One-way Input Tape Research Articles

Computing halting probabilities from other halting probabilities

The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin's omega number, and is the most well known example of a real which is random in the sense of Martin-Löf. Although omega numbers depend on the underlying universal Turing machine, they are robust in the sense that they all have the same Turing degree, namely the degree of the halting problem. This means that, given two universal prefix-free machines U,V, the halting probability ΩU of U computes the halting probability ΩV of V. If this computation uses at most the first n+g(n) bits of ΩU for the computation of the first n bits of ΩV, we say that ΩU computes ΩV with redundancy g.In this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number. We show that for each ϵ>1, any pair of omega numbers compute each other with redundancy ϵlog⁡n. On the other hand, this is not true for ϵ=1. In fact, we show that for each omega number ΩU there exists another omega number which is not computable from ΩU with redundancy log⁡n. This latter result improves an older result of Frank Stephan.

COMPUTATIONS AND DECIDABILITY OF ITERATIVE ARRAYS WITH RESTRICTED COMMUNICATION

Iterative arrays (IAs) are linear arrays of interconnected interacting finite state machines, where one distinguished one is equipped with a one-way read-only input tape. We investigate IAs operating in real time whose inter-cell communication is bounded by a constant number of bits not depending on the number of states. Their capabilities are considered in terms of syntactical pattern recognition. It is known [17] that such devices can recognize rather complicated sets of unary patterns with a minimum amount of communication, namely one-bit communication. Some examples are the sets {a2n | n ≥ 1}, {an2 | n ≥ 1}, and {ap | p is prime}. Here, we consider non-unary patterns and it turns out that the non-unary case is quite different. We present several real-time constructions for certain non-unary syntactical patterns. For example, the sets {anbn | n ≥ 1}, {anbncn | n ≥ 1}, {an(bn)m | n, m ≥ 1}, and {anbamb(ba)n·m | n, m ≥ 1} are recognized in real time by IAs. Moreover, it is shown that real-time one-bit IAs can, in some sense, add and multiply integer numbers. Furthermore, decidability questions of communication restricted IAs are dealt with. Due to the constructions provided, undecidability results can be derived. It turns out that emptiness is still not even semidecidable for one-bit IAs despite their restricted communication. Moreover, also the questions of finiteness, infiniteness, inclusion, and equivalence are non-semidecidable.

Efficient Simulations by Queue Machines

The following simulations by machines equipped with a one-way input tape and additional queue storage are shown: Every nondeterministic single-tape Turing machine (no separate input-tape) with time bound $t(n)$ can be simulated by one queue in $O(t(n))$ time. Every deterministic machine with a one-turn pushdown store can be simulated deterministically by one queue in $O(n\sqrt{n})$ time. Every Turing machine with several multidimensional tapes accepting with time bound $t(n)$ can be simulated by two queues in $O(t(n) \log^2 t(n))$ time. Every deterministic Turing machine with several linear tapes accepting with time bound $t(n)$ can be simulated deterministically in time $O(t(n) \log t(n))$ by a queue and a pushdown store. The first two results appear to be the first subquadratic simulations of other storage devices by one queue.

On composition and lookahead delegation of [formula omitted]-services modeled by automata

Let M be a class of (possibly nondeterministic) language acceptors with a one-way input tape. A system ( A ; A 1 , … , A r ) of automata in M is composable if for every string w = a 1 ⋯ a n of symbols accepted by A , there is an assignment of each symbol a j in w to one of the A i 's such that for each 1 ⩽ i ⩽ r , the subsequence of w assigned to A i is accepted by A i . For a nonnegative integer k , a k - lookahead delegator for ( A ; A 1 , … , A r ) is a deterministic machine D in M which, knowing (a) the current states of A , A 1 , … , A r and the accessible “local” information of each machine (e.g., the top of the stack if each machine is a pushdown automaton, whether a counter is zero or nonzero if each machine is a multicounter automaton, etc.), and (b) the k lookahead symbols to the right of the current input symbol being processed, can uniquely determine the A i to assign the current symbol. Moreover, every string w accepted by A is also accepted by D ; i.e., the subsequence of string w delegated by D to each A i is accepted by A i . Thus, k -lookahead delegation is a stronger requirement than composability, since the delegator D must be deterministic. A system that is composable may not have a k -delegator for any k . We study the decidability of composability and existence of k -delegators for various classes of machines M . Our results generalize earlier ones (and resolve some open questions) concerning composability of deterministic finite automata as e-services to finite automata that are augmented with unbounded storage (e.g., counters and pushdown stacks) and finite automata with discrete clocks (i.e., discrete timed automata). The results have applications to automated composition of e-services.

Teams of pushdown automata

We introduce team pushdown automata (PDAs) as a theoretical framework capable of modelling various communication and cooperation strategies in complex, distributed systems. Team PDAs are obtained by augmenting distributed PDAs with the notion of team cooperation or, alternatively, by augmenting team automata with pushdown memory. In a team PDA, several PDAs work as a team on the input word placed on a common one-way input tape. At any moment in time one team of PDAs, each with the same symbol on top of its stack, is active: each PDA in the active team replaces the topmost symbol of its stack and changes state, while the current input symbol is read from the input tape by a common reading head. The teams are formed according to the team cooperation strategy of the team PDA and may vary from one moment to the other. Based on the notion of competence, we introduce a variety of team cooperation strategies. If all stacks are empty when the input word has been completely read, then this word is part of the language accepted by the team PDA. Here we focus on the accepting capacity of team PDA. †E-mail: csuhaj@sztaki.hu ‡E-mail: mitrana@funinf.cs.unibuc.ro

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

We prove an O(t(n) d (t(n)) ½ / 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, $ d \ge 2 $ . We prove a matching lower bound which holds for the problem of recognizing languages on machines with a separate one-way input tape.

