Relations between diagonalization,proof systems,and complexity gaps

In this paper we study diagonal processes over time bounded computations of one-tape Turing machines by diagonalizing only over those machines for which there exist formal proofs that they operate in the given time bound. This replaces the traditional “clock” in resource bounded diagonalization by formal proofs about running times and establishes close relations between properties of proof systems and existence of sharp time bounds for one-tape Turing machine complexity classes. These diagonalization methods also show that the Gap Theorem for resource bounded computations can hold only for those complexity classes which differ from the corresponding provable complexity classes. Furthermore, we show that there exist recursive time bounds T(n) such that the class of languages for which we can formally prove the existence of Turing machines which accept them in time T(n) differs from the class of languages accepted by Turing machines for which we can prove formally that they run in time T(n). We also investigate the corresponding problems for tape bound computations and discuss the difference time and tapebounded computations.     

Unbounded-error quantum computation with small space bounds

We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound s satisfying s(n)=o(loglogn). For “one-way” Turing machines, where the input tape head is not allowed to move left, the above result holds for s(n)=o(logn). We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.     

The complexity of the membership problem for some extensions of context-free languagest†

The time and tape complexity of some families of languages defined in the literature by altering methods of generation by context-free grammars is considered. Specifically; it is shown that the following families of languages can be recognized by deterministic multitape Turing machines either in polynomial time or within (log n)² tape:

1) the context independent developmental (EOL) languages;

2) the simple matrix languages;

3) the languages generated by derivation restricted state grammars.:

4) the languages generated by linear context-free grammars with certain non-regular control sets;

5) the languages generated by certain classes of vector grammars.

In fact, these languages are of the same tape complexity as context-free languages. Other results indicate the complexity of EDOL languages and the effects on complexity of applying the homomorphic replication operator to regular and context-free languages.     

A Turing machine time hierarchy

The time separation results concerning classes of languages over a single-letter alphabet accepted by multi-tape nondeterministic Turing machines, well-known from Seiferas, Fischer and Meyer (1978), are supplemented. Moreover, via a universal machine a modified time complexity measure UTIME of Turing machines computations which is sensitive to multiplication by constants (i.e., UTIME(t) ? UTIME(kt), where UTIME(t) is the class of languages of complexity not larger than t) is introduced. On the level of this measure, the results concerning languages over one- and two-letter alphabets are refined. The proof tools are versions of a translational diagonalization and of an unpadding technique.     

A space-hierarchy result on two-dimensional alternating Turing machines with only universal states

This paper investigates a space hierarchy of the classes of sets accepted by alternating space-bounded two-dimensional Turing machines which have only universal states and whose input tapes are restricted to square ones and shows that there exists a dense hierarchy for the classes of sets accepted by these Turing machines with spaces of size less than or equal to logm.     

Matching upper and lower bounds for simulations of several linear tapes on one multidimensional tape

We prove an O(t(n) ^d (t(n)) ^? / log t(n)) time bound for the sim-ulation of t(n) steps of a Turing machine using several one-dimensional work tapes on a Turing machine using one d-dimensional work tape, . We prove a matching lower bound which holds for the problem of recognizing languages on machines with a separate one-way input tape. Received: March 1995.     

Decision algorithms for multiplayer noncooperative games of incomplete information

Extending the complexity results of Reif [1,2] for two player games of incomplete information, this paper (see also [3]) presents algorithms for deciding the outcome for various classes of multiplayer games of incomplete information, i.e., deciding whether or not a team has a winning strategy for a particular game. Our companion paper, [4] shows that these algorithms are indeed asymptotically optimal by providing matching lower bounds. The classes of games to which our algorithms are applicable include games which were not previously known to be decidable. We apply our algorithms to provide alternative upper bounds, and new time-space trade-offs on the complexity of multiperson alternating Turing machines [3]. We analyze the algorithms to characterize the space complexity of multiplayer games in terms of the complexity of deterministic computation on Turing machines.In hierarchical multiplayer games, each additional clique (subset of players with the same information) increases the complexity of the outcome problem by a further exponential. We show that an S(n) space bounded k-player game of incomplete information has a deterministic time upper bound of k + 1 repeated exponentials of S(n). Furthermore, S(n) space bounded k-player blindfold games have a deterministic space upper bound of k repeated exponentials of S(n). This paper proves that this exponential blow-up can occur.We also show that time bounded games do not exhibit such hierarchy. A T(n) time bounded blindfold multiplayer game, as well as a T(n) time bounded multiplayer game of incomplete information, has a deterministic space bound of T(n).     

