Input Tape Research Articles

Optimal Semicomputable Approximations to Reachable and Invariant Sets

In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are upper-semicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optimal.

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.

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.

Turing machines, transition systems, and interaction

This paper presents persistent Turing machines (PTMs), a new way of interpreting Turing-machine computation, based on dynamic stream semantics. A PTM is a Turing machine that performs an infinite sequence of “normal” Turing machine computations, where each such computation starts when the PTM reads an input from its input tape and ends when the PTM produces an output on its output tape. The PTM has an additional worktape, which retains its content from one computation to the next; this is what we mean by persistence . A number of results are presented for this model, including a proof that the class of PTMs is isomorphic to a general class of effective transition systems called interactive transition systems ; and a proof that PTMs without persistence ( amnesic PTMs ) are less expressive than PTMs. As an analogue of the Church-Turing hypothesis which relates Turing machines to algorithmic computation, it is hypothesized that PTMs capture the intuitive notion of sequential interactive computation .

Production of Ideas by Means of Ideas: A Turing Machine Metaphor

ABSTRACTIn this paper production of knowledge is modelled in terms of Turing machines or compositions of Turing machines. The input tape and the output tape of the machines are seen as the encoding of ideas generated by Turing machines. This allows us, among other things, to define and measure the complexity of ideas and knowledge in the simple terms of algorithmic and computational complexity. Moreover the very existence of the halting problem allows for the introduction of a non‐stochastic notion of uncertainty. Consequently uncertainty may be modelled not as the outcome of a predefined probability distribution, but in terms of the randomness associated with the halting problem. In this way the ‘frequency distribution of innovations’ is endogenously generated by the choices of the Turing machines. It is shown that a vast variety of alternative dynamic behaviours in knowledge evolution and hence in productivity can be generated. One can observe clusters of emerging innovations, persistent and increasing emergence of new discoveries as well as periods of explosive evolutions followed by stasis periods.

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

A Relationship between the Accepting Powers of Three-Dimensional Finite Automata and Time-Bounded Bottom-Up Pyramid Cellular Acceptors with Three-Dimensional Layers

In this paper, we introduce a new model of acceptor on a three-dimensional pattern, called the time-bounded bottom-up pyramid cellular acceptor with three-dimensional layers whose input tapes are restricted to cubic ones (3-UPC AC), and investigate a relationship between the accepting powers of 3-UPCAc's and three-dimesional finite automata whose input tapes are restricted to cubic ones (3-FAc's). We first show that nondeterministic 3-UPCAc's are more powerful than nondeterministic 3-FAc's. We next show that O ((diameter) 2 × log diameter) time is sufficient for deterministic 3-U PC Ac's to simulate deterministic 3-FAc's, and that O ((diameter) 3) time is sufficient for deterministic 3-UPCAc's to simulate nondeterministic 3-FAc's. Finally, we show that Ω ((diameter) 2 × log diameter) time is necessary for nondeterministic 3-UPCAc's to simulate alternating 3-FAc's.

A note on one-pebble two-dimensional Turing machines

This paper investigates a relationship among the accepting powers of deterministic, nondeterministic, and alternating one-pebble two-dimensional Turing machines, and shows that 1. nondeterminism are more powerful than determinism for o(log n) space-bounded one-pebble two-dimensional Turing machines whose input tapes are restricted to square ones, and 2. alternation are more powerful than nondeterminism for f( m)+ g( n) (resp., f( m)× g( n)) space-bounded one-pebble two-dimensional Turing machines, where f: N→ N is an arbitrary monotonic nondecreasing function space-constructible by a deterministic one-pebble two-dimensional Turing machine, and g: N→ N is an arbitrary function such that g( n)=o(log n) (resp., g(n)= o( logn/ log log n) ).

A mathematical model and heuristic procedure to schedule printed circuit packs on sequencers

This paper considers the NP-hard problem of scheduling printed circuit packs (PCPs) on sequencers to minimize the total number of changeovers of input tapes required to produce PCPs. A mathematical model for this problem is given. However, when the size of the problem increases, the model will not be computationally feasible. An effective and fast heuristic to solve this problem is presented. The heuristic has shown its effectiveness over two previous heuristics found in the literature. It has been tested on a problem with 45 PCPs, two sequencers, each with seven dispensing heads, and a total of 20 input tape types. Comparison results showed that the heuristic reduced the total number of changeovers by 34% and 23% over previous heuristics. In addition, the proposed heuristic provided balanced workload across available sequencers. To investigate the efficiency of our heuristic further, we tested it against data sets from the literature. The proposed heuristic performs reasonably well compared with the reported results. In printed circuit packs' environment, set-up reduction is a main concern. Set-up is composed of the time required to changeover input tapes on sequencers when switching from PCP type to another. The proposed heuristic is more effective in reducing the number of changeovers of input tapes than heuristics found in the literature. The same heuristic can be used to schedule jobs on CNC machines so that total amount of tool switching is minimized.

