Input Tape Research Articles

Алгоритм валидации ограниченно-детерминированного поведения передатчика в канале частичного стирания

In the previous papers, the notions of a partial erasure structure and a partial erasure channel have been introduced. A partial erasure structure is a triplet consisting of an alphabet A, a family of partitions of the alphabet and a set of probabilities assigned to the partitions. A partial erasure channel functions as follows. Alice sends Bob a symbol a ∈ A, but Bob receives only partial information about the symbol. Bob only knows which partition has been choosen and which class of the partition the symbol belongs to. The set of pairs consisting of a partition and a class of the partition, i.e., the signals Bob can receive, is denoted as Bob’s alphabet B. There is a natural correspondence between the characters sent by Alice and received by Bob. For symbols a ∈ A and b ∈ B, a 7→ b is assumed to hold if there exists a partition T in the partial erasure structure such that b = (πT (a), T). The correspondence is also generalized to the words. We assume that a1 . . . an 7→ b1 . . . bn holds, if ai 7→ bi holds for all i = 1 . . . n. Suppose that Alice has an input tape from which she reads symbols of some finite alphabet S. Bob has an output tape, on which he can print symbols of alphabet S and also specially reserved (not in S) erasure symbol “*”. Thus, a problem of transmitting information via the descibed channel (i.e., a problem of construction of Alice’s and Bob’s behaviour) arises. In the previous papers of the author, a formal model of interaction between a transmitter (Alice) and a receiver (Bob), called a protocol, is presented. A protocol is a pair of functions (F, G), where F : S ∗ → A∗ is the Alice behaviour function and G : B∗ → S ∪ {∗,Λ} is the Bob behaviour function. Suppose that Alice has just read another symbol s ∈ S from the input tape, having previously read the word ˆs = s1, . . . , sm. Then we assume that during the following |F(ˆss)| steps Alice sends the word F(ˆss) symbol by symbol via the channel of partial erasure. Bob receives the symbol b ∈ B on each step. After receiving the symbol, he has to decide what to print on the output tape. He can print a symbol of the alphabet S, or the erasure symbol “*”, or nothing. Bob’s decision is determined by the function G. The protocol (F, G) is said to be valid, if Bob prints on the output tape the same content what originally is on the input tape, while, perhaps, with the replacement of the significant symbols (i.e., the symbols of the alphabet S) to the erasure symbol “*”. Among all Alice’s behavior functions, a class of correct functions is defined. A function F is said to be correct, if F(Λ)F(s 1 1 )F(s 1 1 s 1 2 ). . . F(s 1 1 . . . s1 n ) 7→ β1, F(Λ)F(s 2 1 )F(s 2 1 s 2 2 ). . . F(s 2 1 . . . s2 m) 7→ β2 and β1 is a prefix to β2, where β1, β2 ∈ B∗ . The question is investigated, for which functions of Alice’s behavior F there exists a function of Bob’s behavior G such that the pair (F, G) is a correct protocol. The theorem is proved that for this it is necessary and sufficient that F belongs to the class of correct functions. This paper presents an algorithm for verifying that Alice’s behavior function F belongs to the class of correct functions. The complexity of the algorithm is O(L|Q| 3 |S| 4 ), where |Q| is the number of states of the automaton representing the function F, L is the maximum length of the word Alice throws out, |S| is the number of symbols of the alphabet S.

Two-Way Non-Uniform Finite Automata

We consider two-tape automata where one tape contains the input word [Formula: see text] and the other an advice string [Formula: see text]. Such an automaton recognizes a language [Formula: see text] if there is an advice function for which every word on the input tape is correctly classified. This model has been introduced by Küçük et al. with the aim of modeling non-uniform computations on finite automata. So far, most results concerned automata with 1-way tapes, both for the input and the advice. First, we show that making just one of the tapes 2-way already increases the model’s power. Then we consider the case where both tapes are 2-way, which can also be viewed as a restricted version of the non-uniform families of automata used by Ibarra and Ravikumar to define the class NUDSPACE . We show this restriction to be not too significant since, e.g., [Formula: see text]DFA/poly), i.e., the languages recognized by automata with 2-way input and advice tape with polynomial advice, equals [Formula: see text]. Hence, we can show that many interesting problems concerning the state complexity of families of automata carry over to the problems concerning the advice size of non-uniform automata. In particular, the question whether there can be a more than polynomial gap in advice between determinism and nondeterminism is of great interest: The existence of a language that is recognized by a 2-way NFA with a constant number of advice heads and polynomial (resp. logarithmic) advice, while a corresponding 2-head DFA needs exponential (resp. polynomial) advice, would imply [Formula: see text] (resp. [Formula: see text]). We show that for advice of size [Formula: see text] there is no gap between determinism and nondeterminism, and that the gap is at most exponential.

