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Semi-Feasible Algorithms

UR-CS Participating Faculty: Lane A. Hemaspaandra (= Lane A. Hemachandra)

Project Description

Adapted from the introduction of the book “Theory of Semi-Feasible Algorithms,” by Lane Hemaspaandra and Leen Torenvliet:

The focus of complexity theory is the computational complexity of sets. However, it is an underappreciated fact that sets may have various types of complexity, and not all types are harmony with each other. For example, sets that are complex in terms of deterministic time may nonetheless be simple in other natural senses. Unifying and making more widely accessible a vibrant stream of research--semi-feasible computation--that perfectly showcases this point is the primary goal of this book.

The semi-feasible sets, which are most commonly referred to as the P-selective sets, are those sets $L$ for which there is a deterministic polynomial-time algorithm that, given as input any two strings, outputs the one in $L$ if exactly one is in $L$. The reason we say that the semi-feasible sets showcase the above distinction is that it is well-known that the semi-feasible sets are arbitrarily complex in terms of the deterministic time it takes to recognize them, yet they are simple in a wide range of other senses. In particular, they have small circuits, they are in the extended low hierarchy, and they cannot be NP-complete unless P=NP.

We find the semi-feasible sets to be fascinating for many reasons. First, as mentioned above, they showcase the fact that mere deterministic time complexity is not the only potential type of complexity in the world of computation; sets that are complex in terms of deterministic time may nonetheless be simple in many other computationally natural senses. A second reason that the semi-feasible sets are interesting is that they crisply capture the complexity of (left cuts of) real numbers, and recently a refinement of the semi-feasible sets has been shown to capture the complexity of complexity-bounded real numbers. A third and more historical reason for interest in the semi-feasible sets is that they are the complexity-theoretic analog of a key notion from recursive function theory; the semi-feasible sets are exactly what one gets when one alters the definition of the semi-recursive sets by changing the selector function from “recursive” to “polynomial-time computable.” In the late 1960s the semi-recursive sets yielded great insight into distinguishing the power of reductions in the recursion-theoretic context, and in 1979 Selman launched a program that used--successfully, in the context of structural connections to exponential time--semi-feasible sets to understand the structure of polynomial-time reductions. A fourth and somewhat surprising reason to study semi-feasible sets is that semi-feasible sets (in their recently-defined nondeterministic version) have been shown to conditionally resolve Selman's important question as to whether NP machines can cull down to one the large number of potential solutions of satisfiable formulas; in particular, it is now known that NP lacks such “unique solutions” unless the polynomial hierarchy collapses.

Bibliography

1
This is a list of selected journal (except when the work has not yet appeared in journal/book form) papers, from or related to this project, by University of Rochester authors. Essentially all the papers listed below can be found, in their full technical report versions, in the UR-CS Technical Report Archive's theory section. Here is Lane's complete publication list and links to essentially all his conference and journal papers (and also his arXiv.org technical reports) can be found via the “EE” (electronic edition) links at Lane's entry at the DBLP project.
2
R. Beigel, R. Chang, and M. Ogiwara.
A relationship between difference hierarchies and relativized polynomial hierarchies.
Mathematical Systems Theory, 26(3):293-310, 1993.
3
L. Hemaspaandra and S. Jha.
Defying upward and downward separation.
Information and Computation, 121(1):1-13, 1995.
4
E. Hemaspaandra, L. Hemaspaandra, and H. Hempel.
An introduction to query order.
Bulletin of the EATCS, 63:93-107, 1997.
5
E. Hemaspaandra, L. Hemaspaandra, and H. Hempel.
Query order in the polynomial hierarchy.
Journal of Universal Computer Science, 4(6):574-588, 1998.
6
E. Hemaspaandra, L. Hemaspaandra, and H. Hempel.
R ${}^{{\cal S}{\cal N}}_{1\hbox{-}tt}$(NP) distinguishes robust many-one and Turing completeness.
Theory of Computing Systems, 31(3):307-325, 1998.
7
E. Hemaspaandra, L. Hemaspaandra, and H. Hempel.
What's up with downward collapse: Using the easy-hard technique to link boolean and polynomial hierarchy collapses.
SIGACT News, 29(3):10-22, 1998.
8
E. Hemaspaandra, L. Hemaspaandra, and H. Hempel.
A downward collapse within the polynomial hierarchy.
SIAM Journal on Computing, 28(2):383-393, 1999.
9
L. Hemaspaandra, H. Hempel, and G. Wechsung.
Query order.
SIAM Journal on Computing, 28(2):637-651, 1999.
10
E. Hemaspaandra, L. Hemaspaandra, and H. Hempel.
Using the no-search easy-hard technique for downward collapse.
Technical Report TR-752, Department of Computer Science, University of Rochester, Rochester, NY, June 2001.
11
E. Hemaspaandra, L. Hemaspaandra, and H. Hempel.
Extending downward collapse from 1-versus-2 queries to $m$-versus-$m+1$ queries.
SIAM Journal on Computing, 34(6):1352-1369, 2005.
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(Last modified: February 15, 2020.)


Lane A. Hemaspaandra