Deterministic Turing machines in the range between real-time and linear-time

Deterministic k-tape and multitape Turing machines with one-way, two-way and without a separated input tape are considered. We investigate the classes of languages acceptable by such devices with time bounds of the form n+r(n), where r ∈ o(n) is a sublinear function. It is shown that there exist infinite time hierarchies of separated complexity classes in that range. For these classes weak closure properties are proved.

Non-closure property of space-bounded two-dimensional alternating Turing machines

This paper investigates non-closure properties of the classes of sets accepted by space-bounded two-dimensional alternating Turing machines and three-way two-dimensional alternating Turing machines. Let 2-ATM( L( m, n)) (resp., TR2-ATM( L( m, n))) be the class of sets accepted by L( m, n) space-bounded two-dimensional alternating Turing machines (resp., L( m, n) space-bounded three-way two-dimensional alternating Turing machines), where L( m, n): N 2→ N∪{0} ( N denotes the set of all the positive integers) is a function with two variables m (=the number of rows of input tapes) and n (=the number of columns of input tapes). We show that (i) for any function g( n)=o(log n) (resp., g(n)= o( logn/ log logn) ) and any monotonic non-decreasing function f( m) which can be constructed by some one-dimensional deterministic Turing machine, 2-ATM( L( m, n)) and TR2-ATM( L( m, n)) are not closed under column catenation, column +, and projection, and (ii) for any function f( m)=o(log m) (resp., f(m)=o( logm/ log logm) ) and any monotonic non-decreasing function g( n) which can be constructed by some one-dimensional deterministic Turing machine, 2-ATM( L( m, n)) and TR2-ATM( L( m, n)) are not closed under row catenation, row +, and projection, where L( m, n)= f( m)+ g( n) (resp., L( m, n)= f( m)× g( n)).

Space Complexity in Propositional Calculus

We study space complexity in the framework of propositional proofs. We consider a natural model analogous to Turing machines with a read-only input tape and such popular propositional proof systems as resolution, polynomial calculus, and Frege systems. We propose two different space measures, corresponding to the maximal number of bits, and clauses/monomials that need to be kept in the memory simultaneously. We prove a number of lower and upper bounds in these models, as well as some structural results concerning the clause space for resolution and Frege systems.

CLOSURE PROPERTY OF SPACE-BOUNDED TWO-DIMENSIONAL ALTERNATING TURING MACHINES, PUSHDOWN AUTOMATA, AND COUNTER AUTOMATA

This paper investigates closure property of the classes of sets accepted by space-bounded two-dimensional alternating Turing machines (2-atm's) and space-bounded two-dimensional alternating pushdown automata (2-apda's), and space-bounded two-dimensional alternating counter automata (2-aca's). Let L(m, n): N2 → N (N denotes the set of all positive integers) be a function with two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). We show that (i) for any function f(m) = o( log m) (resp. f(m) = o( log m/ log log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional Turing machine (2-Tm) (resp. two-dimensional pushdown automaton (2-pda)), the class of sets accepted by L(m,n) space-bounded 2-atm's (2-apda's) is not closed under row catenation, row + or projection, and (ii) for any function f(m) = o(m/ log ) (resp. for any function f(m) such that log f(m) = o( log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional counter automaton (2-ca), the class of sets accepted by L(m, n) space-bounded 2-aca's is not closed under row catenation, row + or projection, where L(m, n) = f(m) + g(n) (resp. L(m, n) = f(m) × g(n)).

Using acceptors as transducers

We wish to use a given nondeterministic two-way multi-tape acceptor as a transducer by supplying the contents for only some of its input tapes, and asking it to generate the missing contents for the other tapes. We provide here an algorithm for assuring beforehand that this transduction always results in a finite set of answers. We also develop an algorithm for evaluating these answers whenever the previous algorithm indicated their finiteness. Furthermore, our algorithms can be used for speeding up the simulation of these acceptors even when not used as transducers.