Some accepting powers of three-dimensional parallel Turing machines

Informally, the parallel Turing machine (PTM) proposed by Wiedermann is a set of identical usual sequential Turing machines (STMs) cooperating on two common tapes: storage tape and input tape. Moreover, STMs which represent the individual processors of a parallel computer can multiply themselves in the course of computation. On the other hand, during the past 25 years or so, automata on a three-dimensional tape have been proposed as computational models of three-dimensional pattern processing, and several properties of such automata have been obtained. We proposed a three-dimensional parallel Turing machine (3-PTM),¹ and dealt with a hardware-bounded 3-PTM whose inputs are restricted to cubic ones. We believe that this machine is useful in measuring the parallel computational complexity of three-dimensional images. Here, we continue the study of 3-PTM, whose inputs are restricted to cubic ones, and investigate some of its accepting powers. This work was presented in part at the First European Workshop on Artificial Life and Robotics, Vienna, Austria, July 12–13, 2007     

Computational complexity via programming languages: constant factors do matter

The purpose of this work is to promote a programming-language approach to studying computability and complexity, with an emphasis on time complexity. The essence of the approach is: a programming language, with semantics and complexity measure, can serve as a computational model that has several advantages over the currently popular models and in particular the Turing machine. An obvious advantage is a stronger relevance to the practice of programming. In this paper we demonstrate other advantages: certain proofs and constructions that are hard to do precisely and clearly with Turing machines become clearer and easier in our approach, and sometimes lead to finer results. In particular, we prove several time hierarchy theorems, for deterministic and non-deterministic time complexity which show that, in contrast with Turing machines, constant factors do matter in this framework. This feature, too, brings the theory closer to practical considerations. The above result suggests that this framework may be appropriate for studying low complexity classes, such as linear time. As an example we give a problem complete for non-deterministic\/ linear time under deterministic linear-time reductions. Finally, we consider some extensions and modifications of our programming language and their effect on time complexity results. Received: 26 October 1998 / 9 June 2000     

Extracting Kolmogorov complexity with applications to dimension zero-one laws

We apply results on extracting randomness from independent sources to “extract” Kolmogorov complexity. For any α,?>0, given a string x with K(x)>α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y|=Ω(|x|), with K(y)>(1-?)|y|. This result holds for both unbounded and space-bounded Kolmogorov complexity.We use the extraction procedure for space-bounded complexity to establish zero-one laws for the strong dimensions of complexity classes within ESPACE. The unbounded extraction procedure yields a zero-one law for the constructive strong dimensions of Turing degrees.     

On tape bounds for single letter alphabet language processing

In this paper we show that the tape bounded complexity classes of languages over single letter alphabets (sla) are closed under complementation. We then use this results to show by means of diagonalization that there exists an infinite hierarchy of tape bounded complexity classes of sla languages between log log n and log n tape bounds. On the other hand, we show that the power of diagonalization over sla inputs with less than log n tape is very limited by proving that every infinite sla language accepted using less than log n tape contains infinite regular subsets. From this result it immediately follows that the set of primes in unary notation, P, requires exactly log n tape for its recognition and every infinite subset of P requires at least log n tape.     

Natural halting probabilities,partial randomness,and zeta functions

We introduce the zeta number, natural halting probability, and natural complexity of a Turing machine and we relate them to Chaitin’s Omega number, halting probability, and program-size complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent, and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness—which cannot be naturally characterised in terms of plain complexity—asymptotic randomness admits such a characterisation.     

