The bulk of my mathematical research is on reverse mathematics and
on the degrees of unsolvability of mass problems.
Reverse Mathematics: Reverse mathematics is a study of
the logical strength of mathematical theorems in the context of
second-order arithmetic. The goal is to determine what axioms
are necessary and sufficient to prove a given theorem. To show
that certain axioms suffice to prove a given theorem, all we need is
a proof of that theorem from the axioms. To show that certain
axioms are necessary to prove a given theorem, we fix a set of weak
axioms once-and-for-all and show that if the given theorem is added
to these weak axioms, then we can prove each of the stronger axioms
we started with. This means that the strong axioms were indeed
necessary to prove the theorem. The proofs of the strong
axioms from the weak axioms and the given theorem are the "reverse"
in "reverse mathematics."
Degrees of unsolvability of mass problems: A mass
problem is simply a set of functions thought of as an abstract
mathematical problem, namely the problem of finding a function in
that set. The Medvedev and Muchnik degrees provide frameworks
for analyzing the computational relationships among mass problems
just as the classical Turing degrees provide a framework for
analyzing the computational relationships among functions.
Under the interpretation of mass problems as mathematical problems,
the Medvedev and Muchnik degrees formalize the "problem A is computationally harder
than problem B"
Papers (the versions hosted
here differ slightly from the official versions):
- André Nies and Paul Shafer. Randomness notions and
reverse mathematics. preprint. pdf arXiv
- Paul Shafer and Andrea Sorbi. Comparing the degrees
of enumerability and the closed Medvedev degrees.
- Paul Shafer. Honest elementary degrees and degrees
of relative provability without the cupping property.
Annals of Pure and Applied Logic 168 (2017).
1017–1031. pdf arXiv
- Paul Shafer. The reverse mathematics of the Tietze
extension theorem. Proceedings of the American
Mathematical Society 144 (2016). 5359–5370.
- Emanuele Frittaion, Matthew Hendtlass, Alberto Marcone, Paul
Shafer, and Jeroen Van der Meeren. Reverse
mathematics, well-quasi-orders, and Noetherian spaces.
Archive for Mathematical Logic 55 (2016).
- Laurent Bienvenu, Ludovic Patey, and Paul Shafer. On
the logical strengths of partial solutions to mathematical
problems. Transactions of the London Mathematical
Society 4 (2017). 30–71. pdf arXiv
- Rupert Hölzl and Paul Shafer. Universality,
optimality, and randomness deficiency. Annals of
Pure and Applied Logic 166
(2015). 1049–1069. pdf
- François G. Dorais, Jeffry L. Hirst, and Paul Shafer. Comparing
the strength of diagonally non-recursive functions in the
absence of Sigma^0_2 induction. Journal of Symbolic
Logic 80 (2015). 1211–1235. pdf arXiv
- Laurent Bienvenu, Rupert Hölzl, Christopher P. Porter, and
Paul Shafer. Randomness and semi-measures.
Notre Dame Journal of Formal Logic 58 (2017).
- François G. Dorais, Damir D. Dzhafarov, Jeffry L. Hirst,
Joseph R. Mileti, and Paul Shafer. On uniform relationships between
combinatorial problems. Transactions of the
American Mathematical Society 368 (2016).
- François G. Dorais, Jeffry Hirst, and Paul Shafer. Reverse mathematics and algebraic
field extensions. Computability 2
(2013). 75–92. pdf
- François G. Dorais, Jeffry Hirst, and Paul Shafer. Reverse mathematics, trichotomy,
and dichotomy. Journal of Logic and Analysis 4 (2012). 1–14.
- Paul Shafer. Menger's
theorem in Pi^1_1-CA_0. Archive for Mathematical
Logic 51 (2012).
- Paul Shafer. Coding
true arithmetic in the Medvedev degrees of Pi^0_1 classes.
Annals of Pure and Applied Logic 163 (2012). 321–337. pdf
- Paul Shafer. Coding
true arithmetic in the Medvedev and Muchnik degrees.
Journal of Symbolic Logic 76
(2011). 267–288. pdf
- Paul Shafer. Characterizing
the join-irreducible Medvedev degrees. Notre Dame
Journal of Formal Logic 52
(2011). 21–38. pdf
- Paul Shafer. On the
complexity of mathematical problems: Medvedev degrees and
reverse mathematics. PhD thesis. Cornell
University, 2011. pdf
- Reverse mathematics, a Ramsey theorem for graphs, and
complete linear orders. Given at the Workshop on
Ramsey Theory and Computability in Rome, Italy on 9 July 2018.
- Reverse mathematics of Caristi's fixed point theorem and
Ekeland's variational principle. Given at ASL North
American Annual Meeting 2018 in Macomb, IL on May 19, 2018 and
at Workshop: Proofs and Computation in Bonn, Germany on
July 2, 2018.
- The reverse mathematics of Ekeland's variational principle.
Given at Proof Theory, Modal Logic, and Reflection Principles
2017 in Moscow, Russia on October 17, 2017.
- The reverse mathematics of Caristi's fixed point theorem.
Given at the 11th Panhellenic Logic Symposium in
Delphi, Greece on July 13, 2017 and at the University of
Manchester Logic Seminar in Manchester, England on November 29,
- Honest elementary degrees without the cupping property.
Given at CiE 2016 in Paris, France on June 28, 2016, at Proof
Theory, Modal Logic, and Reflection Principles Third
International Wormshop in Tbilisi, Georgia on September 8, 2016,
and at the Conference on Mathematical Logic in Gyolechitsa,
Bulgaria in honor of Dimiter Skordev's 80th birthday on October
- A tour of the mass problems. Given at The
Foundational Impact of Recursion Theory in Storrs, CT in honor
of Steve Simpson's 70th birthday on May 22, 2016.
