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Iterative procedures have many advantages for computation, for instance an approximation to the final answer is provided even if the computation stops abruptly. A model for iterative computation is proposed where a procedure G executed repeatedly "converges" to a function f. A notion of approximation of functions on the integers is used based on limits of densities. Some results on the power...
This paper continues investigations pertaining to recursive bounds on computing resources (such as time or memory) and the amount by which these bounds must be increased if new computations are to occur within the new bound. The paper proves that no recursive operator can increase every recursive bound enough to reach new computations. In other words, given any general recursive operator F[ ], there...
A new type of grammar, called a parallel leveled grammar, is introduced. The families of languages generated by such grammars with contextfree, linear or right-linear subrules are studied. Right-linear parallel finite-leveled languages can be displayed as nested vector expressions, which are extensions of regular expressions. Various hierarchy theorems for these families of languages are obtained...
A new model of abstract automata is presented employing the concept of finite automata on a network. Each normal network n provided with a one-way input tape determines a family of languages nl. A representation theorem, analogous to the Chomsky-Schützenberger representation theorem for context free languages1, is proved for the class nl. One consequence is that nl is a principal full AFL generated...
The problem of assigning a probability to each string of a language L(G) generated by a grammar G is considered. Two methods are considered. One method assigns a probability to each production associated with G and the other assigns the probabilities on the basis of particular features of the language. Several necessary conditions that must be satisfied by these probability assignment techniques if...
This paper develops techniques for establishing a lower bound on the number of arithmetic operations necessary for sets of simple expressions. The techniques are applied to matrix multiplication. A modification of Strassen's algorithm is developed for multiplying n × p matrices by p × q matrices. The techniques are used to prove that this algorithm minimizes the number of multiplications for a few...
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