Unambiguous Computation

This survey paper is designed to review the literature on unambiguous computation. Broadly speaking, studying unambiguous computation is an attempt to determine how much of the power of non-determinism comes from the ability to accept many di↵erent proofs of the same statement. The literature on this subject is of both practical and philosophical interest, and the Isolating lemma of Mulmuley, Vazirani and Vazirani [5] appears beautifully throughout. In the first first, we will define the notion of unambiguous computation and the associated complexity classes we will consider. In the second section, we’ll discuss unambiguous polynomial-time computation. Most of the results in this section are from the ‘80s—in particular we’ll discuss relationships between the complexity classes P, UP, and NP, and prove the Isolating lemma as well as the ValiantVazirani theorem. In the third and final section, we’ll discuss some of the more recent results about unambiguous log-space computation. In particular, we’ll prove UL/poly = NL/poly and discuss the latest attempts to prove the conjecture that NL = UL.