision procedures for classes of logical formulas

14017-405: The problem of deciding whether there exists a model for a set of logical formulas Φ {\displaystyle \Phi } . F i n S a t ( Φ ) {\displaystyle {\rm {{FinSat}(\Phi )}}} is the same problem, but for finite models. The S a t {\displaystyle {\rm {Sat}}} -problem for

14148-402: The quest to begin. But from the very outset Alonzo Church 's attempts began with a debate that continues to this day. Was the notion of "effective calculability" to be (i) an "axiom or axioms" in an axiomatic system, (ii) merely a definition that "identified" two or more propositions, (iii) an empirical hypothesis to be verified by observation of natural events, or (iv) just a proposal for

14279-613: 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

14410-420: 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

14541-559: 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 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)

14672-408: The sake of argument (i.e. a "thesis")? In the course of studying the problem, Church and his student Stephen Kleene introduced the notion of λ-definable functions , and they were able to prove that several large classes of functions frequently encountered in number theory were λ-definable. The debate began when Church proposed to Gödel that one should define the "effectively computable" functions as

14803-413: 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

14934-474: 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

15065-460: The set B, the computability theorist accepts this as proof that the set is indeed recursive. The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the physical Church–Turing thesis states: "All physically computable functions are Turing-computable." The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another. It has been proved for instance that

15196-657: 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

15327-511: 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

15458-453: 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

15589-427: 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

15720-537: 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

15851-459: The subset B={m 0 , m 1 , m 2 ,...} of A, with the property m i < m i+1 . Claim . B is decidable. For, in order to test k in B we must check if k = m i for some i. Since the sequence of m i 's is increasing we have to produce at most k+1 elements of the list and compare them with k. If none of them is equal to k, then k not in B. Since this test is effective, B is decidable and, by Church's thesis , recursive. In order to make

15982-606: The terminology of Church-Kleene lambda definability, to that of Gödel-Kleene recursiveness (partial recursive functions). In this transition, Kleene modified Gödel's general recursive functions to allow for proofs of the unsolvability of problems in the Intuitionism of E. J. Brouwer. In his graduate textbook on logic, "Church's thesis" is introduced and basic mathematical results are demonstrated to be unrealizable. Next, Kleene proceeds to present "Turing's thesis", where results are shown to be uncomputable, using his simplified derivation of

16113-469: 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

16244-423: The thesis as nothing more than a definition is made explicitly by Robert I. Soare , where it is also argued that Turing's definition of computability is no less likely to be correct than the epsilon-delta definition of a continuous function . Other formalisms (besides recursion, the λ-calculus , and the Turing machine ) have been proposed for describing effective calculability/computability. Kleene (1952) adds to

16375-535: The three notions-as-definitions: All three definitions are equivalent, so it does not matter which one is used. Kleene proposes Thesis I : This left the overt expression of a "thesis" to Kleene. In 1943 Kleene proposed his "Thesis I": This heuristic fact [general recursive functions are effectively calculable] ... led Church to state the following thesis. The same thesis is implicit in Turing's description of computing machines. Thesis I. Every effectively calculable function (effectively decidable predicate)

16506-479: The work of Yuri Matiyasevich , Julia Robinson , Martin Davis , and Hilary Putnam , with the final piece of the proof in 1970, also implies a negative answer to the Entscheidungsproblem . Using the deduction theorem , the Entscheidungsproblem encompasses the more general problem of deciding whether a given first-order sentence is entailed by a given finite set of sentences, but validity in first-order theories with infinitely many axioms cannot be directly reduced to

16637-413: The λ-calculus and recursion, stating: Actually the work already done by Church and others carries this identification considerably beyond the working hypothesis stage. But to mask this identification under a definition… blinds us to the need of its continual verification. Rather, he regarded the notion of "effective calculability" as merely a "working hypothesis" that might lead by inductive reasoning to

16768-434: The λ-definable functions. Gödel, however, was not convinced and called the proposal "thoroughly unsatisfactory". Rather, in correspondence with Church (c. 1934–1935), Gödel proposed axiomatizing the notion of "effective calculability"; indeed, in a 1935 letter to Kleene, Church reported that: His [Gödel's] only idea at the time was that it might be possible, in terms of effective calculability as an undefined notion, to state

16899-434: Was heavily influenced by Kurt Gödel 's earlier work on his incompleteness theorem , especially by the method of assigning numbers (a Gödel numbering ) to logical formulas in order to reduce logic to arithmetic. The Entscheidungsproblem is related to Hilbert's tenth problem , which asks for an algorithm to decide whether Diophantine equations have a solution. The non-existence of such an algorithm, established by

17030-411: Was necessary to identify and prove the equivalence of two notions of effective calculability. Equipped with the λ-calculus and "general" recursion, Kleene with help of Church and J. Barkley Rosser produced proofs (1933, 1935) to show that the two calculi are equivalent. Church subsequently modified his methods to include use of Herbrand–Gödel recursion and then proved (1936) that the Entscheidungsproblem

17161-411: 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

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