Predicative functionals and an interpretation of [formula omitted

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.

Cartesian closed Dialectica categories

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 We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: $iI\!\Sigma_1$ together with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 $\mbox{MR}_{PR}$ .

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.

After an introduction to the syntax of Godel systemT, we present its naive denotational semantics in the domain of lazy natural numbers and show an adequacy property relating syntax and semantics. We recall the natural restrictions of systemT delineating primitive recursion as a subsystem. We discuss the weakness of primitive recursion by exhibiting a simple unary algorithm whose denotation is not the semantics of a primitive recursive algorithm. This algorithm can nevertheless be programmed in systemT by using the power of higher-order (functional) definitions. Generalizing this example, we obtain a representation theorem, asserting that every “reasonable” algorithm of typeN →N can be programmed in systemT. We conclude by discussing what is known in the case of higher arities.

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.

Continuous-time computation with restricted integration capabilities

Computability evaluation of RESTful API using Primitive Recursive Function

Infinitely Long Terms of Transfinite Type

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 +.

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The built-in resources of L A – the first-order language of basic arithmetic which we first introduced in Section 5.2 – are minimal: there are no non-logical predicates, and just three primitive functions, successor, addition and multiplication. We have previously noted, though, a few examples of what else L A can express (‘express’ in the sense of Section 5.4). We now radically extend our list of examples by proving that L A can in fact express any primitive recursive function. Starting the proof Our target, then, is to prove Theorem 15.1 Every p.r. function can be expressed in L A . The proof strategy Suppose that the following three propositions are all true: L A can express the initial functions. If L A can express the functions g and h , then it can also express a function f defined by composition from g and h . If L A can express the functions g and h , then it can also express a function f defined by primitive recursion from g and h . Now, any p.r. function f must be specifiable by a chain of definitions by composition and/or primitive recursion, building up from initial functions. So as we follow through the full chain of definitions which specifies f, we start with initial functions which are expressible in L A , by (1). By (2) and (3), each successive definitional move takes us from expressible functions to expressible functions. So, given (1) to (3) are true, f must therefore also be expressible in L A .

This chapter introduces inductively defined sets and simple inductive data types such as natural numbers, words, lists and trees with emphasis on definition patterns. For these sets and data types, recursive equations in Backus-Naur form are introduced and it is shown how, e.g., propositional formulae and simple while programs can be specified in this format. Next it is explained how functions can be defined inductively on such sets and data types. Examples include the factorial function, Harmonic numbers, Fibonacci numbers, addition and multiplication on natural numbers and functions that compute the length of a word or list and the height of a tree. In contrast, recursive functions are defined in a way that does not necessarily lead to terminating evaluations. Recursive procedures are also described, with examples presented of various sorting algorithms and the Towers of Hanoi puzzle.