Input-Driven Double-Head Pushdown Automata

We introduce and study input-driven deterministic and nondeterministic double-head pushdown automata. A double-head pushdown automaton is a slight generalization of an ordinary pushdown automaton working with two input heads that move in opposite directions on the common input tape. In every step one head is moved and the automaton decides on acceptance if the heads meet. Demanding the automaton to work input-driven it is required that every input symbol uniquely defines the action on the pushdown store (push, pop, state change). Normally this is modeled by a partition of the input alphabet and is called a signature. Since our automaton model works with two heads either both heads respect the same signature or each head owes its own signature. This results in two variants of input-driven double-head pushdown automata. The induced language families on input-driven double-head pushdown automata are studied from the perspectives of their language describing capability, their closure properties, and decision problems.

Physical limitations of work extraction from temporal correlations.

Recently proposed information-exploiting systems extract work from a single heat bath by using temporal correlations on an input tape. We study how enforcing time-continuous dynamics, which is necessary to ensure that the device is physically realizable, constrains possible designs and drastically diminishes efficiency. We show that these problems can be circumvented by means of applying an external, time-varying protocol, which turns the device from a "passive," free-running machine into an "actively" driven one.

Finite-state independence and normal sequences

We consider the previously defined notion of finite-state independence and we focus specifically on normal words. We characterize finite-state independence of normal words in three different ways, using three different kinds of asynchronous deterministic finite automata with two input tapes containing infinite words. Based on one of the characterizations we give an algorithm to construct a pair of finite-state independent normal words.

On counting functions and slenderness of languages

We study counting-regular languages — these are languages L for which there is a regular language L′ such that the number of strings of length n in L and L′ are the same for all n. We show that the languages accepted by unambiguous nondeterministic Turing machines with a one-way read-only input tape and a reversal-bounded worktape are counting-regular. Many one-way acceptors are a special case of this model, such as reversal-bounded deterministic pushdown automata, reversal-bounded deterministic queue automata, and many others, and therefore all languages accepted by these models are counting-regular. This result is the best possible in the sense that the claim does not hold for either 2-ambiguous NPDA's, unambiguous NPDA's with no reversal-bound, and other models.We also study closure properties of counting-regular languages, and we study decidability problems in regards to counting-regularity. For example, it is shown that the counting-regularity of even some restricted subclasses of NPDA's is undecidable. Lastly, k-slender languages — where there are at most k words of any length — are also studied. Amongst other results, it is shown that it is decidable whether a language in any semilinear full trio is k-slender.

Selection and Sorting in the “Restore” Model

We consider the classical selection and sorting problems in a model where the initial permutation of the input has to be restored after completing thecomputation. Such algorithms are useful for designing space-efficient algorithms, when one encounters subproblems that have to be solved by subroutines. It is important that these subroutines leave the array in its original state after they finish so that the computation can be properly resumed. Algorithms in this model can also be relevant for saving communication time, in case the data is distributed among several machines and would need to be copied to further machines for execution of the subroutine. Although the requirement of the restoration is stringent compared to the classicalversions of the problems, this model is more relaxed than a read-only memory where the input elements are not allowed to be moved within the input array. We first show that for a sequence of n integers, selection (finding the median or more generally the k -th smallest element for a given k ) can be done in O ( n ) time using O (lg n ) words 1 of extra space in this model. In contrast, no linear-time selection algorithm is known that uses polylogarithmic space in the read-only memory model. For sorting n integers in this model, we first present an O ( n lg n )-time algorithm using O (lg n ) words of extra space that outputs (in a write only tape) the given sequence in sorted order while restoring the order of the original input in the input tape. When the universe size U is polynomial in n , we give a faster O ( n )-time algorithm (analogous to radix sort) that uses O ( n ε ) words of extra space for an arbitrarily small constant ε > 0. More generally, we show how to match the time bound of any word-RAM integer sorting algorithms using O ( n ε ) words of extra space. In sharp contrast, there is an Ω ( n 2 / S )-time lower bound for integer sorting using O ( S ) bits of space in the read-only memory model. Extension of our results to arbitrary input types beyond integers is not possible: for “indivisible” input elements, we can prove the same Ω ( n 2 / S ) lower bound for sorting in our model. We also describe space-efficient algorithms to count the number of inversions in a given sequence in this model. En route, we develop linear-time in-place algorithms to extract leading bits of the input array and to compress and decompress strings with low entropy; these techniques may be of independent interest.

Watson-Crick pushdown automata

A multi-head 1-way pushdown automaton with $k$ heads is a pushdown automaton with $k$ 1-way read heads on the input tape and a stack. It was previously shown that the deterministic variant of the model cannot accept all the context free languages. In this paper, we introduce a 2-tape, 2-head model namely Watson-Crick pushdown automata where the content of the second tape is determined using a complementarity relation, similar to Watson-Crick automata. We show computational powers of nondeterministic two-head pushdown automata and nondeterministic Watson-Crick pushdown automata are same. Moreover, deterministic Watson-Crick pushdown automata can accept all the context free languages.

