Showing papers in "Journal of the ACM in 1966"
TL;DR: The relative merits of performing local operations on ~ digitized picture in parallel or sequentially are discussed and some applications of the connected component and distance functions are presented.
Abstract: The relative merits of performing local operations on ~ digitized picture in parallel or sequentially are discussed. Sequential local operations are described which l~bel the connected components of a given subset of the picture and compute u \"distance\" from every picture element to the subset. In terms of the \"distance\" function, ~ \"skeleton\" subset is defined which, in a certain sense, minimally determines the original subset. Some applications of the connected component and distance functions are also presented.
1,707 citations
TL;DR: An application to the problem of defining a patternless sequence is proposed in terms of the concepts here developed to study the use of Turing machines for calculating finite binary sequences.
Abstract: The use of Turing machines for calculating finite binary sequences is studied from the point of view of information theory and the theory of recursive functions. Various results are obtained concerning the number of instructions in programs. A modified form of Turing machine is studied from the same point of view. An application to the problem of defining a patternless sequence is proposed in terms of the concepts here developed.
1,072 citations
TL;DR: In this report, certain properties of context-free (CF or type 2) Grammars are investigated, like that of Chomsky, and it is shown that this type of grammar is essentially stronger than type 2 grammars and has the advantage over type 1 grammARS that the phrase structure of a grammatical sentence is unique, once the derivation is given.
Abstract: In this report, certain properties of context-free (CF or type 2) grammars are investigated, like that of Chomsky. In particular, questions regarding structure, possible ambiguity and relationship to finite automata are considered. The following results are presented: The language generated by a context-free grammmar is linear in a sense that is defined precisely.The requirement of unambiguity—that every sentence has a unique phrase structure—weakens the grammar in the sense that there exists a CF language that cannot be generated unambiguously by a CF grammar.The result that not every CF language is a finite automaton (FA) language is improved in the following way. There exists a CF language L such that for any L′ ⊆ L, if L′ is FA, an L″ ⊆ L can be found such that L″ is also FA, L′ ⊆ L″ and L″ contains infinitely many sentences not in L′.A type of grammar is defined that is intermediate between type 1 and type 2 grammars. It is shown that this type of grammar is essentially stronger than type 2 grammars and has the advantage over type 1 grammars that the phrase structure of a grammatical sentence is unique, once the derivation is given.
788 citations
IBM1
TL;DR: A method is described for the numerical inversion of Laplace transforms, in which the inverse is obtained as an expansion in terms of orthonormal Laguerre functions.
Abstract: A method is described for the numerical inversion of Laplace transforms, in which the inverse is obtained as an expansion in terms of orthonormal Laguerre functions. In order for this to be accomplished, the given Laplace transform is expanded in terms of the Laplace transforms of the orthonormal Laguerre functions. The latter expansion is then reduced to a cosine series whose approximate expansion coefficients are obtained by means of trigonometric interpolation. The computational steps have been arranged to facilitate automatic digital computation, and numerical illustrations have been given.
327 citations
TL;DR: Two formal systems for the algebraic transformation of regular expressions are developed, one based upon the uniqueness of the solution of certain regular expression equations, and the other based upon some facts concerning the representation theory of regular events.
Abstract: The theory of finite automata is closely linked with the theory of Kleene's regular expressions. In this paper, two formal systems for the algebraic transformation of regular expressions are developed. Both systems are consistent and complete; i.e., the set of equations derivable within the system equals the set of equations between two regular expressions denoting the same event. One of the systems is based upon the uniqueness of the solution of certain regular expression equations, whereas some facts concerning the representation theory of regular events are used in connection with the other.
312 citations
TL;DR: The trade-off relation between number of tapes and speed of computation can be used in a diagonalization argument to show that, if a given function requires computation time T for a k-tape realization, then it requires at most computation time log T log log log for a two-Tape realization.
Abstract: It has long been known that increasing the number of tapes used by a Turing machine does not provide the ability to compute any new functions. On the other hand, the use of extra tapes does make it possible to speed up the computation of certain functions. It is known that a square factor is sometimes required for a one-tape machine to behave as a two-tape machine and that a square factor is always sufficient.The purpose of this paper is to show that, if a given function requires computation time T for a k-tape realization, then it requires at most computation time T log T for a two-tape realization. The proof of this fact is constructive; given any k-tape machine, it is shown how to design an equivalent two-tape machine that operates within the stated time bounds. In addition to being interesting in its own right, the trade-off relation between number of tapes and speed of computation can be used in a diagonalization argument to show that if T(n) and U(n) are two time functions such that inf T(n) log T(n) ÷ U(n) = 0 then there exists a function that can be computed within the time bound U(n) but not within the time bound T(n).
