Infinitely Long Terms of Transfinite Type

As Gödel [6] has pointed out, there is a certain interchangeability between the intuitionistic notion of proof and the notion of constructive functional of finite type. He achieves this interchange in the direction from logic to functionals by his functional interpretation of Heyting arithmetic H in a free variable theory T of primitive recursive functionals of finite type. In the present paper we shall extend Gödel's functional interpretation to the case in which H and T are extended by adding an abstract notion of constructive ordinal. In other words, we obtain the Gödel functional interpretation of an intuitionistic theory U of numbers (i.e., nonnegative integers) and constructive ordinals in a free variable theory V of finite type over both numbers and constructive ordinals. This allows us to obtain an analysis of noniterated positive inductive definitions [8].The notion of constructive ordinal to be treated is as follows. There is given a function J which embeds the nonnegative integers in the constructive ordinals. A constructive ordinal of the form Jn is said to be minimal. There is also given a function δ which associates to each constructive ordinal Z and number n a constructive ordinal δZn which we denote by Zn. When Z is nonminimal, each Zn is called an immediate predecessor of Z. The basic principle for forming constructive ordinals says: for every function f from numbers n to constructive ordinals, there exists a constructive ordinal Z such that Zn = fn for all n. The principle of transfinite induction with respect to constructive ordinals says: if a property Q(Z) of constructive ordinals Z holds for minimal Z, and if ∀nQ(Zn) → Q(Z) holds for all Z, then Q(Z) holds for all Z.

In the first part of this paper we investigate the intuitionistic version $iI\!\Sigma_1$ of $I\!\Sigma_1$ (in the language of $PRA$ ), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for $HA$ and establishes a number of nice properties for $iI\!\Sigma_1$ , e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: $iI\!\Sigma_1$ together with induction over arbitrary prenex formulas is $\Pi_2$ -conservative over $iI\!\Pi_2$ . In the second part of the article we study the relation of $iI\!\Sigma_1$ to $iI\!\Pi_1$ (in the usual arithmetical language). The situation here is markedly different from the classical case in that $iI\!\Pi_1$ and $iI\!\Sigma_1$ are mutually incomparable, while $iI\!\Sigma_1$ is significantly stronger than $iI\!\Pi_1$ as far as provably recursive functions are concerned: All primitive recursive functions can be proved total in $iI\!\Sigma_1$ whereas the provably recursive functions of $iI\!\Pi_1$ are all majorized by polynomials over ${\Bbb N}$ . 0 $iI\!\Pi_1$ is unusual also in that it lacks closure under Markov's Rule $\mbox{MR}_{PR}$ .

The author recently published a paper which claimed that an ordinal interpretation of numbers had limited applicability for cryptography. A further examination of this subject, in particular to what extent an ordinal interpretation is useful for recurrence sequences, is needed. Hilbert favored an interpretation of the natural numbers that placed their ordinal properties prior to their cardinal properties [1] [2]. The author examines ordinal uses of the integers in number theory in order to discuss the possibilities and limitations of this approach. The author hopes this paper will be useful in clarifying or even correcting some matters that were discussed in his paper of January of 2018. I was trained informally in philosophical realism, and while I think idealism too has a place, at this time in my life I believe that the weight of evidence and usefulness is more on the side of philosophical materialism. I hope this discussion will help supplement for my readers the material in Number in Mathematical Cryptography. I still maintain that a lack of clarity on these matters has hindered progress in cryptography; and it has taken time for me to better understand these things. I hope others who have interest and ability will assist in making these matters clearer. My intention was to work in pure mathematics, and the transition to an applied mindset was difficult for me. As a result, I feel most comfortable in a more middle-of-the road attitude, but have had to slowly move to a more precise analysis of the physical quantities involved. I hope my readers will be patient with my terminology, which is still evolving, and my discussion of things which are more indirectly related, and which are necessary for my expression. These are important things for the mathematical community to understand, and I hope smarter and more knowledgeable people will address my errors, and improve upon the things I might have correct. I am discussing sequences which are sometimes a use of both ordinal and cardinal numbers.

i. S. A. Berezin, "An algebra of one-place primitive recursive functions with iteration operation of general form," Kibernetika, No. 3, 12-19 (1976). 2. G. P. Gavrilov, "Functional completeness in a countably valued logic," in: Problems of Cybernetics [in Russian], No. 15 (1965), pp. 5-64. 3. D. A. Zakharov, Recursive Functions [in Russian], Novosibirsk (1970). 4. V. V. Koz'minykh, "One-place primitive recursive functions," Algebra Logika, ~, 75-90 (1968). 5. A. I. ~l'tsev, "Iterative algebras and Post varieties," Algebra Logika, ~, 5-24 (1966). 6. A. I. Mal'tsev, "Constructive algebras, I," Usp. Mat. Nauk, 16, No. 3, 3-60 (1961). 7. S. S. Marchenkov, ,"B°unded. . recursions," Math. Balkanlca," 2, 124-142 (1972).~ 8. S S. ~rchenkov, Ellmlnationof recursion schemes in the--Grzegorczyk class ," Mat. Zametki, 5, No. 5, 561-568 (1969). 9. V, L. Mikheev, "A class of algebras of primitive recursive functions," Mat. Zametki, 14, No. 1 143-156 (1973). i0. S. V. Pakhomov, "A simple syntactical definition of all classes of the Grzegorczyk hierarchy," Zap. Nauch. Sem. Leningr. Otd. ~at. Inst. Akad. Nauk SSSR, 40, 127-130 (1974). Ii. H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill (1967). 12. M. G. Rozinas, "An algebra of many-place primitive recursive functions," Uch. Zap. Ivanovsk. Gos. Pedagog. Inst., i17, 95-111 (1972). 13. A. Cobham, "The intrinsic computational difficulty of functions," in: Logic, Methodology and Philosophy of Science, Amsterdam (1965), pp. 24-30. 14. A. Grzegorczyk, "Some classes of recursive functions," in: Rozprawy Matematiczne, Vol. 4, Warsaw (1953). 15. R. W. Ritchie, "Classes of predictably computable functions," Trans. Am. Math. Soc., 106, 139-173 (1963).