Input-Driven Double-Head Pushdown Automata

We introduce and study input-driven deterministic and nondeterministic double-head pushdown automata. A double-head pushdown automaton is a slight generalization of an ordinary pushdown automaton working with two input heads that move in opposite directions on the common input tape. In every step one head is moved and the automaton decides on acceptance if the heads meet. Demanding the automaton to work input-driven it is required that every input symbol uniquely defines the action on the pushdown store (push, pop, state change). Normally this is modeled by a partition of the input alphabet and is called a signature. Since our automaton model works with two heads either both heads respect the same signature or each head owes its own signature. This results in two variants of input-driven double-head pushdown automata. The induced language families on input-driven double-head pushdown automata are studied from the perspectives of their language describing capability, their closure properties, and decision problems.

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.

The Analysis of Influence of Sheet Properties on the Ironing Process of Thin-Walled Cylindrical Shell Products from Aluminum Alloys

In industrial practice, where the amount of products produced in a minute is calculated in hundreds of units, the ironing process of thin-walled cylindrical shell products is carried out with very high speed. In the multi-stage forming process it is important to ensure the stability of parameters of the raw material. Properties of input sheets are defined in international standards, but after the rolling process the properties of input tapes may be various while still in the acceptable tolerance. Differentiation of properties of sheets has a significant impact on the progress of the examined process. The multi-stage forming process requires raw material with a high susceptibility to deformation and an adequate level of plasticity margin. In this study tapes of equal thickness and similar chemical composition from various suppliers have been analyzed. Selected aluminum alloys sheet are currently used in Europe in production of beverage packaging. On the basis of uniaxial tension test results, plasticity margin for a selected tape has been determined. Deformation range and plasticity margin have been analyzed in relation to the stability of production process Defining the correlation between identified strength parameters and the stability of production line allowed to examine the impact of plasticity margin for effectiveness and efficiency of the process. In mass production, each break in machine work results in loss of millions of units. The analysis allowed to compare the value of plasticity indicators of the raw tape deformability in the complex process of forming thin-walled cylindrical shell products and influence for increase the efficiency and reduction of defects.

FINITE AUTOMATA WITH ADVICE TAPES

We define a model of advised computation by finite automata where the advice is provided on a separate tape. We consider several variants of the model where the advice is deterministic or randomized, the input tape head is allowed real-time, one-way, or two-way access, and the automaton is classical or quantum. We prove several separation results among these variants, demonstrate an infinite hierarchy of language classes recognized by automata with increasing advice lengths, and establish the relationships between this and the previously studied ways of providing advice to finite automata.

Modeling artificial biological cell in fine-grained structure

Two fine-grained models of artificial biological cells are presented. They can be used as elements of computing devices that mimic properties of living organisms: growth, self-reproduction, and self-healing. The models are based on the parallel substitution algorithm that is a spatial model for representing fine-grained parallel algorithms and architectures. An artificial biological cell is constructed based on a genome fed to the input tape. The result is a model of artificial biological cell that contains the phenotype as a set of fixed data and the genotype as a set of mobile data.

Two-way automata making choices only at the endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2nfas) by two-way deterministic automata (2dfas) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2dfas (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2nfas. Here we use an opposite approach, increasing the power of 2dfas to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2dfas. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2nfas and 2dfas can be obtained only for unrestricted 2nfas using capabilities beyond the proposed new model.As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2dfas would imply L≠NL. In the same way, the alternating version of these machines is related to L=?NL=?P, the classical computational complexity problems.

Self-Replicating Structure as an Artificial Multicellular Organism

Two models of an artificial biological cell are constructed in a fine-grained structure in the form of a self-reproducing loop. The models are developed on the basis of a parallel substitution algorithm, i.e., a system for modeling spatial fine-grained parallel algorithms and architectures. A model of a biological cell is constructed from a one-dimensional artificial genome written on the input tape. The proposed model contains a phenotype in the form of a set of fixed data and a genotype in the form of a set of mobile data. Such a cell can be an element of an artificial multicellular organism that simulates the following properties of living organisms: growth, self-reproduction, and self-repair.

