LA can express every p.r. function

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 We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: $iI\!\Sigma_1$ together with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

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.

A natural example of a function algebra is \(\mathcal{R}\)(T), the class of provably total computable functions (p.t.c.f.) of a theory T in the language of first order Arithmetic. In this paper a simple characterization of that kind of function algebras is obtained. This provides a useful tool for studying the class of primitive recursive functions in \(\mathcal{R}\)(T). We prove that this is the class of p.t.c.f. of the theory axiomatized by the induction scheme restricted to (parameter free) Δ1(T)–formulas (i.e. Σ1–formulas which are equivalent in T to Π1–formulas).

We consider the computation of finite semigroups using unbounded fan-in circuits. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function f, circuits of size O(nf−1(n)) and constant depth exist for the semigroup product of n elements. The depth depends upon the choice of the primitive recursive function f. The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two n-bit numbers.

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.

Continuous-time computation with restricted integration capabilities

Analysis of algorithms is a part of computing curriculums throughout the world in both undergraduate and graduate levels. However, students in general find this topic to be difficult subject due its theoretical and mathematical nature. This paper presents BiGO, a tool to support the students to understand and learn to analyze the time complexity of algorithms in rigorous manner. BiGO has four main features: i. it enables students to encode any given primitive recursive function with a specific scripting language in order to ii. automatically compute the upper bounds of the script and to, iii. visualize the behavior of the upper bounds in comparison to the most commonly occurring time complexity classes and, iv. view how long it takes for a computer to process the algorithm based on the different input values. This paper outlines the pedagogical goals of the tool and the system implementation architecture. Furthermore, we discuss future developmental plans for the tool and an evaluation within undergraduate and graduate level courses such as Data structures and algorithms and Design and analysis of algorithms.

The design of programs as the composition of smaller ones is a wide spread approach to programming. In functional programming, this approach raises the necessity of creating a good amount of intermediate data structures with the only aim of passing data from one function to another. Using program fusion techniques, it is possible to eliminate many of those intermediate data structures by an appropriate combination of the codes of the involved functions. In the standard case, no mention to the eliminated data structure remains in the code obtained from fusion. However, there are situations in which parts of that data structure becomes an internal value manipulated by the fused program. This happens, for example, when primitive recursive functions (so-called paramorphisms) are involved. We show, for example, that the result of fusing a primitive recursive function p with another function f may give as result a function that contains calls to f . Moreover, we show that in some cases the result of fusion may be less efficient than the original composition. We also investigate a general recursive version of paramorphism. This study is strongly motivated by the development of a fusion tool for Haskell programs called HFUSION.

Isabelle/HOL has recently acquired new versions of Definitional packages for inductive datatypes and primitive recursive functions. In contrast to its predecessors and most other implementations, Isabelle/HOL datatypes may be mutually and indirect recursive, even infinitely branching. We also support inverted datatype definitions for characterizing existing types as being inductive ones later. All our constructions are fully Definitional according to established HOL tradition. Stepping back from the logical details, we also see this work as a typical example of what could be called Formal-Logic Engineering. We observe that building realistic theorem proving environments involves further issues rather than pure logic only.

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.

A notion of strictly primitive recursive realizability is introduced by Damnjanovic in 1994. It is a kind of constructive semantics of the arithmetical sentences using primitive recursive functions. It is of interest to study the corresponding predicate logic. It was argued by Park in 2003 that the predicate logic of strictly primitive recursive realizability is not arithmetical. Park’s argument is essentially based on a claim of Damnjanovic that intuitionistic logic is sound with respect to strictly primitive recursive realizability, but that claim was disproved by the author of this article in 2006. The aim of this paper is to present a correct proof of the result of Park.

A realizability notion that employs only primitive recursive functions is defined, and, relative to it, the soundness of the fragment of Heyting Arithmetic (HA) in which induction is restricted to formulae is proved. A dual concept of falsifiability is proposed and an analogous soundness result is established for a further restricted fragment of HA.