Showing papers in "International Journal of Computer Mathematics in 1975"

Some notes on ETOL-languages

Abstract: In this paper we investigate the effect of adding some regulated rewriting properties to ETOL-systems. It follows that if we specify the order in which the tables of an ETOL-system must be applied, or if we add a forbidding context to the tables of the system, the generating power of the system is not increased. If a permitting context is added to the tables, then the resulting class of languages generated, coincides with a subclass of the class of context-free programmed languages.

The automatic transformational analysis of english sentences: An implementation

Abstract: We describe a system being developed for the transformational analysis of complex English sentences. The system is designed to be able to serve as a “front-end” for a variety of applications, such as question-answering, information retrieval, and command systems. It is a two-stage system, with the first stage being the Linguistic String Parser previously developed at New York University. The structure of the system and its relation to contemporary transformational parsers are considered. Several transformations, including those for nominalization, are described in detail, and several sentence analyses produced by the program are presented.

Developmental systems with fragmentation

Abstract: The paper introduces a new class of L systems, where it is possible to continue derivations from certain specified subwords of the words obtained. Such L systems (called L systems with fragmentation or just JL systems) are of interest both from biological and formal language theory point of view. The paper deals with JL systems without interactions, discusses the basic properties of the language families obtained, as well as their position in the L hierarchy. Finally, two infinite hierarchies of language families are obtained by limited fragmentation, the notions being analogous to those of ultralineanty and finiteness of index for context-free languages.

Partition-extrapolation methods for numerical quadratures

Abstract: Partition-extrapolation methods for numerical quadratures are based on dividing the range of integration into an infinite number of subdomains, evaluating the subintegrals over a finite number of subdomains numerically, and then estimating the integral by extrapolation of the partial sums. There is considerable flexibility in the choice of subdivisions, method of evaluating the subintegrals and extrapolation procedure. The treatment is heuristic, it being shown that the present methods compare favorably with previous methods for a variety of examples involving semi-infinite ranges or integrands with singularities.

A multiplexing theorem and generalisation of R-transform

Abstract: Analytical properties of Rapid transform are investigated. Transforms of multiplexed patterns are resolved into the transforms of constituent patterns. It is shown that periodicity in pattern domain corresponds to a null subspace in transform domain and a null subspace in pattern domain gives a periodic transform. Further, the nontrivial portion of the transform is related to the transform of the nontrivial portion of the pattern. Based on these properties of R-transform, a generalised R-transformation for rectangular patterns is defined. It displays the translation invariant property of R-transform and reduces to R-transform in case of square and column patterns.

A practical chebyshev collocation method for differential equations

Abstract: This paper describes a method for solving ordinary and partial differential equations in Chebyshev series. The main feature of the method, which is based on the collocation principle, (Lanczos [8]) is that it solves the problem of differentiating a Chebyshev series directly by the use of a stable recurrence relation. As a practical consequence the method is very simple and can easily be coded into a general-purpose program for solving some differential equations.

A cooley-tukey algorithm for the slant transform

Abstract: A Cooley-Tukey algorithm for the slant transform and its inverse is developed It is shown that using this algorithm, one can compute the slant transform by means of a simple modification of the same Cooley-Tukey algorithm which is used to compute the Hadamard ordered Walsh-Hadamard transform

JL systems with non-fragmented axioms: The hierarchy

Abstract: The effect of allowing only non-fragmented axioms in L systems with fragmentation (JL systems) is investigated from the generative power point of view. In the case of JL systems with interaction (JIL systems) there is a trade-off between the number of fragments in axiom and that of the right hand sides of certain productions. This is not case for JIL systems with erasing fragmentation (jIL systems). It is shown that, for a large enough both-sided context, restricting axioms to the non-fragmented ones does not, however, reduce generating capacity jIL systems. For small amounts of context there is a reduction in this capacity. In fact, even the hierarchy of jIL languages is then more complicated than that of other classes of languages generated by L systems with interaction.

Error estimation in the integration of ordinary differential equations

Abstract: Linear initial value problems, particularly involving first order differential equations, can be transformed into systems of higher order and treated as boundary value problems. Finite difference analogues considered for obtaining approximate solutions of these boundary value problems are proved to be fourth order convergent processes, by deriving considerable sharper bounds for the discretization error. Numerical examples are given to demonstrate the usefulness of our error bounds.

An algorithm for tracing live variables based on a straightened program graph

Abstract: An algorithm is given for finding where the variables of a program are active or live. While the algorithm is modeled after that of Kennedy, it is based on the flow graph straightening procedure of Earnest, Balke, and Anderson, rather than on Cocke-Allen intervals. Thus it can be applied to any program, without appeal to any additional mechanism such as node splitting.

