A uniquely parsable unification grammar (UPUG) is a formal grammar with the following features: (1) parsing is performed without backtracking, and (2) each nonterminal symbol can have arguments, and derivation and parsing processes accompany unification of terms as in Prolog (or logic programming). We newly introduce a uniquely parallel parsable unification grammar (UPPUG) by extending the framework of a UPUG so that parallel parsing is also possible. We show that, in UPPUG, parsing can be done without backtracking in both cases of parallel and sequential reductions. We give examples of UPPUGs where a given input string can be parsed in sublinear number of steps of the length of the input by parallel reduction.

We present a tight time-hierarchy theorem for nondeterministic cellular automata by using a recursive padding argument. It is shown that, if t2(n) is a time-constructible function and t2(n) grows faster than t1(n+1), then there exists a language which can be accepted by a t2(n)-time nondeterministic cellular automaton but not by any t1(n)-time nondeterministic cellular automaton.

In this paper we study the problem of recognizing connectedness of three-dimensional patterns. We investigate the upper and lower bounds of space complexity for this problem. These results are compared with two-dimensional case. It reveals that recognizing three-dimensional connectedness is much more difficult than two-dimensional case. We introduce a three-dimensional k-marker automaton (MA(k)) and a three-dimensional S(n) space-bounded Turing machine (TM(S(n))). We prove that a nondeterministic MA(1), a nondeterministic TM(log n) and a deterministic TM((log n)2) can accept connected patterns. We also prove that a deterministic TM((log n)2) can accept patterns of k connected components (k is a constant). Next, a three-dimensional five-way S(n) space-bounded Turing machine (5WTM(S(n))) is introduced. It is a restricted model of TM(S(n)) whose input head can move north, south, east, west and down, but not up. We prove that the space n2 log n is necessary and sufficient amount for a deterministic 5WTM(S(n)) to recognize connected patterns.

Bennett proved that any irreversible Turing machine can be simulated by a reversible one. However, Bennett's reversible machine uses 3 tapes and many tape symbols. Previously, Gono and Morita showed that the number of symbols can be reduced to 2. In this paper, by improving these methods, we give a procedure to convert an irreversible machine into an equivalent 1-tape 2-symbol reversible machine. First, it is shown that the state-degeneration degree" of any Turing machine can be reduced to 2 or less. Using this result and some other techniques, a given irreversible machine is converted into a 1-tape 32-symbol (i.e., 5-track 2-symbol) reversible machine. Finally the 32-symbol machine is converted into a 1-tape 2-symbol reversible machine. From this result, it is seen that a 1-tape 2-symbol reversible Turing machine is computation universal.

A reversible cellular automaton (CA) is a backward deterministic" CA, i.e., every configuration of it has at most one predecessor. Toffoli showed that a two-dimensional reversible cellular automaton is computation universal. He posed an open problem whether a one-dimensional reversible CA is computation universal. In this paper, we solve this problem affirmatively. This result is proved by using the previous result of Morita et al. that a 1-tape reversible Turing machine is computation universal. We give a construction method of a reversible CA which simulates a given 1-tape reversible Turing machine. To do this, we introduce a one-dimensional partitioned cellular automaton" (1-PCA). 1-PCA has the property that the local reversibility (i.e., injectivity of a local function) is equivalent to the global reversibility, and thus it facilitates to design a reversible CA.

A disentanglement puzzle consists of mechanically interlinked pieces, and the puzzle is solved by disentangling one piece from another set of pieces. A cast puzzle is a type of disentanglement puzzle, where each piece is a zinc die-casting alloy. In this paper, we consider the generalized cast puzzle problem whose input is the layout of a finite number of pieces (polyhedrons) in the 3-dimensional Euclidean space. For every integer k ≥ 0, we present a polynomial-time transformation from an arbitrary k-exponential-space Turing machine M and its input x to a cast puzzle c1 of size k-exponential in |x| such that M accepts x if and only if c1 is solvable. Here, the layout of c1 is encoded as a string of length polynomial (even if c1 has size k-exponential). Therefore, the cast puzzle problem of size k-exponential is k-EXPSPACE-hard for every integer k ≥ 0. We also present a polynomial-time transformation from an arbitrary instance f of the SAT problem to a cast puzzle c2 such that f is satisfiable if and only if c2 is solvable.

Shisen-Sho is a tile-based one-player game. The instance is a set of 136 tiles embedded on 817 rectangular grids. Two tiles can be removed if they are labeled by the same number and if they are adjacent or can be connected with at most three orthogonal line segments. Here, line segments must not cross tiles. The aim of the game is to remove all of the 136 tiles. In this paper, we consider the generalized version of Shisen-Sho, which uses an arbitrary number of tiles embedded on rectangular grids. It is shown that deciding whether the player can remove all of the tiles is NP-complete.

A reversible finite automaton (RFA) is a backward deterministic automaton, i.e., it can uniquely retrace its move sequence if the inverse sequence of its outputs is given. In this paper, we show a simple method to construct an RFA from Fredkin gates, which are reversible and bit-conserving logic gates, and unit wires (unit delays). The resulting circuit obtained by this method is garbage-less" in the sense that it has no inputs to which constants must be supplied nor outputs from which garbage signals are put out. We also show that a one-dimensional reversible partitioned cellular automaton, which are known to be computation universal, can be constructed from Fredkin gates and unit wires as a closed (thus garbage-less) infinite circuit.

