Post's theorem

Post's theorem In computability theory Post's theorem, named after Emil Post, describes the connection between the arithmetical hierarchy and the Turing degrees.

Inhalt 1 Hintergrund 2 Post's theorem and corollaries 3 Proof of Post's theorem 3.1 Formalization of Turing machines in first-order arithmetic 3.2 Implementation example 3.3 Recursively enumerable sets 3.4 Oracle machines 3.5 Turing jump 3.6 Higher Turing jumps 4 References Background See also: Arithmetical hierarchy § Relation to Turing machines The statement of Post's theorem uses several concepts relating to definability and recursion theory. This section gives a brief overview of these concepts, which are covered in depth in their respective articles.

The arithmetical hierarchy classifies certain sets of natural numbers that are definable in the language of Peano arithmetic. A formula is said to be {Anzeigestil Sigma _{m}^{0}} if it is an existential statement in prenex normal form (all quantifiers at the front) mit {Anzeigestil m} alternations between existential and universal quantifiers applied to a formula with bounded quantifiers only. Formally a formula {Anzeigestil phi (s)} in the language of Peano arithmetic is a {Anzeigestil Sigma _{m}^{0}} formula if it is of the form {Anzeigestil links(exists n_{1}^{1}exists n_{2}^{1}cdots exists n_{j_{1}}^{1}Rechts)links(forall n_{1}^{2}cdots forall n_{j_{2}}^{2}Rechts)links(exists n_{1}^{3}cdots richtig)cdots left(Qn_{1}^{m}cdots richtig)rho (n_{1}^{1},ldots n_{j_{m}}^{m},x_{1},Punkte ,x_{k})} wo {Anzeigestil rho } contains only bounded quantifiers and Q is {Anzeigestil für alle } if m is even and {displaystyle exists } if m is odd.

A set of natural numbers {Anzeigestil A} is said to be {Anzeigestil Sigma _{m}^{0}} if it is definable by a {Anzeigestil Sigma _{m}^{0}} formula, das ist, if there is a {Anzeigestil Sigma _{m}^{0}} formula {Anzeigestil phi (s)} such that each number {Anzeigestil n} ist in {Anzeigestil A} dann und nur dann, wenn {Anzeigestil phi (n)} hält. It is known that if a set is {Anzeigestil Sigma _{m}^{0}} then it is {Anzeigestil Sigma _{n}^{0}} für alle {displaystyle n>m} , but for each m there is a {Anzeigestil Sigma _{m+1}^{0}} set that is not {Anzeigestil Sigma _{m}^{0}} . Thus the number of quantifier alternations required to define a set gives a measure of the complexity of the set.

Post's theorem uses the relativized arithmetical hierarchy as well as the unrelativized hierarchy just defined. A set {Anzeigestil A} of natural numbers is said to be {Anzeigestil Sigma _{m}^{0}} relative to a set {Anzeigestil B} , written {Anzeigestil Sigma _{m}^{0,B}} , wenn {Anzeigestil A} is definable by a {Anzeigestil Sigma _{m}^{0}} formula in an extended language that includes a predicate for membership in {Anzeigestil B} .

While the arithmetical hierarchy measures definability of sets of natural numbers, Turing degrees measure the level of uncomputability of sets of natural numbers. A set {Anzeigestil A} is said to be Turing reducible to a set {Anzeigestil B} , written {displaystyle Aleq _{T}B} , if there is an oracle Turing machine that, given an oracle for {Anzeigestil B} , computes the characteristic function of {Anzeigestil A} . The Turing jump of a set {Anzeigestil A} is a form of the Halting problem relative to {Anzeigestil A} . Given a set {Anzeigestil A} , the Turing jump {Anzeigestil A'} is the set of indices of oracle Turing machines that halt on input {Anzeigestil 0} when run with oracle {Anzeigestil A} . It is known that every set {Anzeigestil A} is Turing reducible to its Turing jump, but the Turing jump of a set is never Turing reducible to the original set.

Post's theorem uses finitely iterated Turing jumps. For any set {Anzeigestil A} of natural numbers, the notation {Anzeigestil A^{(n)}} indicates the {Anzeigestil n} –fold iterated Turing jump of {Anzeigestil A} . Daher {Anzeigestil A^{(0)}} ist nur {Anzeigestil A} , und {Anzeigestil A^{(n+1)}} is the Turing jump of {Anzeigestil A^{(n)}} .