270 citations
TL;DR: It is shown that no language contained in w, each of which contains w, is inherently ambiguous, and a necessary and sufficient algebraic condition is given for a bounded language to be inherently ambiguous.
Abstract: Four principal results about ambiguity in languages (i.e., context free languages) are proved. It is first shown that the problem of determining whether an arbitrary language is inherently ambiguous is recursively unsolvable. Then a decision procedure is presented for determining whether an arbitrary bounded grammar is ambiguous. Next, a necessary and sufficient algebraic condition is given for a bounded language to be inherently ambiguous. Finally, it is shown that no language contained in w1*w2*, each w1 a word, is inherently ambiguous.
126 citations
TL;DR: In this paper the properties of the generating functions are examined from certain connectivity considerations of directed graphs and their generating functions.
Abstract: Discrete sequential systems, such as sampled data systems, discrete Markov processes, linear shift register generators, computer programs, sequential code generators, and prefixed comma-free codes, can be represented and studied in a uniform manner by directed graphs and their generating functions. In this paper the properties of the generating functions are examined from certain connectivity considerations of these graphs.
80 citations
IBM1
TL;DR: A procedure for index register allocation is described and the rules of this procedure are shown to yield an optimal allocation for “straight line” programs.
Abstract: A procedure for index register allocation is described. The rules of this procedure are shown to yield an optimal allocation for “straight line” programs.
79 citations
TL;DR: An algorithm is developed which, in a finite number of steps, will convert any given limited entry decision table into an “optimal” computer program, one with minimum average processing time.
Abstract: This paper begins with a brief description of desicion tables, and then presents a discussion of alternative expressions for them as sequential testing procedures for computer implementation and as Boolean functions An algorithm is developed which, in a finite number of steps, will convert any given limited entry decision table into an “optimal” computer program, one with minimum average processing time The algorithm is more general than procedures previously developed and guarantees optimality of the resultant computer program Previous procedures required two distinct steps and gave no assurance of overall optimality Computer implementation of the algorithm is also discussed
77 citations
TL;DR: An account is given of some very recent developments toward reduction in the system overhead needed to facilitate time-sharing and one hardware-software scheme designed to implement this reduction is described in some detail.
Abstract: The problem studied is the effect of a time-sharing environment on the structure of programs and on the addressing strategies which may be employed in the hardware. An account is given of some very recent developments toward reduction in the system overhead needed to facilitate time-sharing. One hardware-software scheme designed to implement this reduction is described in some detail.
TL;DR: A method of finding every cycle of an undirected linear graph by computation, rather than search, is presented.
Abstract: A method of finding every cycle of an undirected linear graph by computation, rather than search, is presented. The method consists of three algorithms. The first produces a fundamental set of cycles from which all others can be generated. The second groups these cycles according to the nonseparable subgraphs of the original graph, and produces an ordering among groups that satisfies a condition required for the third algorithm. The third algorithm generates all and only cycles of the graph, without duplicates.
TL;DR: The various subroutines of the package and their functions are described in this paper and numerous examples of pictures that were produced by BE VISION are given.
Abstract: BE VISION is a package of FORTRAN programs for drawing orthographic views of combinations of plane and quadric surfaces. As input, the package takes rectangular coordinate equations specifying the surfaces plus a three-angle specification of the viewing direction. Output is a drawing on the Stromberg Carlson 4020 Microfilm Recorder. Many views of one scene may be obtained simply by changing the viewpoint.The various subroutines of the package and their functions are described in this paper. It also gives numerous examples of pictures that were produced by BE VISION. The package has been in use since April 1964.
TL;DR: These two step methods for the numerical solution of the vector initial value problem can be defined by making evaluations of F similar to those found in a classical Runge-Kutta formula and are shown to be convergent with its unique solution under certain restrictions.
Abstract: A third order two step method and a fourth order two step method for the numerical solution of the vector initial value problem dy ÷ dx=F(y), y(a) = n can be defined by making evaluations of F similar to those found in a classical Runge-Kutta formula. These two step methods are different from classical Runge-Kutta methods in that evaluations of F made at the previous point are used along with those made at the current point in order to obtain the solution at the next point. If the stepsize is fixed, this use of previous computations makes it possible to obtain the solution at the next point by evaluating F two or three times for the third or fourth order method, respectively.These methods are consistent with the initial value problem and are shown to be convergent with its unique solution under certain restrictions. The local truncation error terms are given. Finally, a few numerical results are presented.
TL;DR: The problem of whether a given context-free language is linear is shown to be recursively undecidable.
Abstract: The problem of whether a given context-free language is linear is shown to be recursively undecidable.