Hierarchy Based on Configuration-Reader about k-Neighborhood Template A-Type Three-Dimensional Bounded Cellular Acceptor

Blum and Hewitt first proposed two-dimensional automata as computational models of two-dimensional pattern processing－two-dimensional finite automata and marker automata, and investigated their pattern recognition abilities in 1967. Since then, many researchers in this field have investigated the properties of automata on two- or three-dimensional tapes. On the other hand, the question of whether or not processing four-dimensional digital patterns is more difficult than processing two- or three-dimensional ones is of great interest from both theoretical and practical standpoints. Thus, the study of four-dimensional automata as the computational models of four-dimensional pattern processing has been meaningful. From this point of view, we are interested in four-dimensional computational models, In this paper, we introduce a new four-dimensional computational model, k-neighborhood template A-type three-dimensional bounded cellular acceptor on four-dimensional input tapes, and investigate about hierarchy based on configuration-reader about this model.

Hierarchy Based on Neighborhood Template about k-Neighborhood Template A-Type Three-Dimensional Bounded Cellular Acceptor

Recently, due to the advance in dynamic image processing, computer animation, augmented reality (AR), and so forth, it has become increasingly apparent that the study of four-dimensional pattern processing (three-dimensional pattern processing with time axis) should be very important. Thus, the study of four-dimensional automata as the computational model of four-dimensional pattern processing has been meaningful. From this point of view, we first introduced a four-dimensional automaton in 2002. In the multi-dimensional pattern processing, designers often use a strategy whereby features are extracted by projecting high-dimensional space on low-dimensional space. In this paper, from this viewpoint, we introduce a new computational model, k-neighborhood template A-type three-dimensional bounded cellular acceptor (abbreviated as A -3BCA(k)) on four-dimensional input tapes, and discuss hierarchy based on neighborhood template about A-3BCA(k).

Cellular automata model of an artificial biological cell in the form of a self-replicating structure

Two cellular automata models of an artificial biological cell, which can be elements of computing devices that mimic the properties of living organisms (growth, self-reproduction, and self-repair), are presented. The models are based on the parallel substitution algorithm, which is a spatial model for representing fine-grained parallel algorithms and architectures. An artificial biological cell is constructed from a genome applied to the input tape of the cellular automata structure. The result is a model of an artificial biological cell, which contains a phenotype as a set of fixed data and a genotype as a set of mobile data.

Factoring and testing primes in small space

We discuss how much space is sufficient to decide whether a unary given number n is a prime. We show that O(log log n) space is sufficient for a deterministic Turing machine, if it is equipped with an additional pebble movable along the input tape, and also for an alternating machine, if the space restriction applies only to its accepting computation subtrees. In other words, the language is a prime is in pebble–DSPACE(log log n) and also in accept–ASPACE(log log n). Moreover, if the given n is composite, such machines are able to find a divisor of n. Since O(log log n) space is too small to write down a divisor, which might require Ω(log n) bits, the witness divisor is indicated by the input head position at the moment when the machine halts.

Tradeoff lower lounds for stack machines

A space-bounded Stack Machine is a regular Turing Machine with a read-only input tape, several space-bounded read-write work tapes, and an unbounded stack. Stack Machines with a logarithmic space bound have been connected to other classical models of computation, such as polynomial-time Turing Machines (P) (Cook in J Assoc Comput Mach 18:4---18, 1971) and polynomial size, polylogarithmic depth, bounded fan-in circuits (NC) e.g., Borodin et al. (SIAM J Comput 18, 1989). In this paper, we present significant new lower bounds and techniques for Stack Machines. This comes in the form of a trade-off lower bound between space and number of passes over the input tape. Specifically, we give an explicit permuted inner product function such that any Stack Machine computing this function requires either $${\Omega (N^{1/4 - \epsilon})}$$ space or $${\Omega (N^{1/4 - \epsilon})}$$ number of passes for every constant $${\epsilon > 0}$$ , where N is the input size. In the case of logarithmic space Stack Machines, this yields an unconditional $${\Omega (N^{1/4 - \epsilon})}$$ lower bound for the number of passes. To put this result in perspective, we note that Stack Machines with logarithmic space and a single pass over the input can compute Parity, Majority, as well as certain languages outside NC. The latter follows from Allender (J Assoc Comput Mach 36:912---928, 1989), conditional on the widely believed complexity assumption that PSPACE $${\subsetneq}$$ EXP. Our technique is a novel communication complexity reduction, thereby extending the already wide range of models of computation for which communication complexity can be used to obtain lower bounds. Informally, we show that a k-player number-in-hand (NIH) communication protocol for a base function f can efficiently simulate a space- and pass-bounded Stack Machine for a related function F, which consists of several "permuted" instances of f, bundled together by a combining function h. Trade-off lower bounds for Stack Machines then follow from known communication complexity lower bounds. The framework for this reduction was given by Beame & Huynh-Ngoc (2008), who used it to obtain similar trade-off lower bounds for Turing Machines with a constant number of pass-bounded external tapes. We also prove that the latter cannot efficiently simulate Stack Machines, conditional on the complexity assumption that E $${\not \subset}$$ PSPACE. It is the treatment of an unbounded stack which constitutes the main technical novelty in our communication complexity reduction.