# Richard A. Shore

## Professor of Mathematics

My major research interests have centered around analyzing the structures of
relative complexity of computation of functions on the natural numbers. The
primary measure of such complexity is given by Turing reducibility: *f* is
easier to compute than *g*, *f *__<___{T} g, if
there is a (Turing) machine which can compute *f* if it is given access to the
values of *g*. I have also worked with various other interesting measures of
such complexity that are defined by restricting the resources available primarily
in terms of access to *g*. The general thrust of my work has been to show that
these structures are as complicated as possible both algebraically and logically
(in terms of the complexity of the decision problems for their theories). These
results also allow one to differentiate among different notions of relative
complexity in terms of the orderings they define. Another major theme in my work
has been the relationship between these notions of computational complexity and
ones based on the difficulty of defining functions in arithmetic. Restricting the
computational resources more directly in terms of time or space leads out of
recursion theory and into complexity theory. Relaxing the restrictions by allowing
various infinitary procedures leads instead into generalized recursion theory
or set theory. The methods developed in these investigations are also useful
in determining the effective content of standard mathematical theorems (when
can existence proofs be made effective) and the inherent difficulty of
combinatorial theorems in proof theoretic terms.

### Professional Activities:

- Member of the AMS, ASL, ACM and SIGACT.

- Referee and reviewer for the NSF, the Natural Sciences and Engineering Research
Council of Canada, the US-Israeli Bi-National Science Foundation, the New Zealand
Mathematical Society Research Awards and many journals.

- Editor of the Journal of Symbolic Logic (1989-93).

- Nominating committee and publications committee of the Association of Symbolic
Logic (1993-94).

- Managing editor for the Bull. Symbolic Logic (1993-).

- Council member of the Assn. Symbolic Logic (1984-).

- Editor of Studies in Logic and the Foundations of
Mathematics, North-Holland (1996-).

### Selected Publications:

- alpha-Recursion Theory, in
*Handbook of Mathematical Logic*, J.Barwise ed.,
North-Holland, 1977, 653-680.

- The Homogeneity Conjecture,
*Proceedings of the National Academy of
Sciences* **76** (1979), 4218-4219.

- Definable Degrees and Automorphisms of
*D*, *Bull. Amer. Math.
Soc.* *(NS)* **4** (1981), 97-100
(with L. Harrington).

- The Degrees of Unsolvability: the Ordering of Functions by Relative
Computability, in
*Proc. Inter. Congress of Mathematicians (Warsaw) 1983*,
PWN-Polish Scientific Publishers, Warsaw 1984, Vol. 1, 337-346.

- The Structure of the Degrees of Unsolvability, in
*Recursion Theory,
Proceedings of the Symposia in Pure Mathematics* **42**, A. Nerode and
R.A. Shore, eds., American Mathematical Society,Providence, R.I., 1985,
33-51.

- Recursive Limits on the Hahn-Banach Theorem,
*Contemporary
Mathematics* **39** (1985), 85-91 (with A. Nerode and G. Metakides).

- On the Strength of König's Duality Theorem for Infinite Bipartite
Graphs,
*J. Comb. Theory (B)* **54** (1992), 257-290
(with R. Aharoni and M. Magidor).

- The
*p-T*-degrees of the Recursive Sets: Lattice Embeddings, Extension
of Embeddings and the Two Quantifier Theory, *Theoretical Computer
Science* **97** (1992), 263-284 (with T. Slaman).

* Logic for Applications*, Texts and Monographs in Computer Science,
Springer-Verlag, New York, 1993 (with A. Nerode).

- Definability in the recursively enumerable degrees,
*Bull. Symb. Logic*
**2** (1996), 392-404, (with A. Nies and T. Slaman).

- Computable Isomorphisms, Degree Spectra of Relations, and Scott Families,
*Ann. of Pure Applied Logic*, to appear (with B. Khoussainov).