TL;DR: It is shown that the asymptotic behavior of π (subscrpt) A (n) = 1 is subject to severe restraints if A is regular, and the technique cannot yield a decision procedure for regularity since for every infinite regular set A there is a nonregular set A.
Abstract: When is a set A of positive integers, represented as binary numbers, “regular” in the sense that it is a set of sequences that can be recognized by a finite-state machine? Let p A(n) be the number of members of A less than the integer n. It is shown that the asymptotic behavior of p A(n) is subject to severe restraints if A is regular. These constraints are violated by many important natural numerical sets whose distribution functions can be calculated, at least asymptotically. These include the set P of prime numbers for which p P(n)
TL;DR: An automatic programming system for the M-20 computer at the Computing Centre of the Siberian Division of the USSR Academy of Sciences has been developed and the principal objective was the production of fast, compact object code.
Abstract: An automatic programming system for the M-20 computer at the Computing Centre of the Siberian Division of the USSR Academy of Sciences has been developed. The translator is a compiler which accepts source programs written in ALPHA language (hardware representation of which input language is an extension of ALGOL 60). The language extensions include the ability to handle complex quantities and the ability to consider variables as matrices or vectors. Certain language limitations are imposed concerning recursion and the correspondence between parameters in procedure definitions and calls. The translator itself consists of two phases. The first phase contains 14 blocks and translates a source program to an intermediate language. The second phase which consists of 10 blocks converts the internal language to machine code. The principal objective in designing the compiler was the production of fast, compact object code. Techniques for such achievement are detailed. Since the compiler has been operational for some time, comparative figures are given for a series of programs as compiled by the ALPHA translator and as handwritten by programmers. Further figures compare ALPHA with TA-1 and TA-2 (earlier compilers written for the same computer). A special debugging system operates in conjunction with the ALPHA translator.
TL;DR: A model of a time-sharing system is proposed, based on the assumption that both interarrival and service times possess an exponential distribution and an imbedded Markov chain is analyzed.
Abstract: The concept of time-shared computer operations is briefly described and a model of a time-sharing system is proposed, based on the assumption that both interarrival and service times possess an exponential distribution. Although the process described by this model is non-Markovian, an imbedded Markov chain is analyzed by exploiting the fact that the instants of completion of a “quantum” of service are regeneration points. It is shown that user congestion possesses a limiting distribution, and the method of generating functions is used to derive this distribution. The concept of cycle time is discussed and two measures of cycle time developed for a scheduling discipline employing a single queue. Finally, a number of numerical examples are presented to illustrate the effect of the system parameters upon user congestion, system response time and computer efficiency.
TL;DR: Some new predictors are presented for use with the Adams-Moulton correctors of orders 4 through 8, which have better stability characteristics than the usual ones which employ the Adams Bashforth predictors and at the same time require no additional storage.
Abstract: Some new predictors are presented for use with the Adams-Moulton correctors of orders 4 through 8. The resulting algorithms have better stability characteristics than the usual ones which employ the Adams Bashforth predictors and at the same time require no additional storage. The regions of absolute and relative stability for the methods mentioned above and for the iterated Adams-Moulton correctors are given. Results obtained using the various methods are compared.
TL;DR: A general class of large-scale multiprocessors is outlined, and some problems of hardware and software implementation for computers of this class are discussed.
Abstract: Various classes of machines incorporating parallelism are considered. A general class of large-scale multiprocessors is outlined, and some problems of hardware and software implementation for computers of this class are discussed.
TL;DR: The existence problem (how arbitrarily the input, output and affected regions of an instruction can be specified) is fully settled for countable memory.
Abstract: @, known as instructions. Each instruction I has an input region IR(I), an output region OR(I), and affected regions AR(M′, I), for M′ ⊆ IR(I). For example, let I be the instruction (CLA Y) on the IBM 7094. If L is the location counter and AC is the accumulator, then IR(I) = Y ∪ L and OR(I) = AC ∪ L; if M′ is the address portion of Y, then AR(M′, I) is the address portion of AC. The fundamental properties of all these notions are derived, and computers are related to other models, such as sequential machines. The existence problem (how arbitrarily the input, output and affected regions of an instruction can be specified) is fully settled for countable memory M.
TL;DR: The principal results concern the “dimensional” and “correlational” effects and their degradation of a multimeasurement recognition system.
Abstract: Some problems arising in multicategory (many pattern types) pattern recognition are treated mathematically, and formulas are derived which describe some inherent limitations associated therewith. The principal results concern the “dimensional” and “correlational” effects and their degradation of a multimeasurement recognition system.
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TL;DR: The purpose is to investigate the input structure of automata which have a group-like character and the class of perfect automata investigated by Fleck and Weeg is a proper subclass of those considered here.
