Rice-Style Theorems in Complexity Theory
Lane A. Hemaspaandra and Mayur Thakur
University of Rochester Computer Science
Motivation
Program accepts
string “1000100”
Program accepts a
finite language
Program accepts a
regular language
Rice’s Theorem
Sledgehammer
Rice’s Theorem [Rice 53] is a
very broad and powerful tool
from recursive function
theory that allows one to
automatically prove a broad
class of language properties
of computer programs to be
undecidable. It is thus
natural to seek analogs of
such sledgehammer-like
results in complexity theory.
History
Borchert and Stephan started the
search for complexity-theoretic
versions of Rice’s theorem and
proved that all nontrivial counting
properties of boolean circuits are
hard for unambiguous
nondeterminisim (UP-hard).
Undecidable
Rice’s Theorem also provides a link between the syntactic objects
(programs, Turing machines, boolean circuits) and the semantic
objects (functions, languages).
Hi! I am Ms. Teeny C
Program from Syntaxville.
I believe we have a
common friend in Mr. Rice.
Hello, I am Mr. Finiteness
Property, resident of Semanchester,
and a friend of Mr. Rice.
He has told me interesting things about you
people from Syntaxville.
Did you know that even though you
people can do great things
like beat Kasparov at chess,
you are utterly powerless against
anyone from my town?
Hemaspaandra and Rothe, 2000
Borchert and Stephan, 2000
Hemaspaandra and Rothe raised
the lower bound from hardness for
unambiguous nondeterminism to
hardness for constant-ambiguity
nondeterminism (UPO(1)-hard).
In this project, we have raised the
lower bound on the hardness of
counting properties to
polynomial-ambiguity
nondeterminism, or informally
speaking, from a constant to a
polynomial.
Hemaspaandra and Thakur, 2002
Strongest Known Rice-Style Theorem in Complexity Theory
Theorem [Hemaspaandra and Thakur, 2002]: All nontrivial language property of the boolean circuits
are FewP-hard, in other words, hard for polynomial-ambiguity nondeterminism.
Optimality of our Rice-Style Theorem
Apart from providing the strongest known Rice-style theorem
in complexity theory, a new improved tool for proving a broad
class of properties of boolean circuits to be intractable, we also
provide strong evidence that our result is optimal with respect
to relativizable techniques. In fact, we prove, via oracle
construction, that our result on the hardness of counting
properties of boolean circuits cannot be improved much using
techniques that relativize. Thus, any improvement to our result
would likely require breakthrough techniques since most
mathematical tools and techniques used in complexity theory
are relativizable.
Sigh! I cannot ever beat Hemaspaandra and Thakur’s mark.
Damn these complexity theorists and their theorems.