MSO definable string transductions and two-way finite-state transducers

We extend a classic result of Büchi, Elgot, and Trakhtenbrot: MSO definable string transductions i.e., string-to-string functions that are definable by an interpretation using monadic second-order (MSO) logic, are exactly those realized by deterministic two-way finite-state transducers, i.e., finite-state automata with a two-way input tape and a one-way output tape. Consequently, the equivalence of two mso definable string transductions is decidable. In the nondeterministic case however, MSO definable string tranductions, i.e., binary relations on strings that are mso definable by an interpretation with parameters, are incomparable to those realized by nondeterministic two-way finite-state transducers. This is a motivation to look for another machine model, and we show that both classes of MSO definable string transductions are characterized in terms of Hennie machines, i.e., two-way finite-state transducers that are allowed to rewrite their input tape, but may visit each position of their input only a bounded number of times.

Optimal Simulations between Unary Automata

We consider the problem of computing the costs---{ in terms of states---of optimal simulations between different kinds of finite automata recognizing unary languages. Our main result is a tight simulation of unary n-state two-way nondeterministic automata by $O({{\rm e}^{\sqrt{{n}\ln{n}}}})$-state one-way deterministic automata. In addition, we show that, given a unary n-state two-way nondeterministic automaton, one can construct an equivalent O(n2 )-state two-way nondeterministic automaton performing both input head reversals and nondeterministic choices only at the ends of the input tape. Further results on simulating unary one-way alternating finite automata are also discussed.

A NOTE ON THREE-WAY TWO-DIMENSIONAL PROBABILISTIC TURING MACHINES

This paper introduces a three-way two-dimensional probabilistic Turing machine (tr2-ptm), and investigates several properties of the machine. The tr2-ptm is a two-dimensional probabilistic Turing machine (2-ptm) whose input head can only move left, right, or down, but not up. Let 2-ptms (resp. tr2-ptms) denote a 2-ptm (resp. tr2-ptm) whose input tape is restricted to square ones, and let 2-PTMs(S(n)) (resp. TR2-PTMs(S(n))) denote the class of sets recognized by S(n) space-bounded 2-ptms's (resp. tr2-ptms's) with error probability less than ½, where S(n): N→N is a function of one variable n (= the side-length of input tapes). Let TR2-PTM(L(m,n)) denote the class of sets recognized by L(m,n) space-bounded tr2-ptm's with error probability less than ½, where L(m,n): N2→N is a function of two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). The main results of this paper are: (1) 2-NFAs - TR2-PTMs(S(n))≠ϕ for any S(n)=o(log n), where 2-NFAs denotes the class of sets of square tapes accepted by two-dimensional nondeterministic finite automata, (2) TR2-PTMsS(n)[Formula: see text]2-PTMs(S(n)) for any S(n)=o(log n), and (3) for any function g(n)=o(log n) (resp. g(n)=o(log n/log log n)) and any monotonic nondecreasing function f(m) which can be constructed by some one-dimensional deterministic Turing machine, TR2-PTM(f(m)+g(n)) (resp. TR2-PTM(f(m)×g(n))) is not closed under column catenation, column closure, and projection. Additionally, we show that two-dimensional nondeterministic finite automata are equivalent to two-dimensional probabilistic finite automata with one-sided error in accepting power.

A variant of inductive counting

We present a new version of the inductive counting, accepting the complement of an NSPACE(s(n)) language nondeterministically in space O(s(n)) , independent of whether s(n)⩾ log n , but using an additional “one-way pebble” – a movable marker placed on the input tape. This reduces the space used by inductive counting to log n+ O(s(n)) bits on the binary work tape and gives the weakest known nondeterministic device accepting a co− NSPACE( o( log n)) language.

Parallel RAMs with owned global memory and deterministic context-free language recognition

We identify and study a natural and frequently occurring subclass of Concurrent Read, Exclusive Write Parallel Random Access Machines (CREW-PRAMs). Called Concurrent Read, Owner Write, or CROW-PRAMS, these are machines in which each global memory location is assigned a unique “owner” processor, which is the only processor allowed to write into it. Considering the difficulties that would be involved in physically realizinga full CREW-PRAM model and demonstrate its stability under several definitional changes. Second, we precisely characterize the power of the CROW-PRAM by showing that the class of languages recognizable by it in time O (log n) (and implicity with a polynomial number of processors) is exactly the class LOGDCFL of languages log space reducible to deterministic context-free languages. Third, using the same basic machinery, we show that the recognition problem for deterministic context-free languages can be solved quickly on a deterministic auxilliary pushdown automation having random access to its input tape, a log n space work tape, and pushdown store of small maximum height. For example, time O ( n 1 + ε ) is achievable with pushdown height O (log 2 n ). These result extend and unify work of von Braunmöhl, Cook, Mehlhorn, and Verbeek, Klein and Reif; and Rytter.