A unary representation result for systemT

We present a primitive recursive programinf_with_lists computing the minimum of two natural numbersn andp (written in unary notation) and using primitive recursion on lists. This program has at first sight the required property of visiting simultaneously its inputs, so it is a counterexample to a theorem showing that such a program cannot be written in the language of primitive recursion on natural numbers, in the more general framework of primitive recursion on term algebras. However, its complexity is at leastinf(n,p)2 so it does not implement the algorithm we have in mind to computeinf(n,p).

We present a version of Gödel’s system T in which the types are ramified in the style of Leivant and a system of dependent typing is introduced. The dependent typing allows the definition of recursively defined types, where the recursion is controlled by ramification; these recursively defined types in turn allow the definition of functions by repeated iteration. We then analyze a subsystem of the full system and show that it defines exactly the primitive recursive functions. This result supports the view that when data use is regulated (for example, by ramification), standard function constructions are intimately connected with standard type-theoretic constructions.KeywordsInference RuleProduct RuleObject TypeHigh TypeDependent TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we proved that the function class CFRF and its proper subclass CFPRF are respectively the partial recursive functions and primitive recursive functions of context free languages (CFLs). Also we discussed the relation between them and recursive functions defined on other domains. It is indicated that the functions of natural numbers and/or symbol strings (words) are functions of CFLs. Several frequently used primitive recursive functions on words were given, including logical connectives, conditional expressions. Also the powerful operators (bounded maximization and minimization operators) for constructing primitive recursive functions were defined. Two important nontrivial algorithms, the characteristic function of arbitrary CFL and the parse function of CFL sentences were constructed. Based on them, the method for extending or restricting function domain was described.

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Eleven different definitions for functions to be computable by Petri nets are given and compared. For nine of these definitions it was possible to show that they are not powerful enough to describe all the primitive recursive functions.

Recently, a pseudocode language for set-theoretic algorithms hos been investigated for use In discrete math/structures courses [1-4]. The language is highly intuitive and contains only eight elementary statements. In the language, it is possible to (a) readily express the elementary algorithms of discrete mathematics and (b) develop correctness proofs using set theory and its underlying logic (without resort to logical-invariance proofs). In this paper, it is demonstrated that correct algorithms for the arithmetic operations over natural numbers are essentially corollaries of two easily-proved theorems about primitive recursion. In essence, for a function defined over natural numbers, a correct definition by mathematical induction immediately yields a correct algorithm for computation.

P. Bernays has pointed out that, even in order to prove only the consistency of classical number theory, it is necessary to extend Hilbert’s finitary standpoint. He suggested admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that a certain concept of computable function of finite simple type over the natural numbers can be used instead, where no other procedures of constructing such functions are necessary except primitive recursion by a number variable and definition of a function by an equality with a term containing only variables and/or previously introduced functions beginning with the function +.

We present a size-aware type system for first-order shapely function definitions. Here, a function definition is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely function definitions may be implementations of matrix multiplication and the Cartesian product of two lists. The type system is proved to be sound w.r.t. the operational semantics of the language. The type checking problem is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear or monotonic. Furthermore, we have shown that the type-inference problem is at least semi-decidable (under this restriction). We have implemented a procedure that combines run-time testing and type-checking to automatically obtain size dependencies. It terminates on total typable function definitions.

This paper reports on some recent investigations aimed at separating and clarifying a number of concepts that frequently appear in object-based and object-oriented languages. Concepts considered include are over-loading, encapsulation, the message paradigm, and a limited form of inheritance. We also introduce formulations of the concepts of overload-systems and coercion-systems, and we present a treatment of (+, ×)-recursive classes and the definition of primitive recursive functions over such classes.KeywordsData TypeFunctional LanguageEmpty StringAbstract Data TypeImperative LanguageThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This poster introduces a new open-source tool for visualizing recursive algorithms. Many students struggle with understanding recursive algorithms and the way in which return values propagate from the base call back up to the original call. Teachers often show visualizations to help students improve their understanding, drawing a tower of calls for linear recursive algorithms or a much more complex tree of calls for tree recursive algorithms. We have created a web-based tool for automating the drawing of these call graphs for any recursive Python function, based solely on the function definition and function call. The call graphs show the arguments and return values for each recursive call, and can be stepped through interactively to show the order of calls from start to end. The graphs can also be embedded in web pages, since they're built with accessible web technologies. We have used this tool to help students understand recursion in university CS1 classes, and would like to keep improving it to be a useful tool for CS classes globally.

