The descriptive complexity approach to LOGCFL

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.Comment: 10 pages, 1 figur

The power of writing, a pebble hierarchy and a narrative for the teaching of Automata Theory

In this work we study pebble automata. Those automata constitute an infinite hierarchy of discrete models of computation. The hierarchy begins at the level of finite state automata (0-pebble automata) and approaches the model of onetape Turing machines. Thus, it can be argued that it is a complete hierarchy that covers, in a continuous way, all the models of automata that are important in the theory of computation. We investigate the use of this hierarchy as a narrative for the teaching of automata theory. We also investigate some fundamental questions concerning the power of pebble automata.Sociedad Argentina de Informática e Investigación Operativa (SADIO

A Pumping Lemma Scheme for the Control Language Hierarchy

In [9] Weir introduced control grammars as a model for describing the syntactic structure of natural languages. Informally, a control grammar is a pair {G, C} where G is a context-free grammar whose productions are assigned labels from a finite set of labels II, and C (called the control set) is a set of strings over II. A derivation in a control grammar is similar to that in an ordinary context-free grammar except that the control set C is used to further constrain the set of valid derivations. In particular, if one views a derivation as a tree, then (in a manner to be described later) each edge in such a tree is given a label from II according to the production of G associated with the edge. The derivation tree is considered valid if certain paths in the tree correspond to strings which are in the control set C. The language generated by the control grammar is then the set of strings having at least one derivation tree in the sense just described

Pebbling, Entropy and Branching Program Size Lower Bounds

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et. al. (2012). Proving a super-polynomial lower bound for the size of nondeterministic thrifty branching programs (NTBP) would separate $NL$ from $P$ for thrifty models solving the tree evaluation problem. First, we show that {\em Read-Once NTBPs} are equivalent to whole black-white pebbling algorithms thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of NTBPs called {\em Bitwise Independence}. The best known NTBPs (of size $O(k^{h/2+1})$) for the tree evaluation problem given by Cook et. al. (2012) are Bitwise Independent. As our main result, we show that any Bitwise Independent NTBP solving $TEP_{2}^{h}(k)$ must have at least $\frac{1}{2}k^{h/2}$ states. Prior to this work, lower bounds were known for NTBPs only for fixed heights $h=2,3,4$ (See Cook et. al. (2012)). We prove our results by associating a fractional black-white pebbling strategy with any bitwise independent NTBP solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights. Our main technique is the entropy method introduced by Jukna and Z{\'a}k (2001) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds given by Cook et. al. (2012) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any $k$-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.Comment: 25 Pages, Manuscript submitted to Journal in June 2013 This version includes a proof for tight size bounds for (syntactic) read-once NTBPs. The proof is in the same spirit as the proof for size bounds for bitwise independent NTBPs present in the earlier version of the paper and is included in the journal version of the paper submitted in June 201

The Complexity of Knapsack in Graph Groups

Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In LohreyZ16 the authors proved that for each graph group, the knapsack problem can be solved in NP. Here, we determine the exact complexity of the problem for every graph group. While the problem is TC^0-complete for complete graphs, it is LogCFL-complete for each (non-complete) transitive forest. For every remaining graph, the problem is NP-complete

Alternating and empty alternating auxiliary stack automata

AbstractWe consider variants of alternating auxiliary stack automata and characterize their computational power when the number of alternations is bounded by a constant or unlimited. In this way we get new characterizations of NP, the polynomial hierarchy, PSpace, and bounded query classes like co-DP=NL〈NP[1]〉 and Θ2P=PNP[O(logn)], in a uniform framework

Tree-size bounded alternation

AbstractThe size of an accepting computation tree of an alternating Turing machine (ATM) is introduced as a complexity measure. We present a number of applications of tree-size to the study of more traditional complexity classes. Tree-size on ATMs is shown to closely correspond to time on nondeterministic TMs and on nondeterministic auxiliary pushdown automata. One application of the later is a useful new characterization of the class of languages log-space-reducible to context-free languages. Surprising relationships with parallel-time complexity are also demonstrated. ATM computations using at most space S(n) and tree-size Z(n) (simultaneously) can be simulated in alternating space S(n) and time S(n) · log Z(n) (simultaneously). Several well-known simulations, e.g., Savitch's theorem, are special cases of this result. It also leads to improved parallel complexity bounds for many problems in terms of both time and number of “processors.” As one example we show that context-free language recognition in time O(log2 n) is possible on several parallel models. Further, this bound is achievable with only a polynomial number of processors, in contrast to all previously known sub-linear time CFL recognizers

A taxonomy of problems with fast parallel algorithms

The class NC consists of problems solvable very fast (in time polynomial in log n) in parallel with a feasible (polynomial) number of processors. Many natural problems in NC are known; in this paper an attempt is made to identify important subclasses of NC and give interesting examples in each subclass. The notion of NC1-reducibility is introduced and used throughout (problem R is NC1-reducible to problem S if R can be solved with uniform log-depth circuits using oracles for S). Problems complete with respect to this reducibility are given for many of the subclasses of NC. A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in NC2 (solvable by uniform Boolean circuits of depth O(log2 n) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, in “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York)