Editing Julia Robinson (section)

==Mathematical contributions==

After graduating, Robinson continued in graduate studies at Berkeley. As a graduate student, Robinson was employed as a teaching assistant with the Department of Mathematics and later as a statistics lab assistant by [[Jerzy Neyman]] in the Berkeley Statistical Laboratory, where her work resulted in her first published paper, titled "''A Note on Exact Sequential Analysis"''.<ref name="NAS Memoir" />{{Rp|5}}

Robinson received her [[PhD]] degree in 1948 under [[Alfred Tarski]] with a dissertation on "Definability and Decision Problems in Arithmetic".<ref name=":0" />{{Rp|14}}
Her dissertation showed that the theory of the [[rational number]]s was an [[undecidable problem]], by demonstrating that elementary number theory could be defined in terms of the rationals. (Elementary number theory was already known to be undecidable by [[Kurt Gödel|Gödel's]] first [[incompleteness theorem]].)<ref name="Julia" />

Here is an excerpt from her thesis:<blockquote>"This consequence of our discussion is interesting because of a result of Gödel which shows that the variety of relations between integers (and operations on integers) which are arithmetically definable in terms of addition and multiplication of integers is very great. For instance from Theorem 3.2 and Gödel's result, we can conclude that the relation which holds between three rationals ''A'', ''B'', and ''N'' if and only if ''N'' is a positive integer and ''A''=''B''<sup>''N''</sup> is definable in the arithmetic of rationals."<ref>Robinson, J. (1949). Definability and decision problems in arithmetic. ''[[Journal of Symbolic Logic]],'' ''14''(2), 98-114. {{doi|10.2307/2266510}}</ref></blockquote>

===Hilbert's tenth problem===
[[Hilbert's tenth problem]] asks for an algorithm to determine whether a [[Diophantine equation]] has any solutions in [[integer]]s. Robinson began exploring methods for this problem in 1948 while at the [[RAND Corporation]]. Her work regarding Diophantine representation for exponentiation and her method of using [[Pell's equation]] led to the J.R. hypothesis (named after Robinson) in 1950. Proving this hypothesis would be central in the eventual solution. Her research publications would lead to collaborations with [[Martin Davis (mathematician)|Martin Davis]], [[Hilary Putnam]], and [[Yuri Matiyasevich]].<ref>{{cite book|title=The Decision Problem for Exponential Diophantine Equations|last1=Robinson|first1=Julia|last2=Davis|first2=Martin|last3=Putnam|first3=Hilary|date=1961|publisher=Annals of Mathematics|location=Princeton University}}</ref>

In 1950, Robinson first met Martin Davis, then an instructor at the [[University of Illinois at Urbana-Champaign]], who was trying to show that all sets with listability property were Diophantine in contrast to Robinson's attempt to show that a few special sets—including prime numbers and the powers of 2—were Diophantine. Robinson and Davis started collaborating in 1959 and were later joined by Hilary Putnam, they then showed that the solutions to a “Goldilocks” equation was key to Hilbert's tenth problem.<ref name="sciencenews.org">{{Cite web|url=https://www.sciencenews.org/article/how-julia-robinson-helped-define-limits-mathematical-knowledge|title = How Julia Robinson helped define the limits of mathematical knowledge|date = 22 November 2019}}</ref>

In 1970, the problem was resolved in the negative; that is, they showed that no such algorithm can exist. Through the 1970s, Robinson continued working with Matiyasevich on one of their solution's corollaries, which she once stated that

<blockquote>there is a constant ''N'' such that, given a Diophantine equation with any number of parameters and in any number of unknowns, one can effectively transform this equation into another with the same parameters but in only ''N'' unknowns such that both equations are solvable or unsolvable for the same values of the parameters.<ref name=":0">{{Cite web|url=https://logic.pdmi.ras.ru/~yumat/Julia/|title=My Collaboration with JULIA ROBINSON|website=logic.pdmi.ras.ru|access-date=2018-08-28}}</ref></blockquote>

At the time the solution was first published, the authors established ''N'' = 200. Robinson and Matiyasevich's joint work would produce further reduction to 9 unknowns.<ref name=":0" />

===Game theory ===
During the late 1940s, Robinson spent a year or so at the [[RAND Corporation]] in Santa Monica researching game theory. Her 1949 technical report, "On the Hamiltonian game (a traveling salesman problem),"<ref name="Robinson1949">{{cite report |last1=Robinson |first1=Julia |title=On the Hamiltonian game (a traveling salesman problem) |date=5 December 1949 |type=RAND report RM-303 |url=https://apps.dtic.mil/sti/tr/pdf/AD0204961.pdf |access-date=2024-04-15 |publisher=The Rand Corporation |location=Santa Monica, California|via=DTIC|language=en}}</ref> is the first publication to use the phrase "[[travelling salesman problem]]".<ref name="Schrijver2005">[[Alexander Schrijver]]'s 2005 paper "On the history of combinatorial optimization (till 1960). Handbook of Discrete Optimization ([[Karen Aardal|K. Aardal]], [[George Nemhauser|G.L. Nemhauser]], R. Weismantel, eds.), Elsevier, Amsterdam, 2005, pp. 1–68.[http://homepages.cwi.nl/~lex/files/histco.ps PS], [http://homepages.cwi.nl/~lex/files/histco.pdf PDF]</ref> Shortly thereafter she published a paper called "''An Iterative Method of Solving a Game''" in 1951.<ref name="NAS Memoir">
{{cite encyclopedia|last=Feferman|first=Solomon|author-link=Solomon Feferman|encyclopedia=Biographical Memoirs|title=Julia Bowman Robinson, 1919–1985|url=http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/robinson-julia.pdf|access-date=2008-06-18|year=1994|publisher=National Academy of Sciences|volume=63|location=Washington, DC|isbn=978-0-309-04976-4|pages=1–28}}
</ref>{{Rp|7}} In her paper, she proved that the [[fictitious play]] dynamics converges to the [[mixed strategy]] [[Nash equilibrium]] in two-player [[zero-sum game]]s. This was posed by [[George W. Brown (academic)|George W. Brown]] as a prize problem at [[RAND Corporation]].<ref name="Julia" />{{Rp|59}}