Borchert and Stephan, 2000

1. Program accepts string “1000100” Program accepts a finite language Program accepts a regular language Rice’s Theorem Borchert and Stephan, 2000 Sledgehammer Undecidable Hemaspaandra and Rothe, 2000 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? Hi! I am Ms. Teeny C Program from Syntaxville. I believe we have a common friend in Mr. Rice. 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. Rice-Style Theorems in Complexity Theory Lane A. Hemaspaandra and Mayur Thakur University of Rochester Computer Science Motivation History 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. 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). Hemaspaandra and Rothe raised the lower bound from hardness for unambiguous nondeterminism to hardness for constant-ambiguity nondeterminism (UPO(1)-hard). Rice’s Theorem also provides a link between the syntactic objects (programs, Turing machines, boolean circuits) and the semantic objects (functions, languages). 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. 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.