A note on three-dimensional finite automata

This paper investigates the relationships between the accepting powers of three-dimensional six-way finite automata (3-FA's) and three-dimensional five-way Turing machines (5WTM's), where the input tapes of these automata are restricted to cubic ones. A 3-FA (5WTM) can be considered as a natural extension of the two-dimensional four-way finite automaton (two-dimensional three-way Turing machine) to three dimensions. The main results are: (1) n²logn (n³) space is necessary and sufficient for deterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's; (2) n² (n²) space is necessary and sufficient for nondeterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's.     

A note on deterministic three-way tape-bounded two-dimensional Turing machines

In this paper, we first investigate the relationship between the accepting powers of four-way two-dimensional finite automata and deterministic three-way tape-bounded two-dimensional Turing machines whose input tapes are restricted to square ones. The second part of this paper solves several open problems concerning closure properties of deterministic three-way tape-bounded two-dimensional Turing machines.     

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.     

Comparing Verboseness for Finite Automata and Turing Machines

A language is called (m,n)-verbose if there exists a Turing machine that enumerates for any n words at most m possibilities for their characteristic string. This notion is compared with (m,n)-fa-verboseness, where instead of a Turing machine a finite automaton is used. By use of a new diagonalisation method, where finite automata trick Turing machines, it is shown that all (m,n)-verbose languages are (h,k)-verbose iff all (m,n)-fa-verbose languages are (h,k)-fa-verbose. In other words, Turing machines and finite automata behave exactly the same way with respect to inclusion of verboseness classes. This identical behaviour implies that the nonspeedup theorem also holds for finite automata. As an application of the theoretical framework, a lower bound is derived on the number of bits that need to be communicated to finite automata protocol checkers for nonregular protocols.     

Succinct representation of regular languages by boolean automata

Boolean automata are a generalization of finite automata in the sense that the ‘next state’, i.e. the result of the transition function given a state and a letter, is not just a single state (deterministic automata) or a union of states (nondeterministic automata) but a boolean function of states. Boolean automata accept precisely regular languages; furthermore they correspond in a natural way to certain language equations as well as to sequential networks. We investigate the succinctness of representing regular languages by boolean automata. In particular, we show that for every deterministic automaton A with m states there exists a boolean automaton with [log₂m] states which accepts the reverse of the language accepted by A (m≥1). We also show that for every n≥1 there exists a boolean automation with n states such that the smallest deterministic automaton accepting the same language has 2⁽²ⁿ⁾ states; moreover this holds for an alphabet with only two letters.     

A relationship between two-dimensional finite automata and three-way tape-bounded two-dimensional Turing machines

This note supplements a result of Inoue and Takanami (1980) who showed that for any function L(m) such that (i) L(m)?log m, and (ii) $\text{lim}\text{m→∞}\text{[}\text{L(m)}\text{m}{}^{\mathrm{<mtext>2</mtext>}}\text{]=}\text{0}$$\text{(}\text{resp}\text{lim}\text{m→∞}\text{[}\text{L(m)}\text{m}\text{log}\text{m}\text{] =}\text{0}\text{)}$, the class of sets of square tapes accepted by deterministic three-way L(m) tape-bounded two-dimensional Turing machines is incomparable with the class of sets of square tapes accepted by nondeterministic (resp. deterministic) two-dimensional finite automata.     

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. This research was supported by the National Science Foundation under Grant No. CDA 8822724 while the author was at the University of Rochester. An extended abstract of this paper appeared in Proceedings, Second Latin American Symposium, LATIN ’95: Theoretical Informatics, Valparaiso, Chile, April 1995.     

Three-way two-dimensional alternating finite automata with rotated inputs

The authors have introduced an automaton on a two-dimensional tape, which decides acceptance or rejection of an input tape by scanning the tape from various sides by three-way (deterministic and nondeterministic) finite automata, and have investigated the accepting powers. This paper continues the investigation of this type of automata, which consists of four three-way two-dimensional alternating finite automata (tr2-afa’s). We first investigate a relationship between the accepting powers of ∨-type automata (obtained by combining tr2-afa’s in ‘or’ fashion) and ∧-type automata (obtained by combining tr2-afa’s in ‘and’ fashion), and show that they are incomparable. Then, we investigate a hierarchy of the accepting powers based on the number of tr2-afa’s combined. Finally, we briefly describe a relationship between the accepting powers of automata obtained by combining three-way two-dimensional nondeterministic and alternating finite automata.     

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 ranking simple languages

Ranking is the problem of computing for an input string its lexicographic index in a given (fixed) language. This paper concerns the complexity of ranking. We show that ranking languages accepted by 1-way unambiguous auxiliary pushdown automata operating in polynomial time is inNC ⁽²⁾. We also prove negative results about ranking for several classes of simple languages.C is rankable in deterministic polynomial time iffP=P ^#P , whereC is any of the following six classes of languages: (1) languages accepted by logtime-bounded nondeterministic Turing machines, (2) languages accepted by (uniform) families of unbounded fan-in circuits of constant depth and polynomial size, (3) languages accepted by 2-way deterministic pushdown automata, (4) languages accepted by multihead deterministic finite automata, (5) languages accepted by 1-way nondeterministic logspace-bounded Turing machines, and (6) finitely ambiguous linear context-free languages.This research was partially supported by the National Science Foundation under Grant DCR-8696097. A preliminary version of this paper was presented at the 3rd Annual Structure in Complexity Theory Conference, Washington, DC, June 1988.     

