A relationship between Turing machines and finite automata on four-dimensional input tapes

Due to the advances in computer animation, motion image processing, virtual reality systems, and so forth recently, it is useful for analyzing computation of multi-dimensional information processing to explicate the properties of four-dimensional automata. From this point of view, we first proposed four-dimensional automata in 2002, and investigated their several accepting powers. In this paper, we coutinue the study, and mainly concentrate on investigating the relationship between the accepting powers of four-dimensional finite automata and seven-way four-dimensional tape-bounded Turing Machines. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

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     

Three-dimensional multiinkdot automata

Recently, related to the open problem whether deterministic and nondeterministic space (especially lower-level) complexity classes are separated, inkdot Turing machines were introduced. On the other hand, due to the advances in many application areas about three-dimensional information processing, the research on three-dimensional automata as the computation of three-dimensional pattern processing has been meaningful. From this viewpoint, we propose a three-dimensional multiinkdot automata, and investigate some properties of three-dimensional multiinkdot automata.This work was presented, in part, at the 9th International Symposium on Artificial Life and Robotics, Oita, Japan, January 28–30, 2004     

Path-bounded three-dimensional finite automata

The comparative study of the computational powers of deterministic and nondeterministic computations is one of the central tasks in complexity theory. This paper investigates the computational power of nondeterministic computing devices with restricted nondeterminism. There are only few results measuring the computational power of restricted nondeterminism. In general, there are three possibilities to measure the amount of nondeterminism in computation. In this paper, we consider the possibility to count the number of different nondeterministic computation paths on any input. In particular, we deal with five-way three-dimensional finite automata with multiple input heads operating on three-dimensional input tapes. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

A note on emptiness for alternating finite automata with a one-letter alphabet

We present a new proof of PSPACE-hardness of the emptiness problem for alternating finite automata with a singleton alphabet. This result was shown by Holzer (1995) who used a proof relying on a series of reductions from several papers. The new proof is simple, direct and self-contained.     

Simulating one-reversal multicounter machines by partially blind multihead finite automata

This work is concerned with simulating nondeterministic one-reversal multicounter automata (NCMs) by nondeterministic partially blind multihead finite automata (NFAs). We show that any one-reversal NCM with k$k$ counters can be simulated by a partially blind NFA with k$k$ blind heads. This provides a nearly complete categorization of the computational power of partially blind automata, showing that the power of a (k+1)$(k+1)$-NFA lies between that of a k$k$-NCM and a (k+1)$(k+1)$-NCM.     

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), 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. In this article, 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 12th International Symposium on Artificial Life and Robotics, Oita, Japan, January 25–27, 2007     

Measuring nondeterminism in pushdown automata

The amount of nondeterminism that a pushdown automaton requires to recognize an input string can be measured by the minimum number of guesses that it must make to accept the string, where guesses are measured in bits of information. When this quantity is unbounded, the rate at which it grows as the length of the string increases serves as a measure of the pushdown automaton's “rate of consumption” of nondeterminism. We show that this measure is similar to other complexity measures in that it gives rise to an infinite hierarchy of complexity classes of context-free languages differing in the amount of the resource (in this case, nondeterminism) that they require. In addition, we show that there are context-free languages that can only be recognized by a pushdown automaton whose nondeterminism grows linearly, resolving an open problem in the literature. In particular, the set of palindromes is such a language.     

Inequality of finite behaviors of rational weight finite automata is in R

Inequivalence of finite automata accepting finite languages over a non-unary alphabet is NP-complete. However, the inequality of their behaviors does not appear to have been carefully investigated. In the simplest case, the behavior of a finite automaton is the formal series f such that the coefficient f(w) of a word w is the number of distinct accepting computations on w. This notion will be generalized in the paper to finite automata with rational weights. The main result is that inequality of rational weight finite automata with finite behaviors is in R, random polynomial time.     

Alternation on cellular automata

In this paper we consider several notions of alternation in cellular automata: non-uniform, uniform and weak alternation. We study relations among these notions and with alternating Turing machines. It is proved that the languages accepted in polynomial time by alternating Turing machines are those accepted by alternating cellular automata in polynomial time for all the proposed alternating cellular automata. In particular, this is true for the weak model where the difference between existential and universal states is omitted for all the cells except the first one. It is proved that real time alternation in cellular automata is strictly more powerful than real time alternation in Turing machines, with only one read-write tape. Moreover, it is shown that in linear time uniform and weak models agree.     