- Reverse mathematics and the strong Tietze extension
theorem. Given at the New Challenges in Reverse
Mathematics Workshop in Singapore on January 11, 2016 and at
Computability Theory and Foundations of Mathematics 2016 in
Tokyo, Japan on September 20, 2016.
- Universality, optimality, and randomness deficiency.
Given at Varieties of Algorithmic Information in Heidelberg,
Germany on June 18, 2015 and at Logic Colloquium 2015 in
Helsinki, Finland on August 4, 2015.
- Reverse mathematics, well-quasi-orders, and Noetherian
spaces. Given at the Sets and Computations Workshop
in Singapore on April 20, 2015 and at CiE 2015 in Bucharest,
Romania on July 2, 2015.
- Exploring randomness, diagonally non-recursiveness, and
Ramsey-type combinatorial principles in reverse mathematics.
Given at ASL North American Annual Meeting 2014 in Boulder, CO
on May 21, 2014 and at Proof Theory, Modal Logic, and Reflection
Principles Second International Wormshop in Mexico City, Mexico
on September 30, 2014.
- Studying the role of induction axioms in reverse
mathematics. Given at the Sendai Logic School in
Tokyo, Japan on February 21, 2014.
- Separating the uniformly computably true from the
computably true via strong Weihrauch reducibility.
Given at Computability Theory and Foundations of Mathematics
2014 in Tokyo, Japan on February 17, 2014.
- Examples of problems that cannot be solved by randomness
and examples of problems that can be solved by randomness.
Given at the University of Udine Department of Mathematics and
Computer Science Seminar in Udine, Italy on September 17, 2013
and at the Sendai Logic Seminar in Sendai, Japan on February 14,
- Reverse mathematics and Birkhoff’s problem 111.
Given at the Bundeswehr University Munich Seminar for
Theoretical Computer Science and Mathematical Logic in Munich,
Germany on August 21, 2013.
- Comparing the strength of DNR(k) functions and DNR(2)
functions. Given at Logic Colloquium 2013 in Évora,
Portugal on July 22, 2013.
- A low DNR(k) function that computes no DNR(2) function.
Given at the Workshop on Reverse Mathematics and Type Theory in
Seoul, South Korea on March 26, 2013 and at Journées
Calculabilités 2013 in Nancy, France on April 11, 2013.
- Randomness from an
algorithmic perspective. Given at the ENS Lyon
Semaine Sport-Etudes 2012-2013 in Les 7 Laux, France on January
- Logics and complexity in a
calculus of problems. Given at the Ghent
University Logic and Analysis Seminar in Ghent, Belgium on
October 24, 2012.
- Presenting the effectively
closed Medvedev degrees requires 0'''. Given at
ASL North American Annual Meeting 2012 in Madison, Wisconsin on
March 31, 2012.
- The Medvedev degrees as
semantics for propositional logic. Given at AMS
Spring Eastern Sectional Meeting 2012 in Washington, DC on March
- Complexity in the degrees of
unsolvability of mass problems. Given at ASL
Winter Meeting 2012 in Boston, Massachusetts on January 6, 2012.
- Complexity in the Medvedev
and Muchnik degrees. Given at the Penn State
Logic Seminar in State College, Pennsylvania on October 14,
- Describing the complexity of
the "problem B is
harder than problem A"
relation. Given at the Appalachian State
University Mathematical Sciences Colloquium in Boone, North
Carolina on October 7, 2011.
- Complexity in the Medvedev
degrees of Pi^0_1 classes. Given at Workshop on
Computability Theory 2011, part 2 in Barcelona, Spain on July
- Birkhoff's theorem and
reverse mathematics. Given at AMS Fall Central
Sectional Meeting 2010 in Notre Dame, Indiana on November 7,
- Medvedev degrees:
characterizing the first-order theory and the
join-irreducibles. Given at the Notre Dame Logic
Seminar in Notre Dame, Indiana on November 4, 2010.
- Menger's theorem in
Pi^1_1-CA_0. Given at Logic Colloquium 2010 in
Paris, France on July 27, 2010.
- Coding second-order
arithmetic into the closed Medvedev degrees.
Given at ASL North American Annual Meeting 2010 in Washington,
DC on March 17, 2010.
- The first-order theory of the
Medvedev lattice is third-order arithmetic. Given
at Logic Colloquium 2009 in Sofia, Bulgaria on August 4, 2009.
- A characterization of the
join-irreducible Medvedev degrees. Given at CiE
2009 in Heidelberg, Germany on July 22, 2009.
- Menger's theorem and reverse
mathematics. Given at the MIT Logic Seminar in
Cambridge, Massachusetts on April 2, 2008.
My research in computer science consists of work on several projects
in computational biology and the Biozon
database led by Golan
- Eric O. Williams, Yuanyuan Xiao, Heather M. Sickles, Paul
Shafer, Golan Yona, Jean YH Yang, and David M. Lin. Novel subdomains of the mouse
olfactory bulb defined by molecular heterogeneity in the
nascent external plexiform and glomerular layers.
BMC Developmental Biology 7:48
- Paul Shafer, David M. Lin, and Golan Yona. EST2Prot: Mapping EST sequences to
proteins. BMC Genomics 7:41 (2006).
- Paul Shafer, Timothy Isganitis, and Golan Yona. Hubs of knowledge: using the
functional link structure of Biozon to mine for biologically
significant entities. BMC Bioinformatics 7:71 (2006).
- Paul Shafer and Timothy Isganitis. Hubs of Knowledge: using the
functional link structure in Biozon to mine for biologically
significant entities. Given at ISMB 2005 in
Detroit, Michigan on June 28, 2005.