Post's theorem and corollaries Post's theorem establishes a close connection between the arithmetical hierarchy and the Turing degrees of the form {displaystyle emptyset ^{(n)}} , das ist, finitely iterated Turing jumps of the empty set. (The empty set could be replaced with any other computable set without changing the truth of the theorem.) Post's theorem states: A set {Anzeigestil B} ist {Anzeigestil Sigma _{n+1}^{0}} dann und nur dann, wenn {Anzeigestil B} is recursively enumerable by an oracle Turing machine with an oracle for {displaystyle emptyset ^{(n)}} , das ist, dann und nur dann, wenn {Anzeigestil B} ist {Anzeigestil Sigma _{1}^{0,emptyset ^{(n)}}} . The set {displaystyle emptyset ^{(n)}} ist {Anzeigestil Sigma _{n}^{0}} -complete for every {displaystyle n>0} . This means that every {Anzeigestil Sigma _{n}^{0}} set is many-one reducible to {displaystyle emptyset ^{(n)}} .

Post's theorem has many corollaries that expose additional relationships between the arithmetical hierarchy and the Turing degrees. These include: Fix a set {Anzeigestil C} . A set {Anzeigestil B} ist {Anzeigestil Sigma _{n+1}^{0,C}} dann und nur dann, wenn {Anzeigestil B} ist {Anzeigestil Sigma _{1}^{0,C^{(n)}}} . This is the relativization of the first part of Post's theorem to the oracle {Anzeigestil C} . A set {Anzeigestil B} ist {Anzeigestil Delta _{n+1}} dann und nur dann, wenn {displaystyle Bleq _{T}emptyset ^{(n)}} . Allgemeiner, {Anzeigestil B} ist {Anzeigestil Delta _{n+1}^{C}} dann und nur dann, wenn {displaystyle Bleq _{T}C^{(n)}} . A set is defined to be arithmetical if it is {Anzeigestil Sigma _{n}^{0}} für einige {Anzeigestil n} . Post's theorem shows that, gleichwertig, a set is arithmetical if and only if it is Turing reducible to {displaystyle emptyset ^{(m)}} for some m. Proof of Post's theorem Formalization of Turing machines in first-order arithmetic The operation of a Turing machine {Anzeigestil T} on input {Anzeigestil n} can be formalized logically in first-order arithmetic. Zum Beispiel, we may use symbols {Anzeigestil A_{k}} , {Anzeigestil B_{k}} , und {Anzeigestil C_{k}} for the tape configuration, machine state and location along the tape after {Anzeigestil k} Schritte, beziehungsweise. {Anzeigestil T} 's transition system determines the relation between {Anzeigestil (EIN_{k},B_{k},C_{k})} und {Anzeigestil (EIN_{k+1},B_{k+1},C_{k+1})} ; their initial values (zum {displaystyle k=0} ) are the input, the initial state and zero, beziehungsweise. The machine halts if and only if there is a number {Anzeigestil k} so dass {Anzeigestil B_{k}} is the halting state.

The exact relation depends on the specific implementation of the notion of Turing machine (z.B. their alphabet, allowed mode of motion along the tape, usw.) In case {Anzeigestil T} halts at time {Anzeigestil n_{1}} , the relation between {Anzeigestil (EIN_{k},B_{k},C_{k})} und {Anzeigestil (EIN_{k+1},B_{k+1},C_{k+1})} must be satisfied only for k bounded from above by {Anzeigestil n_{1}} .

Thus there is a formula {Anzeigestil Varphi (n,n_{1})} in first-order arithmetic with no unbounded quantifiers, so dass {Anzeigestil T} halts on input {Anzeigestil n} at time {Anzeigestil n_{1}} at most if and only if {Anzeigestil Varphi (n,n_{1})} is satisfied.

Implementation example For example, for a prefix-free Turing machine with binary alphabet and no blank symbol, we may use the following notations: {Anzeigestil A_{k}} is the 1-ary symbol for the configuration of the whole tape after {Anzeigestil k} Schritte (which we may write as a number with LSB first, the value of the m-th location on the tape being its m-th least significant bit). Im Speziellen {Anzeigestil A_{0}} is the initial configuration of the tape, which corresponds the input to the machine. {Anzeigestil B_{k}} is the 1-ary symbol for the Turing machine state after {Anzeigestil k} Schritte. Im Speziellen, {Anzeigestil B_{0}=q_{ich}} , the initial state of the Turing machine. {Anzeigestil C_{k}} is the 1-ary symbol for the Turing machine location on the tape after {Anzeigestil k} Schritte. Im Speziellen {Anzeigestil C_{0}=0} . {Anzeigestil M(q,b)} is the transition function of the Turing machine, written as a function from a doublet (machine state, bit read by the machine) to a triplet (new machine state, bit written by the machine, +1 oder -1 machine movement along the tape). {displaystyle bit(j,m)} is the j-th bit of a number {Anzeigestil m} . This can be written as a first-order arithmetic formula with no unbounded quantifiers.