Two-way unary automata versus logarithmic space

We show that if L=NL (the classical logarithmic space classes), then each unary 2nfa (a two-way nondeterministic finite automaton) can be converted into an equivalent 2dfa (a deterministic two-way automaton), keeping the number of states polynomial. (Unlike other results of this kind, here the deterministic simulation is valid for inputs of all lengths, not only polynomially long ones.) This shows a connection between the standard logarithmic space complexity and the state complexity of two-way unary automata: it indicates that L could be separated from NL by proving a superpolynomial gap, in the number of states, for the conversion from unary 2nfas to 2dfa. Moreover, without any unproven assumptions, we show that each n-state unary 2nfa can be simulated by an equivalent 2ufa (an unambiguous 2nfa) with a polynomial number of states.     

One way finite visit automata

A one-way preset Turing machine with base L is a nondeterministic on-line Turing machine with one working tape preset to a member of L. FINITEREVERSAL($\text{L}$) (FINITEVISIT ($\text{L}$)) is the class of languages accepted by one-way preset Turing machines with bases in $\text{L}$ which are limited to a finite number of reversals (visits). For any full semiAFL $\text{L}$, FINITEREVERSAL ($\text{L}$) is the closure of $\text{L}$ under homomorphic replication or, equivalently, the closure of $\text{L}$ under iteration of controls on linear context-free grammars while FINITEVISIT ($\text{L}$) is the result of applying controls from $\text{L}$ to absolutely parallel grammars or, equivalently, the closure of $\text{L}$ under deterministic two-way finite state transductions. If $\text{L}$ is a full AFL with $\text{L}$ ≠ FINITEVISIT($\text{L}$), then FINITEREVERSAL($\text{L}$) ≠ FINITEVISIT($\text{L}$). In particular, for one-way checking automata, k + 1 reversals are more powerful than k reversals, k + 1 visits are more powerful than k visits, k visits and k + 1 reversals are incomparable and there are languages definable within 3 visits but no finite number of reversals. Finite visit one-way checking automaton languages can be accepted by nondeterministic multitape Turing machines in space log₂n. Results on finite visit checking automata provide another proof that not all context-free languages can be accepted by one-way nonerasing stack automata.     

One-way simple multihead finite automata

The first half of this paper investigates the accepting powers of various types of simple one-way multihead finite automata. It is shown that(1)for each k?1, simple one-way (k+1)-head finite automata are more powerful than simple one-way k-head finite automata.(2)for each k?2, nondeterministic simple one-way k-head finite automata are more powerful than deterministic ones, and(3)for each k?2, sensing simple one-way k-head finite automata are more powerful than non-sensing ones.In the latter half, closure properties for various types of simple one-way multihead finite automata are investigated.Finally, we demonstrate that languages accepted by nondeterministic sensing simple one-way 2-head finite automata are related to some open problem concerning deterministic and nondeterministic tape-bounded Turing computations.     

Deterministic Autopoietic Automata

This paper studies some issues related to autopoietic automata, a model of evolving interactive systems, where the automata produce other automata of the same kind. It is shown how they relate to interactive Turing machines. All results by Jiri Wiedermann on nondeterministic autopoietic automata are extended to deterministic computations. In particular, nondeterminism is not needed for a single autopoietic automaton to generate all autopoietic automata.     

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.     

Two-dimensional automata with rotated inputs (projection-type)

This paper introduces a new type of automaton on a two-dimensional tape, which decides acceptance or rejection of an input tape x by first scanning the tape x from various sides with parallel/sequential array readers, and by then scanning the pair of the halting state configurations (i.e., projections) generated by these array readers with a multitape finite automaton. We mainly concentrate on investigating the accepting power of two-dimensional automata which consist of one-way parallel/sequential array readers and a multitape finite automaton operating in real time.     

The reactive simulatability (RSIM) framework for asynchronous systems

We define reactive simulatability for general asynchronous systems. Roughly, simulatability means that a real system implements an ideal system (specification) in a way that preserves security in a general cryptographic sense. Reactive means that the system can interact with its users multiple times, e.g., in many concurrent protocol runs or a multi-round game. In terms of distributed systems, reactive simulatability is a type of refinement that preserves particularly strong properties, in particular confidentiality. A core feature of reactive simulatability is composability, i.e., the real system can be plugged in instead of the ideal system within arbitrary larger systems; this is shown in follow-up papers, and so is the preservation of many classes of individual security properties from the ideal to the real systems.A large part of this paper defines a suitable system model. It is based on probabilistic IO automata (PIOA) with two main new features: One is generic distributed scheduling. Important special cases are realistic adversarial scheduling, procedure-call-type scheduling among colocated system parts, and special schedulers such as for fairness, also in combinations. The other is the definition of the reactive runtime via a realization by Turing machines such that notions like polynomial-time are composable. The simple complexity of the transition functions of the automata is not composable.As specializations of this model we define security-specific concepts, in particular a separation between honest users and adversaries and several trust models.The benefit of IO automata as the main model, instead of only interactive Turing machines as usual in cryptographic multi-party computation, is that many cryptographic systems can be specified with an ideal system consisting of only one simple, deterministic IO automaton without any cryptographic objects, as many follow-up papers show. This enables the use of classic formal methods and automatic proof tools for proving larger distributed protocols and systems that use these cryptographic systems.     

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.     

Complementing two-way finite automata

We study the relationship between the sizes of two-way finite automata accepting a language and its complement. In the deterministic case, for a given automaton (2dfa) with n states, we build an automaton accepting the complement with at most 4n states, independently of the size of the input alphabet. Actually, we show a stronger result, by presenting an equivalent 4n-state 2dfa that always halts. For the nondeterministic case, using a variant of inductive counting, we show that the complement of a unary language, accepted by an n-state two-way automaton (2nfa), can be accepted by an O(n⁸)-state 2nfa. Here we also make 2nfa’s halting. This allows the simulation of unary 2nfa’s by probabilistic Las Vegas two-way automata with O(n⁸) states.