Predicative functionals and an interpretation of [formula omitted

With regard to the impact of multimedia digital service transfer delay in computer network, we apply the simplified Smith predictor algorithm combined with the recursive algorithm, and propose a recursive optimization control algorithm based on predictor. The method is easy to implement and has a practical application in multimedia networks. The simulation result shows that this method can adjust the service send rate and queue in buffer rapidly and effectively, can improve the link utilization, and the delay effect is effectively overcome.

We present a general nonlinear Bayesian filter for high-dimensional state estimation using the theory of reproducing kernel Hilbert space (RKHS). By applying the kernel method and the representer theorem to perform linear quadratic estimation in a functional space, we derive a Bayesian recursive state estimator for a general nonlinear dynamical system in the original input space. Unlike existing nonlinear extensions of the Kalman filter where the system dynamics are assumed known, the state-space representation for the Functional Bayesian Filter (FBF) is completely learned online from measurement data in the form of an infinite impulse response (IIR) filter or recurrent network in the RKHS, with universal approximation property. Using a positive definite kernel function satisfying Mercer’s conditions to compute and evolve information quantities, the FBF exploits both the statistical and time-domain information about the signal, extracts higher-order moments, and preserves the properties of covariances without the ill effects due to conventional arithmetic operations. We apply this novel kernel adaptive filtering (KAF) to recurrent network training, chaotic time-series estimation and cooperative filtering using Gaussian and non-Gaussian noises, and inverse kinematics modeling. Simulation results show FBF outperforms existing Kalman-based algorithms.

This article presents some results of the attempt to simplify the writing of arithmetic functions on the computer so that users can apply them without additional operations, such as summing after a set whose elements must be calculated, such as the set of numbers prime. The important role of the remainder function in defining most arithmetic functions is highlighted. Defining algorithms for the prime factorization of natural numbers highlights the possibility of representing natural numbers in a basis as "natural" as possible for natural numbers, namely the basis of prime numbers. The disadvantage of this natural basis is, for the time being, that it is infinitely dimensional. For now, this representation provides advantages but also disadvantages. Among the arithmetic functions proposed in the article, there are also statistical characterizations of the distribution of prime numbers, given with the hope of helping a better knowledge of the set of prime numbers. The importance of the remainder function in the computational definitions of arithmetic functions leads to reflections on the importance of fundamental operations - addition and multiplication - of natural numbers and the importance of inverse functions - subtraction and division. In turn, these operations can be seen as functions of two variables on the set of natural numbers. From here, readers are invited to reflect on the problem of the origin of natural numbers, the origin based on revelation or the origin provided by set theory, although this may also be a revelation.

This research introduces a novel set of mathematical formulae for efficiently determining the frequency of individual digits within the set of natural numbers less than a given integer N. The study aims to derive a general closed-form expression that avoids iteration, based on the digit structure of N, and to develop a new positional operator for identifying digit placement within multi-digit numbers. The methodology is built on place-value analysis and the definition of a digit frequency function f(t;N), which incorporates a universal positional term fn and a helper function G(ai,t). The formula f(t;N) = fn + ∑G(ai,t) is proven to hold across numerical classes and is validated through extensive numerical testing up to 10^18. The research also introduces a novel mathematical operator, Ta = t-1, to determine the exact placement of a digit at the end of a number sequence. The results demonstrate a 92% improvement in computational efficiency compared to enumeration, with broad applications in number theory, coding, and pattern recognition. Additionally, the approach resolves a known non-linear radical system with exact solutions, showcasing the formulae's algebraic utility. In conclusion, this study contributes three new tools to mathematics: a closed-form digit frequency function, a terminal digit positional operator, and a novel solution method for radical equations.