The Power of the Queue

Queues, stacks, and tapes are basic concepts that have direct applications in compiler design and the general design of algorithms. Whereas stacks (pushdown store or last-in–first-out storage) have been thoroughly investigated and are well understood, this is much less the case for queues (first-in-first-out storage). In this paper a comprehensive study comparing queues to stacks and tapes (off-line and with a one-way input tape) is presented. The techniques used rely on Kolmogorov complexity. In particular, one queue and one tape (or stack) are incomparable: (1) Simulating one stack (and hence one tape) by one queue requires $\Omega (n^{4/3} /\log n)$ time in both the deterministic and the nondeterministic cases. A corollary of this lower bound states that for this model of one-queue machines, nondeterministic linear time is not closed under complement. (2) Simulating one queue by one tape requires $\Omega (n^2 )$ time in the deterministic case and requires $\Omega (n^{4/3}/(\log n)^{2/3} )$ in the nonde...

The Power of Alternating One-Reversal Counters and Stacks

The relation between reversals and alternation is studied in two simple models of computation: the 2-counter machine with a one-way input tape whose counters make only one reversal (1-reversal 2CM) and the one-way pushdown automaton whose pushdown store makes only one reversal (1-reversal PDA). The following is shown: (a) alternating 1-reversal 2CM’s accept all recursively enumerable languages; (b) alternating 1-reversal PDA’s accept exactly the languages accepted by exponential time-bounded deterministic TM’s. The first improves on the known result that alternating 1-reversal 4CM’s accept all recursively enumerable languages. The second improves an earlier result that alternating PDA’s with no restrictions on reversals accept exactly the exponential-time languages.

Combinatorial lower bound arguments for deterministic and nondeterministic Turing machines

We introduce new techniques for proving quadratic lower bounds for deterministic and nondeterministic 1 1 -tape Turing machines (all considered Turing machines have an additional one-way input tape). In particular, we derive for the simulation of 2 2 -tape Turing machines by 1 1 -tape Turing machines an optimal quadratic lower bound in the deterministic case and a nearly optimal lower bound in the nondeterministic case. This answers the rather old question whether the computing power of the considered types of Turing machines is significantly increased when more than one tape is used (problem Nos. 1 and 7 in the list of Duris, Galil, Paul, Reischuk [3]). Further, we demonstrate a substantial superiority of nondeterminism over determinism and of co-nondeterminism over nondeterminism for 1 1 -tape Turing machines.

Multiple equality sets and post machines

Equality sets of finite sets of homomorphisms are studied as part of formal language theory. Some particular equality sets, called Merge k ( k-COPY), are investigated. These languages are combinatorially difficult, and are full semiAFL generators of the recursively enumerable sets, and are semiAFL generators of the class MULTI-RESET, provided k ⩾ 3. To accomplish this characterization, equality sets are related to multihead and multitape Post machines operating in real time. A Post machine has a one-way input tape and Post tapes as storage tapes, which in the multihead version are scanned from left to right by a write head and several read heads. By simulating Post machines by multiple reset machines, and vice versa, several new characterisations of the class MULTI-RESET are obtained, and it is shown that for multihead and multitape Post machines linear time is no more powerful than real time, and two Post tapes or, alternatively, three heads on one Post tape are as powerful as any finite number of heads or tapes. Finally, some complexity bounds for equality sets and Post machines are discussed.

Reset machines

A reset tape has one read-write head which moves only left-to-right except that the head can be reset once to the left end and the tape rescanned; a multiple-reset machine has reset tapes as auxiliary storage and a one-way input tape. Linear time is no more powerful than real time for nondeterministic multiple-reset machines and so the family MULTI-RESET of languages accepted in real time by nondeterministic multiple-reset machines is closed under linear erasing. MULTI-RESET is closed under Kleene. It can be characterized as the smallest family of languages containing the regular sets and closed under intersection and linear-erasing homomorphic duplication or as the smallest intersection-closed semiAFL containing COPY = {ww | w in {a, b}∗} . A circular tape is read full-sweep from left-to-right only and then reset to the left, any number of times; a nonwriting circular tape cannot be altered after the first sweep. For nondeterministic machines operating in real time, multiple reset tapes, circular tapes or nonwriting circular tapes have the same power. Languages in MULTI-RESET can be accepted in real time by nondeterministic machines using only three reset tapes or using only one reset tape and one nonwriting circular tape.

Tape-reversal bounded turing machine computations

This paper studies the classification of recursive sets by the number of tape reversals required for their recognition on a two-tape Turing machine with a one-way input tape. This measure yields a rich hierarchy of tape-reversal limited complexity classes and their properties and ordering are investigated. The most striking difference between this and the previously studied complexity measures lies in the fact that the “speed-up” theorem does not hold for slowly growing tape-reversal complexity classes. These differences are discussed, and several relations between the different complexity measures and languages are established.

Sets accepted by one-way stack automata are context sensitive

A stack automaton is a pushdown automaton that can read the interior of its pushdown list without altering it. It is shown that a nondeterministic stack automation with a one-way input tape can be simulated by a deterministic linear bounded automaton. Hence, each nondeterministic one-way stack language is context sensitive.