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.

Minimization and NP multifunctions

Every recursively enumerable set of integers (r.e. set) is enumerable by a primitive recursive function. But if the enumeration is required to be one-one, only a proper subset of all r.e. sets qualify. Starting from a collection of total recursive functions containing the primitive recursive functions, we thus define a sub-computability as an enumeration of the r.e. sets that are themselves one-one enumerable by total functions of the given collection. Notions similar to the classical computability ones are introduced and variants of the classical theorems are shown. We also introduce sub-reducibilities and study the related completeness notions. One of the striking results is the existence of natural (recursive) sets which play the role of low (non-recursive) solutions to Post's problem for these sub-reducibilities. The similarity between sub-computabilities and (complete) computability is surprising, since there are so many missing r.e. sets in sub-computabilities. They can be seen as toy models of computability.

The production environment of coal mine is complex and tough, so once dangerous situation appears it requires coal miners take accurate and effective emergency measures. Actually, the occurrence and further expansion of a large number of mining accidents are associated with coal miners’ mishandling of emergency. Therefore, evaluating coal miners’ emergency response capacity is the significant way to improve the coal miners' emergency response capacity level and pre-control coal mine accidents. Considering the drawbacks of traditional evaluation methods, this paper introduced the principle of catastrophe theory in evaluation, and proposed the evaluation model of coal miner’s emergency response capacity by introducing the triangular fuzzy number theory in order to reduce the uncertainties of underlying index grades made by experts. In this method, the importance and quantitative process of each assessing indices were determined by the intrinsic mechanism of the normalized formula in catastrophe model, thus the problems of estimating the assessing index weights were avoided and the effect of subjective factors was mitigated. Finally, the verification of evaluation was implemented on specific example, and the results show that this evaluation method is reasonable, feasible, exact, and stable.

Continuous-time computation with restricted integration capabilities

Pure patterns of order 2

Computability evaluation of RESTful API using Primitive Recursive Function

In the 1990s Yankov published two papers containing important contributions to proof theory and based on the application of a dialogical interpretation of proofs. In both cases the method is used for providing constructive proofs of important metalogical results concerning classical logic and fundamental mathematical theories. In the first paper it is shown that impredicative extensions of intuitionistic versions of arithmetic, analysis and set theory, enriched with suitable bar induction schemata, are sufficiently strong for proving the consistency of their classical counterparts. In the second paper the same method is applied to provide a constructive proof of the completeness theorem for classical logic. In both cases a version of a one-sided sequent calculus in Schütte-style is used and cut elimination is established in the second case. Although the obtained results are important, and the applied method is original and interesting, they have not received the attention they deserve from the wider community of researchers in proof theory. In this paper we briefly recall the content of both papers. We focus on essential features of Yankov’s approach and provide comparisons with other results of similar character.KeywordsProof theoryDialogical interpretation of proofsConsistency proofsBar inductionArithmeticAnalysisSet theory

We undertake a thorough complexity study of the following fundamental optimization problem, known as the ℓp-norm shortest extended GCD multiplier problem: given a 1,..., a n ∈ ℤ, find an ℓp-norm shortest gcd multiplier for a 1,..., a n, i.e., a vector x ∈ ℤn with minimum \( \left( {\sum\nolimits_{i = 1}^n {|x_i |^p } } \right)^{1/p} \) satisfying \( \sum\nolimits_{i = 1}^n {x_i a_i = \gcd (a_1 ...,a_n )} \). First, we prove that the shortest GCD multiplier problem (in its feasibility recognition form) is NP-complete for every ℓp-norm with p ∈ ℤ. This gives an affirmative answer to a conjecture raised by Havas and Majewski. We then strengthen this negative result by ruling out even polynomial-time algorithms which only approximate an ℓp-norm shortest gcd multiplier within a factor \( n^{1/(p\log ^\gamma n)} \) for γ an arbitrary small positive constant, under the widely accepted complexity theory assumption \( NP \nsubseteq DTIME\left( {n^{poly(\log n)} } \right) \).For positive results we focus on the ℓ2-norm GCD multiplier problem. We show that approximating this problem within a factor of \( \sqrt n \) is very unlikely NP-hard by placing it in NP ⋂ coAM through a simple constant-round interactive proof system. This result is complemented by a polynomial-time algorithm which computes an ℓ2-norm shortest gcd multiplier up to a factor of 2(n−1)/2.This study is motivated by the importance of extended gcd calculations in applications in computational algebra and number theory. Our results rest upon the close connection between the hardness of approximation and the theory of interactive proof systems.Key WordsApproximation algorithmscomputational problems of diophantine equationsextended gcd computationsNP-hardnessprobabilistically checkable proofsinteractive proof systems

THE title of this little work does not indicate, as the English reader might expect, another addition to the ever-growing list of treatises upon geometrical methods of calculation or the graphical solution of ordinary problems. It might rather be described as an essay upon the geometrical interpretation of the theory of numbers. Arithmetique graphique. Les Espaces arithmeliques; leurs Transformations. By Gabriel Arnoux. Pp. xii+84. (Paris: Gauthier-Villars, 1908.) Price 3 francs.

An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.