Input Tape Research Articles

The determination of the value of Rado’s noncomputable function Σ(𝑘) for four-state Turing machines

The well-defined but noncomputable functions Σ ( k ) \Sigma (k) and S ( k ) S(k) given by T. Rado as the "score" and "shift number" for the k-state Turing machine "Busy Beaver Game" were previously known only for k ⩽ 3 k \leqslant 3 . The largest known lower bounds yielding the relations Σ ( 4 ) ⩾ 13 \Sigma (4) \geqslant 13 and S ( 4 ) ⩾ 107 S(4) \geqslant 107 , reported by this author, supported the conjecture that these lower bounds are the actual particular values of the functions for k = 4 k = 4 . The four-state case has previously been reduced to solving the blank input tape halting problem of only 5,820 individual machines. In this final stage of the k = 4 k = 4 case, one appears to move into a heuristic level of higher order where it is necessary to treat each machine as representing a distinct theorem. The remaining set consists of two primary classes in which a machine and its tape are viewed as the representation of a growing string of cellular automata. The proof techniques, embodied in programs, are entirely heuristic, while the inductive proofs, once established by the computer, are completely rigorous and become the key to the proof of the new and original mathematical results: Σ ( 4 ) = 13 \Sigma (4) = 13 and S ( 4 ) = 107 S(4) = 107 .

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) n 2 log n ( n 3) space is necessary and sufficient for deterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's; (2) n 2 ( n 2) space is necessary and sufficient for nondeterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's.

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.

Selection and sorting with limited storage

When selecting from, or sorting, a file stored on a read-only tape and the internal storage is rather limited, several passes of the input tape may be required. We study the relation between the amount of internal storage available and the number of passes required to select the Kth highest of N inputs. We show, for example, that to find the median in two passes requires at least ω(N 1 2 ) and at most O(N 1 2 log N) internal storage. For probabilistic methods, θ(N 1 2 ) internal storage is necessary and sufficient for a single pass method which finds the median with arbitrarily high probability.

Lower bounds on the size of sweeping automata

A sweeping automaton is a deterministic two-way finite automaton which only changes head direction at the ends of the input tape. It is shown that for every n, there is a language which is accepted by an n-state nondeterministic one-way finite automaton, yet which is not accepted by any sweeping automaton with fewer than 2 n states.

Two-dimensional automata with rotated inputs

This paper introduces a new type of automata on a two-dimensional tape, which decides acceptance or rejection of an input tape x by scanning the tape x from various sides by various automata. The accepting power of such an automaton is investigated. This paper mainly concentrates on investigating the accepting power of two-dimensional automata which consist of four parallel/sequential array automata, say M 1, M 2, M 3, and M 4, and which accept an input tape x if and only if x, x R , ( x R ) R , and (( x R ) R ) R are accepted by M 1, M 2, M 3, and M 4, respectively, where for any tape y, y R is the tape obtained by rotating y clockwise 90°.

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.

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.

Three-way two-dimensional multicounter automata

In this paper, we introduce a three-way two-dimensional multicounter automaton which can be considered as an extension of an on-line multicounter machine to two dimensions, and investigate some of its properties. We first investigate the difference between the accepting powers of three-way and four-way two-dimensional multicounter automata, and between the accepting powers of nondeterministic and deterministic three-way two-dimensional multicounter automata. We then show that hierarchies can be obtained by varying the number of counters or the amount of space allowed, for classes of sets accepted by three-way two-dimensional multicounter automata. We finally examine closure properties under several operations. Input tapes in this paper are restricted to square ones.

Closure properties of three-way and four-way tape-bounded two-dimensional turing machines

This paper investigates closure properties (under row and column catenations, row and column closures, row and column cyclic closures, and projection) of the classes of sets accepted by four-way and three-way tape-bounded two-dimensional Turing machines whose input tapes are not restricted to square ones.

Two-dimensional multipass on-line tessellation acceptors

The purpose of this paper is to propose a new type of acceptor on a two-dimensional tape, called a “two-dimensional multipass on-line tessellation acceptor” (denoted by “2-mpota”), and to examine some properties of the 2-mpota. It is shown that (1) nondeterministic 2-mpota's with the property that the sizes of the input and output tapes in each pass are the same, are equivalent to nondeterministic rectangular array bounded automata, and (2) nondeterministic 2-mpota's which are permitted to have an enlarged output tape in each pass are equivalent to nondeterministic Turing rectangular array automata.

