Emil Leon Post
|Emil Leon Post|
|Born||February 11, 1897
Augustów, Suwałki Governorate, Russian Empire
|Died||April 21, 1954
New York City, U.S.
|Alma mater||City College of New York (B.S., Mathematics, 1917)
Columbia University (Ph.D., Mathematics, 1923)
|Known for||Formulation 1,
Post correspondence problem,
completeness-proof of Principia's propositional calculus
After completing his Ph.D. in mathematics at Columbia University, he did a post-doctorate at Princeton University. While at Princeton, he came very close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931. Post then became a high school mathematics teacher in New York City.
In his doctoral thesis, Post proved, among other things, that the propositional calculus of Principia Mathematica was complete: all tautologies are theorems, given the Principia axioms and the rules of substitution and modus ponens. Post also devised truth tables independently of Wittgenstein and C.S. Peirce and put them to good mathematical use. Jean Van Heijenoort's well-known source book on mathematical logic (1966) reprinted Post's classic article setting out these results.
In 1936, Post developed, independently of Alan Turing, a mathematical model of computation that was essentially equivalent to the Turing machine model. Intending this as the first of a series of models of equivalent power but increasing complexity, he titled his paper Formulation 1. This model is sometimes called "Post's machine" or a Post-Turing machine, but is not to be confused with Post's tag machines or other special kinds of Post canonical system, a computational model using string rewriting and developed by Post in the 1920s but first published in 1943. Post's rewrite technique is now ubiquitous in programming language specification and design, and so with Church's lambda-calculus is a salient influence of classical modern logic on practical computing. Post devised a method of 'auxiliary symbols' by which he could canonically represent any Post-generative language, and indeed any computable function or set at all.
In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem. This question, which became known as Post's problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the powerful priority method in recursion theory.
Post made a fundamental and still influential contribution to the theory of polyadic, or n-ary, groups in a long paper published in 1940. His major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n − 1. He also demonstrated that a polyadic group operation on a set can be expressed in terms of a group operation on the same set. The paper contains many other important results.
- Post, Emil Leon (1936). "Finite Combinatory Processes - Formulation 1". Journal of Symbolic Logic 1: 103–105.
- Post, Emil Leon (1940). "Polyadic groups". Transactions of the American Mathematical Society 48: 208–350.
- Post, Emil Leon (1943). "Formal Reductions of the General Combinatorial Decision Problem". American Journal of Mathematics 65: 197–215.
- Post, Emil Leon (1944). "Recursively enumerable sets of positive integers and their decision problems". Bulletin of the American Mathematical Society 50: 284–316. Introduces the important concept of many-one reduction.
- Arithmetical hierarchy
- Functional completeness
- List of multiple discoveries
- Post's inversion formula
- Post's lattice
- Post's theorem
- Urquhart (2008)
- O'Connor, John J.; Robertson, Edmund F., "Emil Leon Post", MacTutor History of Mathematics archive, University of St Andrews.
- Baaz, Matthias, ed. (2011). Kurt Gödel and the Foundations of Mathematics: Horizons of Truth (1st ed.). Cambridge University Press. ISBN 9781139498432.
- Urquhart (2008), p. 430.
- Urquhart, Alasdair (2008). "Emil Post" (PDF). In Gabbay, Dov M.; Woods, John Woods. Logic from Russell to Church. Handbook of the History of Logic 5. Elsevier BV.
- Anshel, Iris Lee; Anshel, Michael (November 1993). "From the Post-Markov Theorem Through Decision Problems to Public-Key Cryptography". The American Mathematical Monthly (Mathematical Association of America) 100 (9): 835–844.
- Dedicated to Emil Post and contains special material on Post. This includes "Post's Relation to the Cryptology and Cryptographists of his Era: ... Steven Brams, the noted game theorist and political scientist, has remarked to us that the life and legacy of Emil Post represents one aspect of New York intellectual life during the first half of the twentieth century that is very much in need of deeper exploration. The authors hope that this paper serves to further this pursuit". (pp. 842–843)
- Davis, Martin, ed. (1993). The Undecidable. Dover. pp. 288–406. ISBN 0-486-43228-9.
- Reprints several papers by Post.
- Davis, Martin (1994). "Emil L. Post: His Life and Work". Solvability, Provability, Definability: The Collected Works of Emil L. Post. Birkhäuser. pp. xi—xxviii.
- A biographical essay.
- Jackson, Allyn (May 2008). "An interview with Martin Davis". Notices of the AMS 55 (5): 560–571.
- Much material on Emil Post from his first-hand recollections.
- Emil Leon Post at the Mathematics Genealogy Project
- Emil Leon Post Papers 1927-1991, American Philosophical Society, Philadelphia, Pennsylvania.