Abstract: The purpose is to investigate the input structure of automata which have a group-like character. The class of perfect automata investigated by Fleck in [2] and Weeg in [6] is a proper subclass of those considered here.
TL;DR: In this paper, some necessary and sufficient conditions for the reverse order property to hold are given.
Abstract: The “reverse order law” related to ordinary inverses of matrix products, i.e., (AB)-1 = B-1A-1, is generally not transferable to the generalized inverse. There are, however, applications in which the reverse order law related to the generalized inverse reveals interesting properties in certain classes of matrices.In this paper, some necessary and sufficient conditions for the reverse order property to hold are given.
TL;DR: A method is proposed for internal sorting which has the property that the expected time required per record, to sort n records with random keys, is a bounded function of n.
Abstract: A method is proposed for internal sorting which has the property that the expected time required per record, to sort n records with random keys, is a bounded function of n. Moreover the storage required in addition to that for the records is only 2n address fields. It appears that the method may be a good choice when the number of records to be sorted is of the order of 10,000 or more.The method consists of partially sorting the records by a radix sort and completing the sort by sifting. In the radix sort, a list type of structure is used to present the distribution of the records.
TL;DR: The results are obtained directly by recognizing that a sequential processing may be viewed as a cyclic queue, and are good only for exponentially distributed computation times.
Abstract: Results are obtained for a model of many processors operating in series. The results are obtained directly by recognizing that a sequential processing may be viewed as a cyclic queue. Exact results are given for two sequential processing stages with a buffer storage of arbitrary size between the stages, and approximate results for the case of 2M (M an integer) stages. The analysis is good only for exponentially distributed computation times.
TL;DR: Neither unambiguity nor inherent ambiguity is preserved by any of the following language preserving operations: (a) one state complete sequential machine; (b) product by a two-element set.
Abstract: Various elementary operations are studied to find whether they preserve on ambiguity and inherent ambiguity of language (“language” means “context-free language”) The following results are established: If L is an unambiguous language and S is a generalized sequential machine, then (a) S(L) is an unambiguous language if S is one-to-one on L, and (b) S-1(L) is an unambiguous language. Inherent ambiguity is preserved by every generalized sequential machine which is one-to-one on the set of all words. The product (either left or right) of a language and a word preserves both unambiguity and inherent ambiguity. Neither unambiguity nor inherent ambiguity is preserved by any of the following language preserving operations: (a) one state complete sequential machine; (b) product by a two-element set; (c) Init(L) = [u ≠ dur in L for some v]; (d) Subw(L) = [w ≠ durr in L for some u, v].
TL;DR: It is shown that any amplitude requency transfer function expressible as an even trigonometric rational polynomial can be synthesized by a real stable linear digital recursive filter.
Abstract: The circumstances and methods of the synthesis of linear digital recursive filters which are both stable and physically realizable are described. It is shown that any amplitude requency transfer function expressible as an even trigonometric rational polynomial can be synthesized by a real stable linear digital recursive filter. The degree of the corresponding difference equation is twice the degree of the denominator of the rational trigonometric polynomial.A class of even rational trigonometric functions which exhibit pointwise convergence to the deal rectangular low or high-pass amplitude frequency transfer function is chosen. A member of this class is shown to approximate more closely the ideal rectangular filter than does the corresponding classical continuous Butterworth filter. This class of filters is then mechanized. The phase and unit-impulse response functions are calculated for the corresponding difference equations of degrees 2 and 4.
TL;DR: A contracting error program for the solution of simultaneous linear equations is described, demonstrating the application of the idea to a wider class of problems.
Abstract: Recent developments in computer design and error analysis have made feasible the use of variable precision arithmetic and the preparation of programs that automatically determine their own precision requirements Such programs enable the user to specify the accuracy he wants, and yield answers guaranteed to lie within the bounds prescribed A class of such programs, called “contracting error programs,” is defined in which the precision is determined by prescribing error bounds on the data A variant of interval arithmetic is defined which enables a limited class of algorithms to be programmed as contracting error programs A contracting error program for the solution of simultaneous linear equations is described, demonstrating the application of the idea to a wider class of problems
TL;DR: The two theorems presented in this paper lay the suspicion to rest that the unambiguous languages might be precisely those languages with context-free complements by providing an inherently ambiguous language with Context-free complement and an unambiguous language without context- free complement.
Abstract: Call a (context-free) language unambiguous if it is not inherently ambiguous. In the absence of evidence to the contrary, the suspicion has arisen that the unambiguous languages might be precisely those languages with context-free complements. The two theorems presented in this paper lay the suspicion to rest by providing (1) an inherently ambiguous language with context-free complement and (2) an unambiguous language without context-free complement. This establishes the independence of inherent ambiguity from complementedness among the context-free languages.