Complexity-based induction systems: Comparisons and convergence theorems

In 1964 the author proposed as an explication of {\em a priori} probability the probability measure induced on output strings by a universal Turing machine with unidirectional output tape and a randomly coded unidirectional input tape. Levin has shown that if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tilde{P}'_{M}(x)</tex> is an unnormalized form of this measure, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P(x)</tex> is any computable probability measure on strings, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> , then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tilde{P}'_{M}\geqCP(x)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> is a constant independent of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> . The corresponding result for the normalized form of this measure, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P'_{M}</tex> , is directly derivable from Willis' probability measures on nonuniversal machines. If the conditional probabilities of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P'_{M}</tex> are used to approximate those of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> , then the expected value of the total squared error in these conditional probabilities is bounded by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-(1/2) \ln C</tex> . With this error criterion, and when used as the basis of a universal gambling scheme, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P'_{M}</tex> is superior to Cover's measure <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b\ast</tex> . When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H\ast\equiv -\log_{2} P'_{M}</tex> is used to define the entropy of a rmite sequence, the equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H\ast(x,y)= H\ast(x)+H^{\ast}_{x}(y)</tex> holds exactly, in contrast to Chaitin's entropy definition, which has a nonvanishing error term in this equation.

ω-Computations on Turing machines

The paper develops the theory of Turing machines as recognizers of infinite (ω-type) input tapes. Various models of ω-type Turing acceptors are considered, varying mainly in their mechanism for recognizing ω-tapes. A comparative study of the models is made. It is shown that regardless of the ω-recognition model considered, non-deterministic ω-Turing acceptors are strictly more powerful than their deterministic counterparts. Canonical forms are obtained for each of the ω-Turing acceptor models. The corresponding families of ω-sets are studied; normal forms and algebraic characterizations are derived for each family.

A note on two-dimensional finite automata

During the past about thirty-five years, many types of twoor three-dimensional automata have been proposed and investigated the properties of them as the computational model of pattern processing. On the other hand, recently, due to the advances in many application areas such as computer animation, motion image processing, and so on, the study of three-dimensional pattern processing with the time axis has been of crucial importance. Thus, we think that it is very useful for analyzing computation of three-dimensional pattern processing with the time axis to explicate the properties of four-dimensional automata. In this paper, we propose a four-dimensional Turing machine and a four-dimensional finite automaton, and show the space complexities necessary and sufficient for seven-way four-dimensional Turing machines to simulate four-dimensional finite automata, where each sidelength of each input tape of these automata is equivalent. Key-Words: Turing machine, finite automaton, on-line tessellation acceptor, four-dimensional input tape, determinism, nondeterminism, space-bound, configuration, computation, space complexity

Determining relative performance in educational evaluation

Determining relative performance in educational evaluation requires a statistical technique for analyzing test results which considers initial individual differences. One such field-tested statistical method (Klein & Erlich, Note I) is based on regression analysis and construction of a band around the regression line to account for errors of estimation of expected scores. In this method the relative performance of the student is classified as being or expectancy level, according to whether his score falls within, or the regression band lines. The bandwidth is a function of the standard error of estimate and the consideration of Type I and Type II errors. Defining bandwidth should depend on the nature of the data and the practical scale used for scoring performance. Once bandwidth is defined and the relative performance of each observation determined, the interpretation of the results should consider the expectancy level as well as the distance of the scores from the band line-defined hereafter as the band residual score. The Program. The purpose of the program is to: classify performance scores as below, at, or above expectancy levels; report the actual, expected, residual, and band residual scores for each observation; and describe for each expectancy level the number of observations, the mean score, and the mean band residual score. Program Calculations. At Step I, the program calculates an expected score, a residual score, an expectancy level, and a band residual score for each observation. At Step 2, the number and percentage of observations falling in each level of expectancy are calculated along with the mean score in each expectancy level, and the mean band residual score in the above and below expectancy levels. Input. The program input includes the following six cards: title card (Columns 1-80); name of the unit of analysis (e.g., student, class, school) (Columns 1-16); total number of variables (maximum 10 variables), numerical order of dependent variable, and number of input tape (312); data input format within parentheses (Columns 1-80); values of the constant term and of the independent variables' (predictors) regression coefficients (IOF8.0); standard error of estimate (SE) and a multiple of SE to define the width of half band (2F8.0). Information for the last two cards can be defined by any regression program available. Output. The program provides the following output: (l) Summary table for each observation, including actual score, expected score, residual score, expectancy score, and band residual score. (2) Summary table that describes, for each expectancy level, the number and percentage of observations, the mean actual score, and the mean band residual scores (only for above and below expectancy levels). Computer and Language. The program has been run on CDC 6600 and IBM 360/91 computers using FORTRAN IV; core requirement is lOOK; execution time is less than 10 sec. Availability. The source listing of the program, its documentation, and a sample application may be obtained at no cost from Oded Erlich, School of Education, Tel-Aviv University, Tel-Aviv, Israel.

