Post's problem

Post's problem
Name	Post's problem
Field	Mathematical logic
Introduced	Emil Post, 1944
Related	Recursion theory, Computability theory, Turing degrees

Post's problem Post's problem asks whether there exist recursively enumerable degrees of unsolvability strictly between the computable degree and the halting problem degree. It originated in work by Emil Post and motivated advances in recursion theory, computability theory, and the study of Turing degrees, influencing research across mathematical logic, set theory, and theoretical aspects of computer science. The problem generated constructions and techniques that connected to figures and institutions such as Alonzo Church, Alan Turing, Kurt Gödel, Stephen Kleene, and research centers like Princeton University and Harvard University.

Introduction

Post formulated his question after contributions by Alonzo Church, Alan Turing, Kurt Gödel, Stephen Kleene, and Emil Post himself to establish the landscape of decidability and undecidability. Early investigations involved scholars at Harvard University, Princeton University, University of California, Berkeley, and the Institute for Advanced Study. The query tied into concepts introduced by Alonzo Church's lambda calculus, Alan Turing's Turing machines, and Stephen Kleene's recursive function theory, prompting comparative work by researchers affiliated with Massachusetts Institute of Technology, Yale University, University of Chicago, and Columbia University.

Historical Background

The historical lineage runs through the Entscheidungsproblem debates involving David Hilbert and Wittgenstein-era discussions, then to negative results by Alonzo Church and Alan Turing that established undecidability. Emil Post posed his intermediate-degree question after studies at institutions like City College of New York and exchanges with contemporaries including John von Neumann, Kurt Gödel, Alonzo Church, and Stephen Kleene. Subsequent decades saw major contributions from researchers such as Richard Friedberg, Albert Muchnik, Emil Post's students, and later from scholars at University of Chicago, University of California, Berkeley, Cornell University, and University of Michigan.

Formal Statement and Definitions

The formal framework employs notions from Turing degree theory, recursively enumerable sets, and reducibility concepts pioneered by Alan Turing and Alonzo Church. Definitions use reducibility relations like many-one reducibility as studied by Stephen Kleene and Emil Post, and oracle computations introduced by Alan Turing and elaborated by John Myhill and M. B. Pour-El. Key objects include the decidable degree (associated with recursive sets) and the degree of the halting problem (often linked to Turing machine computations studied by Alan Turing). Formal statements reference work by Richard Friedberg and Albert Muchnik who provided constructions in the framework developed by Stephen Kleene and Emil Post.

Key Results and Solutions

The independent solutions by Richard M. Friedberg and Albert A. Muchnik in the 1950s produced the first positive answers, demonstrating intermediate recursively enumerable degrees via priority constructions. These results connected to methods used by Emil Post and later refined by researchers such as Robert Soare, R. I. Soare, Gerald Sacks, Donald A. Martin, and Leo Harrington. Later contributions by S. Barry Cooper, Anil Nerode, Richard Shore, and scholars at University of Chicago amplified the theory, while developments in priority arguments involved figures like C. J. Ash and Jockusch.

Methods and Techniques

Techniques central to the problem include priority arguments, finite and infinite injury methods, and forcing adaptations in recursion theory—approaches advanced by Richard Friedberg, Albert Muchnik, Gerald Sacks, S. Barry Cooper, and Robert Soare. The priority method was inspired by combinatorial constructions used by researchers at Harvard University and further elaborated by collaborators and critics such as Anil Nerode, John Myhill, Harold Putnam, and Alonzo Church. Forcing-style techniques adapted from set-theoretic forcing by Paul Cohen found analogues in recursion theory through work by Ash and Jockusch, with subsequent refinements by Stephen Simpson and David Marker.

Impact and Applications

Solutions to the problem reshaped research in mathematical logic, recursion theory, and theoretical computer science at institutions including Princeton University, MIT, Stanford University, University of California, Berkeley, and the Institute for Advanced Study. The methods influenced complexity-theoretic hierarchies studied by researchers like Leslie Valiant, Richard Karp, Michael Rabin, and Noam Nisan, and informed structural investigations in degrees of unsolvability pursued by Gerald Sacks, Robert Soare, and Richard Shore. Applications extended to areas examined by scholars at Bell Labs, IBM Research, Microsoft Research, and in collaborations involving DARPA and academic units at Carnegie Mellon University.

Open Questions and Developments

Ongoing work connects to definability in the lattice of Turing degrees, automorphism problems explored by Richard Shore and Steve Simpson, and structural questions paralleled in research by Harvey Friedman, Hugh Woodin, W. Hugh Woodin, and S. Barry Cooper. Contemporary developments draw on tools from model theory used by Saharon Shelah and Alexei S. Kechris, and descriptive set theory pursued at Princeton University and University of California, Berkeley by researchers like Yiannis N. Moschovakis. Active open questions involve fine structure of degrees studied in seminars at Cambridge University, Oxford University, University of Toronto, and conferences sponsored by Association for Symbolic Logic and European Association for Theoretical Computer Science.

Category:Computability theory