Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages

The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: - The classes of the Boolean hierarchy over level $\Sigma_1$ of the dot-depth hierarchy are decidable in $NL$ (previously only the decidability was known). The same remains true if predicates mod $d$ for fixed $d$ are allowed. - If modular predicates for arbitrary $d$ are allowed, then the classes of the Boolean hierarchy over level $\Sigma_1$ are decidable. - For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level $\Sigma_2$ of the Straubing-Th\'erien hierarchy are decidable in $NL$. This is the first decidability result for this hierarchy. - The membership problems for all mentioned Boolean-hierarchy classes are logspace many-one hard for $NL$. - The membership problems for quasi-aperiodic languages and for $d$-quasi-aperiodic languages are logspace many-one complete for $PSPACE$

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

A SURVEY OF LIMITED NONDETERMINISM IN COMPUTATIONAL COMPLEXITY THEORY

Nondeterminism is typically used as an inherent part of the computational models used incomputational complexity. However, much work has been done looking at nondeterminism asa separate resource added to deterministic machines. This survey examines several differentapproaches to limiting the amount of nondeterminism, including Kintala and Fischer\u27s βhierarchy, and Cai and Chen\u27s guess-and-check model

Logic Meets Algebra: the Case of Regular Languages

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.Comment: 37 page

Self-Specifying Machines

We study the computational power of machines that specify their own acceptance types, and show that they accept exactly the languages that \manyonesharp-reduce to NP sets. A natural variant accepts exactly the languages that \manyonesharp-reduce to P sets. We show that these two classes coincide if and only if \psone = \psnnoplusbigohone, where the latter class denotes the sets acceptable via at most one question to \sharpp followed by at most a constant number of questions to \np.Comment: 15 pages, to appear in IJFC