Theory of ω-languagesI: Characterizations of ω-context-free languages

An ω-language is a set consisting of ω-length strings over some alphabet; an ω-automaton is a device capable of recognizing ω-length input tapes. This paper introduces the basic notions concerning generation of ω-languages by means of ω-grammars and their recognition by ω-automata with various recognition modes. Attention is focused on ω-CFL's, the ω-languages generated by ω-context-free grammars. Two main characterizations of the family of ω-CFL's are obtained. (a) An ω-language is an ω-CFL if and only if it can be represented as Ui=1/nUiVi/ω, where for each i=1,…, n, Ui and Vi are context-free languages. (b) ω-CFL's are precisely the ω-languages recognized by ω-pushdown automata of three distinct types. A few other characterizations and normal forms for the ω-CFL's are obtained and several decidability results are established.

Design Considerations for Electrodeposition onto a High Resistance Electroless Flash in a Continuous Plater

The buildup of copper by electrodeposition onto a high resistance electroless flash has been studied in a roll‐to‐roll continuous plater employing copper fluoborate at ambient temperature. Input to the plater was circuit‐patterned polyimide tape having an initial deposit which ranged from 2.4 μin. electroless copper through 146 μin. electroless plus electroplated copper. Buildup of copper through one (or more) plating loops was traced by atomic absorption analysis of millimeter‐sized pattern sections cut from the tape; this analysis followed an abrupt interruption in steady‐state plating and a rapid ejection of the tape from the plater. By this technique, copper distributions along and across the tape were determined. Thickness buildups to 400 μin. at tape speeds of 2.3 and 3.1 ft/min are reported. The copper distributions have been used to compute maximum and average cathodic current densities, and , respectively, for selected plater configurations and input copper thicknesses. Significant asymmetries in local plating rate due to the high input tape resistance were observed and the ratio was as high as in some cases. Since excessive cathodic current densities can degrade performance of the plated product, the regulation of within definable limits is a critical aspect of plater design. A correlation of to input thickness of the electroless flash and to applied plating currents is presented and its implications with respect to plater design examined.

Spatial complexity of recognition of context-free languages

For the recognition of context-free languages (CF-languages), use is made of Turing machines with two tapes: the input tape and the operational tape. Let n be the length of the input tape. As a measure of complexity we choose the length of the operational tape L(n). We denote by M (respectively, N) the class of languages which are recognized on deterministic (nondeterministic) Turing machines with L(n) ~< log2n. It is known that all CF-languages enter in class N [4, 7] and are recognized by deterministic Turing machines with L(n) ~< (log2n) 2 [3], while class M contains certain subclasses of CF-languages (for example, the Dyck language, bounded CFlanguages [7]). It is also noted in [7] that all the examples of CF-languages known to the authors belong to class M. The following two problems are well known:

Abstract: Strategies in lithography

The functional complexity of integrated circuits has been doubling each year for many years and can be expected to continue increasing at that rate for the forseeable future. The feasibility of larger devices with smaller features is well illustrated by papers in this issue. The motivation to continue this trend is the high cost of interconnecting smaller units and the repeated denonstration that the more complex device soon costs less than the multiple‐chip equivalent. The smaller feature devices also provide faster response and consume less power.Optical systems: Optical techniques for fabricating devices are generally limited to ≳5 μm linewidths in practical production by the diffraction effects. While contact printing can achieve <2 μm resolutions, the defects caused by contacting the wafer limit the yields, especially when device area is large. Projection printing can give 2‐μm features when small areas are exposed at one time with individual focus correction.Alternate wavelengths: Electron beams or ...

Marker automata

This paper deals with finite automata augmented with markers which the automata can move about on their input tapes. The concept of augmenting markers to automata was first introduced by Blum and Hewitt [1] in two-dimensional automata. Kreider, Ritchie, and Springsteel [6, 7, 8, 12] investigated recognition of context-free languages by (one-dimensional) automata with markers. In this paper, we investigate some fundamental properties of marker automata and study their relationships to other types of automata and languages. The main result in this paper is the establishment of an infinite hierarchy of languages recognizable by deterministic and deterministic, halting marker automata. It also turns out that, because of the equivalence of finite marker automata and multi-head automata, the study of three-marker automata becomes very interesting due to results of Hartmanis [4] and Savitch [10].