A uniquely parsable grammar (UPG) introduced by Morita et al. is a kind of generative grammar, in which parsing can be performed without backtracking. It is known that UPGs and their three subclasses form the "deterministic Chomsky hierarchy. " In this paper, we introduce three kinds of normal forms for UPGs, i.e., Type-0, Type-1 and Type-2 UPGs by restricting the forms of rules to very simple ones. We show that the upper three classes in the deterministic Chomsky hierarchy can be exactly characterized by the three types of UPGs.

A number-conserving cellular automaton (NCCA) is a cellular automaton (CA) such that all states of cells are represented by integers and the sum of the cell states is conserved throughout its computing process. It can be thought of as a kind of modelization of the physical conservation law of mass or energy. It is known that the local function of a two-dimensional 45-degree reflection-symmetric von Neumann neighbor NCCA can be represented by linear combinations of a binary function. In spite of the number-conserving constraints, it is possible to design an NCCA with complex rules by employing this representation. In this paper, we study the case in which the binary function depends only on the difference of two cell states, i.e., the case in which the function can be regarded as a unary one and its circuit for applying rules to a cell only need adders and a single value table look up module. Even under this constraint, it is possible to construct a logically universal NCCA.

In this paper, multi-head and finite-state transducers are proposed, and their computing abilities of number-theoretic functions are investigated. A multi-head transducer is an automaton which maps a unary input into a unary output using a finite number of input heads. A finite-state transduccer is one with single input head. First, it is shown that a two-way finite-state transducer is strictly more powerful than a one-way finite-state transducer, but a two-way finite-state transducer can be simulated by a two-scan finite-state transducer. As for the multi-head transducer, the following results are derived. The upper bound of increasing degree of functions computed by two-way multi-head transducers varies with the number of input heads. However, a two-way two-head transducer can compute an arbitrarily slowly increasing monotone total recursive function, so that there exists no lower bound of increasing degree of a function computed by it. Although the class of two-way multi-head transducers forms an infinite hierarchy of computing abilities with respect to the number of input heads, it is shown that a one-way multi-head transducer is equivalent to a two-way finite-state transducer provided that the number of input heads is more than one.

A reversible cellular automaton (RCA) is a computing model having a property analogous to physical reversibility. We investigate the problem of finding simple RCAs in which any circuit composed of rotary elements (REs) can be embedded. Since an RE is known to be a universal reversible logic element, such RCAs are also universal in this respect. In this paper, after giving a survey of known results on RE and its implementation in RCAs, we propose a new RCA model in which REs and some signal routing elements can be embedded. The new model has a simpler local transition function (in the sense it is described by fewer rules) than the previous one, though the number of states is the same. In addition, the patterns realizing an RE and signal routing elements are smaller than those of the previous model.

We study the complexity of finding the minimum number of face guards which can observe the whole surface of a polyhedral terrain. Here, a face guard is allowed to be placed on the faces of a terrain, and the guard can walk around on the allocated face. It is shown that finding the minimum number of face guards is NP-hard.

A reversible (or injective) cellular automaton (RCA) is a backward deterministic" CA, i.e., every configuration of it has at most one predecessor. Margolus has been shown that there is a computation-universal two-dimensional 2-state RCA model. Although his model is very interesting, it differs from a standard CA model because of its somewhat spatial and temporal non-uniformity. In this paper, we present two kinds of simple 16-state computation-universal models using the framework of two-dimensional reversible partitioned CA (PCA). Since PCA can be considered as a subclass of standard CA, we can immediately obtain 16-state standard RCA models from them. For each of these models, we designed a configuration which simulates a Fredkin gate. Since Fredkin gate has been known to be a universal logic element, computation-universality of these two models is concluded.

This paper investigates prefix computations on Iterative Arrays (IAs) with sequential input/output mode. We show that, for any language L accepted by a linear-time IA, there is an IA which, given an infinite string a1a2 ai, generates the values of χL(a1),χL(a1a2),,χL(a1a2 ai), at steps 4,16,,4i2,, respectively. Here, χL:Σ*{0,1} is the characteristic function of the language L Σ*, defined as χL(w) = 1 iff w L. We also construct 2i3-time and i4-time prefix algorithms for languages accepted by quadratic-time and cubic-time IAs, respectively.

A number-conserving cellular automaton (NCCA) is a cellular automaton such that all states of cells are represented by integers and the total number of its configuration is conserved throughout its computing process. In constrast to normal cellular automata, there are infinitely many assignments of states for NCCAs with a constant state number. As for von Neumann neighbor(radius one) NCCAs with rotation-symmetry, a local function can be represented by summation of four binary functions. In this paper, we show that the minimum size of state set of rotation-symmetric von Neumann neighbor NCCA is 5 by using this representation.

We present an Isometric Context-Free Array Grammar (ICFAG) that generates the set of all solid upright rectangles. This is performed by using the property that blank symbols in the rewriting rules enable ICFAGs to sense the local shapes of the host array. Thus ICFAGs are context-sensitive in some sense.

We study the computational complexity of the following two-player game. The instance is a graph G = (V,E), an initial vertex s ∈ V, and a target set T ⊆ V. A “cat” is initially placed on s. Player 1 chooses a vertex in the graph and removes it and its incident edges from the graph. Player 2 moves the cat from the current vertex to one of the adjacent vertices. Players 1 and 2 alternate removing a vertex and moving the cat, respectively. The game continues until either the cat reaches a vertex of T or the cat cannot be moved. Player 1 wins if and only if the cat cannot be moved before it reaches a vertex of T. It is shown that deciding whether player 1 has a forced win on the game on G is PSPACE-complete.