Turing jump
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| Turing jump | |
|---|---|
| Name | Turing jump |
| Field | Mathematical logic |
| Introduced | 1940s–1950s |
| Founders | Alan Turing, Emil Post, Alonzo Church, Stephen Kleene |
| Related | Recursion theory, Degrees of unsolvability, Arithmetical hierarchy, Kleene hierarchy |
Turing jump is an operator in computability theory that assigns to each Turing degree or set of natural numbers a strictly higher degree capturing the halting information relative to that set. It formalizes a step of noncomputability refinement that underlies major structures studied by Alan Turing, Emil Post, Alonzo Church, Stephen Kleene, and John Myhill. The jump connects to classical results involving the halting problem, Post's problem, Rice's theorem, Friedberg–Muchnik theorem, and the development of higher recursion theory by Gerald Sacks and Harvey Friedman.
Definition and basic properties
The Turing jump of a set A of naturals, often denoted A', is defined so that A' encodes the halting behavior of Turing machines with oracle A; this relates to the Halting problem and to oracles studied by Edward F. Moore and Marian Pour-El. Basic properties include that A' is Turing above A, that the jump is monotone relative to Turing reducibility studied by Alfred Tarski and Dana Scott, and that no set of the same degree can compute its own jump, a phenomenon appearing in work by Emil Post and Richard Friedberg. The jump operation yields degrees that are not minimal and interacts with join and meet operations seen in the lattice-theoretic investigations by S. C. Kleene and Rózsa Péter.
Formal construction and equivalent formulations
Formally, A' can be constructed as the set of indices e for which the e-th universal Turing machine with oracle A halts on input e, connecting to universal machines discussed by Marvin Minsky and Donald Knuth. Equivalent formulations include viewing A' as the complete Σ1^A set in the arithmetical hierarchy studied by Stephen Cole Kleene and Gerald Sacks, or as the jump operator defined via functional composition in the framework of degrees by Emil Post and Albert R. Meyer. The jump also admits characterization through partial recursive functionals elaborated by Alan Turing and Alonzo Church in lambda calculus comparisons with work by Haskell Curry.
Degree-theoretic and recursion-theoretic significance
In degree theory, the jump maps a Turing degree a to a' strictly above a, forming central examples in the lattice of Turing degrees examined by Friedberg, Muchnik, and S. C. Kleene. The jump provides invariants used in the solution of Post's problem by Friedberg–Muchnik theorem and underlies the definition of the jump operator on degrees by Gerald Sacks and Andrew Y. Kechris. In recursion theory, the jump is crucial to notions of relative computability and to the study of recursively enumerable sets from the perspectives of Martin Davis, Hilary Putnam, and Julia Robinson. The jump interacts with concepts like highness and lowness defined by Noam Greenberg and Rod Downey in modern computability.
Iterated jumps and the arithmetical hierarchy
Iterating the jump yields transfinite hierarchies: A^(n) for finite n and ordinal-indexed jumps for effective ordinals developed by Kurt Gödel and Gerald Sacks. These iterations correspond to levels Σn and Πn of the arithmetical hierarchy analyzed by Stephen Kleene and Hilbert's program commentators, and to hyperarithmetical hierarchies studied by Spector and Sacks. Iterated jump operators play a role in comparisons with the analytical hierarchy topics by Kreisel and Shoenfield, and in calibrating definability used by Solomon Feferman and Harvey Friedman.
Applications in computability and logic
The jump is applied to prove nondefinability and undecidability results such as variants of the halting problem and relativized incompleteness theorems echoing Kurt Gödel and Alonzo Church. It is instrumental in constructing degrees with prescribed properties in priority arguments from Friedberg and Soare, and in delineating the boundary between decidable theories in work by Alfred Tarski, Julia Robinson, and Dana Scott. In proof theory, the jump informs ordinals used in consistency proofs by Gerald Gentzen and in reverse mathematics classifications by Stephen Simpson and Harvey Friedman.
Historical context and development
The concept emerged from early work on undecidability by Alan Turing, Alonzo Church, and Emil Post in the 1930s and 1940s, with formal jump notions consolidated by S. C. Kleene and John Myhill in the 1950s. Subsequent developments by Richard Friedberg, Albert M. Meyer, and Gerald Sacks expanded its use in degree theory and higher recursion theory, while later researchers such as Rod Downey, Denis R. Hirschfeldt, and Noam Greenberg connected the jump to modern algorithmic randomness and computable analysis. The Turing jump remains central to ongoing research by groups at institutions like Massachusetts Institute of Technology, University of California, Berkeley, University of Cambridge, and Princeton University.