Shigeki Iwata

Classes of Pebble Games and Complete Problems

A “pebble game” is introduced and some restricted pebble games are considered. It is shown that in each of these games the problem to determine whether there is a winning strategy (two-person game) is harder than the solvability problem (one-person game). We also show that each of these problems is complete in well-known complexity classes. Several familiar games are presented whose winning strategy problems are complete in exponential time.

The Othello game on an n × n board is PSPACE-complete

Abstract Given an arbitrary position of the Othello game played on an n × n board, the problem of determining the winner is shown to be PSPACE-complete. It can be reduced from generalized geography played on bipartite graphs with maximum degree 3.

Some combinatorial game problems require Ω( n k ) time

Dans cet article, on considere quelques problemes combinatoires et on etablit des problemes «naturels» dans P, pour la reconnaissance dont les limites inferieures en temps polynomial peuvent etre connues. Introduction du «jeu de galet» et on montre que quelques-uns de ces problemes peuvent etre complets dans certaines classes de complexite

Gradually intractable problems and nondeterministic log-space lower bounds

The paper places five different problems (thek-pebble game problem, two problems aboutk finite automata, the reachability problem for Petri nets withk tokens, and the teachability problem for graphs whose “k-dimensional” edge sets are described by Cartesian products ofk factors) into the hierarchyNLk of problems solvable by nondeterministic Turing machines ink-log2n space (and binary tape alphabet, to avoid tape “speed-up”). The results, when combined with the conjecture thatNLi contains problems that requireO(nk) deterministic time, show that these problems, while inP for every fixed value ofk, have polynomial deterministic time complexities with the degree of the polynomial growing linearly with the parameterk, and hence are, in this sense, “gradually intractable.”

Low level complexity for combinatorial games

There have been numerous attempts to discuss the time complexity of problems and classify them into hierarchical classes such as P, NP, PSPACE, EXP, etc. A great number of familiar problems have been reported which are complete in NP (nondeterministic polynomial time). Even and Tarjan considered generalized Hex and showed that the problem to determine who wins the game if each player plays perfectly is complete in polynomial space. Shaefer derived some two-person game from NP complete problems which are complete in polynomial space. A rough discussion such as to determine whether or not a given problem belongs to NP is independent of the machine model and the way of defining the size of problems, since any of the commonly used machine models can be simulated by any other with a polynomial loss in running time and by no matter what criteria the size is defined, they differ from each other by polynomial order. However, in precise discussion, for example, in the discussion whether the computation of a problem requires O(n k) time or O(nk+l) time, the complexity heavily depends on machine models and the definition of size of problems. From these points, we introduce somewhat stronger notion of the reducibility.

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).

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.

Simultaneous (Poly-time, Log-space) lower bounds

Abstract In this paper we establish a lower bound for the simultaneous complexity of the halting problem for a class of ‘simple programs’, which allow setting variables to constants and if-goto statements. Let HALT( h, k ) be the problem: given a simple program P k with k variables, determine whether P k halts within n h steps, where n is the length of P k . We show that the problem HALT( h, k ) cannot be solved in time less than n (h−4) 2 and space less than 1 4 (k − 17) log 2 n by any Turing machine with one storage tape and binary storage symbols.