Translational lemmas,polynomial time,and (log n)j-space

Translational lemmas are stated in a general framework and then applied to specific complexity classes. Necessary and sufficient conditions are given for every set accepted by a Turing acceptor which operates in linear or polynomial time to be accepted by a Turing acceptor which operates in space (log n)^j for some j ? 1.     

Multihead two-way probabilistic finite automata

We present properties of multihead two-way probabilistic finite automata that parallel those of their deterministic and nondeterministic counterparts. We define multihead probabilistic finite automata withlogspace constructible transition probabilities, and we describe a technique to simulate these automata by standard logspace probabilistic Turing machines. Next, we represent logspace probabilistic complexity classes as proper hierarchies based on corresponding multihead two-way probabilistic finite automata, and we show their (deterministic logspace) reducibility to the second levels of these hierarchies. We obtain a simple formula for the maximum inherent bandwidth of the configuration transition matrices associated with thek-head probabilistic finite automata processing a length-n input string. (The inherent bandwidth of the configuration transition matrices associated with an automaton processing a length-n input string is the smallest bandwidth we can get by changing the enumeration order of the automaton’s configurations.) Partially based on this relation, we find an apparently easier logspace complete problem forPL (the class of languages recognized by logspace unbounded-error probabilistic Turing machines), and we discuss possibilities for a space-efficient deterministic simulation of probabilistic automata.     

One Head Machines from a symbolic approach

We consider the Turing Machine as a dynamical system and we study a particular partition projection of it. In this way, we define a language (a subshift) associated to each machine. The classical definition of Turing Machines over a one-dimensional tape is generalized to allow for a tape in the form of a Cayley Graph. We study the complexity of the language of a machine in terms of realtime recognition by putting it in relation with the structure of its tape. In this way, we find a large set of realtime subshifts some of which are proved not to be deterministic in realtime. Sofic subshifts of this class correspond to machines that cannot make arbitrarily large tours. We prove that these machines always have an ultimately periodic behavior when starting with a periodic initial configuration, and this result is proved for any Cayley Graph.     

Some properties of four-dimensional parallel Turing machines

Informally, the parallel Turing machine (PTM) proposed by Wiedermann is a set of identical usual sequential Turing machines (STMs) cooperating on two common tapes: a storage tape and an input tape. Moreover, STMs which represent the individual processors of a parallel computer can multiply themselves in the course of computation. On the other hand, during the past 7 years or so, automata on a four-dimensional tape have been proposed as computational models of four-dimensional pattern processing, and several properties of such automata have been obtained. We proposed a four-dimensional parallel Turing machine (4-PTM), and dealt with a hardware-bounded 4-PTM in which each side-length of each input tape is equivalent. We believe that this machine is useful in measuring the parallel computational complexity of three-dimensional images. In this work, we continued the study of the 3-PTM, in which each side-length of each input tape is equivalent, and investigated some of its accepting powers.     

On the transition graphs of turing machines

As for pushdown automata, we consider labelled Turing machines with ε-rules. With any Turing machine M and with a rational set C of configurations, we associate the restriction to C of the ϵ-closure of the transition set of M. We get the same family of graphs by using the labelled word rewriting systems. We show that this family is the set of graphs obtained from the binary tree by applying an inverse mapping into F followed by a rational restriction, where F is any family of recursively enumerable languages containing the rational closure of all linear languages. We show also that this family is obtained from the rational graphs by inverse rational mappings. Finally we show that this family is also the set of graphs recognized by (unlabelled) Turing machines with labelled final states, and even if we restrict to deterministic Turing machines.     

Parallel Turing machines on four-dimensional input tapes

The parallel Turing machine (PTM) proposed by Wiedermann is a set of identical usual sequential Turing machines (STMs) cooperating on two common tapes: storage tape and input tape. On the other hand, due to the advances in many application areas such as motion picture processing, computer animation, virtual reality systems, and so forth, it has become increasingly apparent that the study of four-dimensional patterns is of crucial importance. Therefore, we think that the study of four-dimensional automata as a computational model of four-dimensional pattern processing is also meaningful. In this article, we propose a four-dimensional parallel Turing machine (4-PTM), and investigate some of its properties based on hardware complexity.