Derandomizing Isolation in Space-Bounded Settings

We study the possibility of deterministic and randomness-efficient isolation in space-bounded models of computation: Can one efficiently reduce instances of computational problems to equivalent instances that have at most one solution? We present results for the NL-complete problem of reachability on digraphs, and for the LogCFL-complete problem of certifying acceptance on shallow semi-unbounded circuits. A common approach employs small weight assignments that make the solution of minimum weight unique. The Isolation Lemma and other known procedures use Omega(n) random bits to generate weights of individual bitlength O(log(n)). We develop a derandomized version for both settings that uses O(log(n)^{3/2}) random bits and produces weights of bitlength O(log(n)^{3/2}) in logarithmic space. The construction allows us to show that every language in NL can be accepted by a nondeterministic machine that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. Similarly, every language in LogCFL can be accepted by a nondeterministic machine equipped with a stack that does not count towards the space bound, that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. We also show that the existence of somewhat more restricted isolations for reachability on digraphs implies that NL can be decided in logspace with polynomial advice. A similar result holds for certifying acceptance on shallow semi-unbounded circuits and LogCFL

Structure of computations in parallel complexity classes

Issued as Annual report, and Final project report, Project no. G-36-67

Realizable paths and the NL vs L problem

A celebrated theorem of Savitch [Savitch'70] states that NSPACE(S) is contained in DSPACE(S²). In particular, Savitch gave a deterministic algorithm to solve ST-Connectivity (an NL-complete problem) using O({log}²{n}) space, implying NL (non-deterministic logspace) is contained in DSPACE({log}²{n}). While Savitch's theorem itself has not been improved in the last four decades, several graph connectivity problems are shown to lie between L and NL, providing new insights into the space-bounded complexity classes. All the connectivity problems considered in the literature so far are essentially special cases of ST-Connectivity. In this dissertation, we initiate the study of auxiliary PDAs as graph connectivity problems and define sixteen different "graph realizability problems" and study their relationships. The complexity of these connectivity problems lie between L (logspace) and P (polynomial time). ST-Realizability, the most general graph realizability problem is P-complete. 1DSTREAL(poly), the most specific graph realizability problem is L-complete. As special cases of our graph realizability problems we define two natural problems, Balanced ST-Connectivity and Positive Balanced ST-Connectivity, that lie between L and NL. We study the space complexity of SGSLOGCFL, a graph realizability problem lying between L and LOGCFL. We define generalizations of graph squaring and transitive closure, present efficient parallel algorithms for SGSLOGCFL and use the techniques of Trifonov to show that SGSLOGCFL is contained in DSPACE(lognloglogn). This implies that Balanced ST-Connectivity is contained in DSPACE(lognloglogn). We conclude with several interesting new research directions.PhDCommittee Chair: Richard Lipton; Committee Member: Anna Gal; Committee Member: Maria-Florina Balcan; Committee Member: Merrick Furst; Committee Member: William Coo

Arithmetic Branching Programs with Memory

We extend the well known characterization of VPws as the class of polynomials computed by polynomial size arithmetic branching programs to other complexity classes. In order to do so we add additional memory to the computation of branching programs to make them more expressive. We show that allowing different types of memory in branching programs increases the computational power even for constant width programs. In particular, this leads to very natural and robust characterizations of VP and VNP by branching programs with memory. 1

The complexity of the characteristic and the minimal polynomial

AbstractWe investigate the complexity of (1) computing the characteristic polynomial, the minimal polynomial, and all the invariant factors of an integer matrix, and of (2) verifying them, when the coefficients are given as input.It is known that each coefficient of the characteristic polynomial of a matrix A is computable in GapL, and the constant term, the determinant of A, is complete for GapL. We show that the verification of the characteristic polynomial is complete for complexity class C=L (exact counting logspace).We show that each coefficient of the minimal polynomial of a matrix A can be computed in AC0(GapL), the AC0-closure of GapL, and there is a coefficient which is hard for GapL. Furthermore, the verification of the minimal polynomial is in AC0(C=L) and is hard for C=L. The hardness result extends to (computing and verifying) the system of all invariant factors of a matrix