PAC-learnability of probabilistic deterministic finite state automata in terms of variation distance

We consider the problem of PAC-learning distributions over strings, represented by probabilistic deterministic finite automata (PDFAs). PDFAs are a probabilistic model for the generation of strings of symbols, that have been used in the context of speech and handwriting recognition, and bioinformatics. Recent work on learning PDFAs from random examples has used the KL-divergence as the error measure; here we use the variation distance. We build on recent work by Clark and Thollard, and show that the use of the variation distance allows simplifications to be made to the algorithms, and also a strengthening of the results; in particular that using the variation distance, we obtain polynomial sample size bounds that are independent of the expected length of strings.     

Equivalence of two-dimensional multitape automata

The equivalence of multitape automata with multi-dimensional tapes is considered. Their heads move monotonically in all directions (their backward movement is impossible). The special case when the dimensions of tapes are less than or equal to 2 is proved to be solvable. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 3–10, January–February 2008.     

Tax evasion behavior using finite automata: Experiments in Chile and Italy

In this paper, we use a Moore Automata with Binary Stochastic Output Function in order to capture the extensive decision regarding tax evasion made by subjects in experiments run in Chile and Italy. Firstly, we show how an hypothesis about subject behavior is converted into an automaton, and how we compute the probabilities of evading for every state of an automaton. We use this procedure in order to look for the automaton which is able to anticipate the highest number of decisions made by the subjects during the experiments. Finally, we show that automata with few states perform better than automata with many states, and that the bomb-crater effect described in Mittone (2006) is a well identified pattern of behavior in a subset of subjects.     

On cellular automata over Galois rings

In this paper we study the invertibility of one-dimensional cellular automata, determined by a local rule, acting on the space of all doubly-infinite sequences taking values in a finite Galois ring. We also compute the topological entropy of one-dimensional CA generated by additive local rule over a finite Galois ring. We conclude by showing that the topological entropy of an additive invertible CA over a finite Galois ring is equal to its inverse.     

A very simple function that requires exponential size nondeterministic graph-driven read-once branching programs

Branching programs are a well-established computation model for Boolean functions, especially read-once branching programs (BP1s) have been studied intensively. A very simple function f in n² variables is exhibited such that both the function f and its negation ¬f can be computed by Σ³_p-circuits, the function f has nondeterministic BP1s (with one nondeterministic node) of linear size and ¬f has size O(n⁴) for oblivious nondeterministic BP1s but f requires nondeterministic graph-driven BP1s of size . This answers an open question stated by Jukna, Razborov, Savický, and Wegener [Comput. Complexity 8 (1999) 357-370].     

Theory of general linear automata

Linear and affine automata are considered in their general form. The concept of dimensions of a finite automaton is introduced and finite automata of maximal dimensions are shown to be possible. The state reachability problem in monomial form is proved to be undecidable for two-dimensional affine automata. An analogue of Moore's theorem and theorems on homogenous and diagnostic words are also proved. An application of linear automata to mathematical economics is considered. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 10–21, January–February 2009.     

Determinization of weighted finite automata over strong bimonoids

We consider weighted finite automata over strong bimonoids, where these weight structures can be considered as semirings which might lack distributivity. Then, in general, the well-known run semantics, initial algebra semantics, and transition semantics of an automaton are different. We prove an algebraic characterization for the initial algebra semantics in terms of stable finitely generated submonoids. Moreover, for a given weighted finite automaton we construct the Nerode automaton and Myhill automaton, both being crisp-deterministic, which are equivalent to the original automaton with respect to the initial algebra semantics, respectively, the transition semantics. We prove necessary and sufficient conditions under which the Nerode automaton and the Myhill automaton are finite, and we provide efficient algorithms for their construction. Also, for a given weighted finite automaton, we show sufficient conditions under which a given weighted finite automaton can be determinized preserving its run semantics.     

Complexity of weak acceptance conditions in tree automata

Weak acceptance conditions for automata on infinite words or trees are defined in terms of the set of states that appear in the run. This is in contrast with, more usual, strong conditions that are defined in terms of states appearing infinitely often on the run. Weak conditions appear in the context of model-checking and translations of logical formalisms to automata. We study the complexity of the emptiness problem for tree automata with weak conditions. We also study the translations between automata with weak and strong conditions.