For a prefix-free Turing machine we may use, for input n, the initial tape configuration {Anzeigestil t(n)=cat(2^{Decke(log_{2}n)}-1,0,n)} where cat stands for concatenation; daher {Anzeigestil t(n)} ist ein {displaystyle log(n)-} length string of {displaystyle 1-s} followed by {Anzeigestil 0} and then by {Anzeigestil n} .

The operation of the Turing machine at the first {Anzeigestil n_{1}} steps can thus be written as the conjunction of the initial conditions and the following formulas, quantified over {Anzeigestil k} für alle {Anzeigestil k existential quantifiers followed by a negation of a formula in {Anzeigestil Sigma _{1}^{0}} ; the latter formula can be enumerated by a Turing machine and can thus be checked immediately by an oracle for {displaystyle emptyset ^{(1)}} .

We may thus enumerate the {Anzeigestil k_{1}} –tuples of natural numbers and run an oracle machine with an oracle for {displaystyle emptyset ^{(1)}} that goes through all of them until it finds a satisfaction for the formula. This oracle machine halts on precisely the set of natural numbers satisfying {Anzeigestil Varphi (n)} , and thus enumerates its corresponding set.

Higher Turing jumps More generally, suppose every set that is recursively enumerable by an oracle machine with an oracle for {displaystyle emptyset ^{(p)}} ist in {Anzeigestil Sigma _{p+1}^{0}} . Then for an oracle machine with an oracle for {displaystyle emptyset ^{(p+1)}} , {displaystyle psi ^{Ö}(m)=exists m_{1}:psi _{H}(m,m_{1})} ist in {Anzeigestil Sigma _{p+1}^{0}} .

Seit {displaystyle psi ^{Ö}(m)} is the same as {Anzeigestil Varphi (n)} for the previous Turing jump, it can be constructed (as we have just done with {Anzeigestil Varphi (n)} Oben) so dass {Anzeigestil psi _{H}(m,m_{1})} in {displaystyle Pi _{p}^{0}} . After moving to prenex formal form the new {Anzeigestil Varphi (n)} ist in {Anzeigestil Sigma _{p+2}^{0}} .

Durch Induktion, every set that is recursively enumerable by an oracle machine with an oracle for {displaystyle emptyset ^{(p)}} , ist in {Anzeigestil Sigma _{p+1}^{0}} .

The other direction can be proven by induction as well: Suppose every formula in {Anzeigestil Sigma _{p+1}^{0}} can be enumerated by an oracle machine with an oracle for {displaystyle emptyset ^{(p)}} .

Now Suppose {Anzeigestil Varphi (n)} is a formula in {Anzeigestil Sigma _{p+2}^{0}} mit {Anzeigestil k_{1}} existential quantifiers followed by {Anzeigestil k_{2}} universal quantifiers etc. Äquivalent, {Anzeigestil Varphi (n)} hat {Anzeigestil k_{1}} > existential quantifiers followed by a negation of a formula in {Anzeigestil Sigma _{p+1}^{0}} ; the latter formula can be enumerated by an oracle machine with an oracle for {displaystyle emptyset ^{(p)}} and can thus be checked immediately by an oracle for {displaystyle emptyset ^{(p+1)}} .

We may thus enumerate the {Anzeigestil k_{1}} –tuples of natural numbers and run an oracle machine with an oracle for {displaystyle emptyset ^{(p+1)}} that goes through all of them until it finds a satisfaction for the formula. This oracle machine halts on precisely the set of natural numbers satisfying {Anzeigestil Varphi (n)} , and thus enumerates its corresponding set.

References Rogers, H. Die Theorie rekursiver Funktionen und effektiver Berechenbarkeit, MIT Press. ISBN 0-262-68052-1; ISBN 0-07-053522-1 Sonne, R. Rekursiv aufzählbare Mengen und Grade. Perspektiven in der mathematischen Logik. Springer-Verlag, Berlin, 1987. ISBN 3-540-15299-7 Kategorien: Theorems in the foundations of mathematicsComputability theoryMathematical logic hierarchies