Trading Determinism for Time in Space Bounded Computations

Savitch showed in $1970$ that nondeterministic logspace (NL) is contained in deterministic $\mathcal{O}(\log^2 n)$ space but his algorithm requires quasipolynomial time. The question whether we can have a deterministic algorithm for every problem in NL that requires polylogarithmic space and simultaneously runs in polynomial time was left open. In this paper we give a partial solution to this problem and show that for every language in NL there exists an unambiguous nondeterministic algorithm that requires $\mathcal{O}(\log^2 n)$ space and simultaneously runs in polynomial time.Comment: Accepted in MFCS 201

Arithmetic Circuits and the Hadamard Product of Polynomials

Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity testing. Our main results are the following. 1. We show that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class \ceql, and over fields of characteristic $p$ the problem is in \ModpL/\Poly. 2.We show an exponential lower bound for expressing the Raz-Yehudayoff polynomial as the Hadamard product of two monotone multilinear polynomials. In contrast the Permanent can be expressed as the Hadamard product of two monotone multilinear formulas of quadratic size.Comment: 20 page

The Complexity of Reasoning for Fragments of Autoepistemic Logic

Autoepistemic logic extends propositional logic by the modal operator L. A formula that is preceded by an L is said to be "believed". The logic was introduced by Moore 1985 for modeling an ideally rational agent's behavior and reasoning about his own beliefs. In this paper we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of counting the number of stable expansions of a given knowledge base. To the best of our knowledge this is the first paper analyzing the counting problem for autoepistemic logic

Computing Bits of Algebraic Numbers

We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like \pi, in Logspace. Our main result is that computing a bit of a fixed real algebraic number is in C=NC1\subseteq Logspace when the bit position has a verbose (unary) representation and in the counting hierarchy when it has a succinct (binary) representation. Our tools are drawn from elementary analysis and numerical analysis, and include the Newton-Raphson method. The proof of our main result is entirely elementary, preferring to use the elementary Liouville's theorem over the much deeper Roth's theorem for algebraic numbers. We leave the possibility of proving non-trivial lower bounds for the problem of computing the bits of an algebraic number given the bit position in binary, as our main open question. In this direction we show very limited progress by proving a lower bound for rationals

The Complexity of Bisimulation and Simulation on Finite Systems

In this paper the computational complexity of the (bi)simulation problem over restricted graph classes is studied. For trees given as pointer structures or terms the (bi)simulation problem is complete for logarithmic space or NC$^1$, respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and S\'antha. Furthermore, if only one of the input graphs is required to be a tree, the bisimulation (simulation) problem is contained in AC$^1$ (LogCFL). In contrast, it is also shown that the simulation problem is P-complete already for graphs of bounded path-width

Longest paths in Planar DAGs in Unambiguous Logspace

We show via two different algorithms that finding the length of the longest path in planar directed acyclic graph (DAG) is in unambiguous logspace UL, and also in the complement class co-UL. The result extends to toroidal DAGs as well