 
Summary: New Surprises from SelfReducibility
Eric Allender
Department of Computer Science, Rutgers University, Piscataway, NJ 08855,
allender@cs.rutgers.edu
Abstract. Selfreducibility continues to give us new angles on attacking some of
the fundamental questions about computation and complexity.
1 Introduction
Perhaps the most surprising thing about SelfReducibility is its longevity. Who would
have suspected that this simple idea would continue to play a key role in important
developments in our evolving understanding of computation and complexity over a span
of four decades? Yet recent developments demonstrate that selfreducibility is still able
to lead us to new insights, both in computability theory and in complexity theory.
Trakhtenbrot introduced the notion of autoreducibility in a paper published forty
years ago [28]. Briefly, a set A is autoreducible if A is accepted by an oracle Turing
machine M that has A as an oracle, with the restriction that M, on input x, does not
ask its oracle about x. Already in his 1970 paper, Trakhtenbrot studied autoreducibil
ity in the context of resourcebounded computation, although in early years the notion
was studied primarily in the context of computability [23,20]. Polynomialtime au
toreducibility per se seems to have been studied first by AmbosSpies [5], although
some types of polynomialtime selfreducibility (corresponding to restricted versions of
