Inductive and Recursive Definitions

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

Ordinal Recursion and Inductive Definitions

In this paper, we have determined the limit of ratio of (n+1)th term to the nth term of famous sequences in mathematics like Fibonacci Sequence, Fibonacci – Like Sequence, Pell's Sequence, Generalized Fibonacci Sequence, Padovan Sequence, Generalized Padovan Sequence, Narayana Sequence, Generalized Narayana Sequence, Generalized Recurrence Relations of Fibonacci – Type sequence, Polygonal Numbers, Catalan Sequence, Cayley numbers, Harmonic Numbers and Partition Numbers. We define this ratio as limiting ratio of the corresponding sequence. Sixteen different classes of special sequences are considered in this paper and we have determined the limiting ratios for each one of them. In particular, we have shown that the limiting ratios of Fibonacci sequence and Fibonacci – Like sequence is the fascinating real number called Golden Ratio which is 1.618 approximately. We have shown that the limiting ratio of Pell's sequence is a real number called Silver Ratio and the limiting ratios for generalized Fibonacci sequence are metallic ratios. We have also obtained the limiting ratios of Padovan and generalized Padovan sequence. The limiting ratio of Narayana sequence happens to be a number called super Golden Ratio which is 1.4655 approximately. We have shown that the limiting ratios of Generalized Narayana sequence are the numbers known as super Metallic Ratios. We have also shown that the limiting ratio of generalized recurrence relation of Fibonacci type is 2 and that of Polygonal numbers and Harmonic numbers are 1. We have proved that the limiting ratio of the famous Catalan sequence and Cayley numbers are 4. Finally, assuming Rademacher's Formula, we have shown that the limiting ratio of Partition numbers is the natural logarithmic base e. We have proved fourteen theorems to derive limiting ratios of various well known sequences in this paper. From these limiting ratio values, we can understand the asymptotic behavior of the terms of all these amusing sequences of numbers in mathematics. The limiting ratio values also provide an opportunity to apply in lots of counting and practical problems.

Let (Fn)n≥0 be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z:Z≥1→Z≥1 is defined as z(n):=min{k≥1:Fk≡0(modn)}. In this paper, among other things, we prove that z(n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1).

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

In 2016, Heule, Kullmann and Marek solved the Boolean Pythagorean Triples problem: is there a binary coloring of the natural numbers such that every Pythagorean triple contains an element of each color? By encoding a finite portion of this problem as a propositional formula and showing its unsatisfiability, they established that such a coloring does not exist. Subsequently, this answer was verified by a correct-by-construction checker extracted from a Coq formalization, which was able to reproduce the original proof. However, none of these works address the question of formally addressing the relationship between the propositional formula that was constructed and the mathematical problem being considered. In this work, we formalize the Boolean Pythagorean Triples problem in Coq. We recursively define a family of propositional formulas, parameterized on a natural number n, and show that unsatisfiability of this formula for any particular n implies that there does not exist a solution to the problem. We then formalize the mathematical argument behind the simplification step in the original proof of unsatisfiability and the logical argument underlying cube-and-conquer, obtaining a verified proof of Heule et al.’s solution.

Sabadini, Walters and others have developed a categorical, machine based theory of concurrency in which there are four essential aspects: a distributive category of data-types; a bicategory Mach whose objects are data types, and whose arrows are input-output machines built from data types; a semantic category (or categories) Sem, suitable to contain the behaviors of machines, and a functor, “behavior”: Mach→Sem. Suitable operations on machines and semantics are found so that the behavior functor preserves these operations. Then, if each machine is decomposable into primitive machines using these operations, the behavior of a general machine is deducible from the behavior of its parts. The theory of non-deterministic finite state automata provides an example of the paradigm and also throws some light on the classical theory of finite state automata. We describe a bicategory whose objects are natural numbers, in which an arrow M: n→p is a finite state automaton with n input states, p output states, and some additional internal states; we require that no transitions begin at output states or end at input states. A machine is represented by an q+n by q+p matrix. The bicategory supports additional operations: non-deterministic choice, parallel interleaving, and feedback. Enough operations are imposed on machines to show that each machine may be obtained from some atomic ones by means of the operations. The semantic category is the (Bloom-Esik) iteration theory Mat (X✶ whose objects are natural numbers and whose arrows from n to p are n×p matrices with entries in the semiring of languages. The behavior functor associates to a machine M: n→p a matrix |M| of languages, one language to each pair of input and output states. Behavior preserves composition, feedback, takes non-deterministic choice to union, and parallel-interleaving to shuffle. Thus, behavior gives a compositional semantics to a primitive notion of concurrent processes.

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.

An object knowledge canonical form for knowledge reuse

To perform the numerous tasks of a computer algebra system, a multitude of simple and complex data types, operators and statements are necessary. Important elementary data types are the various numerical data types like floating point numbers, complex numbers, rational numbers and integers of any length as well as identifiers and character strings. Additionally indexed identifiers are necessary to enable access to tables, arrays, lists and expression sequences. To control, for instance, the execution of statements, it must be possible to evaluate expressions to boolean values. In this context, the boolean constants TRUE and FALSE are important. Further available constants are E, EULER and PI. More complex data types allow objects to be combined to form a new object. These data types include lists, sets, tables and arrays and domains.

Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system.<br /> <br />Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided.<br /> <br />The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily.

Some Series Involving Products Between the Harmonic Numbers and the Fibonacci Numbers

In this paper, we make use of the probabilistic method to calculate the moment representation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characteriza tions and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established.Â

It is proved that if $k$ and $d$ are positive integers such that the product of any two distinct elements of the set \[ \{F_{2k},\, 5F_{2k},\, 4F_{2k+2},\, d\} \] increased by $4$ is a perfect square, than $d=4L_{2k}F_{4k+2}$. This is a generalization of the results of Kedlaya, Mohanty and Ramasamy for $k=1$.

In this paper, we intend to establish the theory of multi-type recursive functions (MTRF). The aim is to present a theoretical prototype of functional language to be implemented on computer. MTRF functions are computable, whose domain and range include: natural numbers, Boolean values, words on nite alphabet, context free languages, and parse trees of context free grammars. It is indicated that the natural number recursive functions and the MTRF functions are equivalent. Finally, an MTRF function eval is constructed, which is an evaluator to compute arbitrary MTRF functions, and can be implemented on the computer.