Etsuro Moriya

A grammatical characterization of alternating pushdown automata

Abstract The notion of an alternating context-free grammar is introduced and it is shown that the class of alternating context-free languages is equal to the class of languages accepted by alternating pushdown automata. Some properties on alternating context-free languages are also studied.

On state-alternating context-free grammars

State-alternating context-free grammars are introduced, and the language classes obtained from them are compared to the classes of the Chomsky hierarchy as well as to some well-known complexity classes. In particular, state-alternating context-free grammars are compared to alternating context-free grammars (Theoret. Comput. Sci. 67 (1989) 75-85) and to alternating pushdown automata. Further, various derivation strategies are considered, and their influence on the expressive power of (state-) alternating context-free grammars is investigated.

On two-way tree automata

As is well known there are two models for defining the behavior of a tree automaton on a labeled tree, the root-to-frontier (i.e., top-down) model and the frontier-to-root (i.e., bottom-up) model [6]. There is some difference of recognition power between these two models for deterministic devices, but no difference for nondeterministic ones [6]. Combining these two models together, in this paper we consider two-way tree automata. It is shown that the two-way nondeterministic tree automata are equivalent to the ordinary one-way nondeterministic ones in their recognition abilities. In [2] and [5] Guessarian, Schimpf and Gallier introduced pushdown tree automata. We show that the two-way pushdown tree automata are strictly more powerful than the one-way ones.

Two Ways of Introducing Alternation into Context-Free Grammars and Pushdown Automata

Two ways of introducing alternation for context-free grammars and pushdown automata are compared. One is the usual way which combines “states” with alternation [1], [4], [7], and the other is the way used in [6] to define the alternating context-free grammar, i.e., alternation is governed by the variables of the grammar. In this paper the latter way is taken over to define a new type of alternating pushdown automaton by combining the “pushdown symbols” of the pushdown automaton with alternation. We have derived a characterization of the original alternating context-free grammars in terms of such a new type of alternating pushdown automaton without states. It is also shown that, if (non-alternating) states are introduced as an additional feature for this type of pushdown automaton, then the resulting alternating pushdown automaton has exactly the same expressive power as the original alternating pushdown automaton.

Optimally fast shortest path algorithms for some classes of graphs

Two algorithms for shortest path problems are presented. One is to find the all-pairs shortest paths (APSP) that runs in O(n 2logn + nm) time for n-vertex m-edge directed graphs consisting of strongly connected components with O(logn) edges among them. The other is to find the single-source shortest paths (SSSP) that runs in O(n) time for graphs reducible to the trivial graph by some simple transformations. These algorithms are optimally fast for some special classes of graphs in the sense that the former achieves O(n 2) which is a lower bound of the time necessary to find APSP, and that the latter achieves O(n) which is a lower bound of the time necessary to find SSSP. The latter can be used to find APSP, also achieving the running time O(n 2).

A note on some simultaneous relations among time, space, and reversal for singlework tape nondeterministic turing machines

Simultaneous resource bounded complexity classes for nondeterministic single worktape off-line Turing machines are considered such as time-space bounded classes, denoted by NTISP 1 ( T, S ), reversal-space bounded classes, denoted by NRESP 1 ( R, S ), and time-reversal bounded classes, denoted by NTIRE 1 ( T, R ). It is shown that NRESP 1 ( R ( n ), S ( n )) contains NTISP 1 ( S ( n ), R ( n )) and is contained in NTISP 1 ( R ( n ) S ( n ) n 2 log n , R ( n ) log n ). The following corollaries follow: (1) the affirmative solution to the nondeterministic single worktape version of the NC = ? SC problem, NTIRE 1 (poly, polylog) = NTISP 1 (poly, polylog), and (2) a reversal-space trade-off, NRESP 1 (polylog, poly) = NRESP 1 (poly, polylog).

ON ALTERNATING PHRASE-STRUCTURE GRAMMARS

The concepts of alternation and of state alternation are extended from context-free grammars to context-sensitive and arbitrary phrase-structure grammars. For the resulting classes of alternating grammars the expressive power is investigated with respect to the leftmost derivation mode and with respect to the unrestricted derivation mode. In particular new grammatical characterizations for the class of languages that are accepted by alternating pushdown automata are obtained in this way.

On Alternating Phrase-Structure Grammars

We study several extensions of the notion of alternation from context-free grammars to context-sensitive and arbitrary phrase-structure grammars. Thereby new grammatical characterizations are obtained for the class of languages that are accepted by alternating pushdown automata.

On The Space Complexity Of Turn Bounded Pushdown Automata

A pushdown automaton is said to make a turn at a given instant if it changes at that instant from stack increasing to stack decreasing. Let \hbox{NPDA-TURN}(\,f(n)) and \hbox{DPDA-TURN}(\,f(n)) denote the classes of languages accepted by nondeterministic and deterministic pushdown automata respectively that make at most f v ( n ) turns for any input of length n . In this paper the following inclusions that express the space complexity of turn bounded pushdown automata are given: \hbox{DPDA-TURN}(\,f(n)) \subseteq {\bf DSPACE}(\log f(n)\log n) , and \hbox{NPDA-TURN}(\,f(n)) \subseteq {\bf NSPACE} (\log f(n)\log n) . In particular, it follows that finite-turn pushdown automata are logarithmic space bounded: \hbox{DPDA-TURN}(O(1))\subseteq {\bf DL} and \hbox{NPDA-TURN}(O(1))\subseteq {\bf NL} , from which two corollaries follow: one is that the class of metalinear context-free languages is complete for NL , and the other is that a more tight inclusion \hbox{NPDA-TURN}(\,f(n)) \subseteq {\bf DSPACE}(\log^2 f(n)\log n) cannot be derived unless {\bf DL}={\bf NL} , though \hbox{NPDA-TURN} (\,f(n)) \subseteq {\bf DSPACE}(\log^2 f(n)\log^2n) holds.

Relations among simultaneous complexity classes of nondeterministic and alternating Turing machines

Ruzzo [Tree-size bounded alternation, J. Comput. Syst. Sci. 21] introduced the notion of tree-size for alternating Turing machines (ATMs) and showed that it is a reasonable measure for classification of complexity classes. We establish in this paper that computations by tree-size and space simultaneously bounded ATMs roughly correspond to computations by time and space simultaneously bounded nondeterministic TMs (NTMs).We also show that not every polynomial time bounded and sublinear space simultaneously bounded NTM can be simulated by any deterministic TM with a slightly increased time bound and a slightly decreased space bound simultaneously.