A note on a family of acceptors for some families of developmental languages

Abstract: In a previous paper Rozenberg (1974) introduced the PDAC acceptors (pushdown-array-of-counters acceptors) and showed that the restricted PDACA characterize the EOL languages. In this present note the PDAPDA (pushdown-array-of-pushdowns acceptors) are introduced and it is shown that the restricted PDA2 characterize the ETOL languages.

On the Sethi-Ullman algorithm

Abstract: A directed acyclic graph (dag) can be used to represent an arithmetic expression consisting of sequences of binary operations on arguments. The Sethi-Ullman algorithm (Journal of the ACM, Vol. 17 No. 4, pp. 715-728) generates optimal object code for a machine with N≧1 registers and unlimited memory capacity when the dag is a binary tree. We first describe a dag labeling algorithm which gives the minimum number of registers required to evaluate any node without any stores. We then define a subclass of dags, the 1-load binary dags, employing a tree-like grammar, and modify the Sethi-Ullman algorithm to include this subclass but only when N = 2 registers. The proof of optimality relies on the use of syntax directed translation schemas. Modification of the algorithm to include all dags, or to allow N > 2 registers, appears difficult.

On efficiently computing the product of two binary relations

Abstract: We precisely analyse ideas of M. A. Kronrod for the efficient composition of binary relations on finite domains. It shows that the “Four Russians” algorithm for Boolean matrix multiplication behaves no different in average and worst cases, which therefore makes it less preferable for use than is often claimed.

A multilevel flow control statement

Abstract: The premise of this report is that the nested ‘if-then-else’ construct occurs frequently and that these constructs if nested deeply are detrimental to program readability and maintenance. To mitigate this situation we propose and describe a control structure which provides for the description of such a multilevel decision process via a binary tree. Examples are presented to promote our viewpoint, and a formal inference rule is supplied.

AEPL: An extensible programming language†

Abstract: This paper presents an extensible programming language (AEPL) which has been designed as a tool for the implementation of a large class of languages for specific applications. AEPL includes a powerful data definition facility which enables one to define data structures and new types of data elements as well as new operators to manipulate both new and old data elements. A syntax driven parsing scheme derived from the Markov Algorithm makes it possible to control the syntax of the language dynamically, thus allowing one to define new language structures such as expressions and statements. As a result of the method used for syntactic definition, languages obtained by extension of AEPL belong to the general class of phrase structure languages, i.e. they are not restricted to the class of context-free languages. AEPL was extended to produce a language for the manipulation of linear graphs. Our experience indicates that such extensions are fairly simple and, in many cases, straight-forward.

On artificial machine learning: some ideas in search of a theory†

Abstract: In this paper an algorithm for machine learning is defined and justified heuristically and empirically. “Consistency” properties of perfect evaluation functions are derived and these are used to select the best of a family of evaluation functions, that is, that evaluation function that is most consistent. Methods peculiar to a particular game such as rote learning, looking ahead are generally eschewed; “learning” consists of finding the evaluation which is most consistent. In the game of Nim, in which the winning strategy is well-known, we show that these principles are sufficient to derive a perfect evaluation function (under appropriate conditions) and so arrive at a winning behavioral strategy. In the Mod(6) game, for which a winning strategy is also known, we use the algorithm to deduce an evaluation function and evaluate its effectiveness. Finally, in the game of Hex we match the algorithm with a random player and observe its success.

On the characterization of certain classes of developmental languages by automata

Abstract: A new family of automata, called orderless contraction automata of complexity (0, ∞), is introduced, and it is proved that this family of automata accepts precisely the class of propagating, normal, limited, extended, tabled 0-Lindenmayer languages (Rosenberg [4], Herman [2]). Furthermore a subfamily of this new family of automata accepts precisely the class of propagating, normal, limited, extended 0-Lindenmayer languages.

Dynamic programming solutions of mildly non-linear elliptic problems over irregular regions

Abstract: A numerical method for solving midly non-linear elliptic problems over irregular regions is proposed in the paper. The given irregular region is imbedded in a region of simple geometry. The original non-linear problem is replaced by a sequence of linear problems by using the technique of quasi-linearisation. The method of dynamic programming is then used for computing the required solutions. The method is illustrated by solving a given problem. It is found that the proposed method is fast and can be applied to problems over complex regions.

A method for pogram analysis and Its applications to program-correctness problems

Abstract: Described in this paper is a program-analysis method that can be used to effectively determine the logical structure of a program, explicate the computation a program will perform, and show the equivalence of programs. Applications to the problem areas in test-case generation, proving program correctness, and translation of “GOTO” programs into “GOTO-less” programs are discussed.

Necessary and sufficient conditions for a power language to be deterministic

Abstract: Necessary and sufficient conditions for a power language to be an LR language [LR(k) for some k] are given in terms of the words of the power language. Further, the conditions for a power language to be an LR-regular language are obtained.